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5,100 | 5,617 | Large Scale Canonical Correlation Analysis with
Iterative Least Squares
Yichao Lu
University of Pennsylvania
[email protected]
Dean P. Foster
Yahoo Labs, NYC
[email protected]
Abstract
Canonical Correlation Analysis (CCA) is a widely used statistical tool with both
well established theory and favorable performance for a wide range of machine
learning problems. However, computing CCA for huge datasets can be very slow
since it involves implementing QR decomposition or singular value decomposition of huge matrices. In this paper we introduce L-CCA , a iterative algorithm
which can compute CCA fast on huge sparse datasets. Theory on both the asymptotic convergence and finite time accuracy of L-CCA are established. The experiments also show that L-CCA outperform other fast CCA approximation schemes
on two real datasets.
1
Introduction
Canonical Correlation Analysis (CCA) is a widely used spectrum method for finding correlation
structures in multi-view datasets introduced by [15]. Recently, [3, 9, 17] proved that CCA is able
to find the right latent structure under certain hidden state model. For modern machine learning
problems, CCA has already been successfully used as a dimensionality reduction technique for
the multi-view setting. For example, A CCA between the text description and image of the same
object will find common structures between the two different views, which generates a natural
vector representation of the object. In [9], CCA is performed on a large unlabeled dataset in order
to generate low dimensional features to a regression problem where the size of labeled dataset is
small. In [6, 7] a CCA between words and its context is implemented on several large corpora to
generate low dimensional vector representations of words which captures useful semantic features.
When the data matrices are small, the classical algorithm for computing CCA involves first a
QR decomposition of the data matrices which pre whitens the data and then a Singular Value
Decomposition (SVD) of the whitened covariance matrix as introduced in [11]. This is exactly how
Matlab computes CCA. But for huge datasets this procedure becomes extremely slow. For data
matrices with huge sample size [2] proposed a fast CCA approach based on a fast inner product
preserving random projection called Subsampled Randomized Hadamard Transform but it?s still
slow for datasets with a huge number of features. In this paper we introduce a fast algorithm for
finding the top kcca canonical variables from huge sparse data matrices (a single multiplication with
these sparse matrices is very fast) X 2 n ? p1 and Y 2 n ? p2 the rows of which are i.i.d samples
from a pair of random vectors. Here n
p1 , p2
1 and kcca is relatively small number like 50
since the primary goal of CCA is to generate low dimensional features. Under this set up, QR
decomposition of a n ? p matrix cost O(np2 ) which is extremely slow even if the matrix is sparse.
On the other hand since the data matrices are sparse, X> X and Y> Y can be computed very fast.
1
1
So another whitening strategy is to compute (X> X) 2 , (Y> Y) 2 . But when p1 , p2 are large this
3 3
takes O(max{p1 , p2 }) which is both slow and numerically unstable.
1
The main contribution of this paper is a fast iterative algorithm L-CCA consists of only QR decomposition of relatively small matrices and a couple of matrix multiplications which only involves
huge sparse matrices or small dense matrices. This is achieved by reducing the computation of CCA
to a sequence of fast Least Square iterations. It is proved that L-CCA asymptotically converges to
the exact CCA solution and error analysis for finite iterations is also provided. As shown by the
experiments, L-CCA also has favorable performance on real datasets when compared with other
CCA approximations given a fixed CPU time.
It?s worth pointing out that approximating CCA is much more challenging than SVD(or PCA).
As suggested by [12, 13], to approximate the top singular vectors of X, it suffices to randomly
sample a small subspace in the span of X and some power iteration with this small subspace will
automatically converge to the directions with top singular values. On the other hand CCA has
to search through the whole X Y span in order to capture directions with large correlation. For
example, when the most correlated directions happen to live in the bottom singular vectors of the
data matrices, the random sample scheme will miss them completely. On the other hand, what
L-CCA algorithm doing intuitively is running an exact search of correlation structures on the top
singular vectors and an fast gradient based approximation on the remaining directions.
2
2.1
Background: Canonical Correlation Analysis
Definition
Canonical Correlation Analysis (CCA) can be defined in many different ways. Here we use the definition in [9, 17] since this version naturally connects CCA with the Singular Value Decomposition
(SVD) of the whitened covariance matrix, which is the key to understanding our algorithm.
Definition 1. Let X 2 n ? p1 and Y 2 n ? p2 where the rows are i.i.d samples from a pair of
random vectors. Let x 2 p1 ? p1 , y 2 p2 ? p2 and use x,i , y,j to denote the columns of
x , y respectively. X x,i , Y y,j are called canonical variables if
?
di if i = j
>
>
=
x,i X Y y,j
0 if i 6= j
?
?
1 if i = j
1 if i = j
>
>
>
>
X
X
=
Y
Y
=
x,j
y,j
x,i
y,i
0 if i 6= j
0 if i 6= j
X
2.2
x,i , Y y,i
is the ith pair of canonical variables and di is the ith canonical correlation.
CCA and SVD
First introduce some notation. Let
Cxx = X> X Cyy = Y> Y Cxy = X> Y
For simplicity assume Cxx and Cyy are full rank and Let
1
1
? xy = Cxx2 Cxy Cyy2
C
The following lemma provides a way to compute the canonical variables by SVD.
? xy = UDV> be the SVD of C
? xy where ui , vj denote the left, right singular
Lemma 1. Let C
1
1
vectors and di denotes the singular values. Then XCxx2 ui , YCyy2 vj are the canonical variables
of the X, Y space respectively.
1
1
Proof. Plug XCxx2 ui , YCyy2 vj into the equations in Definition 1 directly proves lemma 1
As mentioned before, we are interested in computing the top kcca canonical variables where kcca ?
p1 , p2 . Use U1 , V1 to denote the first kcca columns of U, V respectively and use U2 , V2 for the
1
remaining columns. By lemma 1, the top kcca canonical variables can be represented by XCxx2 U1
1
and YCyy2 V1 .
2
Algorithm 1 CCA via Iterative LS
Input : Data matrix X 2 n ? p1 ,Y 2 n ? p2 . A target dimension kcca . Number of orthogonal
iterations t1
Output : Xkcca 2 n ? kcca , Ykcca 2 n ? kcca consist of top kcca canonical variables of X and Y.
1.Generate a p1 ? kcca dimensional random matrix G with i.i.d standard normal entries.
2.Let X0 = XG
3.
for t = 1 to t1 do
Yt = HY Xt 1 where HY = Y(Y> Y) 1 Y>
Xt = HX Yt where HX = X(X> X) 1 X>
end for
4.Xkcca = QR(Xt1 ), Ykcca = QR(Yt1 )
Function QR(Xt ) extract an orthonormal basis of the column space of Xt with QR decomposition
3
Compute CCA by Iterative Least Squares
? xy which can
Since the top canonical variables are connected with the top singular vectors of C
be compute with orthogonal iteration [10] (it?s called simultaneous iteration in [21]), we can also
compute CCA iteratively. A detailed algorithm is presented in Algorithm1:
The convergence result of Algorithm 1 is stated in the following theorem:
1
2
Theorem 1. Assume |d1 | > |d2 | > |d3 |... > |dkcca +1 | and U>
1 Cxx G is non singular (this will hold
with probability 1 if the elements of G are i.i.d Gaussian). The columns of Xkcca and Ykcca will
converge to the top kcca canonical variables of X and Y respectively if t1 ! 1.
Theorem 1 is proved by showing it?s essentially an orthogonal iteration [10, 21] for computing the
? xy C
? > . A detailed proof is provided in the supplementary materials.
top kcca eigenvectors of A = C
xy
3.1
A Special Case
When X Y are sparse and Cxx , Cyy are diagonal (like the Penn Tree Bank dataset in the experiments), Algorithm 1 can be implemented extremely fast since we only need to multiply with sparse
matrices or inverting huge but diagonal matrices in every iteration. QR decomposition is performed
not only in the end but after every iteration for numerical stability issues (here we only need to QR
with matrices much smaller than X, Y). We call this fast version D-CCA in the following discussions.
When Cxx , Cyy aren?t diagonal, computing matrix inverse becomes very slow. But we can still run
D-CCA by approximating (X> X) 1 , (Y> Y) 1 with (diag(X> X)) 1 , (diag(Y> Y)) 1 in algorithm 1 when speed is a concern. But this leads to poor performance when Cxx , Cyy are far from
diagonal as shown by the URL dataset in the experiments.
3.2
General Case
Algorithm 1 reduces the problem of CCA to a sequence of iterative least square problems. When
X, Y are huge, solving LS exactly is still slow since it consists inverting a huge matrix but fast
LS methods are relatively well studied. There are many ways to approximate the LS solution by
optimization based methods like Gradient Descent [1, 23], Stochastic Gradient Descent [16, 4] or
by random projection and subsampling based methods like [8, 5]. A fast approximation to the top
kcca canonical variables can be obtained by replacing the exact LS solution in every iteration of Algorithm 1 with a fast approximation. Here we choose LING [23] which works well for large sparse
design matrices for solving the LS problem in every CCA iteration.
The connection between CCA and LS has been developed under different setups for different purposes. [20] shows that CCA in multi label classification setting can be formulated as an LS problem.
[22] also formulates CCA as a recursive LS problem and builds an online version based on this
observation. The benefit we take from this iterative LS formulation is that running a fast LS ap3
Algorithm 2 LING
Input : X 2 n ? p ,Y 2 n ? 1. kpc , number of top left singular vectors selected. t2 , number of
iterations in Gradient Descent.
Output : Y? 2 n ? 1, which is an approximation to X(X > X) 1 X > Y
1. Compute U1 2 n ? kpc , top kpc left singular vector of X by randomized SVD (See supplementary materials for detailed description).
2. Y1 = U1 U1> X.
3.Compute the residual. Yr = Y Y1
4.Use gradient descent initial at the 0 vector (see supplementary materials for detailed description)
to approximately solve the LS problem min r 2Rp kX r Yr k2 . Use r,t2 to denote the solution
after t2 gradient iterations.
5. Y? = Y1 + X r,t2 .
proximation in every iteration will give us a fast CCA approximation with both provable theoretical
guarantees and favorable experimental performance.
4
Algorithm
In this section we introduce L-CCA which is a fast CCA algorithm based on Algorithm 1.
4.1
LING: a Gradient Based Least Square Algorithm
First we need to introduce the fast LS algorithm LING as mentioned in section 3.2 which is used in
every orthogonal iteration of L-CCA .
Consider the LS problem:
?
= arg minp {kX
Y k2 }
2R
for X 2 n ? p and Y 2 n ? 1. For simplicity assume X is full rank. X ? = X(X > X) 1 X > Y is
the projection of Y onto the column space of X. In this section we introduce a fast algorithm LING
to approximately compute X ? without formulating (X > X) 1 explicitly which is slow for large p.
The intuition of LING is as follows. Let U1 2 n ? kpc (kpc ? p) be the top kpc left singular vectors
of X and U2 2 n ? (p kpc ) be the remaining singular vectors. In LING we decompose X ? into
two orthogonal components,
X ? = U1 U1> Y + U2 U2> Y
the projection of Y onto the span of U1 and the projection onto the span of U2 . The first term can
be computed fast given U1 since kpc is small. U1 can also be computed fast approximately with the
randomized SVD algorithm introduced in [12] which only requires a few fast matrix multiplication
and a QR decomposition of n ? kpc matrix. The details for finding U1 are illustrated in the supplementary materials. Let Yr = Y
U1 U1> Y be the residual of Y after projecting onto U1 . For the
second term, we compute it by solving the optimization problem
minp {kX
r
r 2R
Yr k2 }
with Gradient Descent (GD) which is also described in detail in the supplementary materials. A
detailed description of LING are presented in Algorithm 2.
In the above discussion Y is a column vector. It is straightforward to generalize LING to fit into
Algorithm 1 where Y have multiple columns by applying Algorithm 2 to every column of Y .
In the following discussions, we use LING (Y, X, kpc , t2 ) to denote the LING output with corresponding inputs which is an approximation to X(X > X) 1 X > Y .
The following theorem gives error bound of LING .
Theorem 2. Use
i
to denote the ith singular value of X. Consider the LS problem
minp {kX
2R
4
Y k2 }
Algorithm 3 L-CCA
Input : X 2 n ? p1 ,Y 2 n ? p2 : Data matrices.
kcca : Number of top canonical variables we want to extract.
t1 : Number of orthogonal iterations.
kpc : Number of top singular vectors for LING
t2 : Number of GD iterations for LING
Output : Xkcca 2 n ? kcca , Ykcca 2 n ? kcca : Top kcca canonical variables of X and Y.
1.Generate a p1 ? kcca dimensional random matrix G with i.i.d standard normal entries.
? 0 = QR(X0 )
2.Let X0 = XG, X
3.
for t = 1 to t1 do
? t 1 , Y, kpc , t2 ), Y
? t = QR(Yt )
Yt = LING(X
?
?
Xt = LING(Yt , X, kpc , t2 ), Xt = QR(Xt )
end for
? t , Yk = Y
?t
4.Xk = X
cca
1
cca
1
for X 2 n ? p and Y 2 n ? 1. Let Y ? = X(X > X)
space of X and Y?t2 = LING (Y, X, kpc , t2 ). Then
kY ?
for some constant C > 0 and r =
2
kpc +1
2
kpc +1 +
2
p
2
p
1
X > Y be the projection of Y onto the column
Y?t2 k2 ? Cr2t2
(1)
<1
The proof is in the supplementary materials due to space limitation.
Remark 1. Theorem 2 gives some intuition of why LING decompose the projection into two components. In an extreme case if we set kpc = 0 (i.e. don?t remove projection on the top principle
2
2
components and directly apply GD to the LS problem), r in equation 1 becomes 12 + 2p . Usually
p
1
1 is much larger than p , so r is very close to 1 which makes the error decays slowly. Removing
projections on kpc top singular vector will accelerate error decay by making r smaller. The benefit
of this trick is easily seen in the experiment section.
4.2
Fast Algorithm for CCA
Our fast CCA algorithm L-CCA are summarized in Algorithm 3:
There are two main differences between Algorithm 1 and 3. We use LING to solve Least squares
approximately for the sake of speed. We also apply QR decomposition on every LING output for
numerical stability issues mentioned in [21].
4.3
Error Analysis of L-CCA
This section provides mathematical results on how well the output of L-CCA algorithm approximates the subspace spanned by the top kcca true canonical variables for finite t1 and t2 . Note that the
asymptotic convergence property of L-CCA when t1 , t2 ! 1 has already been stated by theorem
1. First we need to define the distances between subspaces as introduced in section 2.6.3 of [10]:
Definition 2. Assume the matrices are full rank. The distance between the column space of matrix
W1 2 n ? k and Z1 2 n ? k is defined by
dist(W1 , Z1 ) = kHW1
W1 (W1> W1 ) 1 W1> ,
where HW1 =
HZ1
the matrix norm is the spectrum norm. Easy
invertible k ? k matrix R1 , R2 .
HZ1 k2
>
= Z1 (Z1 Z1 ) 1 Z>
1
to see dist(W1 , Z1 )
are projection matrices. Here
= dist(W1 R1 , Z1 R2 ) for any
1
We continue to use the notation defined in section 2. Recall that XCxx2 U1 gives the top kcca canon1
ical variables from X. The following theorem bounds the distance between the truth XCxx2 U1 and
? t , the L-CCA output after finite iterations.
X
1
5
Theorem 3. The distance between subspaces spanned top kcca canonical variables of X and the
subspace returned by L-CCA is bounded by
?
?2t1
1
d2kcca
dkcca +1
2
?
dist(Xt1 , XCxx U1 ) ? C1
+ C2 2
r2t2
dkcca
dkcca d2kcca +1
where C1 , C2 are constants. 0 < r < 1 is introduced in theorem 2. t1 is the number of power
iterations in L-CCA and t2 is the number of gradient iterations for solving every LS problem.
The proof of theorem 3 is in the supplementary materials.
5
Experiments
In this section we compare several fast algorithms for computing CCA on large datasets. First let?s
introduce the algorithms we compared in the experiments.
? RPCCA : Instead of running CCA directly on the high dimensional X Y, RPCCA computes CCA only between the top krpcca principle components (left singular vector) of X
and Y where krpcca ? p1 , p2 . For large n, p1 , p2 , we use randomized algorithm introduced
in [12] for computing the top principle components of X and Y (see supplementary material for details). The tuning parameter that controls the tradeoff between computational
cost and accuracy is krpcca . When krpcca is small RPCCA is fast but fails to capture the
correlation structure on the bottom principle components of X and Y. When krpcca grows
larger the principle components captures more structure in X Y space but it takes longer
to compute the top principle components. In the experiments we vary krpcca .
? D-CCA : See section 3.1 for detailed descriptions. The advantage of D-CCA is it?s extremely fast. In the experiments we iterate 30 times (t1 = 30) to make sure D-CCA
achieves convergence. As mentioned earlier, when Cxx and Cyy are far from diagonal
D-CCA becomes inaccurate.
? L-CCA : See Algorithm 3 for detailed description. We find that the accuracy of LING
in every orthogonal iteration is crucial to finding directions with large correlation while a
small t1 suffices. So in the experiments we fix t1 = 5 and vary t2 . In both experiments
we fix kpc = 100 so the top kpc singular vectors of X, Y and every LING iteration can be
computed relatively fast.
? G-CCA : A special case of Algorithm 3 where kpc is set to 0. I.e. the LS projection in
every iteration is computed directly by GD. G-CCA does not need to compute top singular
vectors of X and Y as L-CCA . But by equation 1 and remark 1 GD takes more iterations
to converge compared with LING . Comparing G-CCA and L-CCA in the experiments
illustrates the benefit of removing the top singular vectors in LING and how this can affect
the performance of the CCA algorithm. Same as L-CCA we fix the number of orthogonal
iterations t1 to be 5 and vary t2 , the number of gradient iterations for solving LS.
RPCCA , L-CCA , G-CCA are all "asymptotically correct" algorithms in the sense that if we
spend infinite CPU time all three algorithms will provide the exact CCA solution while D-CCA is
extremely fast but relies on the assumption that X Y both have orthogonal columns. Intuitively,
given a fixed CPU time, RPCCA dose an exact search on krpcca top principle components of X
and Y. L-CCA does an exact search on the top kpc principle components (kpc < krpcca ) and an
crude search over the other directions. G-CCA dose a crude search over all the directions. The
comparison is in fact testing which strategy is the most effective in finding large correlations over
huge datasets.
Remark 2. Both RPCCA and G-CCA can be regarded as special cases of L-CCA . When t1 is
large and t2 is 0, L-CCA becomes RPCCA and when kpc is 0 L-CCA becomes G-CCA .
In the following experiments we aims at extracting 20 most correlated directions from huge data
matrices X and Y. The output of the above four algorithms are two n ? 20 matrices Xkcca and Ykcca
the columns of which contains the most correlated directions. Then a CCA is performed between
Xkcca and Ykcca with matlab built-in CCA function. The canonical correlations between Xkcca and
Ykcca indicates the amount of correlations captured from the the huge X Y spaces by above four
6
algorithms. In all the experiments, we vary krpcca for RPCCA and t2 for L-CCA and G-CCA to
make sure these three algorithms spends almost the same CPU time ( D-CCA is alway fastest). The
20 canonical correlations between the subspaces returned by the four algorithms are plotted (larger
means better).
We want to make to additional comments here based on the reviewer?s feedback. First, for the two
datasets considered in the experiments, classical CCA algorithms like the matlab built in function
takes more than an hour while our algorithm is able to get an approximate answer in less than 10
minutes. Second, in the experiments we?ve been focusing on getting a good fit on the training
datasets and the performance is evaluated by the magnitude of correlation captured in sample. To
achieve better generalization performance a common trick is to perform regularized CCA [14] which
easily fits into our frame work since it?s equivalent to running iterative ridge regression instead of
OLS in Algorithm 1. Since our goal is to compute a fast and accurate fit, we don?t pursue the
generalization performance here which is another statistical issue.
5.1
Penn Tree Bank Word Co-ocurrence
CCA has already been successfully applied to building a low dimensional word embedding in [6, 7].
So the first task is a CCA between words and their context. The dataset used is the full Wall Street
Journal Part of Penn Tree Bank which consists of 1.17 million tokens and a vocabulary size of 43k
[18]. The rows of X matrix consists the indicator vectors of the current word and the rows of Y
consists of indicators of the word after. To avoid sample sparsity for Y we only consider 3000 most
frequent words, i.e. we only consider the tokens followed by 3000 most frequent words which is
about 1 million. So X is of size 1000k ? 43k and Y is of size 1000k ? 3k where both X and
Y are very sparse. Note that every row of X and Y only has a single 1 since they are indicators
of words. So in this case Cxx , Cyy are diagonal and D-CCA can compute a very accurate CCA
in less than a minute as mentioned in section 3.1. On the other hand, even though this dataset can
be solved efficiently by D-CCA , it is interesting to look at the behavior of other three algorithms
which do not make use of the special structure of this problem and compare them with D-CCA
which can be regarded as the truth in this particular case. For RPCCA L-CCA G-CCA we try
three different parameter set ups shown in table 1 and the 20 correlations are shown in figure 1.
Among the three algorithms L-CCA performs best and gets pretty close to D-CCA as CPU time
increases. RPCCA doesn?t perform well since a lot correlation structure of word concurrence exist
in low frequency words which can?t be captured in the top principle components of X Y. Since the
most frequent word occurs 60k times and the least frequent words occurs only once, the spectral of
X drops quickly which makes GD converges very slowly. So G-CCA doesn?t perform well either.
Table 1: Parameter Setup for Two Real Datasets
id
1
2
3
5.2
PTB word co-occurrence
krpcca
t2
t2
RPCCA L-CCA G-CCA
300
7
17
500
38
51
800
115
127
CPU
time
170
460
1180
id
1
2
3
krpcca
RPCCA
600
600
600
URL features
t2
t2
L-CCA G-CCA
4
7
11
16
13
17
CPU
time
220
175
130
URL Features
The second dataset is the URL Reputation dataset from UCI machine learning repository. The
dataset contains 2.4 million URLs each represented by 3.2 million features. For simplicity we only
use first 400k URLs. 38% of the features are host based features like WHOIS info, IP prefix and
62% are lexical based features like Hostname and Primary domain. See [19] for detailed information
about this dataset. Unfortunately the features are anonymous so we pick the first 35% features as our
X and last 35% features as our Y. We remove the 64 continuous features and only use the Boolean
features. We sort the features according to their frequency (each feature is a column of 0s and 1s,
the column with most 1s are the most frequent feature). We run CCA on three different subsets of
X and Y. In the first experiment we select the 20k most frequent features of X and Y respectively.
7
CPU time: 170 secs
PTB Word Occurrence
CPU time: 460 secs
PTB Word Occurrence
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.6
0.5
0.4
Correlation
1
Correlation
Correlation
PTB Word Occurrence
1
0.7
0.6
0.5
L?CCA
D?CCA
RPCCA
G?CCA
0.4
0.3
10
Index
15
20
0.7
0.6
0.5
L?CCA
D?CCA
RPCCA
G?CCA
0.4
0.3
5
CPU time: 1180 secs
L?CCA
D?CCA
RPCCA
G?CCA
0.3
5
10
Index
15
20
5
10
Index
15
20
Figure 1: PTB word co-ocurrence: Canonical correlations of the 20 directions returned by four
algorithms. x axis are the indices and y axis are the correlations.
In the second experiment we select 20k most frequent features from X Y after removing the top
100 most frequent features of X and 200 most frequent features of Y. In the third experiment we
remove top 200 most frequent features from X and top 400 most frequent features of Y. So we are
doing CCA between two 400k ? 20k data matrices in these experiments. In this dataset the features
within X and Y has huge correlations, so Cxx and Cyy aren?t diagonal anymore. But we still run
D-CCA since it?s extremely fast. The parameter set ups for the three subsets are shown in table 1
and the 20 correlations are shown in figure 2.
For this dataset the fast D-CCA doesn?t capture largest correlation since the correlation within X
and Y make Cxx , Cyy not diagonal. RPCCA has best performance in experiment 1 but not as good
in 2, 3. On the other hand G-CCA has good performance in experiment 3 but performs poorly in 1,
2. The reason is as follows: In experiment 1 the data matrices are relatively dense since they includes
some frequent features. So every gradient iteration in L-CCA and G-CCA is slow. Moreover, since
there are some high frequency features and most features has very low frequency, the spectrum of
the data matrices in experiment 1 are very steep which makes GD in every iteration of G-CCA
converges very slowly. These lead to poor performance of G-CCA . In experiment 3 since the
frequent features are removed data matrices becomes more sparse and has a flat spectrum which is
in favor of G-CCA . L-CCA has stable and close to best performance despite those variations in
the datasets.
URL2 CPU time: 175secs
URL3 CPU time: 130secs
1
0.98
0.96
0.96
0.96
0.94
0.94
0.94
0.92
0.92
0.92
0.9
0.88
0.86
0.84
0.82
Correlation
1
0.98
Correlation
Correlation
URL1 CPU time: 220secs
1
0.98
0.9
0.88
0.86
0.84
L?CCA
D?CCA
RPCCA
G?CCA
0.82
0.8
0.86
0.84
L?CCA
D?CCA
RPCCA
G?CCA
0.82
0.8
5
10
Index
15
20
0.9
0.88
L?CCA
D?CCA
RPCCA
G?CCA
0.8
5
10
Index
15
20
5
10
Index
15
20
Figure 2: URL: Canonical correlations of the 20 directions returned by four algorithms. x axis are
the indices and y axis are the correlations.
6
Conclusion and Future Work
In this paper we introduce L-CCA , a fast CCA algorithm for huge sparse data matrices. We
construct theoretical bound for the approximation error of L-CCA comparing with the true CCA
solution and implement experiments on two real datasets in which L-CCA has favorable performance. On the other hand, there are many interesting fast LS algorithms with provable guarantees
which can be plugged into the iterative LS formulation of CCA. Moreover, in the experiments we
focus on how much correlation is captured by L-CCA for simplicity. It?s also interesting to use
L-CCA for feature generation and evaluate it?s performance on specific learning tasks.
8
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9
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5,101 | 5,618 | Cone-constrained Principal Component Analysis
Yash Deshpande
Electrical Engineering
Stanford University
Andrea Montanari
Electrical Engineering and Statistics
Stanford University
Emile Richard
Electrical Engineering
Stanford University
Abstract
Estimating a vector from noisy quadratic observations is a task that arises naturally in many contexts, from dimensionality reduction, to synchronization and
phase retrieval problems. It is often the case that additional information is available about the unknown vector (for instance, sparsity, sign or magnitude of its
entries). Many authors propose non-convex quadratic optimization problems that
aim at exploiting optimally this information. However, solving these problems is
typically NP-hard.
We consider a simple model for noisy quadratic observation of an unknown vector
v0 . The unknown vector is constrained to belong to a cone C 3 v0 . While
optimal estimation appears to be intractable for the general problems in this class,
we provide evidence that it is tractable when C is a convex cone with an efficient
projection. This is surprising, since the corresponding optimization problem is
non-convex and ?from a worst case perspective? often NP hard. We characterize
the resulting minimax risk in terms of the statistical dimension of the cone ?(C).
This quantity is already known to control the risk of estimation from gaussian
observations and random linear measurements. It is rather surprising that the same
quantity plays a role in the estimation risk from quadratic measurements.
1
Introduction
In many statistical estimation problems, observations can be modeled as noisy quadratic functions
of an unknown vector v0 = (v0,1 , v0,2 , . . . , v0,n )T ? Rn . For instance, in positioning and graph
localization [5, 24], one is given noisy measurements of pairwise distances (v0,i ? v0,j )2 (where
?for simplicity? we consider the case in which the underlying geometry is one-dimensional). In
principal component analysis (PCA) [15], one is given a data matrix X ? Rn?p , and tries to reduce
its dimensionality by postulating an approximate factorization X ? u0 v0 T . Hence Xij can be
interpreted as a noisy observation of the quadratic function u0,i v0,j . As a last example, there has
been significant interest recently in phase retrieval problems [11, 6]. In this case, the unknown vector
v0 is ?roughly speaking? an image, and the observations are proportional to the square modulus of
a modulated Fourier transform |Fv0 |2 .
In several of these contexts, a significant effort has been devoted to exploiting additional structure
of the unknown vector v0 . For instance, in Sparse PCA, various methods have been developed to
exploit the fact that v0 is known to be sparse [14, 25]. In sparse phase retrieval [13, 18], a similar
assumption is made in the context of phase retrieval.
All of these attempts face a recurring dichotomy. One the hand, additional information on v0 can
increase dramatically the estimation accuracy. On the other, only a fraction of this additional information is exploited by existing polynomial time algorithms. For instance in sparse PCA, if it is
known that only k entries of the vector v0 are non-vanishing, an optimal estimator is successful in
identifying them from roughly k samples (neglecting logarithmic factors) [2]. On the other hand,
known polynomial-time algorithms require about k 2 samples [16, 7].
1
This fascinating phenomenon is however poorly understood so far. Classifying estimation problems
as to whether optimal estimation accuracy can be achieved or not in polynomial time is an outstanding challenge. In this paper we develop a stylized model to study estimation from quadratic
observations, under additional constraints. Special choices of the constraint set yield examples for
which optimal estimation is thought to be intractable.
However we identify a large class of constraints for which estimation appears to be tractable, despite the corresponding maximum likelihood problem is non-convex. This shows that computational
tractability is not immediately related to simple considerations of convexity or worst-case complexity.
Our model assumes v0 ? Cn with Cn ? Rn a closed cone. Observations are organized in a symmetric matrix X = (Xij )1?i,j?n defined by
X = ? v0 v0 T + Z .
(1)
Here Z is a symmetric noise matrix with independent entries (Zij )i?j with Zij ? N(0, 1/n) for
i < j and Zii ? N(0, 2/n). We assume, without loss of generality, kv0 k2 = 1, and hence ? is the
signal to noise ratio. We will assume ? to be known to avoid non-essential complications.
b : Rn?n ? Sn?1 ? {v ? Rn : kvk2 = 1},
We consider estimators that return normalized vectors v
and will characterize such an estimator through the risk function
1
RCn (b
v; v0 ) = E min(kb
v(X) ? v0 k22 , kb
v(X) ? v0 k22 ) = 1 ? E{|hb
v(X), v0 i|} .
(2)
2
The corresponding worst-case risk is R(b
v; Cn ) ? supv0 ?Cn RCn (b
v; v0 ), and the minimax risk
R(Cn ) = inf vb R(b
v; Cn ).
Remark 1.1. Let Cn = Sn,k be the cone of vectors v0 that have at most k non-zero entries, all
positive, and with equal magnitude. The problem of testing whether ? = 0 or ? ? ?0 within
the model (1) coincides with the problem of detecting a non-zero mean submatrix in a Gaussian
matrix. For the latter, Ma and Wu [20] proved that it cannot be accomplished in polynomial time
unless an algorithm exists for the so-called planted clique problem in a regime in which the latter is
conjectured to be hard.
This suggests that the problem of estimating v0 with rate-optimal minimax risk is hard for the
constraint set Cn = Sn,k .
We next summarize our results. While ?as shown by the last remark? optimal estimation is generically intractable for the model (1) under the constraint v0 ? Cn , we show that ?roughly speaking? it
is instead tractable if Cn is a convex cone. Note that this does not follow from elementary convexity
considerations. Indeed, the maximum likelihood problem
maximize hv, Xvi ,
(3)
subject to v ? Cn , kvk2 = 1 ,
is non-convex. Even more, solving exactly this optimization problem is NP-hard even for simple
choices of the convex cone Cn . For instance, if Cn = Pn ? {v ? Rn : v ? 0} is an orthant, then
solving the above is equivalent to copositive programming, which is NP-hard by reduction from
maximum independent sets [12, Chapter 7].
Our results naturally characterize the cone Cn through its statistical dimension [1]. If PCn denotes
the orthogonal projection on Cn , then the fractional statistical dimension of Cn is defined as
2
1
(4)
?(Cn ) ? E
PCn (g)
2 ,
n
where expectation is with respect to g ? N(0, In?n ). Note that ?(Cn ) ? [0, 1] can be significantly
smaller than 1. For instance, if Cn = Mn ? {v ? Rn+ : ?i , vi+1 ? vi } is the cone of nonnegative, monotone increasing sequences, then [9, Lemma 4.2] proves that ?(Cn ) ? 20(log n)2 /n.
Below is an informal summary of our results, with titles referring to sections where these are established.
Information-theoretic p
limits. We prove that in order to estimate accurately v0 , it is necessary
to havep? & ?(Cn ). Namely, there exist universal constants c1 , c2 > 0 such that, if
? ? c1 ?(Cn ), then R(Cn ) ? c2 .
2
b ML (X) be thep
Maximum likelihood estimator. Let v
maximum-likelihood estimator, i.e. any solution of Eq. (3). We then prove that, for ? ? ?(Cn )
p
4 ?(Cn )
ML
R(b
v ; Cn ) ?
.
(5)
?
Low-complexity iterative estimator. In the special case Cn = Rn , the solution of the optimization problem (3) is given by the eigenvector with the largest eigenvalue. A standard
low-complexity approach to computing the leading eigenvector is provided by the power
method. We consider a simple generalization that ?starting from the initialization v0 ? alternates between projection onto Cn and multiplication by (X + ?In ) (? > 0 is added to
improve convergence):
PCn (ut )
,
kPCn (ut )k2
ut = (X + ?In )b
vt .
b t+1 =
v
(6)
(7)
We prove that, for t & log n iterations, this algorithm
yields an estimate with R(b
v t ; Cn ) .
p
p
?(Cn )/?, and hence order optimal, for ? & ?(Cn ). (Our proof technique requires the
initialization to have a positive scalar product with v0 .)
As a side result of our analysis of the maximum likelihood estimator, we prove a new, elegant, upper bound on the value of the optimization problem (3), denoted by ?1 (Z; Cn ) ?
maxv?Cn ?Sn?1 hv, Zvi. Namely
p
E?1 (Z; Cn ) ? 2 ?(Cn ) .
(8)
In the special case Cn = Rn , ?1 (Z; Rn ) is the largest eigenvalue of Z, and the above inequality
shows that this is bounded in expectation by 2. In this case, the bound is known to be asymptotically
tight [10]. In the supplementary material, we prove that it is tight for certain other examples such
as the nonnegative orthant and for circular cones (a.k.a. ice-cream cones). We conjecture that this
inequality is asymptotically tight for general convex cones.
Unless stated otherwise, in the following we will defer proofs to the Supplementary Material.
2
Information-theoretic limits
We use an information-theoretic argument to show
p that, under the observation model (1), then the
minimax risk can be bounded below for ? . ?(Cn ). As is standard, our bound employs the
so-called packing number of Cn .
Definition 2.1. For a cone Cn ? Rn , we define its packing number N (Cn , ?) as the size of the
maximal subset X of Cn ? Sn?1 such that for every x1 , x2 ? Cn ? Sn?1 , kx1 ? x2 k ? ?.
We then have the following.
Theorem 1. There exist universal constants C1 , C2 > 0 such that for any closed convex cone Cn
with ?(Cn ) ? 3/n:
p
C2 ?(Cn )
? ? C1 ?(Cn ) ? R(Cn ) ?
.
(9)
log(1/?(Cn ))
Notice that the last expression for the lower bound depends on the cone width, as it is to be expected:
even for ? = 0, it is possible to estimate v0 with risk going to 0 if the cone Cn ?shrinks? as n ? ?.
The proof of this theorem is provided in Section 2 of the supplement.
3
Maximum likelihood estimator
Under the Gaussian noise model for Z, cf. Eq. (1), the likelihood of observing X under a hypothesis
v is proportional to exp(?kX ? vvT k2F /2). Using the constraint that kvk = 1, it follows that any
solution of (3) is a maximum likelihood estimator.
3
p
b ML (X) to the
Theorem 2. Consider the model as in (1). Then, when ? ? ?(Cn ), any solution v
maximum likelihood problem (3) satisfies
( p
)
4 ?(Cn ) 16
ML
RCn (b
v ; Cn ) ? min
, 2 .
(10)
?
?
p
p
Thus, for ? & ?(Cn ), the risk of the maximum likelihood estimator decays as ?(Cn )/? while
for ? & 1, it shifts to a faster decay of 1/? 2 . We have made no attempt to optimize the constants in
the statement of the theorem, though we believe that the correct leading constant in either case is 1.
Note that without the cone constraint (or with Cn = Rn ) the maximum likelihood estimator reduces
b PC of X. Recent results in random matrix theory [10] and
to computing the principal eigenvector v
statistical decision theory [4] prove that in the case of principal eigenvector, a nontrivial risk (i.e.
RCn (b
v PC ; Cn ) < 1 asymptotically)
is obtained only when ? > 1. Our result shows that this threshold
p
is, instead, reduced to ?(Cn ), which can be significantly smaller than 1. The proof of this theorem
is provided in Section 3 of the supplement.
4
Low-complexity iterative estimator
Sections 2 and 3 provide theoretical insight into the fundamental limits of estimation of v0 from
quadratic observations of the form ?v0 v0 T + Z. However, as previously mentioned, the maximum
likelihood estimator of Section 3 is NP-hard to compute, in general. In this section, we propose a
simple iterative algorithm that generalizes the well-known power iteration to compute the principal
eigenvector of a matrix. Furthermore, we prove that, given an initialization with positive scalar
product with v0 , this algorithm achieves the same risk of the maximum likelihood estimator up to
constants. Throughout, the cone Cn is assumed to be convex.
b PC of X. This is
Our starting point is the power iteration to compute the principal eigenvector v
t+1
t
t
b
given by letting, for t ? 0: v
= Xb
v /kXb
v k. Under our observation model, we have X =
?v0 v0 T + Z with v0 ? Cn . We can incorporate this information by projecting the iterates on to the
cone Cn (see e.g. [19] for related ideas):
bt =
v
PCn (ut )
,
kPCn (ut )k
ut+1 = Xvt + ?vt .
(11)
The projection is defined in the standard way:
PCn (x) ? arg min ky ? xk2 .
y?Cn
(12)
If Cn is convex, then the projection is unique. We have implicitly assumed that the operation of
projecting to the cone Cn is available to the algorithm as a simple primitive. This is the case for many
convex cones of interest, such as the orthant Pn , the monotone cone Mn , and ice-cream cones the
projection is easy to compute. For instance, if Cn = Pn is the non-negative orthant PCn (x) = (x)+
is the non-negative part of x. For the monotone cone, the projection can be computed efficiently
through the pool-adjacent violators algorithm.
The memory term ?vt is necessary for our proof technique to go through. It is straightforward to
see that adding ?In to the data X does not change the optimizers of the problem (3). The following
theorem provides deterministic conditions under which the distance between the iterative estimator
and the vector v0 can be bounded.
b t be the power iteration estimator (11). Assume ? > ? and that the noise matrix
Theorem 3. Let v
Z satisfies:
max |hx, Zyi| : x, y ? Cn ? Sn?1 ? ? .
(13)
b 0 ? Cn ? Sn?1 satisfies hb
If ? > 4?, and the initial point v
v0 , v0 i ? 2?/?, then there exits
t0 = t0 (?/?, ?/?) < ? independent of n such that, for all t ? t0
kb
v t ? v0 k ?
4
4?
.
?
(14)
We can apply this theorem to the Gaussian noise model to obtain the following bound on the risk of
the power iteration estimator.
p
Corollary 4.1. Under the model (1) let ?n = 8 log n/n. Assume that hb
v0 , v0 i > 0 and
p
? > 2( ?(Cn ) + ?n ) max 2, hb
v0 , v0 i?1 .
(15)
Then
R(b
v t , Cn ) ?
2?(Cn ) + ?n
.
?
(16)
In other words, power iteration has risk within a constant from the maximum likelihood estimator,
provided an initialization is available whose scalar product with v0 is bounded away from zero. The
proofs of Theorem 3 and Corollary 4.1 are provided in Section 4 of the supplement.
5
A case study: sharp asymptotics and minimax results for the orthant
In this section, we will be interested in the example in which the cone Cn is the non-negative orthant
Cn = Pn . Non-negativity constraints within principal component analysis arise in non-negative
matrix factorization (NMF). Initially introduced in the context of chemometrics [23, 22], NMF attracted considerable interest because of its applications in computer vision and topic modeling. In
particular, Lee and Seung [17] demonstrated empirically that NMF successfully identifies parts of
images, or topics in documents? corpora.
Note that the in applications of NMF to computer vision or topic modeling the setting is somewhat
different from the model studied here: X is rectangular instead of symmetric, and the rank is larger
than one. Such generalizations of our analysis will be the object of future work.
Here we will use the positive orthant to illustrate the results in previous sections. Further, we will
show that stronger results can be proved in this case, thanks to the separable structure of this cone.
Namely, we derive sharp asymptotics and we characterize the least-favorable vectors for the maximum likelihood estimator.
We denote by ?+ (X) = ?1 (X; Cn = Pn ) the value of the optimization problem (3). Our first result
yields the asymptotic value of this quantity for ?pure noise,? confirming the general conjecture put
forward above.
p
?
Theorem 4. We have almost surely limn?? ?+ (Z) = 2 ?(Pn ) = 2.
Next we characterize the risk phase transition: this result confirms and strengthen Theorem 2.
Theorem
? 5. Consider estimation in the non-negative orthant Cn = Pn under the model (1). If
? ? 1/ 2, then there exists a sequence of vectors {v0 (n)}n?0 , such that almost surely
lim R(v ML ; v0 (n)) = 1 .
n??
(17)
?
?
For ? > 1/ 2, there exists a function ? 7? R+ (?) with R+ (?) < 1 for all ? > 1/ 2, and
R+ (?) ? 1 ? 1/2? 2 , such that the following happens. For any sequence of vectors {v0 (n)}n?0 ,
we have, almost surely
lim sup R(v ML ; v0 (n)) ? R+ (?) .
(18)
n??
In other words, in the high-dimensional limit, the
estimator is positively correp maximum likelihood
?
lated with the signal v0 (n) if and only if ? > ?(Cn ) = 1/ 2.
Explicit (although non-elementary) expressions for R+ (?) can be computed, along with the limit
value of the risk R(v ML ; v0 (n)) for sequences of vectors {v0 (n)}n?1 whose entries empirical distribution converges. These results go beyond the scope of the present paper (but see Fig. 1 below for
illustration).
As a byproduct of our analysis, we can characterize the least-favorable choice of the signal v0 .
Namely
for k ? [1, n], wee let u(n, k) denote a vector with bkc non-zero entries, all equal to
p
1/ bkc. Then we can prove that the asymptotic minimax risk is achieved along sequences of
vectors of this type.
5
Theorem 6. Consider estimation in the non-negative orthant
Cn = Pn under the model (1), and let
?
R+ (?) be the same function as in Theorem 5. If ? ? 1/ 2 then there exists kn = o(n) such that
lim R(v ML ; u(n, kn )) = 1 .
n??
(19)
?
If ? > 1/ 2 then there exists ?# = ?# (?) ? (0, 1] such that
lim R(v ML ; u(n, n?# )) = R+ (?) .
n??
(20)
We refer the reader to [21] for a detailed analysis of the case of nonnegative PCA and the full proofs
of Theorems 4, 5 and 6.
5.1
Approximate Message Passing
The next question is whether, in the present example Cn = Pn , the risk of the maximum likelihood
estimator can be achieved by a low-complexity iterative algorithm. We prove that this is indeed the
case (up to an arbitrarily small error), thus confirming Theorem 3. In order to derive an asymptotically exact analysis, we consider an ?approximate message passing? modification of the power
iteration.
Let f (x) = (x)+ /k(x)
? + k2 denote the normalized projector. We consider the iteration defined by
v0 = (1, 1, . . . , 1)T / n, v?1 = (0, 0, . . . , 0)T , and for t ? 0,
?
vt+1 = Xf (vt ) ? bt f (vt?1 ) and bt ? k(vt )+ k0 /{ nk(vt )+ k2 }
AMP
The algorithm AMP is a slight modification of the projected power iteration algorithm up to adding
at each step the ?memory term? ?bt f (vt?1 ). As shown in [8, 3] this term plays a crucial role in
allowing for an exact high-dimensional characterization. At each step the estimate produced by the
b t = (vt )+ /k(vt )+ k2 . We have the following
sequence is v
Theorem 7. Let X be generated as in (1). Then we have, almost surely,
ML
v , Xv ML i ? hb
vt , Xb
vt i = 0 .
(21)
lim lim hb
t?? n??
5.2
Numerical illustration: comparison with classical PCA
We performed numerical experiments on synthetic data generated according to the model (1) and
with signal v0 = u(n, n?) as defined in the previous section. We provide in the Appendix formulas
b ML i, which correspond to continuous black lines in the Figure 1. We
for the value of limn?? hv0 , v
compare these predictions with empirical values obtained by running AMP.
We generated samples of size n = 104 , sparsity level ? ? {0.001, 0.1, 0.8}, and signal-to-noise
ratios ? ? {0.05, 0.10, . . . , 1.5}. In each case we run AMP for t = 50 iterations and plot the
empirical average of hb
vt , v0 i over 32 instances. Even for such moderate values of n, the asymptotic
predictions are remarkably accurate.
Observe that sparse vectors (small ?) correspond to the least favorable signal for small signal-tonoise ratio ?, while the situation is reverted for large values
? of ?. In dashed green we represented
the theoretical prediction for ? ? 0. The value ? = 1/ 2 corresponds to the phase transition. At
b ML i for a grid of values of ? and
the bottom the images correspond to values of the correlation hv0 , v
?. The top left-hand frame in Figure 1 is obtained by repeating the experiment for a grid of values
of n, and fixed ? = 0.05 and several value of ?. For each point we plot the average of hb
vt , v0 i after
ML
?b
t = 50 iteration, over 32 instances. The data suggest hb
v , v0 i + A n ? limn?? hv0 , v+ i with
b ? 0.5.
6
Polyhedral cones and convex relaxations
A polyhedral cone Cn is a closed convex cone that can be represented in the form Cn = {x ? Rn :
Ax ? 0} for some matrix A ? Rm?n . In section 5 we considered the non-negative orthant, which
is an example of polyhedral cone with A = In . A number of other examples of practical interest fall
within this category of cones. For instance, monotonicity or convexity of a vector v = (v1 , . . . , vn )
6
Non?negative PCA
0
0.9
0.8
0.7
? = .800
0.6
< v0, v+ >
Deviation from asymptotic
10
0.5
?1
10
0.4
? = .100
0.3
Empirical
n?1/2
0.2
? = .001
0.1
n
?2
10
0
1
2
10
3
10
4
10
10
2?1/ 2
Theory Prediction
?
1
Empirical (n = 1000)
1
1
0.9
0.8
0.9
0.8
0.8
0.7
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.6
0.5
?
?
0.5
0.5
0.4
0.4
0.4
0.3
0.4
0.3
0.3
0.3
0.2
0.5
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.2
0.4
0.6
|
0.8
21/2
1
?
1.2
1.4
0.2
0.4
0.6
|
0.8
21/2
1
?
1.2
1.4
Figure 1: Numerical simulations with the model 1 for the positive orthant cone Cn = Pn . Topleft: empirical deviation from asymptotic prediction. Top-right: black lines represent the theoretical
predictions of Theorem 5, and dots represent empirical values of hb
vt , v0 i for the AMP estimator
(in red) and hv1 , v0 i for standard PCA (in blue). Bottom: a comparison of theoretical asymptotic
values (left frame) and empirical values (right frame) of hv0 , v ML i for a range of ? and ?.
an be enforced ?in their discrete version? through inequality constraints (respectively vi+1 ? vi ? 0
and vi+1 ? 2vi + vi?1 ? 0), and hence give rise to polyhedral cones. Furthermore, it is possible to
approximate any convex cone Cn with a sequence of increasingly accurate polyhedral cones.
For a polyhedral cone, the maximum likelihood problem (3) reads:
maximize hv, Xvi
subject to: Av ? 0; kvk = 1.
(22)
The modified power iteration (11), can be specialized to this case, via the appropriate projection.
The projection remains computationally feasible provided the matrix A is not too large. Indeed, it
is easy to show using convex duality that PCn (u) is given by:
PCn (u) = arg min kAx + uk2 , x ? 0 .
This reduces the projection onto a general polyhedral cone to a non-negative least squares problem,
for which efficient routines exist. In special cases such as the orthant, the projection is closed form.
In the case of polyhedral cones, it is possible to relax this problem (22) using a natural convex
surrogate. To see this, we introduce the variable V = vvT and write the following equivalent
version of problem 22:
maximize hX, Vi
subject to: AVAT ? 0; Tr(V) = 1;
V0; rank(V) = 1.
Here the constraint AVAT ? 0 is to be interpreted as entry-wise non-negativity, while we write
V0 to denote that V is positive semidefinite. We can now relax this problem by dropping the rank
constraint:
maximize hX, Vi
(23)
T
subject to: AVA ? 0; Tr(V) = 1; V0.
Note that this increases the number of variables from n to n2 , as V ? Rn?n , which results in a
significant cost increase for standard interior point methods, over the power iteration (11). Furtherb . On
more, if the solution V is not rank one, it is not clear how one can use it to form an estimate v
the other hand, this convex relaxation yields a principled approach to bounding the sub-optimality
7
4.5
Power Iteration
Proposed dual witness
4
Exact dual witness
?1(X + Y)
3.5
3
2.5
2
1.5
1
0
0.5
1
1.5
2
2.5
3
3.5
4
?
Figure 2: Value of the maximum likelihood problem (3) for Cn = Pn , as approximated by power
iteration. The red line is the value achieved by power iteration, and the blue points the upper bound
obtained by dual witness (25). The gap at small ? is due to the suboptimal choice of the dual witness,
since solving exactly Problem (24) yields the dual witness with value given by the teal circles. As
can be seen, they match exactly the value obtained by power iteration, showing zero duality gap!
The simulation is for n = 50 and 40 Monte Carlo iterations.
of the estimate provided by the power iteration. It is straightforward to derive the dual program of
(23):
minimize ?1 (X + AT YA)
subject to: Y ? 0,
(24)
where Y is the decision variable, the constraint is interpreted as entry-wise nonnegativity as above,
and ?1 ( ? ) denotes the largest eigenvalue. If one can construct a dual witness Y ? 0 such that
b , then this estimator is the maximum likelihood
?1 (X + AT YA) = hb
v, Xb
vi for any estimator v
b=v
b t , such a dual witness can provide
estimator. In particular, using the power iteration estimator v
a certificate of convergence of the power iteration (11).
We next describe a construction of dual witness that we found empirically successful at large enough
signal-to-noise ratio. Assume that a heuristic (for instance, the modified power iteration (11)) has
b that is a local maximizer of the problem (3). It is is proved in the Supproduced an estimate v
plementary Material, that such a local maximizer must satisfy the modified eigenvalue equation:
Xb
v = ?b
v ? AT ?, with ? ? 0 and hb
v, AT ?i = 0.
We then suggest the witness
Y(b
v) =
1 T T
T
?b
v
A
+
Ab
v
?
.
kAb
vk2
(25)
Note that Y(b
v) is non-negative by construction and hence dual feasible. A direct calculation shows
b is an eigenvector of the matrix X + AT YA with eigenvalue ? = hb
that v
v, Xb
vi. We then obtain
the following sufficient condition for optimality.
b be a local maximizer of the problem (3). If v
b is the principal eigenvector of
Proposition 6.1. Let v
b is a global maximizer.
X + AT Y(b
v)A, then v
In Figure 2 we plot the average value of the objective function over 50 instances of the problem for
Cn = Pn , n = 100. We solved the maximum likelihood problem using the power iteration heuristics
(11), and used the above construction to compute an upper bound via duality. It is possible to show
that this upper bound cannot be tight unless ? > 1, but appears to be quite accurate. We also solve
the problem (24) directly for case of nonnegative PCA, and (rather surprisingly) the dual is tight for
every ? > 0.
8
References
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9
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5,102 | 5,619 | Improved Distributed Principal Component Analysis
Vandana Kanchanapally
School of Computer Science
Georgia Institute of Technology
[email protected]
Maria-Florina Balcan
School of Computer Science
Carnegie Mellon University
[email protected]
David Woodruff
Almaden Research Center
IBM Research
[email protected]
Yingyu Liang
Department of Computer Science
Princeton University
[email protected]
Abstract
We study the distributed computing setting in which there are multiple servers,
each holding a set of points, who wish to compute functions on the union of their
point sets. A key task in this setting is Principal Component Analysis (PCA), in
which the servers would like to compute a low dimensional subspace capturing as
much of the variance of the union of their point sets as possible. Given a procedure for approximate PCA, one can use it to approximately solve problems such
as k-means clustering and low rank approximation. The essential properties of an
approximate distributed PCA algorithm are its communication cost and computational efficiency for a given desired accuracy in downstream applications. We give
new algorithms and analyses for distributed PCA which lead to improved communication and computational costs for k-means clustering and related problems.
Our empirical study on real world data shows a speedup of orders of magnitude,
preserving communication with only a negligible degradation in solution quality.
Some of these techniques we develop, such as a general transformation from a
constant success probability subspace embedding to a high success probability
subspace embedding with a dimension and sparsity independent of the success
probability, may be of independent interest.
1
Introduction
Since data is often partitioned across multiple servers [20, 7, 18], there is an increased interest in
computing on it in the distributed model. A basic tool for distributed data analysis is Principal
Component Analysis (PCA). The goal of PCA is to find an r-dimensional (affine) subspace that
captures as much of the variance of the data as possible. Hence, it can reveal low-dimensional
structure in very high dimensional data. Moreover, it can serve as a preprocessing step to reduce
the data dimension in various machine learning tasks, such as Non-Negative Matrix Factorization
(NNMF) [15] and Latent Dirichlet Allocation (LDA) [3].
In the distributed model, approximate PCA was used by Feldman et al. [9] for solving a number
of shape fitting problems such as k-means clustering, where the approximation is in the form of a
coreset, and has the property that local coresets can be easily combined across servers into a global
coreset, thereby providing an approximate PCA to the union of the data sets. Designing small
coresets therefore leads to communication-efficient protocols. Coresets have the nice property that
their size typically does not depend on the number n of points being approximated. A beautiful
property of the coresets developed in [9] is that for approximate PCA their size also only depends
linearly on the dimension d, whereas previous coresets depended quadratically on d [8]. This gives
the best known communication protocols for approximate PCA and k-means clustering.
1
Despite this recent exciting progress, several important questions remain. First, can we improve the
communication further as a function of the number of servers, the approximation error, and other
parameters of the downstream applications (such as the number k of clusters in k-means clustering)?
Second, while preserving optimal or nearly-optimal communication, can we improve the computational costs of the protocols? We note that in the protocols of Feldman et al. each server has to
run a singular value decomposition (SVD) on her local data set, while additional work needs to be
performed to combine the outputs of each server into a global approximate PCA. Third, are these algorithms practical and do they scale well with large-scale datasets? In this paper we give answers to
the above questions. To state our results more precisely, we first define the model and the problems.
Communication Model. In the distributed setting, we consider a set of s nodes V = {vi , 1 ? i ?
s}, each of which can communicate with a central coordinator v0 . On each node vi , there is a local
data matrix Pi ? Rni ?d having ni data points in d dimension
data PP
? Rn?d
i > d). The global
(n
s
>
>
>
>
is then a concatenation of the local data matrix, i.e. P = P1 , P2 , . . . , Ps and n = i=1 ni .
Let pi denote the i-th rowP
of P. Throughout the paper, we assume that the data points are centered
n
to have zero mean, i.e.,
i=1 pi = 0. Uncentered data requires a rank-one modification to the
algorithms, whose communication and computation costs are dominated by those in the other steps.
P
Approximate PCA and `2 -Error Fitting. For a matrix A = [aij ], let kAk2F = i,j a2ij be its
Frobenius norm, and let ?i (A) be the i-th singular value of A. Let A(t) denote the matrix that
contains the first t columns of A. Let LX denote the linear subspace spanned by the columns of X.
For a point p, let ?L (p) be its projection onto subspace L and let ?X (p) be shorthand for ?LX (p).
For a point p ? Rd and a subspace L ? Rd , we denote the squared distance between p and L by
d2 (p, L) := min kp ? qk22 = kp ? ?L (p)k22 .
q?L
Definition 1. The linear (or affine) r-Subspace k-Clustering on P ? Rn?d is
n
X
min d2 (P, L) :=
min d2 (pi , L)
L
i=1
L?L
(1)
where P is an n ? d matrix whose rows are p1 , . . . , pn , and L = {Lj }kj=1 is a set of k centers, each
of which is an r-dimensional linear (or affine) subspace.
PCA is a special case when k = 1 and the center is an r-dimensional subspace. This optimal rdimensional subspace is spanned by the top r right singular vectors of P, also known as the principal
components, and can be found using the singular value decomposition (SVD). Another special case
of the above is k-means clustering when the centers are points (r = 0). Constrained versions of this
problem include NNMF where the r-dimensional subspace should be spanned by positive vectors,
and LDA which assumes a prior distribution defining a probability for each r-dimensional subspace.
We will primarily be concerned with relative-error approximation algorithms, for which we would
like to output a set L0 of k centers for which d2 (P, L0 ) ? (1 + ) minL d2 (P, L).
For approximate distributed PCA, the following protocol is implicit in [9]: each server i computes
its top O(r/) principal components Yi of Pi and sends them to the coordinator. The coordinator
stacks the O(r/) ? d matrices Yi on top of each other, forming an O(sr/) ? d matrix Y, and
computes the top r principal components of Y, and returns these to the servers. This provides a
relative-error approximation to the PCA problem. We refer to this algorithm as Algorithm disPCA.
Our Contributions. Our results are summarized as follows.
Improved Communication: We improve the communication cost for using distributed PCA for kmeans clustering and similar `2 -fitting problems. The best previous approach is to use Corollary 4.5
in [9], which shows that given a data matrix P, if we project the rows onto the space spanned by
the top O(k/2 ) principal components, and solve the k-means problem in this subspace, we obtain a
(1 + )-approximation. In the distributed setting, this would require first running Algorithm disPCA
with parameter r = O(k/2 ), and thus communication at least O(skd/3 ) to compute the O(k/2 )
global principal components. Then one can solve a distributed k-means problem in this subspace,
and an ?-approximation in it translates to an overall ?(1 + ) approximation.
Our Theorem 3 shows that it suffices to run Algorithm disPCA while only incurring O(skd/2 )
communication to compute the O(k/2 ) global principal components, preserving the k-means solution cost up to a (1 + )-factor. Our communication is thus a 1/ factor better, and illustrates that
2
for downstream applications it is sometimes important to ?open up the box? rather than to directly
use the guarantees of a generic PCA algorithm (which would give O(skd/3 ) communication). One
feature of this approach is that by using the distributed k-means algorithm in [2] on the projected
data, the coordinator can sample points from the servers proportional to their local k-means cost
solutions, which reduces the communication roughly by a factor of s, which would come from each
server sending their local k-means coreset to the coordinator. Furthermore, before applying the
above approach, one can first run any other dimension reduction to dimension d0 so that the k-means
cost is preserved up to certain accuracy. For example, if we want a 1+ approximation factor, we can
set d0 = O(log n/2 ) by a Johnson-Lindenstrauss transform; if we want a larger 2+ approximation
factor, we can set d0 = O(k/2 ) using [4]. In this way the parameter d in the above communication
cost bound can be replaced by d0 . Note that unlike these dimension reductions, our algorithm for
projecting onto principal components is deterministic and does not incur error probability.
Improved Computation: We turn to the computational cost of Algorithm disPCA, which to the best
of our knowledge has not been addressed. A major bottleneck is that each player is computing
a singular value decomposition (SVD) of its point set Pi , which takes min(ni d2 , n2i d) time. We
change Algorithm disPCA to instead have each server first sample an oblivious subspace embedding
(OSE) [22, 5, 19, 17] matrix Hi , and instead run the algorithm on the point set defined by the rows
of Hi Pi . Using known OSEs, one can choose Hi to have only a single non-zero entry per column
and thus Hi Pi can be computed in nnz(Pi ) time. Moreover, the number of rows of Hi is O(d2 /2 ),
which may be significantly less than the original ni number of rows. This number of rows can be
further reducted to O(d logO(1) d/2 ) if one is willing to spend O(nnz(Pi ) logO(1) d/) time [19].
We note that the number of non-zero entries of Hi Pi is no more than that of Pi .
One technical issue is that each of s servers is locally performing a subspace embedding, which
succeeds with only constant probability. If we want a single non-zero entry per column of Hi , to
achieve success probability 1 ? O(1/s) so that we can union bound over all s servers succeeding,
we naively would need to increase the number of rows of Hi by a factor linear in s. We give a
general technique, which takes a subspace embedding that succeeds with constant probability as a
black box, and show how to perform a procedure which applies it O(log 1/?) times independently
and from these applications finds one which is guaranteed to succeed with probability 1 ? ?. Thus,
in this setting the players can compute a subspace embedding of their data in nnz(Pi ) time, for
which the number of non-zero entries of Hi Pi is no larger than that of Pi , and without incurring
this additional factor of s. This may be of independent interest.
It may still be expensive to perform the SVD of Hi Pi and for the coordinator to perform an SVD
on Y in Algorithm disPCA. We therefore replace the SVD computation with a randomized approximate SVD computation with spectral norm error. Our contribution here is to analyze the error in
distributed PCA and k-means after performing these speedups.
Empirical Results: Our speedups result in significant computational savings. The randomized techniques we use reduce the time by orders of magnitude on medium and large-scal data sets, while
preserving the communication cost. Although the theory predicts a new small additive error because
of our speedups, in our experiments the solution quality was only negligibly affected.
Related Work A number of algorithms for approximate distributed PCA have been proposed [21,
14, 16, 9], but either without theoretical guarantees, or without considering communication. Most
closely related to our work is [9, 12]. [9] observes the top singular vectors of the local data is its
summary and the union of these summaries is a summary of the global data, i.e., Algorithm disPCA.
[12] studies
Ps algorithms in the arbitrary partition model in which each server holds a matrix Pi and
P = i=1 Pi . More details and more related work can be found in the appendix.
2
Tradeoff between Communication and Solution Quality
Algorithm disPCA for distributed PCA is suggested in [21, 9], which consists of a local stage and a
global stage. In the local stage, each node performs SVD on its local data matrix, and communicates
the first t1 singular values ?i (t1 ) and the first t1 right singular vectors Vi (t1 ) to the central coordinator. Then in the global stage, the coordinator concatenates ?i (t1 ) (Vi (t1 ) )> to form a matrix Y,
and performs SVD on it to get the first t2 right singular vectors.
To get some intuition, consider the easy case when the data points actually lie in an r-dimensional
subspace. We can run Algorithm disPCA with t1 = t2 = r. Since Pi has rank r, its projection to
3
?
?
?
Local PCA
P1
??????
? .. ?
..
P=? . ?
.
Local PCA
Ps
??????
?
?
?
?
?
(t )
?1 1
(t )
..
.
?s 1
> ?
?
?
Y1
?
? ? . ?
Global PCA
?=? . ?=Y?
?????? V(t2 )
.
?
> ?
Ys
(t1 )
(t )
V1 1
Vs
Figure 1: The key points of the algorithm disPCA.
b i = Ui ?i (r) (Vi (r) )> , is identical
the subspace spanned by its first t1 = r right singular vectors, P
b
b i . Observing that P
b = UY
e where
to Pi . Then we only need to do PCA on P, the concatenation of P
(r)
(r)
e is orthonormal, it suffices to compute SVD on Y, and only ?i Vi needs to be communicated.
U
In the general case when the data may have rank higher than r, it turns out that one needs to set t1
b i approximates Pi well enough and does not introduce too much error
sufficiently large, so that P
into the final solution. In particular, the following close projection property about SVD is useful:
b = AV(t) (V(t) )> denote its SVD truncation.
Lemma 1. Suppose A has SVD A = U?V and let A
If t = O(r/), then for any d ? r matrix X with orthonormal columns,
2
2
2
2
2
b
b
0 ? kAX ? AXk
F ? d (A, LX ), and 0 ? kAXkF ? kAXkF ? d (A, LX ).
b and A on any r-dimensional subspace are close, when the
This means that the projections of A
projected dimension t is sufficiently large compared to r. Now, note that the difference between
> 2
b 2 = P [kPi Xk2 ?
b ? PXX
b
kF is only related to kPXk2F ? kPXk
kP ? PXX> k2F and kP
F
F
i
b i Xk2 ]. Each term in which is bounded by the lemma. So we can use P
b as a proxy for P in
kP
F
b is equivalent to computing SVD on Y, as done in
the PCA task. Again, computing PCA on P
Algorithm disPCA. These lead to the following theorem, which is implicit in [9], stating that the
algorithm can produce a (1 + )-approximation for the distributed PCA problem.
Theorem 2. Suppose Algorithm disPCA takes parameters t1 ? r + d4r/e ? 1 and t2 = r. Then
kP ? PV(r) (V(r) )> k2F ? (1 + ) min kP ? PXX> k2F
X
where the minimization is over d?r orthonormal matrices X. The communication is O( srd
) words.
2.1 Guarantees for Distributed `2 -Error Fitting
Algorithm disPCA can also be used as a pre-processing step for applications such as `2 -error fitting.
In this section, we prove the correctness of Algorithm disPCA as pre-processing for these applications. In particular, we show that by setting t1 , t2 sufficiently large, the objective value of any solu? = PV(t2 ) (V(t2 ) )> .
tion merely changes when the original data P is replaced the projected data P
Therefore, the projected data serves as a proxy of the original data, i.e., any distributed algorithm
can be applied on the projected data to get a solution on the original data. As the dimension is lower,
the communication cost is reduced. Formally,
Theorem 3. Let t1 = t2 = O(rk/2 ) in Algorithm disPCA for ? (0, 1/3). Then there exists a
constant c0 ? 0 such that for any set of k centers L in r-Subspace k-Clustering,
? L) + c0 ? (1 + )d2 (P, L).
(1 ? )d2 (P, L) ? d2 (P,
? is a (1 + 3)?The theorem implies that any ?-approximate solution L on the projected data P
?
approximation on the original data P. To see this, let L denote the optimal solution. Then
? L) + c0 ? ?d2 (P,
? L? ) + c0 ? ?(1 + )d2 (P, L? )
(1 ? )d2 (P, L) ? d2 (P,
which leads to d2 (P, L) ? (1 + 3)?d2 (P, L? ). In other words, the distributed PCA step only
introduces a small multiplicative approximation factor of (1 + 3).
The key to prove the theorem is the close projection property of the algorithm (Lemma 4): for any
? on the subspace are close. In
low dimensional subspace spanned by X, the projections of P and P
4
Algorithm 1 Distributed k-means clustering
Input: {Pi }si=1 , k ? N+ and ? (0, 1/2), a non-distributed ?-approximation algorithm A?
1: Run Algorithm disPCA with t1 = t2 = O(k/2 ) to get V, and send V to all nodes.
2: Run the distributed k-means clustering algorithm in [2] on {Pi VV> }si=1 , using A? as a subroutine, to get k centers L.
Output: L.
particular, we choose X to be the orthonormal basis of the subspace spanning the centers. Then the
? can be decomposed into two terms depending
difference between the objective values of P and P
? 2 and kPXk2 ?kPXk
? 2 respectively, which are small as shown by the lemma.
only on kPX? PXk
F
F
F
The complete proof of Theorem 3 is provided in the appendix.
Lemma 4. Let t1 = t2 = O(k/) in Algorithm disPCA. Then for any d?k matrix X with orthonor? 2 ? d2 (P, LX ), and 0 ? kPXk2 ? kPXk
? 2 ? d2 (P, LX ).
mal columns, 0 ? kPX ? PXk
F
F
F
Proof Sketch: We first introduce some auxiliary variables for the analysis, which act as intermediate
? Imagine we perform two kinds of projections: first project Pi to
connections between P and P.
(t1 )
(t1 ) >
b
b i to Pi = P
b i V(t2 ) (V(t2 ) )> . Let P
b denote the vertical
Pi = Pi Vi (Vi ) , then project P
b i and let P denote the vertical concatenation of Pi . These variables are designed
concatenation of P
b and that between P
b and P are easily bounded.
so that the difference between P and P
Our proof then proceeds by first bounding these differences, and then bounding that between P and
? In the following we sketch the proof for the second statement, while the other statement can be
P.
proved by a similar argument. See the appendix for details.
i h
i
h
i h
b 2 + kPXk
b 2 ? kPXk2 + kPXk2 ? kPXk
? 2F .
? 2F = kPXk2F ? kPXk
kPXk2F ? kPXk
F
F
F
F
i
Ps h
b i Xk2 , each of which can be bounded by Lemma 1,
The first term is just i=1 kPi Xk2F ? kP
F
b
since Pi is the SVD truncation of P. The second term can be bounded similarly. The more difficult
(t2 )
(t2 ) >
b
?
part is the third term. Note that
h Pi = Pi Z, Pi =i Pi Z where Z := V (V ) X, leading to
P
s
b i Zk2 ? kPi Zk2 . Although Z is not orthonormal as required by
? 2 =
kPXk2 ? kPXk
kP
F
F
i=1
F
F
Lemma 1, we prove a generalization (Lemma 7 in the appendix) which can be applied to show that
the third term is indeed small.
Application to k-Means Clustering To see the implication, consider the k-means clustering problem. We can first perform any other possible dimension reduction to dimension d0 so that the kmeans cost is preserved up to accuracy , and then run Algorithm disPCA and finally run any
distributed k-means clustering algorithm on the projected data to get a good approximate solution.
For example, in the first step we can set d0 = O(log n/2 ) using a Johnson-Lindenstrauss transform,
or we can perform no reduction and simply use the original data.
As a concrete example, we can use original data (d0 = d), then run Algorithm disPCA, and finally
run the distributed clustering algorithm in [2] which uses any non-distributed ?-approximation algorithm as a subroutine and computes a (1 + )?-approximate solution. The resulting algorithm is
presented in Algorithm 1.
Theorem 5. With probability at least 1 ? ?, Algorithm 1 outputs a (1 + )2 ?-approximate solution
sk
for distributed k-means clustering. The total
communication cost of Algorithm 1 is O( 2 ) vectors
in Rd plus O
3
1 k2
4 ( 2
+ log 1? ) + sk log
sk
?
2
vectors in RO(k/ ) .
Fast Distributed PCA
Subspace Embeddings One can significantly improve the time of the distributed PCA algorithms
by using subspace embeddings, while keeping similar guarantees as in Lemma 4, which suffice for
l2 -error fitting. More precisely, a subspace embedding matrix H ? R`?n for a matrix A ? Rn?d
has the property that for all vectors y ? Rd , kHAyk2 = (1 ? )kAyk2 . Suppose independently,
5
each node vi chooses a random subspace embedding matrix Hi for its local data Pi . Then, they run
Algorithm disPCA on the embedded data {Hi Pi }si=1 instead of on the original data {Pi }si=1 .
The work of [22] pioneered subspace embeddings. The recent fast sparse subspace embeddings [5]
and its optimizations [17, 19] are particularly suitable for large scale sparse data sets, since their
running time is linear in the number of non-zero entries in the data matrix, and they also preserve
the sparsity of the data. The algorithm takes as input an n?d matrix A and a parameter `, and outputs
an ` ? d embedded matrix A0 = HA (the embedded matrix H does need to be built explicitly). The
embedded matrix is constructed as follows: initialize A0 = 0; for each row in A, multiply it by +1
or ?1 with equal probability, then add it to a row in A0 chosen uniformly at random.
The success probability is constant, while we need to set it to be 1 ? ? where ? = ?(1/s). Known
results which preserve the number of non-zero entries of H to be 1 per column increase the dimension of H by a factor of s. To avoid this, we propose an approach to boost the success probability
by computing O(log 1? ) independent embeddings, each with only constant success probability, and
then run a cross validation style procedure to find one which succeeds with probability 1 ? ?. More
precisely, we compute the SVD of all embedded matrices Hj A = Uj ?j Vj> , and find a j ? [r]
such that for at least half of the indices j 0 6= j, all singular values of ?j Vj> Vj 0 ?>
j 0 are in [1 ? O()]
(see Algorithm 4 in the appendix). The reason why such an embedding Hj A succeeds with high
probability is as follows. Any two successful embeddings Hj A and Hj 0 A, by definition, satisfy
that kHj Axk22 = (1 ? O())kHj 0 Axk22 for all x, which we show is equivalent to passing the test
on the singular values. Since with probability at least 1 ? ?, 9/10 fraction of the embeddings are
successful, it follows that the one we choose is successful with probability 1 ? ?.
Randomized SVD The exact SVD of an n ? d matrix is impractical in the case when n or d
is large. Here we show that the randomized SVD algorithm from [11] can be applied to speed
up the computation without compromising the quality of the solution much. We need to use their
specific form of randomized SVD since the error is with respect to the spectral norm, rather than the
Frobenius norm, and so can be much smaller as needed by our applications.
The algorithm first probes the row space of the ` ? d input matrix A with an ` ? 2t random matrix
? and orthogonalizes the image of ? to get a basis Q (i.e., QR-factorize A> ?); projects the data to
this basis and computes the SVD factorization on the smaller matrix AQ. It also performs q power
iterations to push the basis towards the top t singular vectors.
Fast Distributed PCA for l2 -Error Fitting We modify Algorithm disPCA by first having each
node do a subspace embedding locally, then replace each SVD invocation with a randomized SVD
invocation. We thus arrive at Algorithm 2. For `2 -error fitting problems, by combining approximation guarantees of the randomized techniques with that of distributed PCA, we are able to prove:
Theorem 6. Suppose Algorithm 2 takes ? (0, 1/2], t1 = t2 = O(max k2 , log ?s ), ` =
2
O( d2 ), q = O(max{log d , log sk
}) as input, and sets the failure probability of each local sub0
? = PVV> . Then with probability at least 1 ? ?, there exists
space embedding to ? = ?/2s. Let P
a constant c0 ? 0, such that for any set of k points L,
? L) + c0 ? (1 + )d2 (P, L) + kPXk2F
(1 ? )d2 (P, L) ? kPXk2F ? d2 (P,
2
where X is an orthonormal
matrix whose
h 3 columns
i span L. Thetotal communication is O(skd/ )
2 2
and the total time is O nnz(P) + s d4k + k 6d log d log sk
? .
? enjoys the close projection property as in Lemma 4, i.e.,
Proof Sketch: It suffices to show that P
? 2 ? 0 and kPXk2 ? kPXk
? 2 ? 0 for any orthonormal matrix whose columns
kPX ? PXk
F
F
F
span a low dimensional subspace. Note that Algorithm 2 is just running Algorithm disPCA (with
? enjoys
randomized SVD) on TP where T = diag(H1 , H2 , . . . , Hs ), so we first show that TP
this property. But now exact SVD is replaced with randomized SVD, for which we need to use
the spectral error bound to argue that the error introduced is small. More precisely, for a matrix A
b computed by randomized SVD, it is guaranteed that the spectral norm of
and its SVD truncation A
b
b
A ? A is small, then k(A ? A)Xk
F is small for any X with small Frobenius norm, in particular,
?
the orthonormal basis spanning a low dimensional subspace. This then suffices to guarantee TP
? enjoys this property as
enjoys the close projection property. Given this, it suffices to show that P
? which follows from the definition of a subspace embedding.
TP,
6
Algorithm 2 Fast Distributed PCA for l2 -Error Fitting
Input: {Pi }si=1 ; parameters t1 , t2 for Algorithm disPCA; `, q for randomized techniques.
1: for each node vi ? V do
2:
Compute subspace embedding P0i = Hi Pi .
3: end for
4: Run Algorithm disPCA on {P0i }si=1 to get V, where the SVD is randomized.
Output: V.
4
Experiments
Our focus is to show the randomized techniques used in Algorithm 2 reduce the time taken significantly without compromising the quality of the solution. We perform experiments for three tasks:
rank-r approximation, k-means clustering and principal component regression (PCR).
Datasets We choose the following real world datasets from UCI repository [1] for our experiments.
For low rank approximation and k-means clustering, we choose two medium size datasets NewsGroups (18774 ? 61188) and MNIST (70000 ? 784), and two large-scale Bag-of-Words datasets:
NYTimes news articles (BOWnytimes) (300000 ? 102660) and PubMed abstracts (BOWpubmed)
(8200000 ? 141043). We use r = 10 for rank-r approximation and k = 10 for k-means clustering. For PCR, we use MNIST and further choose YearPredictionMSD (515345 ? 90), CTslices
(53500 ? 386), and a large dataset MNIST8m (800000 ? 784).
Experimental Methodology The algorithms are evaluated on a star network. The number of nodes
is s = 25 for medium-size datasets, and s = 100 for the larger ones. We distribute the data over
the nodes using a weighted partition, where each point is distributed to the nodes with probability
proportional to the node?s weight chosen from the power law with parameter ? = 2.
For each projection dimension, we first construct the projected data using distributed PCA. For low
rank approximation, we report the ratio between the cost of the obtained solution to that of the
solution computed by SVD on the global data. For k-means, we run the algorithm in [2] (with
Lloyd?s method as a subroutine) on the projected data to get a solution. Then we report the ratio
between the cost of the above solution to that of a solution obtained by running Lloyd?s method
directly on the global data. For PCR, we perform regression on the projected data to get a solution.
Then we report the ratio between the error of the above solution to that of a solution obtained by
PCR directly on the global data. We stop the algorihtm if it takes more than 24 hours. For each
projection dimension and each algorithm with randomness, the average ratio over 5 runs is reported.
Results Figure 2 shows the results for low rank approximation. We observe that the error of the fast
distributed PCA is comparable to that of the exact solution computed directly on the global data.
This is also observed for distributed PCA with one or none of subspace embedding and randomized
SVD. Furthermore, the error of the fast PCA is comparable to that of normal PCA, which means
that the speedup techniques merely affects the accuracy of the solution. The second row shows the
computational time, which suggests a significant decrease in the time taken to run the fast distributed
PCA. For example, on NewsGroups, the time of the fast distributed PCA improves over that of
normal distributed PCA by a factor between 10 to 100. On the large dataset BOWpubmed, the
normal PCA takes too long to finish and no results are presented, while the speedup versions produce
good results in reasonable time. The use of the randomized techniques gives us a good performance
improvement while keeping the solution quality almost the same.
Figure 3 and Figure 4 show the results for k-means clustering and PCR respectively. Similar to
that for low rank approximation, we observe that the distributed solutions are almost as good as that
computed directly on the global data, and the speedup merely affects the solution quality. We again
observe a huge decrease in the running time by the speedup techniques.
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, and AFOSR grant FA9550-091-0538. David Woodruff would like to acknowledge 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.
7
1.14
1.2
Fast_PCA
Only_Subspace
Only_Randomized
Normal_PCA
1.12
1.08
1.14
1.07
1.12
1.16
1.06
1.1
1.05
1.12
1.08
1.08
1.04
1.06
1.06
1.08
1.03
1.04
1.04
1.02
1.04
1.02
1.02
1.01
1
5
10
15
20
1
14
25
(a) NewsGroups
24
34
44
54
10
(b) MNIST
4
25
1
10
30
15
20
25
30
(d) BOWpubmed
5
10
10
Fast_PCA
Only_Subspace
Only_Randomized
Normal_PCA
20
(c) BOWnytimes
3
10
15
4.9
10
3
2
10
10
4.8
10
4
10
2
1
10
10
4.7
10
1
10
5
0
10
15
20
25
10
14
(e) NewsGroups
3
24
34
44
10
10
54
(f) MNIST
15
20
25
10
30
(g) BOWnytimes
15
20
25
30
(h) BOWpubmed
Figure 2: Low rank approximation. First row: error (normalized by baseline) v.s. projection
dimension. Second row: time v.s. projection dimension.
1.14
1.1
Fast_PCA
Only_Randomized
Only_Subspace
Normal_PCA
1.08
1.135
1.1
1.115
1.08
1.095
1.06
1.075
1.04
1.055
1.02
1.12
1.1
1.08
1.06
1.06
1.04
1.04
1.02
1.02
5
10
15
20
1
14
25
(a) NewsGroups
24
34
44
1.035
10
54
(b) MNIST
20
25
1
10
30
(c) BOWnytimes
3
4
15
20
25
30
(d) BOWpubmed
4
10
10
15
10
Fast_PCA
Only_Subspace
Only_Randomized
Normal_PCA
3
3
10
10
2
10
4
10
2
2
10
10
1
10
5
1
1
10
15
20
25
10
14
(e) NewsGroups
24
34
44
10
10
54
(f) MNIST
15
20
25
10
30
(g) BOWnytimes
15
20
25
30
(h) BOWpubmed
Figure 3: k-means clustering. First row: cost (normalized by baseline) v.s. projection dimension.
Second row: time v.s. projection dimension.
1.012
Fast_PCA
Only_Subspace
Only_Randomized
Normal_PCA
1.01
1.008
1.12
1.014
1.11
1.012
1.1
1.01
1.09
1.008
1.08
1.006
1.07
1.004
1.06
1.002
1.003
1.0025
1.002
1.006
1.0015
1.004
1.002
14
24
34
44
54
(a) MNIST
1.05
10
15
20
25
1
10
30
(b) YearPredictionMSD
20
25
30
1.001
14
(c) CTslices
34
44
54
4
10
10
24
(d) MNIST8m
2
3
3
10
15
10
Fast_PCA
Only_Subspace
Only_Randomized
Normal_PCA
2
2
10
10
1
3
10
1
10
0
10
14
10
1
10
34
(e) MNIST
44
54
10
10
2
0
0
24
15
20
25
10
10
30
(f) YearPredictionMSD
15
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25
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30
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14
24
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(h) MNIST8m
Figure 4: PCR. First row: error (normalized by baseline) v.s. projection dimension. Second row:
time v.s. projection dimension.
8
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9
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5,103 | 562 | A Neural Network for Motion Detection of
Drift-Balanced Stimuli
Hilary Tunley*
School of Cognitive and Computer Sciences
Sussex University
Brighton, England.
Abstract
This paper briefly describes an artificial neural network for preattentive
visual processing. The network is capable of determiuing image motioll in
a type of stimulus which defeats most popular methods of motion detect.ion
- a subset of second-order visual motion stimuli known as drift-balanced
stimuli(DBS). The processing st.ages of the network described in this paper
are integratable into a model capable of simultaneous motion extractioll.
edge detection, and the determination of occlusion.
1
INTRODUCTION
Previous methods of motion detection have generally been based on one of
two underlying approaches: correlation; and gradient-filter. Probably the best
known example of the correlation approach is th(! Reichardt movement detEctor
[Reiehardt 1961]. The gradient-filter (GF) approach underlies the work of AdElson
and Bergen [Adelson 1985], and Heeger [Heeger L9H8], amongst others.
These motion-detecting methods eannot track DBS, because DBS Jack essential
componellts of information needed by such methods. Both the correlation and
GF approaches impose constraints on the input stimuli. Throughout the image
sequence, correlation methods require information that is spatiotemporally correlatable; and GF motion detectors assume temporally constant spatial gradi,'nts.
"Current address:
University.
714
Experimental Psychology, School of Biological Sciences, Sussex
A Neural Network for Motion Detection of Drift-Balanced Stimuli
The network discussed here does not impose such constraints. Instead, it extracts
motion energy and exploits the spatial coherence of movement (defined more formally in the Gestalt theory of common fait [Koffka 1935]) to achieve tracking.
The remainder of this paper discusses DBS image sequences, then correlation methods, then GF methods in more detail, followed by a qualitative description of this
network which can process DBS.
2
SECOND-ORDER AND DRIFT-BALANCED STIMULI
There has been a lot of recent interest in second-order visual stimuli , and DBS in
particular ([Chubb 1989, Landy 1991]). DBS are stimuli which give a clear percept
of directional motion, yet Fourier analysis reveals a lack of coherent motion energy,
or energy present in a direction opposing that of the displacement (hence the term
'drift-balanced '). Examples of DBS include image sequences in which the contrast
polarity of edges present reverses between frames.
A subset of DBS, which are also processpd by the network, are known as microbalanced stimuli (MBS). MBS cont,ain no correlatable features and are driftbalanced at all scales. The MBS image sequences used for this work were created
from a random-dot image in which an area is successively shifted by a constant
displacement between each frame and sim ultaneously re-randomised.
3
3.1
EXISTING METHODS OF MOTION DETECTION
CORRELATION METHODS
Correlation methods perform a local cross-correlation in image space: the matching
of features in local neighbourhoods (depending upon displacement/speed) between
image frames underlies the motion detection. Examples of this method include
[Van Santen 1985J. Most correlation models suffer from noise degradation in that
any noise features extracted by the edge detection are available for spurious correlation .
There has been much recent debate questioning the validity of correlation methods
for modelling human motion detection abilit.ies. In addition to DBS, there is also
increasing psychophysical evidence ([Landy 1991, Mather 1991]) which correlation
methods cannot account for.
These factors suggest that correlation techniques are not suitable for low-level motion processing where no information is available concerning what is moving (as
with MBS). However, correlation is a more plausible method when working with
higher level constructs such as tracking in model-based vision (e.g . [Bray 1990]),
3.2
GRADIENT-FILTER (GF) METHODS
GF methods use a combination of spatial filtering to determine edge positions and
temporal filtering to determine whether such edges are moving. A common assumption used by G F methods is that spatial gradients are constant. A recent method by
Verri [Verri 1990], for example, argu es that flow det.ection is based upon the notion
715
716
Tunley
-
??
?
??
?
? ? ?? ??
~
.
??? ?
??
?
?
T ??
Model
R:
Receptor UnIts - Detect temporal
changes In IMage intensit~
(polarIty-independent)
M:
Motion Units - Detect
distribution of change
iniorMtlon
0:
OcclusIon Units - Detect
changes In .otlon
dIstribution
E:
Edge Units - Detect edges
dlrectl~ from occluslon
Figure 1: The Network (Schematic)
of tracking spatial gradient magnitude and/or direction, and that any variation in
the spatial gradient is due to some form of motion deformation - i.e. rotation,
expansion or shear. Whilst for scenes containing smooth surfaces this is a valid
approximation, it is not the case for second-order stimuli such as DBS.
4
THE NETWORK
A simplified diagram illustrating the basic structure of the network (based upon
earlier work ([Tunley 1990, Tunley 1991a, Tunley 1991b]) is shown in Figure 1
( the edge detection stage is discussed elsewhere ([Tunley 1990, Tunley 1991 b,
Tunley 1992]).
4.1
INPUT RECEPTOR UNITS
The units in the input layer respond to rectified local changes in image intensity
over time. Each unit has a variable adaption rate, resulting in temporal sensitivity
- a fast adaption rate gives a high temporal filtering rate. The main advantages for
this temporal averaging processing are:
? Averaging removes the D.C. component of image intensity. This eliminates problematic gain for motion in high brightness areas of the image.
[Heeger 1988] .
? The random nature of DBS/MBS generation cannot guarantee that each pixel
change is due to local image motion. Local temporal averaging smooths the
A Neural Network for Motion Detection of Drift-Balanced Stimuli
moving regions, thus creating a more coherently structured input for the motion
units.
The input units have a pointwise rectifying response governed by an autoregressive
filter of the following form:
(1 )
where a E [0,1] is a variable which controls the degree of temporal filtering of the
change in input intensity, nand n - 1 are successive image frames, and Rn and In
are the filter output and input, respectively.
The receptor unit responses for two different a values are shown in Figure 2. C\' can
thus be used to alter the amount of motion blur produced for a particular frame
rate, effectively producing a unit with differing velocity sensitivity.
( a)
(b)
Figure 2: Receptor Unit Response: (a) a = 0.3; (b) a = 0.7.
4.2
MOTION UNITS
These units determine the coherence of image changes indicated by corresponding
receptor units. First-order motion produces highly-tuned motion activity - i.e. a
strong response in a particular direction - whilst second-order motion results in less
coherent output.
The operation of a basic motion detector can be described by:
(2)
where !vI is the detector, (if, j') is a point in frame n at a distance d from (i, j),
a point in frame n - 1, in the direction k. Therefore, for coherent motion (i.e.
first-order), in direction k at a speed of d units/frame, as n ---- 00:
(3)
717
718
Tunley
The convergence of motion activity can be seen using an example. The stimulus
sequence used consists of a bar of re-randomising texture moving to the right in
front of a leftward moving background with the same texture (i.e. random dots).
The bar motion is second-order as it contains no correlatable features, whilst the
background consists of a simple first-order shifting of dots between frames. Figures 3, 4 and 5 show two-dimensional images of the leftward motion activity for the
stimulus after 3,4 and 6 frames respectively. The background, which has coherent
leftward movement (at speed d units/frame) is gradually reducing to zero whilst
the microbalanced rightwards-moving bar, remains active. The fact that a non-zero
response is obtained for second-order motion suggests, according to the definition
of Chubb and Sperling [Chubb 1989], that first-order detectors produce no response
to MBS, that this detector is second-order with regard to motion detection.
Figure 3: Leftward Motion Response to Third Frame in Sequence.
HfOL(tlyllmh ~ .4)
.. '
Figure 4: Leftward Motion Response to Fourth Frame.
Hf Ol (llyrlnh ~. 6)
Figure 5: Leftward Motion Response to Sixth Frame.
The motion units in this model are arranged on a hexagonal grid. This grid is
known as a flow web as it allows information to flow, both laterally between units
of the same type, and between the different units in the model (motion, occlusion
or edge). Each flow web unit is represented by three variables - a position (a, b)
and a direction k, which is evenly spaced between 0 and 360 degrees. In this model
each k is an integer between 1 and kmax - the value of kmax can be varied to vary
the sensitivity of the units.
A way of using first-order techniques to discriminate between first and secondorder motions is through the concept of coherence. At any point in the motionprocessed images in Figures 3-5, a measure of the overall variation in motion activity
can be used to distinguish between the motion of the micro-balanced bar and its
background. The motion energy for a detector with displacement d, and orientation
A Neural Network for Motion Detection of Drift-Balanced Stimuli
k, at position (a, b), can be represented by Eabkd. For each motion unit, responding
over distance d, in each cluster the energy present can be defined as:
E
_
abkdn -
mink(Mabkd)
AI
(4)
abkd
where mink(xk) is the minimum value of x found searching over k values. If motion
is coherent, and of approximately the correct speed for the detector M, then as
n -+ 00:
(5)
where k m is in the actual direction of the motion. In reality n need only approach
around 5 for convergence to occur. Also, more importantly, under the same convergence conditions:
(6)
This is due to the fact that the minimum activation value in a group of first-order
detectors at point (a, b) will be the same as the actual value in the direction, km .
By similar reasoning, for non-coherent motion as n -+ 00:
Eabkdn -
(7)
1 'Vk
in other words there is no peak of activity in a given direction . The motion energy
is ambiguous at a large number of points in most images, except at discontinuities
and on well-textured surfaces.
A measure of motion coherence used for the motion units can now be defined as:
Mc( abkd)
=
. Eabkd
E
",", k max
L...k=l
For coherent motion in direction k m as n
(8)
abkd
-+ 00:
(9)
Whilst for second-order motion, also as n -
00:
(10)
Using this approach the total Me activity at each position - regardless of coherence,
or lack of it - is unity. Motion energy is the same in all moving regions, the difference
is in the distribution, or tuning of that energy.
Figures 6, 7 and 8 show how motion coherence allows the flow web structure to
reveal the presence of motion in microbalanced areas whilst not affecting the easily
detected background motion for the stimulus.
719
720
Tunley
Figure 6: Motion Coherence Response to Third Frame
Figure 7: Motion Coherence Response to Fourth Frame
Figure 8: Motion Coherence Response to Sixth Frame
4.3
OCCLUSION UNITS
These units identify discontinuities in second-order motion which are vitally important when computing the direction of that motion . They determine spatial and
temporal changes in motion coherence and can process single or multiple motions at
each image point . Established and newly-activated occlusion units work, through
a gating process, to enhance continuously-displacing surfaces, utilising the concept
of visual inertia.
The implementation details of the occlusion stage of this model are discussed elsewhere [Tunley 1992], but some output from the occlusion units to the above secondorder stimulus are shown in Figures 9 and 10. The figures show how the edges of
the bar can be determined.
References
[Adelson 1985)
E.H. Adelson and J .R. Bergen . Spatiotemporal energy models for
the perception of motion. J. Opt. Soc. Am. 2, 1985 .
[Bray 1990)
A.J . Bray. Tracking objects using image disparities. Image and
Vision Computin,q, 8, 1990.
[Chubb 1989)
C. Chubb and G. Sperling. Second-order motion perception:
Space/time separable mechanisms. In Proc. Workshop on Visual
Motion, Irvine, CA , USA, 1989.
A Neural Network for Motion Detection of Drift-Balanced Stimuli
Figure 9: Occluding Motion Information: Occlusion activity produced by an increase in motion coherence activity.
O( IlynnlJsl . 1")
Figure 10: Occluding Motion Information: Occlusion activity produced by a decrease in motion activity at a point. Some spurious activity is produced due to the
random nature of the second-order motion information.
D.J. Heeger. Optical Flow using spatiotemporal filters. Int. J.
Camp. Vision, 1, 1988.
K. Koffka. Principles of Gestalt Psychology. Harcourt Brace,
[Koffka 1935]
1935.
M.S. Landy, B.A. Dosher, G. Sperling and M.E. Perkins. The
[Landy 1991]
kinetic depth effect and optic flow II: First- and second-order
motion. Vis. Res. 31, 1991.
G. Mather. Personal Communication.
[Mather 1991]
[Reichardt 1961] W. Reichardt. Autocorrelation, a principle for the evaluation of
sensory information by the central nervous system. In W. Rosenblith, editor, Sensory Communications. Wiley NY, 1961.
[Van Santen 1985] J .P.H. Van Santen and G. Sperling. Elaborated Reichardt detectors. J. Opt. Soc. Am. 2, 1985.
H. Tunley. Segmenting Moving Images. In Proc. Int. Neural Net[Tunley 1990]
work Conf (INN C9 0) , Paris, France, 1990.
H. Tunley. Distributed dynamic processing for edge detection. In
[Tunley 1991a]
Proc. British Machine Vision Conf (BMVC91), Glasgow, Scotland, 1991.
H. Tunley. Dynamic segmentation and optic flow extraction. In.
[Tunley 1991b]
Proc. Int. Joint. Conf Neural Networks (IJCNN91) , Seattle,
USA, 1991.
H. Tunley. Sceond-order motion processing: A distributed ap[Tunley 1992]
proach. CSRP 211, School of Cognitive and Computing Sciences,
University of Sussex (forthcoming).
A. Verri, F. Girosi and V. Torre. Differential techniques for optic
[Verri 1990]
flow. J. Opt. Soc. Am. 7, 1990.
[Heeger 1988]
721
Recurrent Eye Tracking Network Using a
Distributed Representation of Image Motion
P. A. Viola
Artificial Intelligence Laboratory
Massachusetts Institute of Technology
S. G. Lisberger
Department of Physiology
W.M. Keck Foundation Center for Integrative Neuroscience
Neuroscience Graduate Program
University of California, San Francisco
T. J. Sejnowski
Salk Institute, Howard Hughes Medical Institute
Department of Biology
University of California, San Diego
Abstract
We have constructed a recurrent network that stabilizes images of a moving
object on the retina of a simulated eye. The structure of the network
was motivated by the organization of the primate visual target tracking
system. The basic components of a complete target tracking system were
simulated, including visual processing, sensory-motor interface, and motor
control. Our model is simpler in structure, function and performance than
the primate system, but many of the complexities inherent in a complete
system are present.
380
Recurrent Eye Tracking Network Using a Distributed Representation of Image Motion
Visual
Processing
Images
V
Target
Estimate of
Retinal
Velocity
Retinotopic
Maps
Motor
Interface
!
Eye
Velocity
I
Motor
Control
V
Eye
->
Figure 1: The overall structure of the visual tracking model.
1
Introduction
The fovea of the primate eye has a high density of photoreceptors. Images that fall
within the fovea are perceived with high resolution. Perception of moving objects
poses a particular problem for the visual system. If the eyes are fixed a moving
image will be blurred. When the image moves out the of the fovea, resolution
decreases. By moving their eyes to foveate and stabilize targets, primates ensure
maximum perceptual resolution. In addition, active target tracking simplifies other
tasks, such as spatial localization and spatial coordinate transformations (Ballard,
1991).
Visual tracking is a feedback process, in which the eyes are moved to stabilize and
foveate the image of a target. Good visual tracking performance depends on accurate estimates of target velocity and a stable feedback controller. Although many
visual tracking systems have been designed by engineers, the primate visual tracking
system has yet to be matched in its ability to perform in complicated environments,
with unrestricted targets, and over a wide variety of target trajectories. The study
of the primate oculomotor system is an important step toward building a system
that can attain primate levels of performance. The model presented here can accurately and stably track a variety of targets over a wide range of trajectories and is
a first step toward achieving this goal.
Our model has four primary components: a model eye, a visual processing network, a motor interface network, and a motor control network (see Figure 1). The
model eye receives a sequence of images from a changing visual world, synthetically
rendered, and generates a time-varying output signal. The retinal signal is sent to
the visual processing network which is similar in function to the motion processing
areas of the visual cortex. The visual processing network constructs a distributed
representation of image velocity. This representation is then used to estimate the
velocity of the target on the retina. The retinal velocity of the target forms the input to the motor control network that drives the eye. The eye responds by rotating,
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Viola, Lisberger, and Sejnowski
Motion Energy Output Unit
Combination
Layer
Space-time
Separable Units
Figure 2: The structure of a motion energy unit. Each space-time separable unit
has a receptive field that covers 16 pixels in space and 16 steps in time (for a total
of 256 inputs). The shaded triangles denote complete projections.
which in turn affects incoming retinal signals.
If these networks function perfectly, eye velocity will match target velocity. Our
model generates smooth eye motions to stabilize smoothly moving targets. It makes
no attempt to foveate the image of a target. In primates, eye motions that foveate
targets are called saccades. Saccadic mechanisms are largely separate from the
smooth eye motion system (Lisberger et. al. 1987). We do not address them here.
In contrast with most engineered systems, our model is adaptive. The networks
used in the model were trained using gradient descent l . This training process
circumvented the need for a separate calibration of the visual tracking system.
2
Visual Processing
INetwork simulations were carried out with the SN2 neural network simulator.
Recurrent Eye Tracking Network Using a Distributed Representation of Image Motion
The middle temporal cortex (area MT) contains cells that are selective for the
direction of visual motion. The neurons in MT are organized into a retinotopic
map and small lesions in this area lead to selective impairment of visual tracking
in the corresponding regions of the visual field (Newsome and Pare, 1988). The
visual processing networks in our model contain directionally-selective processing
units that are arranged in a retinotopic map. The spatio-temporal motion energy
filter of Adelson and Bergen (Adelson and Bergen, 1985) has many of the properties of directionally-selective cortical neurons; it is used as the basis for our visual
processing network. We constructed a four layer time-delay neural network that
implements a motion energy calculation.
A single motion-energy unit can be constructed from four intermediate units having separable spatial and temporal filters. Adelson and Bergen demonstrate that
two spatial filters (of even and odd symmetry) and two temporal filters (temporal
derivatives for fast and slow speeds) are sufficient to detect motion. The filters
are combined to construct 4 intermediate units which project to a single motion
energy unit. Because the spatial and temporal properties of the receptive field are
separable, they can be computed separately and convolved together to produce the
final output. The temporal response is therefore the same throughout the extent of
the spatial receptive field.
In our model, motion energy units are implemented as backpropagation networks.
These units have a receptive field 16 pixels wide over a 16 time step window. Because
the input weights are shared, only 32 parameters were needed for each space-time
separable unit. Four space-time separable units project through a 16 unit combination layer to the output unit (see Figure 2). The entire network can be trained
to approximate a variety of motion-energy filters.
We trained the motion energy network in two different ways: as a single multilayered
network and in stages. Staged training proceded first by training intermediate units,
then, with the intermediate units fixed, by training the three layer network that
combines the intermediate units to produce a single motion energy output. The
output unit is active when a pattern in the appropriate range of spatial frequencies
moves through the receptive field with appropriate velocity. Many such units are
required for a range of velocities, spatial frequencies, and spatial locations. We
use six different types of motion energy units - each tuned to a different temporal
frequency - at each of the central 48 positions of a 64 pixel linear retina. The 6
populations form a distributed, velocity-tuned representation of image motion for
a total of 288 motion energy units.
In addition to the motion energy filters, static spatial frequency filters are also
computed and used in the interface network, one for each band and each position
for a total of 288 units.
We chose an adaptive network rather than a direct motion energy calculation because it allows us to model the dynamic nature of the visual signal with greater
:flexibility. However, this raises complications regarding the set of training images.
Assuming 5 bits of information at each retinal position, there are well over 10 to
the 100th possible input patterns. We explored sine waves, random spots and a
variety of spatial pre-filters, and found low-pass filtered images of moving random
spots worked best. Typically we began the training process from a plausible set of
383
384
Viola, Lisberger, and Sejnowski
weights, rather than from random values, to prevent the network from settling into
an initial local minima. Training proceeded for days until good performance was
obtained on a testing set.
Krauzlis and Lisberger (1989) have predicted that the visual stimulus to the visual
tracking system in the brain contains information about the acceleration and impulse of the target as well as the velocity. Our motion energy networks are sensitive
to target acceleration, producing transients for accelerating stimuli.
3
The Interface Network
The function of the interface is to take the distributed representation of the image
motion and extract a single velocity estimate for the moving object. We use a
relatively simple method that was adequate for tracking single objects without other
moving distractolS. The activity level of a single motion energy unit is ambiguous.
First, it is necessary for the object to have a feature that is matched to the spatial
frequency ba.ndpass of the motion energy unit. Second, there is an a.llay of units
for each spatial frequency and the object will stimulate only a few of these at any
given time. For instance, a large white object will have no features in its interior; a
unit with its receptive field located in the interior can detect no motion. Conversely,
detectors with receptive fields on the border between the object and the background
will be strongly stimulated.
We use two stages of processing to extract a velocity. In the first stage, the motion
energy in each spatial frequency band is estimated by summing the outputs of the
motion energy filters across the retina weighted by the spatial frequency filter at
each location. The six populations of spatial frequency units each yield one value.
Next, a 6-6-1 feedforward network, trained using backpropagation, predicts target
velocity from these values.
4
The Motor Control Network
In comparison with the visual processing network, the motor control network is quite
small (see Figure 3). The goal of the network is to move the eye to stabilize the
image of the object. The visual processing and interface networks convert images
of the moving target into an estimate for the retinal velocity of the target. This
retinal velocity can be considered a motor error. One approach to reducing this
error is a simple proportional feedback controller, which drives the eye at a velocity
proportional to the error. There is a large, 50-100 ms delay that occurs during
visual processing in the primate visual system. In the presence of a large delay a
proportional controller will either be inaccurate or unstable. For this reason simple
proportional feedback is not sufficient to control tracking in the primate. Tracking
can be made stable and accurate by including an internal positive feedback pathway
to prevent instability while preserving accuracy (Robinson, 1971).
The motor control network was based on a model of the primate visual tracking
motor control system by Lisberger and Sejnowski (1992). This recurrent artificial
neural network includes both the smooth visual tracking system and the vestibuloocular system, which is important for compensating head movements. We use a
Recurrent Eye Tracking Network Using a Distributed Representation of Image Morion
Flocculus
Target
Retinal
Velocity
-?~~-?~lggl? ~~
l
Igl--t. . ~ -.. . .
--1...
Brain
Stem
Eye Velocity
Motor
Neurons
Figure 3: The structure of the recurrent network. Each circle is a unit. Units within
a box are not interconnected and all units between boxes were fully interconnected
as indicated by the arrows.
simpler version of that model that does not have vestibular inputs. The network is
constructed from units with continuous smooth temporal responses. The state of a
unit is a function of previous inputs and previous state:
Bj(t + ~t) = (1 -
T~t)Bj(t)
+ IT~t
where Bj(t) is the state of unit j at time t, T is a time constant and I is the
sigmoided sum of the weighted pre-synaptic activities. The resulting network is
capable of smooth responses to inputs.
The motor control network has 12 units, each with a time constant of 5 ms (except
for a few units with longer delay). There is a time delay of 50 ms between the
interface network and control network. (see Figure 3). The input to the network
is retinal target velocity, the output is eye velocity. The motor control network is
trained to track a target in the presence of the visual delay.
The motor control network contains a positive feedback loop that is necessary to
maintain accurate tracking even when the error signal falls to zero. The overall
control network also contains a negative feedback loop since the output of the network affects subsequent inputs. The gradient descent optimization procedure uses
the relationship between the output and the input during training-this relationship can be considered a model of the plant. It should be possible to use the same
approach with more complex plants.
The control network was trained with the visual processing network frozen. A
training example consists of an object trajectory and the goal trajectory for the
eye. A standard recurrent network training paradigm is used to adjust the weights
to minimize the error between actual outputs and desired outputs for step changes
in target velocity.
385
386
Viola, Lisberger, and Sejnowski
I
I
I
,,,
,,
-
--..... --,~
~----
....
--.
I
I
J
Seconds
Figure 4: Response of the eye to a step in target velocity of 30 degrees per second.
The solid line is target velocity, the dashed line is eye velocity. This experiment
was performed with a target that did not appear in the training set.
5
Performance
After training the network on a set of trajectories for a single target, the tracking
performance was equally good on new targets. TYacking is accurate and stable with little tendency to ring (see Figure 4). This good performance is surprising in
the presence of a 50 millisecond delay in the visual feedback signal 2 ? Stable tracking
is not possible without the positive internal feedback loop in the model (eye velocity
signal to the flocculus in Figure 3).
6
Limitations
The system that we have designed is a relatively small one having a one-dimensional
retina only 64 pixels wide. The eye and the target can only move in one dimensionalong the length of the retina. The visual analysis that is performed is not, however,
limited to one dimension. Motion energy filters are easily generalized to a twodimensional retina. Our approach should be extendable to the two-dimensional
tracking problem.
The backgrounds of images that we used for tracking were featureless. The current system cannot distinguish target features from background features. Also, the
interface network was designed to track a single object in the absence of moving
distractors. The next step is to expand this interface to model the attentional
phenomena observed in primate tracking, especially the process of initial target
2We selected time constants, delays, and sampling rates throughout the model to
roughly approximate the time course of the primate visual tracking response. The model
runs on a workstation taking approximately thirty times real-time to complete a processing
step.
Recurrent Eye Tracking Network Using a Distributed Representation of Image Motion
acquisition.
7
Conclusion
In simulations, our eye tracking model performed well. Many additional difficulties
must be addressed, but we feel this system can perform well under real-world realtime constraints. Previous work by Lisberger and Sejnowski (1992) demonstrates
that this visual tracking model can be integrated with inertial eye stabilization-the
vestibulo-ocular reflex. Ultimately, it should be possible to build a physical system
using these design principles.
Every component of the system was designed using network learning techniques.
The visual processing, for example, had a variety of components that were trained
separately and in combinations. The architecture of the networks were based on
the anatomy and physiology of the visual and oculomotor systems. This approach
to reverse engineering is based on the existing knowledge of the flow of information
through the relevant brain pathways.
It should also be possible to use the model to develop and test theories about the
nature of biological visual tracking. This is just a first step toward developing a
realistic model of the primate oculomotor system, but it has already provided useful
predictions for the possible sites of plasticity during gain changes of the vestibuloocular reflex (Lisberger and Sejnowski, 1992).
References
[1] E. H. Adelson and J. R. Bergen. Spatiotemporal energy models of the perception
of motion. Journal of the Optical Society of America, 2(2):284-299, 1985.
[2] D. H. Ballard. Animate vision. Artificial Intelligence, 48:57-86, 1991.
[3] R.J. Krauzlis and S. G. Lis berger. A control systems model of smooth pursuit
eye movements with realistic emergent properties. Neural Computation, 1:116122, 1992.
[4] S. G. Lisberger, E. J. Morris, and L. Tychsen. Ann. Rev. Neurosci., 10:97-129,
1987.
[5] S.G. Lisberger and T.J. Sejnowski. Computational analysis suggests a new
hypothesis for motor learning in the vestibulo-ocular reflex. Submitted for publication., 1992.
[6] W.T. Newsome and E. B. Pare. A selective impairment of motion perception
following lesions of the middle temporal visual area (MT). J. Neuroscience,
8:2201-2211, 1988.
[7] D. A. Robinson. Models of oculomotor neural organization. In P. Bach y Rita
and C. C. Collins, editors, The Control of Eye Movements, page 519. Academic,
New York, 1971.
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5,104 | 5,620 | Generalized Unsupervised Manifold Alignment
Zhen Cui1,2
Hong Chang1
Shiguang Shan1
Xilin Chen1
Key Lab of Intelligent Information Processing of Chinese Academy of Sciences (CAS),
Institute of Computing Technology, CAS, Beijing, China
2
School of Computer Science and Technology, Huaqiao University, Xiamen, China
{zhen.cui,hong.chang}@vipl.ict.ac.cn; {sgshan,xlchen}@ict.ac.cn
1
Abstract
In this paper, we propose a Generalized Unsupervised Manifold Alignment (GUMA) method to build the connections between different but correlated datasets
without any known correspondences. Based on the assumption that datasets of the
same theme usually have similar manifold structures, GUMA is formulated into
an explicit integer optimization problem considering the structure matching and preserving criteria, as well as the feature comparability of the corresponding points
in the mutual embedding space. The main benefits of this model include: (1)
simultaneous discovery and alignment of manifold structures; (2) fully unsupervised matching without any pre-specified correspondences; (3) efficient iterative
alignment without computations in all permutation cases. Experimental results on
dataset matching and real-world applications demonstrate the effectiveness and
the practicability of our manifold alignment method.
1
Introduction
In many machine learning applications, different datasets may reside on different but highly correlated manifolds. Representative scenarios include learning cross visual domains, cross visual views,
cross languages, cross audio and video, and so on. Among them, a key problem in learning with such
datasets is to build connections cross different datasets, or align the underlying (manifold) structures.
By making full use of some priors, such as local geometry structures or manually annotated counterparts, manifold alignment tries to build or strengthen the relationships of different datasets and
ultimately project samples into a mutual embedding space, where the embedded features can be
compared directly. Since samples from different (even heterogeneous) datasets are usually located
in different high dimensional spaces, direct alignment in the original spaces is very difficult. In
contrast, it is easier to align manifolds of lower intrinsic dimensions.
In recent years, manifold alignment becomes increasingly popular in machine learning and computer
vision community. Generally, existing manifold alignment methods fall into two categories, (semi)supervised methods and unsupervised methods. The former methods [15, 26, 19, 33, 28, 30] usually
require some known between-set counterparts as prerequisite for the transformation learning, e.g.,
labels or handcrafted correspondences. Thus they are difficult to generalize to new circumstances,
where the counterparts are unknown or intractable to construct.
In contrast, unsupervised manifold alignment learns from manifold structures and naturally avoids the above problem. With manifold structures characterized by local adjacent weight matrices ,
Wang et al. [29] define the distance between two points respectively from either manifold as the
minimum matching scores of the corresponding weight matrices in all possible structure permutations. Therefore, when K neighbors are considered, the distance computation for any two points
needs K! permutations, a really high computational cost even for a small K. To alleviate this problem, Pei et al. [21] use a B-spline curve to fit each sorted adjacent weight matrix and then compute
matching scores of the curves across manifolds for the subsequent local alignment. Both methods
1
in [29] and [21] divide manifold alignment into two steps, the computation of matching similarities of data points across manifolds and the sequential counterparts finding. However, the two-step
approaches might be defective, as they might lead to inaccurate alignment due to the evolutions of
neighborhood relationships, i.e., the local neighborhood of one point computed in the first step may
change if some of its original neighbors are not aligned in the second step. To address this problem,
Cui et al. [7] propose an affine-invariant sets alignment method by modeling geometry structures
with local reconstruction coefficients.
In this paper, we propose a generalized unsupervised manifold alignment method, which can globally discover and align manifold structures without any pre-specified correspondences, as well as learn
the mutual embedding subspace. In order to jointly learn the transforms into the mutual embedding
space and the correspondences of two manifolds, we integrate the criteria of geometry structure
matching, feature matching and geometry preserving into an explicit quadratic optimization model
with 0-1 integer constraints. An efficient alternate optimization on the alignment and transformations is employed to solve the model. In optimizing the alignment, we extend the Frank-Wolfe (FW)
algorithm [9] for the NP-hard integer quadratic programming. The algorithm approximately seeks
for optima along the path of global convergence on a relaxed convex objective function. Extensive
experiments demonstrate the effectiveness of our proposed method.
Different from previous unsupervised alignment methods such as [29] and [21], our method can
(i) simultaneously discover and align manifold structures without predefining the local neighborhood structures; (ii) perform structure matching globally; and (iii) conduct heterogeneous manifold
alignment well by finding the embedding feature spaces. Besides, our work is partly related to other methods such as kernelized sorting [22], latent variable model [14], etc. However, they mostly
discover counterparts in a latent space without considering geometric structures, although to some
extend the constrained terms used in our model are formally similar to theirs.
2
Problem Description
We first define the notations used in this paper. A lowercase/uppercase letter in bold denotes
a vector/matrix, while non-bold letters denote scalars. Xi? (X?i ) represents the ith row (column) of matrix X. xij or [X]ij denotes the element at the ith row and j th column of matrix
X. 1m?n , 0m?n ? Rm?n are matrices of ones and zeros. In ? Rn?n is an identity matrix.
The superscript | means the transpose of a vector or matrix. tr(?) represents the trace norm.
?X?2F = tr(X| X) designates the Frobenius norm. vec(X) denotes the vectorization of matrix
X in columns. diag(X) is the diagonalization on matrix X, and diag(x) returns a diagonal matrix
of the diagonal elements x. X ? Z and X ? Z denote the Kronecker and Hadamard products,
respectively.
Let X ? Rdx ?nx and Z ? Rdz ?nz denote two datasets, residing in two different manifolds Mx
and Mz , where dx (dz ) and nx (nz ) are respectively the dimensionalities and cardinalities of the
datasets. Without loss of generality, we suppose nx ? nz . The goal of unsupervised manifold
alignment is to build connections between X and Z without any pre-specified correspondences. To
this end, we define a 0-1 integer matrix F ? {0, 1}nx ?nz to mark the correspondences between X
and Z. [F]ij = 1 means that the ith point of X and the j th point of Z are a counterpart. If all
counterparts are limited to one-to-one, the set of integer matrices F can be defined as
? = {F|F ? {0, 1}nx ?nz , F1nz = 1nx , 1|nx F ? 1|nz , nx ? nz }.
(1)
nx ?= nz means a partial permutation. Meanwhile, we expect to learn the lower dimensional intrinsic
representations for both datasets through explicit linear projections, Px ? Rd?dx and Pz ? Rd?dz ,
from the two datasets to a mutual embedding space M. Therefore, the correspondence matrix F
as well as the embedding projections Px and Pz are what we need to learn to achieve generalized
unsupervised manifold alignment.
3
The Model
Aligning two manifolds without any annotations is not a trivial work, especially for two heterogeneous datasets. Even so, we can still make use of the similarities between the manifolds in geometry
structures and intrinsic representations to build the alignment. Specifically, we have three intuitive
2
observations to explore. First, manifolds under the same theme, e.g., the same action sequences of
different persons, usually imply a certain similarity in geometry structures. Second, the embeddings
of corresponding points from different manifolds should be as close as possible. Third, the geometry
structures of both manifolds should be preserved respectively in the mutual embedding space. Based
on these intuitions, we proposed an optimization objective for generalized unsupervised manifold
alignment.
Overall objective function
Following the above analysis, we formulate unsupervised manifold alignment into an optimization
problem with integer constraints,
min
Es + ?f Ef + ?p Ep
(2)
Px ,Pz ,F
s.t.
F ? ?, Px , Pz ? ?,
where ?f , ?p are the balance parameters, ? is a constraint to avoid trivial solutions for Px and Pz ,
Es , Ef and Ep are three terms respectively measuring the degree of geometry matching, feature
matching and geometry preserving, which will be detailed individually in the following text.
Es : Geometry matching term
To build correspondences between two manifolds, they should be first geometrically aligned. Therefore, discovering the geometrical structure of either manifold should be the first task. For this propose, graph with weighted edges can be exploited to characterize the topological structure of manifold, e.g., via graph adjacency matrices Kx , Kz of datasets X and Z, which are usually non-negative
and symmetric if not considering directions of edges. In the literatures of manifold learning, many
methods have been proposed to construct these adjacency matrices locally, e.g., via heat kernel function [2]. However, in the context of manifold alignment, there might be partial alignment cases, in
which some points on one manifold might not be corresponded to any points on the other manifold.
Thus these unmatched points should be detected out, and not involved in the computation of the
geometry relationship. To address this problem, we attempt to characterizes the global manifold
geometry structure by computing the full adjacency matrix, i.e., [K]ij = d(X?i , X?j ), where d is
geodesic distance for general cases or Euclidean distance for flat manifolds. Note that, in order to
reduce the effect of data scales, X and Z are respectively normalized to have unit standard deviation.
The degree of manifold matching in global geometry structures is then formulated as the following
energy term,
Es = ?Kx ? FKz F| ?2F ,
(3)
where F ? ? is the (partial) correspondence matrix defined in Eqn.(1).
Ef : Feature matching term
Given two datasets X and Z, the aligned data points should have similar intrinsic feature representations in the mutual embedding space M. Thus we can formulate the feature matching term
as,
Ef = ?P|x X ? P|z ZF| ?2F ,
(4)
where Px and Pz are the embedding projections respectively for X and Z. They can also be extended to implicit nonlinear projections through kernel tricks. This term penalizes the divergence of
intrinsic features of aligned points in the embedding space M.
Ep : Geometry preserving term
In unrolling the manifold to the mutual embedding space, the local neighborhood relationship of
either manifold is not expected to destroyed. In other words, the local geometry of either manifold
should be well preserved to avoid information loss. As done in many manifold learning algorithms
[23, 2], we construct the local adjacency weight matrices Wx and Wz respectively for the datasets
X and Z. Then, the geometry preserving term is defined as
?
?
x
z
Ep =
?P|x (xi ?xj )?2 wij
+
?P|z (zi ?zj )?2 wij
= tr(P|x XLx X| Px + Pz ZLz Z| Pz ), (5)
i,j
i,j
x
z
wij
(wij
)
where
is the weight between the ith point?
and the j th ?
point in X (Z), Lx and Lz
x
are the graph Laplacian matrices, with Lx = diag([ j w1j
, . . . , j wnxx j ]) ? Wx and Lz =
? z
?
diag([ j w1j
, . . . , j wnz z j ]) ? Wz .
3
4
Efficient Optimization
Solving the objective function (2) is difficult due to multiple indecomposable variables and integer
constraints. Here we propose an efficient approximate solution via alternate optimization. Specifically, the objective function (2) is decomposed into two submodels, corresponding to the optimizations of the integer matrix F and the projection matrices Px , Pz , respectively. With Px and
Pz fixed, we can get a submodel by solving a non-convex quadratic integer programming, whose
approximate solution is computed along the gradient-descent path of a relaxed convex model by
extending the Frank-Wolfe algorithm [9]. When fixing F, an analytic solution can be obtained for
Px and Pz . The two submodels are alternately optimized until convergence to get the final solution.
4.1
Learning Alignment
When fixing Px Pz , the original problem reduces to minimize the following function,
min ?0 (F) = Es + ?f Ef .
F??
(6)
b = P| X and Z
b = P| Z denote the transformed features. After a series of derivation, the
Let X
z
x
objective function can be rewritten as
min ?0 (F) = ?Kx F ? FKz ?2F + tr(F| 11| FKzz ) + tr(F| B),
F??
(7)
b ? Z)
b ? 2X
b | Z)
b ? 11| Kzz . This quadratic alignment
where Kzz = Kz ? Kz and B = ?f (11| (Z
problem is NP-hard with n! enumerations under an exhaustive search strategy. To get effective and
efficient solution, we relax this optimization problem under the framework of Frank-Wolfe (FW)
algorithm [9], which is designed for convex models over a compact convex set. Concretely, we have
following two modifications:
(i) Relax ? into a compact convex set. As the set of 0-1 integer matrices ? is not closed, we can
relax it to a compact closed set by using right stochastic matrices [3] as
?? = {F|F ? 0, F1nz = 1nx , 1|nx F ? 1|nz , nx ? nz }.
(8)
Obviously, ?? is a compact convex set.
(ii) Relax the objective function ?0 into a convex function. As ?0 is non-convex, its optimization
easily suffers from local optima. To avoid local optima in the optimization, we can incorporate an
auxiliary function ?(F) = ? tr(F| F), with ? = nx ? max{? min(eig (Kzz )), 0}, into ?0 and get
the new objective as
?(F)
=
?Kx F ? FKz ?2 + tr(F| 11| FKzz + ?F| F) + tr(F| B).
(9)
In Eqn.(9), the first term is positive definite quadratic form for variable vec(F), and the Hessian
matrix of the second term is 2(K|zz ? (11| ) + ?I) which is also positive definite. Therefore, the
new objective function ? is convex over the convex set ?? . Moreover, the solutions from minimizing
?0 and ? over the integer constraint F ? ? are consistent because ?(F) is a constant.
The extended FW algorithm is summarized in Alg.1, which iteratively projects the one-order approximate solution of ? into ?. In step (4), the optimized solution is obtained using the KuhnCMunkres (KM) algorithm in the 0-1 integer space [20], which makes the solution of the relaxed
objective function ? equal to that of the original objective ?0 . Meanwhile, the continuous solution
path is gradually descending in steps (5)?(11) due to the convexity of function ?, thus local optima
is avoided unlike the original non-convex function over the integer space ?. Furthermore, it can
be proved that the objective value ?(Fk ) is non-increasing at each iteration and {F1 , F2 , . . .} will
converge into a fixed point.
4.2
Learning Transformations
When fixing F, the embedding transforms can be obtained by minimizing the following function,
Ec +?p Ep = tr (P|x X(?f I+?p Lx )X| Px +P|z Z(?f F| F+?p Lz )Z| Pz ?2?f P|x XFZ| Pz ) . (10)
4
Algorithm 1 Manifold alignment
b Z,
b F0
Input: Kx , Kz , X,
1: Initialize: F? = F0 , k = 0.
2: repeat
3:
Computer the gradient of ? w.r.t. Fk :
?(Fk ) = 2(K|x Kx Fk + Fk Kz K|z ? 2K|x Fk Kz + 11| Fk Kzz + ?Fk ) + B;
4:
Find an optimal alignment at the current solution Fk by minizing one-order Taylor expansion
of the objective function ?, i.e., H = arg min tr(?(Fk )| F) using the KM algorithm;
F??
5:
6:
7:
8:
if ?(H) < ?(Fk ) then
F? = Fk+1 = H;
else
Find the optimal step ? = arg min ?(Fk + ?(H ? Fk ));
0???1
9:
10:
Fk+1 = Fk + ?(H ? Fk );
F? = arg min ?(F);
F?{H,F? }
11:
end if
12:
k = k + 1;
13: until ??(Fk+1 ) ? ?(Fk )? < ?.
Output: F? .
To avoid trivial solutions of Px , Pz , we centralize X, Z and reformulate the optimization problem
by considering the rotation-invariant constraints:
max
Px ,Pz
s.t.
tr (P|x XFZ| Pz ) ,
(11)
P|x X(?f I + ?p Lx )X| Px = I,
P|z Z(?f F| F + ?p Lz )Z| Pz = I.
The above problem can be solved analytically by eigenvalue decomposition like Canonical Correlation Analysis (CCA) [16].
4.3
Algorithm Analysis
By looping the above two steps, i.e., alternating optimization on the correspondence matrix F and the
embedding maps Px , Pz , we can reach a feasible solution just like many block-coordinate descent
methods. The computational cost mainly lies in learning alignment, i.e., the optimization steps in
Alg.1. In Alg.1, the time complexity of KM algorithm for linear integer optimization is O(n3z ). As
the Frank-Wolfe method has a convergence rate of O(1/k) with k iterations, the time cost of Alg.1
is about O( 1? n3z ), where ? is the threshold in step (13) of Alg.1. If the whole GUMA algorithm
(please see the auxiliary file) needs to iterate t times, the cost of whole algorithm will be O( 1? tn3z ).
In our experiments, only a few t and k iterations are required to achieve the satisfactory solution.
5
Experiment
To validate the effectiveness of the proposed manifold alignment method, we first conduct two manifold alignment experiments on dataset matching, including the alignment of face image sets across
different appearance variations and structure matching of protein sequences. Further applications are also performed on video face recognition and visual domain adaptation to demonstrate the
practicability of the proposed method.
The main parameters of our method are the balance parameters ?f , ?p , which are simply set to 1.
In the geometry preserving term, we set the nearest neighbor number K = 5 and the heat kernel
parameter to 1. The embedding dimension d is set to the minimal rank of two sets minus 1.
5.1
GUMA for Set Matching
First, we perform alignment of face image sets containing different appearance variations in poses,
expressions, illuminations and so on. In this experiment, the goal is to connect corresponding face
5
images of different persons but with the same poses/expression. Here we use Multi-PIE database
[13]. We choose totally 29,400 face images of the first 100 subjects in the dataset, which cover 7
poses with yaw within [?45? , +45? ](15? intervals), different expressions and illuminations across 3
sessions. These faces are cropped and normalized into 60?40 pixels with eyes at fixed locations. To
accelerate the alignment, their dimensions are further reduced by PCA with 90% energy preserved.
The quantitative matching results1 on pose/expression matching are shown in Fig.1, which contains
the matching accuracy2 of poses (Fig.1(a)), expressions (Fig.1(b)) and their combination (Fig.1(c)).
We compare our method with two state-of-the-art methods, Wang?s [29] and Pei?s [21]. Moreover,
the results of using only feature matching or structure matching are also reported respectively, which
are actually special cases of our method. Here we briefly name them as GUMA(F)/GUMA(S), respectively corresponding to the feature/structure matching. As shown in Fig.1, we have the following observations:
(1) Manifold alignment benefits from manifold structures as well as sample features. Although
features contribute more to the performance in this dataset, manifold structures also play an
important role in alignment. Actually, their relative contributions may be different with different
datasets, as the following experiments on protein sequence alignment indicate that manifold
structures alone can achieve a good performance. Anyway, combining both manifold structures
and sample features promotes the performance more than 15%.
(2) Compared with the other two manifold alignment methods, Wang?s [29] and Pei?s [21], the proposed method achieves better performance, which may be attributed to the synergy of global
structure matching and feature matching. It is also clear that Wang?s method achieves relatively
worse performance, which we conjecture can be ascribed to the use of only the geometric similarity. This might also account for its similar performance to GUMA(S), which also makes uses
of structure information only.
(3) Pose matching is easier than expression matching in the alignment task of face image sets. This
also follows our intuition that poses usually vary more dramatic than subtle face expressions.
Further, the task combining poses and expressions (as shown in Fig.1(c)) is more difficult than
either single task.
60
40
Wang?s
Pei?s
GUMA(F)
GUMA(S)
GUMA
20
0
(a) Pose matching
80
70
70
60
60
50
40
30
Wang?s
Pei?s
GUMA(F)
GUMA(S)
GUMA
20
10
0
matching accuracy (%)
80
matching accuracy (%)
matching accuracy (%)
100
50
40
30
Wang?s
Pei?s
GUMA(F)
GUMA(S)
GUMA
20
10
0
(b) Exp. matching
(c) Pose & exp. matching
Figure 1: Alignment accuracy of face image sets from Multi-PIE [13].
Besides, we also compare with two representative semi-supervised alignment methods [15, 28] to
investigate?how much user labeling is need to reach a performance comparable to our GUMA
method??. In semi-supervised cases, we randomly choose some counterparts from two given sets
as labeled data, and keep the remaining samples unlabeled. For both methods, 20%?30% data is
required to be labeled in pose matching, and 40%?50% is required in expression and union matching. The high proportional labeling for the latter case may be attributed to the extremely subtle face
expressions, for which first-order feature comparisons in both methods are not be effective enough.
Next we illustrate how our method works by aligning the structures of two manifolds. We choose
manifold data from bioinformatics domain [28]. The structure matching of Glutaredoxin protein
PDB-1G7O is used to validate our method, where the protein molecule has 215 amino acids. As
shown in Fig.2, we provide the alignment results in 3D subspace of two sequences, 1G7O-10 and
1G7O-21. More results can be found in the auxiliary file. Wang?s method [29] reaches a decent
matching result by only using local structure matching, but our method can achieve even better
performance by assorting to sample features and global manifold structures.
1
2
Some aligned examples can be found in the auxiliary file.
Matching accuracy = #(correct matching pairs in testing) #(ground-truth matching pairs).
6
3D
3D
3D
20
20
20
10
10
10
0
0
0
?10
?20
20
?10
?20
20
?10
20
20
50
0
0
?20 ?50
20
0
0
0
0
?20 ?20
?20 ?20
(a) Pei?s[21]
(b) Wang?s[29]
(c) GUMA
Figure 2: The structure alignment results of two protein sequences, 1G7O-10 and 1G7O-21.
5.2
GUMA for Video-based Face Verification
In the task of video face verification, we need to judge whether a pair of videos are from the same
person. Here we use the recent published YouTube faces dataset [32], which contains 3,425 clips
downloaded from YouTube. It is usually used to validate the performance of video-based face
recognition algorithms. Following the settings in [5], we normalize the face region sub-images to
40?24 pixels and then use histogram equalization to remove some lighting effect. For verification,
we first align two videos by GUMA and then accumulate Euclidean distances of the counterparts as
their dissimilarity. This method, without use of any label information, is named as GUMA(un). After
alignment, CCA may be used to learn discriminant features by using training pairs, which is named
as GUMA(su). Besides, we compare our algorithms with some classic video-based face recognition
methods, including MSM[34], MMD[31], AHISD[4], CHISD[4], SANP[17] and DCC[18]. For the
implementation of these methods, we use the source codes released by the authors and report the best
results with parameter tuning as described in their papers. The accuracy comparisons are reported
in Table 1. In the ?Unaligned? case, we accumulate the similarities of all combinatorial pairs across
two sequences as the distance. We can observe that the alignment process promotes the performance
to 65.74% from 61.80%. In the supervised case, GUMA(su) significantly surpasses the most related
DCC method, which learns discriminant features by using CCA from the view of subspace.
Table 1: The comparisons on YouTube faces dataset (%).
Method
MSM[34] MMD[31] AHISD[4] CHISD[4] SANP[17] DCC[18] Unaligned GUMA(un) GUMA(su)
Mean Accuracy 62.54
64.96
66.50
66.24
63.74
70.84
61.80
65.74
75.00
Standard Error ?1.47
?1.00
?2.03
?1.70
?1.69
?1.57
?2.29
?1.81
?1.56
5.3
GUMA for Visual Domain Adaptation
To further validate the proposed method, we also apply it to visual domain adaptation task, which
generally needs to discover the relationship between the samples of the source domain and those of
the target domain. Here we consider unsupervised domain adaptation scenario, where the labels of
all the target examples are unknown. Given a pair of source domain and target domain, we attempt
to use GUMA to align two domains and meanwhile find their embedding space. In the embedding
space, we classify the unlabeled examples of the target domain.
We use four public datasets, Amazon, Webcam, and DSLR collected in [24], and Caltech-256 [12].
Following the protocol in [24, 11, 10, 6], we extract SURF features [1] and encode each image with
800-bin token frequency feature by using a pre-trained codebook from Amazon images. The features
are further normalized and z-scored with zero mean and unit standard deviation per dimension. Each
dataset is regarded as one domain, so in total 12 settings of domain adaptation are formed. In the
source domain, 20 examples (resp. 8 examples) per class are selected randomly as labeled data from
Amazon, Webcam and Caltech (resp. DSLR). All the examples in the target domain are used as
unlabeled data and need to predict their labels as in [11, 10]. For all the settings, we conduct 20
rounds of experiments with different randomly selected examples.
We compare the proposed method with five baselines, OriFea, Sampling Geodesic Flow (SGF) [11],
Geodesic Flow Kernel (GFK) [10], Information Theoretical Learning (ITL) [25] and Subspace
Alignment (SA) [8]. Among them, the latter four methods are the state-of-the-art unsupervised
domain adaptation methods proposed recently. OriFea uses the original features; SGF and its extended version GFK try to learn invariant features by interpolating intermediate domains between
source and target domains; ITL is a recently proposed unsupervised domain adaptation method; and
7
38
45
45
36
40
34
40
OriFea
SGF(PCA)
SGF(PLS)
GFK(PCA)
GFK(PLS)
ITL
SA
GUMA
30
25
20
35
30
OriFea
SGF(PCA)
SGF(PLS)
GFK(PCA)
GFK(PLS)
ITL
SA
GUMA
25
20
(a) C?A
35
30
OriFea
SGF(PCA)
SGF(PLS)
GFK(PCA)
GFK(PLS)
ITL
SA
GUMA
25
20
(b) C?W
Accuracy(%)
35
Accuracy(%)
32
Accuracy(%)
Accuracy(%)
40
30
28
OriFea
SGF(PCA)
SGF(PLS)
GFK(PCA)
GFK(PLS)
ITL
SA
GUMA
26
24
22
20
18
(c) C?D
(d) A?C
35
36
35
35
34
30
24
22
20
20
15
(e) A?W
15
36
38
34
36
OriFea
SGF(PCA)
SGF(PLS)
GFK(PCA)
GFK(PLS)
ITL
SA
GUMA
55
50
45
(i) W?D
25
OriFea
SGF(PCA)
SGF(PLS)
GFK(PCA)
GFK(PLS)
ITL
SA
GUMA
20
15
(h) W?A
85
80
34
30
Accuracy(%)
60
Accuracy(%)
65
30
(g) W?C
32
70
OriFea
SGF(PCA)
SGF(PLS)
GFK(PCA)
GFK(PLS)
ITL
SA
GUMA
20
(f) A?D
75
Accuracy(%)
OriFea
SGF(PCA)
SGF(PLS)
GFK(PCA)
GFK(PLS)
ITL
SA
GUMA
25
25
Accuracy(%)
OriFea
SGF(PCA)
SGF(PLS)
GFK(PCA)
GFK(PLS)
ITL
SA
GUMA
26
30
28
26
OriFea
SGF(PCA)
SGF(PLS)
GFK(PCA)
GFK(PLS)
ITL
SA
GUMA
24
22
20
18
32
30
OriFea
SGF(PCA)
SGF(PLS)
GFK(PCA)
GFK(PLS)
ITL
SA
GUMA
28
26
24
22
(j) D?C
(k) D?A
Accuracy(%)
28
Accuracy(%)
30
Accuracy(%)
Accuracy(%)
32
75
70
OriFea
SGF(PCA)
SGF(PLS)
GFK(PCA)
GFK(PLS)
ITL
SA
GUMA
65
60
55
(l) D?W
Figure 3: Performance comparisons in unsupervised domain adaptation. (A: Amazon, C: Caltech,
D: DSLR, W: Webcam)
SA tries to align the principal directions of two domains by characterizing each domain as a subspace. Except ITL, we use the source codes released by the original authors. For fair comparison,
the best parameters are tuned to report peak performance for all comparative methods. To compare
intrinsically, we use the NN classifier to predict the sample labels of target domain. Note SGF(PLS)
and GFK(PLS) use partial least square (PLS) to learn discriminant mappings according to their papers. In our method, to obtain stable sample points from space of high-dimensionality, we perform
clustering on the data of each class for source domain, and then cluster all unlabeled samples of target domain, to get the representative points for subsequent manifold alignment, where the number
of clusters is estimated using Jump method [27].
All comparisons are reported in Fig.3. Compared with the other methods, our method achieves more
competitive performance, i.e., the best results in 9 out of 12 cases, which indicates manifold alignment can be properly applied to domain adaptation. It also implies that it can reduce the difference
between domains by using manifold structures rather than the subspaces as in SGF, GFK and SA.
Generally, domain adaptation methods are better than OriFea. In the average accuracy, our method
is better than the second best result, 44.98% for ours v.s. 43.68% for GFK(PLS).
6
Conclusion
In this paper, we propose a generalized unsupervised manifold alignment method, which seeks for
the correspondences while finding the mutual embedding subspace of two manifolds. We formulate
unsupervised manifold alignment as an explicit 0-1 integer optimization problem by considering
the matching of global manifold structures as well as sample features. An efficient optimization
algorithm is further proposed by alternately solving two submodels, one is learning alignment with
integer constraints, and the other is learning transforms to get the mutual embedding subspace. In
learning alignment, we extend Frank-Wolfe algorithm to approximately seek for optima along the
descent path of the relaxed objective function. Experiments on set matching, video face recognition
and visual domain adaptation demonstrate the effectiveness and practicability of our method. Next
we will further generalize GUMA by relaxing the integer constraint and explore more applications.
Acknowledgments
This work is partially supported by Natural Science Foundation of China under contracts Nos.
61272319, 61222211, 61202297, and 61390510.
8
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9
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5,105 | 5,621 | On Prior Distributions and Approximate Inference
for Structured Variables
Rajiv Khanna
ECE Dept., UT Austin
[email protected]
Oluwasanmi Koyejo
Psychology Dept., Stanford
[email protected]
Russell A. Poldrack
Psychology Dept., Stanford
[email protected]
Joydeep Ghosh
ECE Dept., UT Austin
[email protected]
Abstract
We present a general framework for constructing prior distributions with structured variables. The prior is defined as the information projection of a base distribution onto distributions supported on the constraint set of interest. In cases
where this projection is intractable, we propose a family of parameterized approximations indexed by subsets of the domain. We further analyze the special case
of sparse structure. While the optimal prior is intractable in general, we show
that approximate inference using convex subsets is tractable, and is equivalent to
maximizing a submodular function subject to cardinality constraints. As a result, inference using greedy forward selection provably achieves within a factor
of (1-1/e) of the optimal objective value. Our work is motivated by the predictive
modeling of high-dimensional functional neuroimaging data. For this task, we
employ the Gaussian base distribution induced by local partial correlations and
consider the design of priors to capture the domain knowledge of sparse support.
Experimental results on simulated data and high dimensional neuroimaging data
show the effectiveness of our approach in terms of support recovery and predictive
accuracy.
1
Introduction
Data in scientific and commercial disciplines are increasingly characterized by high dimensions and
relatively few samples. For such cases, a-priori knowledge gleaned from expertise and experimental
evidence are invaluable for recovering meaningful models. In particular, knowledge of restricted
degrees of freedom such as sparsity or low rank has become an important design paradigm, enabling the recovery of parsimonious and interpretable results, and improving storage and prediction
efficiency for high dimensional problems. In Bayesian models, such restricted degrees of freedom
can be captured by incorporating structural constraints on the design of the prior distribution. Prior
distributions for structured variables can be designed by combining conditional distributions - each
capturing portions of the problem structure, into a hierarchical model. In other cases, researchers
design special purpose prior distributions to match the application at hand. In the case of sparsity,
an example of the former approach is the spike and slab prior [1, 2], and an example of the latter
approach is the horseshoe prior [3].
We describe a framework for designing prior distributions when the a-priori information include
structural constraints. Our framework follows the maximum entropy principle [4, 5]. The distribution is chosen as one that incorporates known information, but is as difficult as possible to discriminate from the base distribution with respect to relative entropy. The maximum entropy approach
1
has been especially successful with domain knowledge expressed as expectation constraints. In such
cases, the solution is given by a member of the exponential family [6, 7] e.g. quadratic constraints
result in the Gaussian distribution. Our work extends this framework to the design of prior distributions when a-priori information include domain constraints.
Our main technical contributions are as follows:
? We show that under standard assumptions, the information projection of a base density to
domain constraints is given by its restriction (Section 2).
? We show the equivalence between relative entropy inference with data observation constraints and Bayes rule for continuous variables
? When such restriction is intractable, we propose a family of parameterized approximations
indexed by subsets of the domain (Section 2.1).
We consider approximate inference in the special case of sparse structure:
? We characterize the restriction precisely, showing that it is given by a conditional distribution (Section 3).
? We show that the approximate sparse support estimation problem is submodular. As a
result, greedy forward selection is efficient and guarantees (1- 1e ) factor optimality (Section 3.1).
Our work is motivated by the predictive modeling of high-dimensional functional neuroimaging
data, measured by cognitive neuroscientists for analyzing the human brain. The data are represented using hundreds of thousands of variables. Yet due to real world constraints, most experimental datasets contain only a few data samples [8]. The proposed approach is applied to predictive
modeling of simulated data and high-dimensional neuroimaging data, and is compared to Bayesian
hierarchical models and non-probabilistic sparse predictive models, showing superior support recovery and predictive accuracy (Section 4). Due to space constraints, all proofs are provided in the
supplement.
1.1
Preliminaries
This section includes notation and a few basic definitions. Vectors are denoted by lower case x and
matrices by capital X. xi,j denotes the (i, j)th entry of the matrix X. xi,: denotes the ith row of
X and x:,j denotes the j th column. Let |X| denote the determinant of X. Sets are denoted by sans
serif e.g. S. The reals are denoted by R. [n] denotes the set of integers {1, . . . , n}, and ?(n) denotes
the power set of [n]. Let X be either a countable set, or a complete separable metric space equipped
with the standard Borel ?-algebra of measurable Rset. Let P denote the set of probability densities
on X i.e. positive functions P = {p : X 7? [0, 1] , X p(x) = 1}. For the remainder of this paper, we
make the following assumption:
Assumption 1. All distributions P are absolutely continuous with respect to the dominating measure ? so there exists a density p ? P that satisfies dP = pd?.
To simplify notation, we use use the standard d? = dx. As a consequence of Assumption 1, the
relative entropy is given in terms of the densities as:
Z
q(x)
KL(qkp) =
q(x) log
dx.
p(x)
X
The relative entropy is strictly convex with respect to its first argument. The information projection
of a probability density p to a constraint set A is given by the solution of:
inf KL(qkp) s.t. q ? A.
q?P
We will only consider projections where A is a closed convex set so the infimum
is achieved.
R
The
delta
function,
denoted
by
?
?
(x)f
(x)dx =
(?) , is a generalized set function that satisfies
A
X
R
R
f
(x)dx,
and
?
(x)dx
=
1,
for
some
some
A
?
X.
The
set
of
domain
restricted
densiA
X A
ties, denoted by FA for A ? X, is the set of probability density functions supported on A i.e.
2
FA = {q ? P | q(x) = 0 ? x ?
/ A} ? {?{x} ? x ? A} ? FA ? P = FX . Further, note that
FA is closed and convex for any A ? X (including nonconvex A).
Restriction is a standard approach for defining distributions on subsets A ? X. An important special
case we will consider is when A is a measure zero subset of X. The common conditional density is
one such example, the existence of which follows from the disintegration theorem [9]. Restrictions
of measure require extensive technical tools in the general case [10]. We will employ the following
simplifying condition for the remainder of this manuscript:
Condition 2. The sample space X is a subset of Euclidean space with ? given by the Lebesgue
measure. Alternatively, X is a countable set with ? given by the counting measure.
Let P be a probability distribution on X. Under Assumption 1 and Condition 2, the restriction of
the density p to the set A ? X is given by:
(
R p(x)
x ? A,
p(x)dx
A
q(x) =
0
otherwise.
2
Priors for structured variables
We assume a-priori information identifying the structure of X via the sub-domain A ? X. We also
assume a pre-defined base distribution P with associated density p. Without loss of generality, let
p have support everywhere1 on X i.e. p(x) > 0 ? x ? X. Following the principle of minimum
discrimination information, we select the prior as the information projection of the base density p
to FA . Our first result identifies the equivalence between information projection subject to domain
constraints and density restriction.
Theorem 3. Under Condition 2, the information projection of the density p to the constraint set FA
is the restriction of p to the domain A.
Theorem 3 gives principled justification for the domain restriction approach to structured prior design. Examples of density restriction in the literature include the truncated Gaussian, Beta and
Gamma densities [11], and the restriction of the matrix-variate Gaussian to the manifold of low
rank matrices [12]. Various properties of the restriction, such as its shape, and tail behavior (up to
re-scaling) follow directly from the base density. Thus the properties of the resulting prior are more
amenable to analysis when the base measure is well understood. Next, we consider a corollary of
Theorem 3 that was introduced by Williams [13].
Corollary 4. Consider the product space X = W ? Y. Let domain constraint be given by W ? {?
y}
for some y? ? Y. Under Condition 2, the information projection of p to FW?{y}
is
given
by
p(w|?
y
)?
?
y
?.
In the Bayesian literature, p(w) is known as the prior, p(y|w) is the likelihood and p(w|?
y ) is the
posterior density given the observation y = y?. Corollary 4 considers the information projection
of the joint density p(w, y) given observed data, and shows that the solution recovers the Bayesian
posterior. Williams [13] considered a generalization of Corollary 4, but did not consider projection
to data constraints2 . While Corollary 4 has been widely applied in the literature e.g. [14], to the best
of our knowledge, the presented result is the first formal proof.
2.1
Approximate inference for structured variables via tractable subsets
For many structural constraints of interest, restriction requires the computation of an intractable
normalization constant. In theory, rejection sampling and Markov Chain Monte Carlo (MCMC)
inference methods [15] do not require normalized probabilities. However, as many structured subdomains are measure zero sets with respect to the dominating measure, randomly generated samples
generated from the base distribution are unlikely to lie in the constrained domains e.g. random
samples from a multivariate Gaussian are not sparse. Hence rejection sampling fails, and MCMC
suffers from low acceptance probabilities. As a result, inference on such structured sub-domains
1
When this condition is violated, we simply redefine X as the subdomain supporting p.
Specifically, Williams [13] noted ?Relative information has been defined only for unconditional distributions, which say nothing about the relative probabilities of events of probability zero.?
2
3
pA
FA
pA?C
FC
p
P
(b) Sequential projections
(a) Gaussian restriction
Figure 1: (a) Gaussian density and restriction to diagonal line shown. (b) Illustration of Theorem 5;
sequence of information projections P ? FA ? FC and P ? FA?C are equivalent.
typically requires specialized methods e.g. [11, 12]. In the following, we propose a class of variational approximations based on an inner representation of the structured subdomain. Let {Si ? A}
represent a (possibly overlapping) partitioning of A into subsets. WeSdefine the domain restricted
density sets generated by these partitions as FSi , and their union D = FSi . Note that by definition
each FSi ? D ? FA ? FX . Our approach is to approximate the optimization over densities in FA by
optimizing over D - a smaller subset of tractable densities.
Approximate inference is generally most successful when the approximation accounts for observed
data. Inspired by the results of Corollary 4, we consider such a projection. Let pA (w, y) be the
information projection of the joint distribution p(x, y) to the set FA?{y}
? . We propose approximate
inference via the following rule:
pS? ,y? = arg min KL(q(w, y)kpA (w, y)) = arg min
min KL(q(w, y)kpA (w, y)) . (1)
q?D?F{y}
?
S
q?FS?{y}
?
Our proposed approach may be decomposed into two steps. The inner step is solved by estimating
a parameterized set of prior densities {qS } corresponding to choices of S, and the outer step is
solved by the selection of the optimal subset S? . The solution is given by pS? ,y? (w, y) = pS? (w|?
y )?y?
(Corollary 4) with the associated approximate posterior given by pS? (w|?
y ).
The following theorem considers the effect of a sequence of domain constrained information projections (see Fig. 1b), which will useful for subsequent results.
Theorem 5. Let ? : [n] 7? [n] be a permutationTfunction and {C?(i) | C?(i) ? X} represent a
sequence of sets with non empty intersection B = Ci 6= ?. Given a base density p, let q0 = p, and
define the sequence of information projections:
qi = arg min KL(qkqi?1 ).
q?FC?(i)
Under Condition 2, q? = qN is independent of ?. Further q? = min KL(qkp).
q?FB
We apply Theorem 5 to formulate equivalent solutions of (1) that may be simpler to solve.
Corollary 6. Let pS? ,y? (w, y) be the solution of (1), then the posterior distribution pS? (w|?
y ) is given
by:
pS? (w|?
y ) = arg min KL(q(w)kpA (w|?
y )) = arg min KL(q(w)kp(w|?
y )).
(2)
q?D
q?D
Corollary 6 implies that we can estimate the approximate structured posterior directly as the information projection of the unstructured posterior distribution p(w|?
y ). Upon further examination,
Corollary 6 also suggests that the proposed approximation is most useful when there exist subsets
of A such that the restriction of the base density to each subset leads to tractable inference. Further,
the result is most accurate when one of the subsets S? ? A captures most of the posterior probability mass. When the optimal subset S? is known, the structured prior density associated with the
structured posterior can be computed as shown in the following corollary.
4
Corollary 7. Let pS? ,y? (w, y) be the solution of (1). Define the density pS? (w) as:
pS? (w) = arg min KL(q(w)kpA (w)) = arg min KL(q(w)kp(w)).
q?FS?
(3)
q?FS?
then pS? (w) is the prior distribution corresponding to the Bayesian posterior pS? (w|?
y ).
3
Priors for sparse structure
We now consider a special case of the proposed framework for sparse structured variables. A d
dimensional variable x ? X is k-sparse if d ? k of its entries take a default value of ci i.e |{i | xi =
ci }| = d?k. In Euclidean space X = Rd and in most cases, ci = 0 ? i. Similarly, the distribution P
on the domain X is k-sparse if all random variables X ? P are at most k-sparse. The support of x ?
X is the set supp(x) = {i | xi 6= ci } ? ?(d). Let S ? X denote the set of variables with support s
i.e. S = {x ? X s.t. supp(x) = s}. We will use the notation xS = {xi | i ? s}, and its complement
xS0 = {xi | i ? s0 }, where s0 = [d]\s. The domain
of k sparse vectors is given by the union of all
S
d!
possible (d?k)!k!
sparse support sets as A = Si . While the sparse domain A is non-convex, each
subset S is a convex set, in fact given by linear subspaces with basis {ei | i ? s}. Further, while the
information projection of a base density p to A is generally intractable, the information projection to
its convex subsets S turn out to be computationally tractable. We investigate the application of the
proposed approximation scheme using these subsets.
3
Consider
S the information projection of an arbitrary probability measure P with density p to the set
D = FSi given by:
min KL(qkp) = min min KL(qkp) = min KL(pS kp).
q?D
S?{Si } q?FS
S?{Si }
Applying Theorem 3, we can compute that pS = p(x)?S (x)/Z, where Z is a normalization factor:
Z
Z
Z
Z = p(x) =
p(xS , xS0 )?S (x) =
p(xS |xS0 )p(xS0 )?S (x) = p(xS0 = cS0 ).
S
X
X
Thus, the normalization factor is a marginal density at xS0 = cS0 . We may now compute the restriction explicitly:
p(xS |xS0 )p(xS0 )?S (x)
= p(xS |xS0 = cS0 )?S (x).
(4)
pS (x) =
p(xS0 = cS0 )
In other words, the information projection to a sparse support domain is the density of xS conditioned
on xS0 = cS0 . The resulting gap is:
Z
Z
p(x)
pS (x)
= pS (x) log
= ? log p(xS0 = cS0 ).
KL(pS kp) = pS (x) log
p(x)
p(x)p(xS0 = cS0 )
S
S
Thus, for a given target sparsity k, we solve:
s? = arg max J(s),
where J(s) = log p(xS0 = cS0 ).
(5)
|s|=k
3.1
Submodularity and Efficient Inference
In this section, we show that the cost function J(s) is monotone submodular, and describe the greedy
forward selection algorithm for efficient inference. Let F : ?(d) 7? R represent a set function. F
is normalized if F (?) = 0. A bounded F can be normalized as F? (s) = F (s) ? F (?) with no
effect on optimization. F is monotonic, if for all subsets u ? v ? ?(d) it holds that F (u) ? F (v).
F is submodular, if for all subsets u, v ? m it holds that F (u ? v) + F (u ? v) ? F (u) + F (v).
Submodular functions have a diminishing returns property [16] i.e. the marginal gain of adding
elements decreases with the size of the set.
? = J(s) ? J(?), then
Theorem 8. Let J : ?(d) 7? R, J(s) = log p(xS0 = cS0 ), and define J(s)
?
J(s) is normalized and monotone submodular.
3
Where p may represent the conditional densities as in Section 2.1. To simplify the discussion, we suppress
the dependence on y?.
5
While constrained maximization of submodular functions is generally NP-hard, a simple greedy
forward selection heuristic has been shown to perform almost as well as the optimal in practice, and
is known to have strong theoretical guarantees.
Theorem 9 (Nemhauser et al. [16]). In the case of any normalized, monotonic submodular function
F, the set s? obtained by the greedy algorithm achieves at least a constant
fraction 1 ? 1e of the
objective value obtained by the optimal solution i.e. F (s? ) = 1 ? 1e max F (s).
|s|?k
In addition, no polynomial time algorithm can provide a better approximation guarantee unless P =
NP [17]. An additional benefit of the greedy approach is that it does not require the decision of the
support size k to be made at training time. As an anytime algorithm, training can be stopped at any k
based on computational constraints, while still returning meaningful results. An interesting special
case occurs when the base density takes a product form.
Corollary 10. Let J(s) be defined as in Theorem 8 and suppose the base density is product form i.e.
Qd
p(x) = i=1 p(xi ), then J(s) is linear.
In particular, define h = {p(xi = 0) ? i ? [d]}, then the solution of (5) is given by set of dimensions
associated with the smallest k values of h.
4
Experiments
We present experimental results comparing the proposed sparse approximate inference projection to
other sparsity inducing models. We performed experiments to test the models ability to estimate the
support of the reconstructed targets and the predictive regression accuracy.
TheP
regression accuracy
P
was measured using the coefficient of determination R2 = 1 ? (?
y ? y)2 / (y ? y?)2 where y
is the target response with sample mean y? and y? is the predicted response. R2 measures the gain in
predictive accuracy compared to a mean model and has a maximum value of 1. The support recovery
was measured using the AUC of the recovered support with respect to the true s? .
The baseline models are: (i) regularized least squares (Ridge), (ii) least absolute shrinkage and selection (Lasso) [18], (iii) automatic relevance determination (ARD) [19], (iv) Spike and Slab [1, 2].
Ridge and Lasso were optimized using implementations from the scikit-learn python package [20].
While Ridge does not return sparse weights, it was included as a baseline for regression performance.
We implemented ARD using iterative re-weighted Lasso as suggested by Wipf and Nagarajan [19].
The noise variance hyperparameter for Ridge and ARD were selected from the set 10{?4,?3,...,4} .
Lasso was evaluated using the default scikit-learn implementation where the hyperparameter is selected from 100 logarithmically spaced values based on the maximum correlation between the features and the response. For each of these models, the hyperparameter was selected in an inner 5-fold
cross validation loop. For speed and scalability, we used a publicly available implementation of
Spike and Slab [21], which uses a mean field variational approximation. In addition to the weights,
Spike and Slab estimates the probability that each dimension is non zero. As Spike and Slab does
not return sparse estimates, sparsity was estimated by thresholding this posterior at 0.5 for each dimension (SpikeSlab0.5 ), we also tested the full spike and slab posterior prediction for regression
performance alone (SpikeSlabFull).
The proposed projection approach is designed to be applicable to any probabilistic model. Thus, we
applied the projection approach as additional post-processing for the two Bayesian model baselines.
The first method is a projection of the standard Gaussian regression posterior (Sparse-G ) (more
details in supplement). The second is a projection of the spike and spike and slab approximate
posterior (SpikeSlabKL). We note that since the spike and slab approximate posterior uses the mean
field approximation, the posterior distribution is in product form and the projection is straightforward
using Corollary 10. Support size selection: The selection of the hyperparameter k - specifying the
sparsity, can be solved by standard model selection routines such as cross-validation. We found that
support size selection using sequential Bayes factors [22] was particularly effective, thus the support
size was selected as the first k where log p(y|Sk+1 ) ? log p(y|Sk ) < .
6
1.0
0.9
0.8
0.8
0.6
R2
Support AUC
1.0
0.4
0.7
Sparse-G
Lasso
ARD
SpikeSlabKL
SpikeSlab0.5
0.6
0.5
5
10
15
Sparse-G
Lasso
Ridge
ARD
SpikeSlabFull
SpikeSlab0.5
SpikeSlabKL
0.2
0.0
5
20
10
Sparse-G
Lasso
SpikeSlab0.5
SpikeSlabKL
ARD
1.0
0.8
0.6
0.8
R2
Support AUC
0.9
0.4
0.7
0.6
0.2
0.5
0.0
40
30
20
10
Signal-to-Noise Ratio(dB)
0
20
(b) R2 as a function of n:k ratio
(a) AUC as a function of n:k ratio
1.0
15
n:k
n:k
40
-10
Sparse-G
Lasso
Ridge
SpikeSlab0.5
SpikeSlabKL
ARD
SpikeSlabFull
30
20
10
Signal-to-Noise Ratio(dB)
0
-10
(d) R2 as a function of SNR
(c) AUC as a function of SNR
Figure 2: Simulated data performance: support recovery (AUC ) and regression (R2 ).
4.1
Simulated Data
We generated random high dimensional feature vectors ai ? Rd with ai,j ? N (0, 1). The response was generated
as yi = w> ai + ?i where ?i represents independent additive noise with
2
?i ? N 0, ? for all i ? [n]. We set ? 2 implicitly via the signal to noise ration (SNR) as
SNR = var(y)/? 2 , where var(y) is the variance of y. In each experiment, we sampled a sparse
weight vector w by sampling k dimensions at random with from [d], then we sampled values
wi ? N (0, 1) and set other dimensions to zero. We performed a series of tests to investigate the
performance of the model in different scenarios. Each experiment was run 10 times with separate
training and test sets. We present the average results on the test set.
Our first experiment tested the performance of all models with limited samples. Here we set
k = 20, d = 10, 000 and an SNR of 20dB. The number of training values was varied from
n = 100, . . . , 400 with 200 test samples. Fig. 2a shows the model performance in terms of support recovery. With limited training samples, Sparse-G outperformed all the baselines including
Lasso. We also found that SpikeSlabKL consistently outperformed SpikeSlab0.5. We speculate that
the significant gap between Sparse-G and SpikeSlabKL may be partly due to the mean field assumption in the underlying Spike and Slab. Fig. 2b shows the corresponding regression performance.
Again, we found that Sparse-G outperformed all other baselines, with Ridge achieving the worst
performance.
Our second experiment tested the performance of all models with high levels of noise. Here we
set k = 20, d = 10, 000 and n = 200 with 200 test samples. We varied the SNR from 40dB to
?10dB (note that ? 2 increases as SNR is decreased). Fig. 2c shows the support recovery performance of the different models. We found a performance gap between Sparse-G and Lasso, more
pronounced than in the small sample test. The SpikeSlab0.5 was the worst performing model, but
the performance was improved by SpikeSlabKL . Only Sparse-G achieved perfect support recovery
at low noise (high SNR ) levels. The regression performance is shown in Fig. 2d. While ARD and
Lasso matched Sparse-G at low noise levels (high SNR), their performance degraded much faster at
higher noise levels (low SNR).
4.2
Functional Neuroimaging Data
Functional magnetic resonance imaging (fMRI) is an important tool for non-invasive study of brain
activity. fMRI studies involve measurements of blood oxygenation (which are sensitive to the
7
Figure 3: Support selected by Sparse-G applied to fMRI data with 100,000 voxels. Slices are across
the vertical dimension. Selected voxels are in red.
amount of local neuronal activity) while the participant is presented with a stimulus or cognitive
task. Neuroimaging signals are then analyzed to identify which brain regions which exhibit a systematic response to the stimulation, and thus to infer the functional properties of those brain regions
[23]. Functional neuroimaging datasets typically consist of a relatively small number of correlated
high dimensional brain images. Hence, capturing the inherent structural properties of the imaging
data is critical for robust inference.
FMRI data were collected from 126 participants while the subjects performed a stop-signal task [24].
For each subject, contrast images were computed for ?go? trials and successful ?stop? trials using a
general linear model with FMRIB Software Library (FSL), and these contrast images were used for
regression against estimated stop-signal reaction times. We used the normalized Laplacian of the 3dimensional spatial graph of the brain image voxels to define the precision matrix. This corresponds
to the observation that nearby voxels tend to have similar functional activation. We present the 10fold cross validation performance of all models tested on this data. We tested all models using the
high dimensional 100,000 voxel brain image and measured average predictive R2 . The results are:
Sparse-G (0.051), Lasso (-0.271), Ridge (-0.473), ARD (-0.478). The negative test R2 for baseline
models show worse predictive performance than the test mean predictor, and indicate the difficulty
of this task. Even with the mean field variational inference, the Spike and Slab models did not
scale to this dataset. Only Sparse-G achieved a positive R2 . The support selected by Sparse-G with
all 100,000 voxels is shown in Fig. 3, sliced across the vertical dimension. The recovered voxels
show biologically plausible brain locations including the orbitofrontal cortex, dorsolateral prefrontal
cortex, putamen, anterior cingulate, and parietal cortex, which are correlated with the observed response. Further neuroscientific interpretation and validation will be included in an extended version
of the paper.
5
Conclusion
We present a principled approach for enforcing structure in Bayesian models via structured prior selection based on the maximum entropy principle. The prior is defined by the information projection
of the base measure to the set of distributions supported on the constraint domain. We focus on the
case of sparse structure. While the optimal prior is intractable in general, we show that approximate
inference using selected convex subsets is equivalent to maximizing a submodular function subject
to cardinality constraints, and propose an efficient greedy forward selection procedure which is guaranteed to achieve within a (1 ? 1e ) factor of the global optimum. For future work, we plan to explore
applications of our approach with other structural constraints such as low rank and structured sparsity for matrix-variate sample spaces. We also plan to explore more complicated base distributions
on other samples spaces.
Acknowledgments: fMRI data was provided by the Consortium for Neuropsychiatric Phenomics (NIH
Roadmap for Medical Research grants UL1-DE019580, RL1MH083269, RL1DA024853, PL1MH083271).
8
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9
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5,106 | 5,622 | Real-Time Decoding of an Integrate and Fire Encoder
Shreya Saxena and Munther Dahleh
Department of Electrical Engineering and Computer Sciences
Massachusetts Institute of Technology
Cambridge, MA 02139
{ssaxena,dahleh}@mit.edu
Abstract
Neuronal encoding models range from the detailed biophysically-based Hodgkin
Huxley model, to the statistical linear time invariant model specifying firing rates
in terms of the extrinsic signal. Decoding the former becomes intractable, while
the latter does not adequately capture the nonlinearities present in the neuronal
encoding system. For use in practical applications, we wish to record the output
of neurons, namely spikes, and decode this signal fast in order to act on this signal,
for example to drive a prosthetic device. Here, we introduce a causal, real-time
decoder of the biophysically-based Integrate and Fire encoding neuron model. We
show that the upper bound of the real-time reconstruction error decreases polynomially in time, and that the L2 norm of the error is bounded by a constant that
depends on the density of the spikes, as well as the bandwidth and the decay of
the input signal. We numerically validate the effect of these parameters on the
reconstruction error.
1
Introduction
One of the most detailed and widely accepted models of the neuron is the Hodgkin Huxley (HH)
model [1]. It is a complex nonlinear model comprising of four differential equations governing
the membrane potential dynamics as well as the dynamics of the sodium, potassium and calcium
currents found in a neuron. We assume in the practical setting that we are recording multiple neurons
using an extracellular electrode, and thus that the observable postprocessed outputs of each neuron
are the time points at which the membrane voltage crosses a threshold, also known as spikes. Even
with complete knowledge of the HH model parameters, it is intractable to decode the extrinsic
signal applied to the neuron given only the spike times. Model reduction techniques are accurate in
certain regimes [2]; theoretical studies have also guaranteed an input-output equivalence between a
multiplicative or additive extrinsic signal applied to the HH model, and the same signal applied to
an Integrate and Fire (IAF) neuron model with variable thresholds [3].
Specifically, take the example of a decoder in a brain machine interface (BMI) device, where the
decoded signal drives a prosthetic limb in order to produce movement. Given the complications
involved in decoding an extrinsic signal using a realistic neuron model, current practices include
decoding using a Kalman filter, which assumes a linear time invariant (LTI) encoding with the extrinsic signal as an input and the firing rate of the neuron as the output [4?6]. Although extremely
tractable for decoding, this approach ignores the nonlinear processing of the extrinsic current by
the neuron. Moreover, assuming firing rates as the output of the neuron averages out the data and
incurs inherent delays in the decoding process. Decoding of spike trains has also been performed
using stochastic jump models such as point process models [7, 8], and we are currently exploring
relationships between these and our work.
1
f (t)
IAF Encoder
{ti }i:|ti |?t
Real-Time Decoder
f?t (t)
Figure 1: IAF Encoder and a Real-Time Decoder.
We consider a biophysically inspired IAF neuron model with variable thresholds as the encoding
model. It has been shown that, given the parameters of the model and given the spikes for all
time, a bandlimited signal driving the IAF model can be perfectly reconstructed if the spikes are
?dense enough? [9?11]. This is a Nyquist-type reconstruction formula. However, for this theory
to be applicable to a real-time setting, as in the case of BMI, we need a causal real-time decoder
that estimates the signal at every time t, and an estimate of the time taken for the convergence
of the reconstructed signal to the real signal. There have also been some approaches for causal
reconstruction of a signal encoded by an IAF encoder, such as in [12]. However, these do not show
the convergence of the estimate to the real signal with the advent of time.
In this paper, we introduce a causal real-time decoder (Figure 1) that, given the parameters of the
IAF encoding process, provides an estimate of the signal at every time, without the need to wait for
a minimum amount of time to start decoding. We show that, under certain conditions on the input
signal, the upper bound of the error between the estimated signal and the input signal decreases
polynomially in time, leading to perfect reconstruction as t ! 1, or a bounded error if a finite
number of iterations are used. The bounded input bounded output (BIBO) stability of a decoder is
extremely important to analyze for the application of a BMI. Here, we show that the L2 norm of the
error is bounded, with an upper bound that depends on the bandwidth of the signal, the density of
the spikes, and the decay of the input signal.
We numerically show the utility of the theory developed here. We first provide example reconstructions using the real-time decoder and compare our results with reconstructions obtained using
existing methods. We then show the dependence of the decoding error on the properties of the input
signal.
The theory and algorithm presented in this paper can be applied to any system that uses an IAF
encoding device, for example in pluviometry. We introduce some preliminary definitions in Section
2, and then present our theoretical results in Section 3. We use a model IAF system to numerically
simulate the output of an IAF encoder and provide causal real-time reconstruction in Section 4, and
end with conclusions in Section 5.
2
Preliminaries
?
We first define the subsets of the L2 space that we consider. L?
2 and L2, are defined as the following.
n
o
?(!) = 0 8! 2
L?
=
f
2
L
|
f
/
[
?,
?]
(1)
2
2
n
o
L?
=
f g 2 L2 | f?(!) = 0 8! 2
/ [ ?, ?]
(2)
2,
, where g (t) = (1+|t|) and f?(!) = (Ff )(!) is the Fourier transform of f . We will only consider
signals in L?
0.
2, for
Next, we define sinc? (t) and
of signals.
[a,b] (t),
both of which will play an integral part in the reconstruction
sinc? (t) =
[a,b] (t)
=
(
?
sin(?t)
?t
1
t 6= 0
t=0
(3)
1
0
t 2 [a, b]
otherwise
(4)
Finally, we define the encoding system based on an IAF neuron model; we term this the IAF Encoder.
We consider that this model has variable thresholds in its most general form, which may be useful if
2
Rt
it is the result of a model reduction technique such as in [3], or in approaches where tii+1 f (? )d?
can be calculated through other means, such as in [9]. A typical IAF Encoder is defined in the
following way: given the thresholds {qi } where qi > 0 8i, the spikes {ti } are such that
Z ti+1
f (? )d? = ?qi
(5)
ti
Rt
This signifies that the encoder outputs a spike at time ti+1 every time the integral ti f (? )d? reaches
the threshold qi or qi . We assume that the decoder has knowledge of the value of the integral
as well as the time at which the integral was reached. For a physical representation with neurons
whose dynamics can faithfully be modeled using IAF neurons, we can imagine two neurons with
the same input f ; one neuron spikes when the positive threshold is reached while the other spikes
when the negative threshold is reached. The decoder views the activity of both of these neurons
and, with knowledge of the corresponding thresholds, decodes the signal accordingly. We can also
take the approach of limiting ourselves to positive fn(t). In order tooremain general in the following
Rt
treatment, we assume that we have knowledge of tii+1 f (? )d? , as well as the corresponding
spike times {ti }.
3
Theoretical Results
The following is a theorem introduced in [11], which was also applied to IAF Encoders in [10,13,14].
We will later use the operators and concepts introduced in this theorem.
Theorem 1. Perfect Reconstruction: Given a sampling set {ti }i2Z and the corresponding samples
R ti+1
?
f (? )d? , we can perfectly reconstruct f 2 L?
ti ) = for some < ?
.
2 if supi2Z (ti+1
ti
Moreover, f can be reconstructed iteratively in the following way, such that
? ?k+1
?
k
kf f k2 ?
kf k2
(6)
?
, and limk!1 f k = f in L2 .
f0
f1
=
=
Af
(I A)f 0 + Af = (I
fk
=
(I
A)f k
1
+ Af =
A)Af + Af
k
X
n=0
, where the operator Af is defined as the following.
1 Z ti+1
X
Af =
f (? )d? sinc? (t
i=1
and si =
ti +ti+1
,
2
A)n Af
(I
si )
(7)
(8)
(9)
(10)
ti
the midpoint of each pair of spikes.
Proof. Provided in [11].
The above theorem requires an infinite number of spikes in order to start decoding. However, we
would like a real-time decoder that outputs the ?best guess? at every time t in order for us to act on
the estimate of the signal. In this paper, we introduce one such decoder; we first provide a high-level
description of the real-time decoder, then a recursive algorithm to apply in the practical case, and
finally we will provide error bounds for its performance.
Real-Time Decoder
At every time t, the decoder outputs an estimate of the input signal f?t (t), where f?t (t) is an estimate
of the signal calculated using all the spikes from time 0 to t. Since there is no new information
between spikes, this is essentially the same as calculating an estimate after every spike ti , f?ti (t),
and using this estimate till the next spike, i.e. for time t 2 [ti , ti+1 ] (see Figure 2).
3
f (t)
f?t1 (t)
f?t2 (t) = f?t1 (t) + gt2 (t)
f?t (t)
f?t3 (t)
0
t0
t1
t3
t2
t4
t5
t6
t7
t
Figure 2: A visualization of the decoding process. The original signal f (t) is shown in black and the
spikes {ti } are shown in blue. As each spike ti arrives, a new estimate f?ti (t) of the signal is formed
(shown in green), which is modified after the next spike ti+1 by the innovation function gti+1 . The
P
output of the decoder f?t (t) = i2Z f?ti (t) [ti ,ti+1 ) (t) is shown in red.
We will show that we can calculate the estimate after every spike f?ti+1 as the sum of the previous
estimate f?ti and an innovation gti+1 . This procedure is captured in the algorithm given in Equations
11 and 12.
Recursive Algorithm
f?t0i+1
=
f?t0i + gt0i+1
f?tki+1
=
f?tki + gtki+1 = f?tki + gtki+11 + gt0i+1
Here, f?t00 = 0, and gt0i+1 (t) =
?R
ti+1
ti
?
?
f (? )d? sinc(t
Ati+1 gtki+11
?
(11)
(12)
si ). We denote f?ti (t) = limk!1 f?tki (t) and
gti+1 (t) = limk!1 gtki+1 (t). We define the operator AT f used in Equation 12 as the following.
X Z ti+1
AT f =
f (? )d? sinc? (t si )
(13)
i:|ti |?T
ti
P
The output of our causal real-time decoder can also be written as f?t (t) = i2Z f?ti (t) [ti ,ti+1 ) (t).
In the case of a decoder that uses a finite number of iterations K at every step, i.e. calculates f?tKi
P
after every spike ti , the decoded signal is f?tK (t) = i2Z f?tKi (t) [ti ,ti+1 ) (t). {f?tki }k are stored after
every spike ti , and thus do not need to be recomputed at the arrival of the next spike. Thus, when a
new spike arrives at ti+1 , each f?tki can be modified by adding the innovation functions gtki+1 .
Next, we show an upper bound on the error incurred by the decoder.
Theorem 2. Real-time reconstruction: Given a signal f 2 L?
2, passed through an IAF encoder
with known thresholds, and given that the spikes satisfy a certain minimum density supi2Z (ti+1
ti ) = for some < ?
? , we can construct a causal real-time decoder that reconstructs a function
?
ft (t) using the recursive algorithm in Equations 11 and 12, s.t.
|f (t)
f?t (t)| ?
c
1
?
?
4
kf k2, (1 + t)
(14)
, where c depends only on , ? and .
Moreover, if we use a finite number of iterations K at every step, we obtain the following error.
? ?K+1
? K+1
1
1 + ??
?
?
|f (t) f?tK (t)| ? c
kf
k
(1
+
t)
+
kf k2
(15)
2,
?
?
?
1
1
?
?
Proof. Provided in the Appendix.
Theorem 2 is the main result of this paper. It shows that the upper bound of the real-time reconstruction error using the decoding algorithm in Equations 11 and 12, decreases polynomially as a function
of time. This implies that the approximation f?t (t) becomes more and more accurate with the passage
of time, and moreover, we can calculate the exact amount of time we would need to record to have a
given level of accuracy. Given a maximum allowed error ?, these bounds can provide a combination
(t, K) that will ensure |f (t) f?tK (t)| ? ? if f 2 L?
2, , and if the density constraint is met.
We can further show that the L2 norm of the reconstruction remains bounded with a bounded input (BIBO stability), by bounding the L2 norm of the error between the original signal and the
reconstruction.
Corollary 1. Bounded L2 norm: The causal decoder provided in Theorem 2, with the same assumptions and
the case of K ! 1, constructs a signal f?t (t) s.t. the L2 norm of the error
qin
p
R1
kf f?t k2 =
|f (t) f?t (t)|2 dt is bounded: kf f?t k2 ? c/ 2 ? 1 kf k2, where c is the same
0
1
constant as in Theorem 2.
?
Proof.
sZ
1
0
|f (t)
f?t (t)|2 dt
?
v
uZ
u
t
1
0
c
1
?
?
!2
kf k22, (1 + t)
2
dt =
p
c/ 2
1
1
?
?
kf k2,
(16)
Here, the first inequality is due to Theorem 2, and all the constants are as defined in the same.
Remark
q R 1: This result also implies that we have a decay in the root-mean-square (RMS) error, i.e.
T !1
1 T
f?t (t)|2 dt
! 0. For the case of a finite number of iterations K < 1, the RMS
T 0 |f (t)
error converges to a non-zero constant
? K+1 1+
?
1
?
?
?
?
kf k2 .
Remark 2: The methods used
in Corollary 1 also provide a bound on the error in the weighted L2
p
c/
?
norm, i.e. kf f k2, ? 1 ? 1 kf k2, for
2, which may be a more intuitive form to use for a
?
subsequent stability analysis.
4
Numerical Simulations
We simulated signals f (t) of the following form, for t 2 [0, 100], using a stepsize of 10
P50
wk (sinc? (t dk ))
f (t) = i=1 P50
i=1 wk
2
.
(17)
Here, the wk ?s and dk ?s were picked uniformly at random from the interval [0, 1] and [0, 100] respectively. Note that f 2 L2,? . All simulations were performed using MATLAB R2014a. For each
simulation experiment, at every time t we decoded using only the spikes before time t.
We first provide example reconstructions using the Real-Time Decoder for four signals in Figure 3,
using constant thresholds, i.e. qi = q 8i. We compare our results to those obtained using a Linear
Firing Rate (FR) Decoder, i.e. we let the reconstructed signal be a linear function of the number
of spikes in the past seconds, being the window size. We can see that there is a delay in the
reconstruction with this decoding approach. Moreover, the reconstruction is not as accurate as that
using the Real-Time Decoder.
5
0.1
0.08
0.08
Amplitude
Amplitude
0.1
0.06
0.04
0.04
0.02
0.02
0
0
20
40
60
Time (s)
0
0
80
0.1
0.1
0.08
0.08
0.06
0.04
0.02
0
0
20
40
60
Time (s)
0.04
0
0
80
20
40
60
Time (s)
80
(d) ? = 0.3?; Linear FR Decoder
0.1
0.08
0.08
Amplitude
Amplitude
80
0.02
(c) ? = 0.3?; Real-Time Decoder
0.06
0.04
0.02
0.06
0.04
0.02
20
40
60
Time (s)
0
0
80
(e) ? = 0.4?; Real-Time Decoder
0.07
0.07
0.06
0.06
0.05
0.05
Amplitude
0.08
0.04
0.03
80
0.03
0.02
0.01
0.01
40
60
Time (s)
40
60
Time (s)
0.04
0.02
20
20
(f) ? = 0.4?; Linear FR Decoder
0.08
0
0
40
60
Time (s)
0.06
0.1
0
0
20
(b) ? = 0.2?; Linear FR Decoder
Amplitude
Amplitude
(a) ? = 0.2?; Real-Time Decoder
Amplitude
0.06
0
0
80
(g) ? = 0.5?; Real-Time Decoder
20
40
60
Time (s)
80
(h) ? = 0.5?; Linear FR Decoder
Figure 3: (a,c,e,g) Four example reconstructions using the Real-Time Decoder, with the original
signal f (t) in black solid and the reconstructed signal f?t (t) in red dashed lines. Here, [ , K] =
[2, 500], and qi = 0.01 8i. (b,d,f,h) The same signal was decoded using a Linear Firing Rate (FR)
Decoder. A window size of = 3s was used.
6
?4
?4
x 10
3
2.5
x 10
2
!f ? f?t! 2
!f ! 2,?
!f ? f?t! 2
!f ! 2,?
2
1.5
1
1
0.5
0
0.1pi
0.2pi
?
0.3pi
(a) ? is varied; [ , , K] = [2,
0
0.6
0.4pi
0.8
1
1.2
1.4
1.6
?
(b) is varied; [?, , K] = [0.3?, 2, 500]
?
, 500]
2?
?4
2
x 10
?4
?6
10
!f ? f?t! 2
!f ! 2,?
!f ? f?t! 2
!f ! 2,?
10
1
?8
10
?10
10
2
2.5
3
3.5
4
4.5
0
0
5
?
(c)
is varied; [?, , K] = [0.3?,
100
200
300
400
500
K
1
0.3
(d) K is varied; [?, , ] = [0.3?, 53 , 2]
, 500]
Figure 4: Average error for 20 different signals while varying different parameters.
Next, we show the decay of the real-time error by averaging out the error for 20 different input
signals, while varying certain parameters, namely ?, , and K (Figure 4). The thresholds qi were
chosen to be constant a priori, but were reduced to satisfy the density constraint wherever necessary.
According to Equation 14 (including the effect of the constant c), the error should decrease as ? is
decreased. We see this effect in the simulation study in Figure 4a. For these simulations, we chose
such that ?? < 1, thus was decreasing as ? increased; however, the effect of the increasing ?
dominated in this case.
In Figure 4b we see that increasing while keeping the bandwidth constant does indeed increase the
error, thus the algorithm is sensitive to the density of the spikes. In this figure, all the values of
satisfy the density constraint, i.e. ?? < 1.
Increasing is seen to have a large effect, as seen in Figure 4c: the error decreases polynomially
in (note the log scale on the y-axis). Although increasing in our simulations also increased
the bandwidth of the signal, the faster decay had a larger effect on the error than the change in
bandwidth.
In Figure 4d, the effect of increasing K is apparent; however, this error flattens out for large values
of K, showing convergence of the algorithm.
7
5
Conclusions
We provide a real-time decoder to reconstruct a signal f 2 L?
2, encoded by an IAF encoder. Under
Nyquist-type spike density conditions, we show that the reconstructed signal f?t (t) converges to f (t)
polynomially in time, or with a fixed error that depends on the computation power used to reconstruct
the function. Moreover, we get a lower error as the spike density increases, i.e. we get better results
if we have more spikes. Decreasing the bandwidth or increasing the decay of the signal both lead to
a decrease in the error, corroborated by the numerical simulations. This decoder also outperforms
the linear decoder that acts on the firing rate of the neuron. However, the main utility of this decoder
is that it comes with verifiable bounds on the error of decoding as we record more spikes.
There is a severe need in the BMI community for considering error bounds while decoding signals
from the brain. For example, in the case where the reconstructed signal is driving a prosthetic, we are
usually placing the decoder and machine in an inherent feedback loop (where the feedback is visual
in this case). A stability analysis of this feedback loop includes calculating a bound on the error
incurred by the decoding process, which is the first step for the construction of a device that robustly
tracks agile maneuvers. In this paper, we provide an upper bound on the error incurred by the realtime decoding process, which can be used along with concepts in robust control theory to provide
sufficient conditions on the prosthetic and feedback system in order to ensure stability [15?17].
Acknowledgments
Research supported by the National Science Foundation?s Emerging Frontiers in Research and Innovation Grant (1137237).
References
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8
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9
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5,107 | 5,623 | Spike Frequency Adaptation Implements Anticipative
Tracking in Continuous Attractor Neural Networks
Yuanyuan Mi
State Key Laboratory of Cognitive Neuroscience & Learning,
Beijing Normal University,Beijing 100875,China
[email protected]
C. C. Alan Fung,
K. Y. Michael Wong
Department of Physics, The Hong Kong University of Science and Technology, Hong Kong
[email protected], [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
To extract motion information, the brain needs to compensate for time delays that
are ubiquitous in neural signal transmission and processing. Here we propose a
simple yet effective mechanism to implement anticipative tracking in neural systems. The proposed mechanism utilizes the property of spike-frequency adaptation (SFA), a feature widely observed in neuronal responses. We employ continuous attractor neural networks (CANNs) as the model to describe the tracking
behaviors in neural systems. Incorporating SFA, a CANN exhibits intrinsic mobility, manifested by the ability of the CANN to support self-sustained travelling
waves. In tracking a moving stimulus, the interplay between the external drive
and the intrinsic mobility of the network determines the tracking performance. Interestingly, we find that the regime of anticipation effectively coincides with the
regime where the intrinsic speed of the travelling wave exceeds that of the external
drive. Depending on the SFA amplitudes, the network can achieve either perfect
tracking, with zero-lag to the input, or perfect anticipative tracking, with a constant leading time to the input. Our model successfully reproduces experimentally
observed anticipative tracking behaviors, and sheds light on our understanding of
how the brain processes motion information in a timely manner.
1
Introduction
Over the past decades, our knowledge of how neural systems process static information has advanced considerably, as is well documented by the receptive field properties of neurons. The equally
important issue of how neural systems process motion information remains much less understood.
A main challenge in processing motion information is to compensate for time delays that are pervasive in neural systems. For instance, visual signal transmitting from the retina to the primary visual
cortex takes about 50-80 ms [1], and the time constant for single neurons responding to synaptic
input is of the order 10-20 ms [2]. If these delays are not compensated properly, our perception of a
fast moving object will lag behind its true position in the external world significantly, impairing our
vision and motor control.
1
A straightforward way to compensate for time delays is to anticipate the future position of a moving
object, covering the distance the object will travel through during the delay period. Experimental
data has suggested that our brain does employ such a strategy. For instance, it was found that in spatial navigation, the internal head-direction encoded by anterior dorsal thalamic nuclei (ADN) cells
in a rodent was leading the instant position of the rodent?s head by ? 25 ms [3, 4, 5], i.e., it was the
direction the rodent?s head would turn into ? 25 ms later. Anticipation also justifies the well-known
flash-lag phenomenon [6], that is, the perception that a moving object leads a flash, although they
coincide with each other at the same physical location. The reason is due to the anticipation of our
brain for the future position of the continuously moving object, in contrast to the lack of anticipation for intermittent flashes. Although it is clear that the brain do have anticipative response to the
animal?s head direction, it remains unclear how neural systems implement appropriate anticipations
against various forms of delays.
Depending on the available information, the brain may employ different strategies to implement
anticipations. In the case of self-generated motion, the brain may use an efference copy of the
motor command responsible for the motion to predict the motion consequence in advance [7]; and
in the case when there is an external visual cue, such as the speed of a moving object, the neural
system may dynamically select a transmission route which sends the object information directly to
the future cortical location during the delay [8]. These two strategies work well in their own feasible
conditions, but they may not compensate for all kinds of neural delays, especially when the internal
motor command and visual cues are not available. Notably, it was found that when a rodent was
moving passively, i.e., a situation where no internal motor command is available, the head-direction
encoded by ADN cells was still leading the actual position of the rodent?s head by around ? 50ms,
even larger than that in a free-moving condition [5]. Thus, extra anticipation strategies may exist in
neural systems.
Here, we propose a novel mechanism to generate anticipative responses when a neural system is
tracking a moving stimulus. This strategy does not depend on the motor command information
nor external visual cues, but rather relies on the intrinsic property of neurons, i.e., spike-frequency
adaptation (SFA). SFA is a dynamical feature commonly observed in the activities of neurons when
they have experienced prolonged firing. It may be generated by a number of mechanisms [10]. In
one mechanism, neural firing elevates the intracellular calcium level of a neuron, which induces an
inward potassium current and subsequently hyperpolarizes the neuronal membrane potential [11].
In other words, strong neuronal response induces a negative feedback to counterbalance itself. In the
present study, we use continuous attractor neural networks (CANNs) to model the tracking behaviors
in neural systems. It was known that SFA can give rise to travelling waves in CANNs [12] analogous
to the effects of asymmetric neuronal interactions; here we will show that its interplay with external
moving stimuli determines the tracking performance of the network. Interestingly, we find that when
the intrinsic speed of the network is greater than that of the external drive, anticipative tracking
occurs for sufficiently weak stimuli; and different SFA amplitude results in different anticipative
times.
2
2.1
The Model
Continuous attractor neural networks
We employ CANNs as the model to investigate the tracking behaviors in neural systems. CANNs
have been successfully applied to describe the encoding of continuous stimuli in neural systems,
including orientation [13], head-direction [14], moving direction [15] and self location [16]. Recent experimental data strongly indicated that CANNs capture some fundamental features of neural
information representation [17].
Consider a one-dimensional continuous stimulus x encoded by an ensemble of neurons (Fig. 1).
The value of x is in the range of (??, ?] with the periodic condition imposed. Denote U (x, t) as the
synaptic input at time t of the neurons whose preferred stimulus is x, and r(x, t) the corresponding
firing rate. The dynamics of U (x, t) is determined by the recurrent input from other neurons, its own
relaxation and the external input I ext (x, t), which is written as
?
dU (x, t)
= ?U (x, t) + ?
J(x, x? )r(x? , t)dx? + I ext (x, t),
(1)
?
dt
x?
2
delay
U(x,t)
vext
I
ext
(x,t)
J(x,x?)
Figure 1: A CANN encodes a continuous stimulus, e.g., head-direction. Neurons are aligned in
the network according to their preferred stimuli. The neuronal interaction J(x, x? ) is translationinvariant in the space of stimulus values. The network is able to track a moving stimulus, but
the response bump U (x, t) is always lagging behind the external input I ext (x, t) due to the neural
response delay.
where ? is the synaptic
of the order 2 ? 5 ms, ? is the neural density and
[ time constant, typically
]
0
J(x, x? ) = ?J2?a
exp ?(x ? x? )2 /(2a2 ) is the neural interaction from x? to x, where the Gaussian
width a controls the neuronal interaction range. We will consider a ? ?. Under this condition, the
neuronal responses are localized and we can effectively treat ?? < x < ? in the following
analysis.
The nonlinear relationship between r(x, t) and U (x, t) is given by
U (x, t)2
?
,
r(x, t) =
1 + k? x? U (x? , t)2 dx?
(2)
where the divisive normalization could be realized by shunting inhibition. r(x, t) first increases
with U (x, t) and then saturates gradually when the total network activity is sufficiently large. The
parameter k controls the strength of divisive normalization. This choice of global normalization can
simplify our analysis and should not alter our main conclusion if localized inhibition is considered.
It can be checked that when I ext = 0, the network supports a continuous family of Gaussian-shaped
stationary states, called bumps, which are,
[
]
[
]
(x ? z)2
(x ? z)2
U (x) = U0 exp ?
,
r(x)
=
r
exp
?
, ?z
(3)
0
4a2
2a2
?
where?the peak position of the bump z is a free parameter. r0 =
2U0 /(?J0 ) and U0 =
?
?
2
[1 + 1 ? 8 2?ak/(?J0 )]/(2 2?ak?). The bumps are stable for 0 < k < kc , with kc =
?
?J02 /(8 2?a).
The bump states of a CANN form a sub-manifold in the state space of the network, on which the
network is neutrally stable. This property enables a CANN to track a moving stimulus smoothly,
provided that the stimulus speed is not too large [18]. However, during the tracking, the network
bump is always lagging behind the instant position of the moving stimulus due to the delay in
neuronal responses (Fig. 1).
2.2
CANNs with the asymmetrical neuronal interaction
It is instructive to look at the dynamical properties of a CANN when the asymmetrical neuronal
interaction is included. In an influential study [14], Zhang proposed an idea of adding asymmetrical
interactions between neurons in a CANN, such that the network can support travelling waves, i.e.,
spontaneously moving bumps. The modified model well describes the experimental finding that in
tracking the rotation of a rodent, the internal representation of head-direction constructed by ADN
cells also rotates and the bump of neural population activity remains largely invariant in the rotating
frame.
By including the asymmetrical neuronal interaction, the CANN model presented above also supports
a travelling wave state. The new neuronal recurrent interaction is written as
[
]
[
]
J0
(x ? x? )2
J0
(x ? x? )2
?
?
?
J(x, x ) = ?
exp ?
+ ?? ?
(x ? x ) exp ?
,
(4)
2a2
2a2
2?a
2?a3
3
where ? is a constant controlling the strength of asymmetrical interaction.
It is straightforward{ to check that the network
} supports the following
{ traveling wave solution,
}
2
2
U (x, t) = U0 exp ? [x ? (z + vt)] /(4a2 ) , r(x, t) = r0 exp ? [x ? (z + vt)] /(2a2 ) ,
where v is the speed of the travelling wave, and v = ?, i.e., the asymmetrical interaction strength
determines the speed of the travelling wave (see Supplementary Information).
2.3
CANNs with SFA
The aim of the present study is to explore the effect of SFA on the tracking behaviors of a CANN.
Incorporating SFA, the dynamics of a CANN is written as
?
dU (x, t)
J(x, x? )r(x? , t)dx? ? V (x, t) + I ext (x, t),
= ?U (x, t) + ?
?
(5)
dt
?
x
where the synaptic current V (x, t) represents the effect of SFA, whose dynamics is given by [12]
?v
dV (x, t)
= ?V (x, t) + mU (x, t),
dt
(6)
where ?v is the time constant of SFA, typically of the order
? t 40 ? 120 ms. The parameter m controls
the SFA amplitude. Eq. (6) gives rise to V (x, t) = m ?? exp [?(t ? t? )/?v ] U (x, t? )dt? /?v , that
is, V (x, t) is the integration of the neuronal synaptic input (and hence the neuronal activity) over
an effective period of ?v . The negative value of V (x, t) is subsequently fed back to the neuron to
suppress its response (Fig. 2A). The higher the neuronal activity level is, the larger the negative
feedback will be. The time constant ?v ? ? indicates that SFA is slow compared to neural firing.
A
B
0.025
0.02
U(x,t)
?v
j
int
r
ij j
0.015
0.01
v
?J
-V(x,t) : SFA
0.005
I ext ( x, t )
0
?0.005
0
? / ?v
0.02
0.04
0.06
0.08
m
Figure
2: A. The inputs to a single neuron in a CANN with SFA, which include a recurrent input
?
ext
(x, t) containing the stimulus information, and a
j Jij rj from other neurons, an external input I
negative feedback current ?V (x, t) representing the SFA effect. The feedback of SFA is effectively
delayed by time ?v . B. The intrinsic speed of the network vint (in units of 1/? ) increases with the
SFA amplitude m. The network starts to have a travelling wave state at m = ? /?v . The parameters
are: ? = 1, ?v = 60, a = 0.5. Obtained by Eq. (13).
3
Travelling Wave in a CANN with SFA
We find that SFA has the same effect as the asymmetrical neuronal interaction on retaining travelling
waves in a CANN. The underlying mechanism is intuitively understandable. Suppose that a bump
emerges at an arbitrary position in the network. Due to SFA, those neurons which are most active
receive the strongest negative feedback, and their activities will be suppressed accordingly. Under
the competition (mediated by recurrent connections and divisive normalization) from the neighboring neurons which are less affected by SFA, the bump tends to shift to the neighborhood; and at the
new location, SFA starts to destabilize neuronal responses again. Consequently, the bump will keep
moving in the network like a travelling wave.
The condition for the network to support a travelling wave state can be theoretically analyzed. In
simulations, we observe that in a traveling wave state, the profiles of U (x, t) and V (x, t) have approximately a Gaussian shape, if m is small enough. We therefore consider the following Gaussian
4
ansatz for the travelling wave state,
}
2
[x ? z(t)]
,
Au exp ?
4a2
{
}
2
[x ? z(t)]
Ar exp ?
,
2a2
{
}
2
[x ? (z(t) ? d)]
Av exp ?
,
4a2
{
U (x, t)
=
r(x, t)
=
V (x, t)
=
(7)
(8)
(9)
where dz(t)/dt is the speed of the travelling wave and d is the separation between U (x, t) and
V (x, t). Without loss of generality, we assume that the bump moves from left to right, i.e.,
dz(t)/dt > 0. Since V (x, t) lags behind U (x, t) due to slow SFA, d > 0 normally holds.
To solve the network dynamics, we utilize an important property of CANNs, that is, the dynamics of
a CANN are dominated by a few motion modes corresponding to different distortions in the shape
of a bump [18]. We can project the network dynamics onto these dominating modes and simplify the
network dynamics significantly. The first two dominating motion modes used in the present study
correspond to the
bump, which
[ distortions in the] height and position of the Gaussian
[
] are given by
2
2
?0 (x|z) = exp ?(x ? z)2 /(4a2 ) and ?1 (x|z) = (x ? z)
exp
?(x
?
z)
/(4a
)
. By projecting
?
?
a function f (x) onto a mode ?n (x), we mean computing x f (x)?n (x)dx/ x ?n (x)2 dx.
Applying the projection method, we solve the network dynamics and obtain the travelling wave state.
The speed of the travelling wave and the bumps? separation are calculated to be (see Supplementary
Information)
?
?
?
?
?
m?v
dz(t)
2a m?v
d = 2a 1 ?
, vint ?
=
?
.
(10)
m?v
dt
?v
?
?
The speed of the travelling wave reflects the intrinsic mobility of the network, and its value is fully
determined by the network parameters (see Eq. (10)). Hereafter, we call it the intrinsic speed of the
network, referred to as vint . vint increases with the SFA amplitude m (Fig. 2B). The larger the value
of vint , the higher the mobility of the network.
From the above equations, we see that the condition for the network to support a travelling wave
state is m > ? /?v . We note that SFA effects can reduce the firing rate of neurons significantly [11].
Since the ratio ? /?v is small, it is expected that this condition can be realistically fulfilled.
3.1
Analogy to the asymmetrical neuronal interaction
Both SFA and the asymmetrical neuronal interaction have the same capacity of generating a travelling wave in a CANN. We compare their dynamics to unveil the underlying cause.
Consider that the network state is given by Eq. (8). The contribution of the asymmetrical neuronal
interaction can be written as (substituting the asymmetrical component in Eq. (4) into the second
term on the right-hand side of Eq. (1)),
?
2
(x?x? )2
(x? ?z)2
?J0 r0 ?? (x ? z) ? (x?z)
J0 ??? r0
?
?
(x ? x? )e? 2a2 e? 2a2 dx? =
(11)
e 4a2 .
2?a3 x?
2 2a2
In a CANN with SFA, when the separation d is sufficiently small, the synaptical current induced by
SFA can be approximately expressed as (the 1st-order Taylor expansion; see Eq. (9)),
]
[
]
[
x?z
(x ? z)2
(x ? z)2
+
dA
exp
?
,
(12)
?V (x, t) ? ?Av exp ?
v
4a2
2a2
4a2
which consists of two terms: the first one has the same form as U (x, t) and the second one has the
same form as the contribution of the asymmetrical interaction (compared to Eq. (11)). Thus, SFA
has the similar effect as the asymmetrical neuronal interaction on the network dynamics.
The notion of the asymmetrical neuronal interaction is appealing for retaining a travelling wave in a
CANN, but its biological basis has not been properly justified. Here, we show that SFA may provide
5
an effective way to realize the effect of the asymmetrical neuronal interaction without recruiting the
hard-wired asymmetrical synapses between neurons. Furthermore, SFA can implement travelling
waves in either direction, whereas, the hard-wired asymmetrical neuronal connections can only
support a travelling wave in one direction along the orientation of the asymmetry. Consequently, a
CANN with the asymmetric coupling can only anticipatively track moving objects in one direction.
4
Tracking Behaviors of a CANN with SFA
SFA induces intrinsic mobility of the bump states of a CANN, manifested by the ability of the network to support self-sustained travelling waves. When the network receives an external input from a
moving stimulus, the tracking behavior of the network will be determined by two competing factors:
the intrinsic speed of the network (vint ) and the speed of the external drive (vext ). Interestingly, we
find that when vint > vext , the network bump leads the instant position of the moving stimulus for
sufficiently weak stimuli, achieving anticipative tracking.
Without
we set the external input to be I ext (x, t)
=
{ loss of generality,
}
2
2
? exp ? [x ? z0 (t)] /(4a ) , where ? represents the input strength, z0 (t) is the stimulus
at time t and the speed of the moving stimulus is vext = dz0 (t)/dt.
Define s = z(t) ? z0 (t) to be the displacement of the network bump relative to the external drive.
We consider that the network is able to track the moving stimulus, i.e., the network dynamics will
reach a stationary state with dz(t)/dt = dz0 (t)/dt and s a constant. Since we consider that the
stimulus moves from left to right, s > 0 means that the network tracking is leading the moving
input; whereas s < 0 means the network tracking is lagging behind.
Using the Gaussian ansatz for the network state as given by Eqs. (7-9) and applying the projection
method, we solve the network dynamics and obtain (see Supplementary Information),
?
?a + a2 + (vext ?v )2
d = 2a
,
(13)
vext ?v
(
)
(
)
s2
1
?
md2
2
s exp ? 2
(14)
=
Au
? vext
.
8a
?
vext ? ?v
2
= md2 /(? ?v ), which
Combining Eqs. (10, 13, 14), it can be checked that when vext = vint , vext
2
2
gives s = 0 ; and when vext < vint , vext < md /(? ?v ), which gives s > 0, i.e., the bump is leading
the external drive (For detail, see Supplementary Information).
Fig. 3A presents the simulation result. There is a minor discrepancy between the theoretical prediction and the simulation result: the separation s = 0 happens at the point when the stimulus
speed vext is slightly smaller than the intrinsic speed of the network vint . This discrepancy arises
from the distortion of the bump shape from Gaussian when the input strength is strong, the stimulus
speed is high and m is large, and hence the Gaussian ansatz on the network state is not accurate.
Nevertheless, for sufficiently weak stimuli, the theoretical prediction is correct.
4.1
Perfect tracking and perfect anticipative tracking
As observed in experiments, neural systems can compensate for time delays in two different ways:
1) perfect tracking, in which the network bump has zero-lag with respect to the external drive, i.e.,
s = 0; and 2) perfect anticipative tracking, in which the network bump leads the external drive by
approximately a constant time tant = s/vext . In both cases, the tracking performance of the neural
system is largely independent of the stimulus speed. We check whether a CANN with SFA exhibits
these appealing properties.
Define a scaled speed variable v ext ? ?v vext /a. In a normal situation, v ext ? 1. For instance,
taking the biologically plausible parameters ?v = 100 ms and a = 50o , v ext = 0.1 corresponds to
vext = 500o /s, which is a rather high speed for a rodent rotating its head in ordinary life.
By using the scaled speed variable, Eq. (14) becomes
[
]
?
(
)
s2
1
(?1 + 1 + v 2ext )2
?
s exp ? 2 = Au a 4m
? v ext .
8a
?
?v
v 3ext
6
(15)
B
A
d
s>0
0.06
V(x)
S
0.03
-?
0
?0.06
0
U(x)
- ?/2
-
C
?0.03
Iext(x)
0
d
?/2
?
s<0
vint
0.004
0.008
0.012
0.016
v
V(x)
ext
-?
ext
U(x)
- ?/2
I (x)
0
?/2
?
Figure 3: A. The separation s vs. the speed of the external input vext . Anticipative tracking s > 0
occurs when vext < vint . The simulation was done with a network of N = 1000 neurons. The
parameters are: J0 = 1, k = 0.1, a = 0.5, ? = 1, ?v = 60, ? = 0.5 and m = 2.5? /?v . B. An
example of anticipative tracking in the reference frame of the external drive. C. An example of
delayed tracking. In both cases, the profile of V (x, t) is lagging behind the bump U (x, t) due to
slow SFA.
?
In the limit of v ext ? 1 and consider s/(2 2a) ? 1 (which is true in practice), we get s ?
Au ?v vext (m ? ??v )/?. Thus, we have the following two observations:
? Perfect tracking. When m ? ? /?v , s ? 0 holds, and perfect tracking is effectively
achieved. Notably, when there is no stimulus, m = ? /?v is the condition for the network
starting to have a traveling wave state.
? Perfect anticipative tracking. When m > ? /?v , s increases linearly with vext , and the
anticipative time tant is approximately a constant.
These two properties hold for a wide range of stimulus speed, as long as the approximation v ext ? 1
is applicable. We carried out simulations to confirm the theoretical analysis, and the results are
presented in Fig. 4. We see that: (1) when SFA is weak, i.e., m < ? /?v , the network tracking is
lagging behind the external drive, i.e. s < 0 (Fig. 4A); (2) when the amplitude of SFA increases to a
critical value m = ? /?v , s becomes effectively zero for a wide range of stimulus speed, and perfect
tracking is achieved (Fig. 4B); (3) when SFA is large enough satisfying m > ? /?v , s increases
linearly with vext for a wide range of stimulus speeds, achieving perfect anticipative tracking (Fig.
4C); and (4) with the increasing amplitude of SFA, the anticipative time of the network also increases
(Fig. 4D). Notably, by choosing the parameters properly, our model can replicate the experimental
finding on a constant leading time of around 25 ms when a rodent was tracking head-direction by
ADN cells (the red points in Fig. 4D for ? = 5 ms) [19].
5
Conclusions
In the present study, we have proposed a simple yet effective mechanism to implement anticipative
tracking in neural systems. The proposed strategy utilizes the property of SFA, a general feature
in neuronal responses, whose contribution is to destabilize spatially localized attractor states in a
network. Analogous to asymmetrical neuronal interactions, SFA induces self-sustained travelling
wave in a CANN. Compared to the former, SFA has the advantage of not requiring the hard-wired
asymmetrical synapses between neurons. We systematically explored how the intrinsic mobility of
a CANN induced by SFA affects the network tracking performances, and found that: (1) when the
intrinsic speed of the network (i.e., the speed of the travelling wave the network can support) is larger
than that of the external drive, anticipative tracking occurs; (2) an increase in the SFA amplitude can
enhance the capability of a CANN to achieve an anticipative tracking with a longer anticipative time
and (3) with the proper SFA amplitudes, the network can achieve either perfect tracking or perfect
anticipative tracking for a wide range of stimulus speed.
The key point for SFA achieving anticipative tracking in a CANN is that it provides a negative feedback modulation to destabilize strong localized neuronal responses. Thus, other negative feedback
7
A
B
0.01
?0.02
0
S=0
m = 0.5 ?/?
v
?0.01
S
S
0
m = ?/?
v
?0.01
?0.015
?0.02
0
0.1
0.2
0.3
?0.02
0
0.4
0.1
C
0.3
0.4
D
20
0.04
0.03
t
0.02
t
t
an
ex
15
m = 2.5 ?/?
10
m = 2.0 ?/?
5
m = 1.5 ?/?
v
[?]
S=
v
m = 2.5 ?/?
v
ant
v
t
S
0.2
v ext
v ext
0.01
0
0
0.1
0.2
0.3
0
0
0.4
v
0.1
0.2
0.3
0.4
v ext
v ext
Figure 4: Tracking performances of a CANN with SFA. A. An example of delayed tracking for
m < ? /?v ; B. An example of perfect tracking for m = ? /?v . s = 0 roughly holds for a wide range
of stimulus speed. C. An example of perfect anticipative tracking for m > ? /?v . s increases linearly
with vext for a wide range of stimulus speed. D. Anticipative time increases with the SFA amplitude
m. The other parameters are the same as those in Fig. 3.
modulation processes, such as short-term synaptic depression (STD) [20, 21] and negative feedback
connections (NFC) from other networks [22], should also be able to realize anticipative tracking.
Indeed, it was found in the previous studies that a CANN with STD or NFC can produce leading
behaviors in response to moving inputs. The three mechanisms, however, have different time scales
and operation levels: SFA has a time scale of one hundred milliseconds and functions at the single
neuron level; STD has a time scale of hundreds to thousands of milliseconds and functions at the
synapse level; and NFC has a time scale of tens of milliseconds and functions at the network level.
The brain may employ them for different computational tasks in conjunction with brain functions.
It was known previously that a CANN with SFA can retain travelling wave [12]. But, to our knowledge, our study is the first one that links this intrinsic mobility of the network to the tracking performance of the neural system. We demonstrate that through regulating the SFA amplitude, a neural
system can implement anticipative tracking with a range of anticipative times. Thus, it provides
a flexible mechanism to compensate for a range of delay times, serving different computational
purposes, e.g., by adjusting the SFA amplitudes, neural circuits along the hierarchy of a signal transmission pathway can produce increasing anticipative times, which compensate for the accumulated
time delays. Our study sheds light on our understanding of how the brain processes motion information in a timely manner.
Acknowledgments
This work is supported by grants from National Key Basic Research Program of China
(NO.2014CB846101, S.W.), and National Foundation of Natural Science of China (No.11305112,
Y.Y.M.; No. 31261160495, S.W.), and the Fundamental Research Funds for the central Universities
(No. 31221003, S. W.), and SRFDP (No.20130003110022, S.W), and Research Grants Council
of Hong Kong (Nos. 605813, 604512 and N HKUST606/12, C.C.A.F. and K.Y.W), and Natural
Science Foundation of Jiangsu Province BK20130282.
8
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9
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5,108 | 5,624 | Analysis of Brain States
from Multi-Region LFP Time-Series
Kyle Ulrich 1 , David E. Carlson 1 , Wenzhao Lian 1 , Jana Schaich Borg 2 ,
Kafui Dzirasa 2 and Lawrence Carin 1
1
Department of Electrical and Computer Engineering
2
Department of Psychiatry and Behavioral Sciences
Duke University, Durham, NC 27708
{kyle.ulrich, david.carlson, wenzhao.lian, jana.borg,
kafui.dzirasa, lcarin}@duke.edu
Abstract
The local field potential (LFP) is a source of information about the broad patterns
of brain activity, and the frequencies present in these time-series measurements
are often highly correlated between regions. It is believed that these regions may
jointly constitute a ?brain state,? relating to cognition and behavior. An infinite
hidden Markov model (iHMM) is proposed to model the evolution of brain states,
based on electrophysiological LFP data measured at multiple brain regions. A
brain state influences the spectral content of each region in the measured LFP.
A new state-dependent tensor factorization is employed across brain regions, and
the spectral properties of the LFPs are characterized in terms of Gaussian processes (GPs). The LFPs are modeled as a mixture of GPs, with state- and regiondependent mixture weights, and with the spectral content of the data encoded in
GP spectral mixture covariance kernels. The model is able to estimate the number
of brain states and the number of mixture components in the mixture of GPs. A
new variational Bayesian split-merge algorithm is employed for inference. The
model infers state changes as a function of external covariates in two novel electrophysiological datasets, using LFP data recorded simultaneously from multiple
brain regions in mice; the results are validated and interpreted by subject-matter
experts.
1
Introduction
Neuroscience has made significant progress in learning how activity in specific neurons or brain areas correlates with behavior. One of the remaining mysteries is how to best represent and understand
the way whole-brain activity relates to cognition: in other words, how to describe brain states [1].
Although different brain regions have different functions, neural activity across brain regions is often highly correlated. It has been proposed that the specific way brain regions are correlated at any
given time may represent a ?state? designed specifically to optimize neural computations relevant
to the behavioral context an organism is in [2]. Unfortunately, although there is great interest in the
concept of global brain states, little progress has been made towards developing methods to identify
or characterize them.
The study of arousal is an important area of research relating to brain states. Arousal is a hotly
debated topic that generally refers to the way the brain dynamically responds to varying levels of
stimulation [3]. One continuum of arousal used in the neuroscience literature is sleep (low arousal) to
wakefulness (higher arousal). Another is calm (low arousal) to excited or stressed (high arousal) [4].
A common electrophysiological measurement used to determine arousal levels is local field potentials (LFPs), or low-frequency (< 200 Hz) extracellular neural oscillations that represent coordinated
1
Example LFP Data
..
.
Spectral Density
(a)
sw?1
(ar)
zw?1
(ar)
yw?1
Normalized Potential
2
?
R
F
??
1
0
(a)
(ar)
zw
(ar)
k` (? )
yw
(a)
(ar)
zw+1
y1
R
..
.
A
y2
1
1.5
Seconds
WK
0
4.2
2
8.3
Frequency, Hz
12.5
16.7
(a)
?`
(ar)
yw+1
SWS
Figure 1: Left: Graphical representation of our state space model.
R
sw+1
0.04
0.02
0.5
sw
REM
0.06
Classify yw
?1
?2
0.08
L
We first assign a sequence of brain states, {sw }W
1 , with Markovian
(a)
dynamics to animal a. Given state sw , each region is assigned to
(ar)
(ar)
a cluster, zw = ` ? {1, . . . , L}, and the data yw is generated
from a Gaussian process with covariance function k(? ; ?` , ?). Top:
Example of two windows of an LFP time-series; we wish to classify
each window based on spectral content. Spectral densities of known
sleep states (REM, SWS, WK) in the hippocampus are shown.
neural activity across distributed spatial and temporal scales. LFPs are useful for describing overall
brain states since they reflect activity across many neural networks. We examine brain states under
different levels of arousal by recording LFPs simultaneously in multiple regions of the mouse brain,
first, as mice pass through different stages of sleep, and second, as mice are moved from a familiar
environment to a novel environment to induce interest and exploration.
In neuroscience, the analysis of electrophysiological time-series data is largely centered around dynamic causal modeling (DCM) [5], where continuous state-space models are formulated based on
differential equations that are specifically crafted around knowledge of underlying neurobiological
processes. However, DCM is not suitable for exploratory analysis of data, such as inferring unknown arousal levels, for two reasons: because the differential equations are driven by inputs of
experimental conditions, and the analysis is dependent on a priori hypotheses about which neuronal
populations and interactions are important. This work focuses on methods suitable for exploratory
analysis.
Previously published neuroscience studies distinguished between slow-wave sleep (SWS), rapideye-movement (REM), and wake (WK) using proportions of high-frequency (33-55 Hz) gamma
oscillations and lower frequency theta (4-9 Hz) oscillations in a brain area called the Hippocampus [6, 7]. As an alternative approach, recent statistical methods for tensor factorization [8] can be
applied to short time Fourier transform (STFT) coefficients by factorizing a 3?way LFP tensor, with
dimensions of brain region, frequency band and time. Distinct sleep states may then be revealed by
clustering the inferred sequence of time-varying score vectors.
Although good first steps, the above two methods have several shortcomings: 1) They do not consider the time dependency of brain activity, and therefore cannot capture state-transition properties.
2) They cannot directly work on raw data, but require preprocessing that only considers spectral
content in predefined frequency bins, thus leading to information loss. 3) They do not allow for individual brain regions to take on their own set of sub-state characteristics within a given global brain
state. 4) Finally, they cannot leverage the shared information of LFP data across multiple animals.
In this paper we overcome the shortcomings of previously published brain-state methods by defining
a sequence of brain states over a sliding window of raw, filtered LFP data, where we impose an
infinite hidden Markov model (iHMM) [9] on these state assignments. Conditioned on this brain
state, each brain region is assigned to a cluster in a mixture model. Each cluster is associated with a
specific spectral content (or density) pattern, manifested through a spectral mixture kernel [10] of a
Gaussian process. Each window of LFP data is generated as a draw from this mixture of Gaussian
processes. Thus, all animals share an underlying brain state space, of which, all brain regions share
the underlying components of the mixture model.
2
Model
For each animal a ? {1, . . . A}, we have time-series of the LFP in R different regions, mea(ar)
sured simultaneously. These time-series are split into sequential, sliding windows, yw ? RN
for w ? {1, . . . , W }, such that windows are common across regions. These windows are chosen
to be overlapping, thereby sharing data points between consecutive windows; nonoverlapping win2
dows may also be used. Each window is considered as a single observation vector, and we wish to
(ar)
model the generative process of these observations, {yw }.
The proposed model aims to describe the spectral content in each of these LFP signals, as a function
of brain region and time. This is done by first assigning a joint ?brain state? to each time window,
(a)
(a)
{s1 , . . . , sW }, shared across all brain regions {1, . . . , R}. The brain state is assumed to evolve in
time as a latent Markov process. The LFP data from a particular brain region is assumed drawn from
a mixture of Gaussian processes. The characteristics of each mixture component are shared across
brain states and brain regions, with mixture weights that are dependent on these two entities.
2.1
Brain state assignment
Within the generative process, each animal has a latent brain state for every time window, w. This
(a)
brain state is represented through a categorical latent variable sw , and an infinite hidden Markov
model (iHMM) is placed on the state dynamics [9, 12]. This process is formulated as
s(a)
w ? Categorical(?
(a)
(a)
sw?1
),
?(a)
g ? DP(?0 ?),
? ? GEM(?0 ),
(1)
Qh?1
where GEM is the stick-breaking process ?h = ?h0 i=1 (1 ? ?i0 ) with ?h0 ? Beta(1, ?0 ). Here,
{?h }H
h=1 represents global transition probabilities to each state in a potentially infinite state space.
For the stick-breaking process, H ? ?, but in a finite collection of data only a finite number of
state transitions will be used and H can be efficiently truncated. Since the state space is shared
(a)
(a)
across animals, we cannot predefine initial state assignments, s1 . To remedy this, we allow s1 ?
Categorical(? (a) ) and place a discrete uniform prior on ? (a) over the truncated state space.
Each animal is given a transition matrix ?(a) , where each row of this matrix is a transition proba(a)
(a)
bility vector ?g such that the transition from state g to state h for animal a is ?gh , each centered
around the global transition vector ?. Because each animal?s brain can be structured differently
(e.g., as an extreme case, consider a central nervous system disorder), we allow ?(a) to vary from
animal to animal.
2.2
Assigning brain regions to clusters
For each brain state, mixture weights are drawn to define the distribution over clusters independently
for each region r, centered around a global mixture ? using a hierarchical Dirichlet Process [12]:
(r)
?h ? DP(?1 ?),
? ? GEM(?1 ),
(2)
(r)
where ?h` is the probability of assigning region r of a window with brain state h to cluster `. This
cluster assignment can be written as
(r)
(ar) (a)
zw
|sw ? Categorical(?
(a)
sw
).
(3)
For each cluster ` there is a set of parameters, ?` , describing a Gaussian process (GP), detailed in
Section 2.3. One could consider the joint probability over cluster assignments for all brain regions
as an extension of a latent nonnegative PARAFAC tensor decomposition [11, 13]. We refer to the
Supplemental Material for details. Our clustering model differs from the infinite tensor factorization
(ITF) model of [11] in three significant ways: we place Markovian dynamics on state assignments
for each animal, we model separate draws from the prior jointly for each animal, and we share
cluster atoms across all regions through use of an HDP.
2.3
2.3.1
Infinite mixture of Gaussian processes
Gaussian processes and the spectral mixture kernel
(ar)
For a single window of data, yw ? RN , we wish to model the data in the limit of a continuoustime function (allowing N ? ?), motivating a GP formulation, and we are interested in the spectral
properties of the LFP signal in this window. Previous research has established a link between the
kernel function of a GP and its spectral properties [10]. We write a distribution over the time-series:
y(t) ? GP(m(t), k(t, t0 )),
3
(4)
where m(t) is known as the mean function, and k(t, t0 ) is the covariance function [14]. This framework provides a flexible, structured method to model time-series data. The structure of observations
in the output space, y, is defined through a careful choice of the covariance function. Since this work
aims to model the spectral content of the LFP signal, we set the mean function to 0, and use a recently proposed spectral mixture (SM) kernel [10]. This kernel is defined through a spectral domain
representation, S(s), of the stationary kernel, represented by a mixture of Q Gaussian components:
XQ
1
?(s) =
?q N (s; ?q , ?q ),
S(s) = [?(s) + ?(?s)],
(5)
q=1
2
where ?(s) is reflected about the origin to obtain a valid spectral density, and ?q , ?q , and ?q respectively define the mean, variance, and relative weight of the q-th Gaussian component in the spectral
domain. Priors may be placed on these parameters; for example, we use the uninformative priors
?q ? Uniform(?min , ?max ), ?q ? Uniform(0, ? max ) and ?q ? Gamma(e0 , f0 ). A bandpass filter
is applied to the LFP signal from ?min to ?max Hz as a preprocessing step, so this prior knowledge
is justified. Also, ? max is set to prevent overfitting, and e0 and f0 are set to manifest a broad prior.
We assume that only a noisy version of the true function is observed, so the kernel is defined as the
Fourier transform of the spectral density S(s) plus white Gaussian noise:
XQ
f (? ; ?) =
?q exp{?2? 2 ? 2 ?q } cos(2?? ?q ),
k(? ; ?, ?) = f (? ; ?) + ? ?1 ?? ,
(6)
q=1
where the set of parameters ? = {?, ?, ?} and ? define the covariance kernel, ? = |t ? t0 |, and ?? is
the Kronecker delta function which equals one if ? = 0. We set the prior ? ? Gamma(e1 , f1 ) where
the hyperparemeters e1 and f1 are chosen to manifest a broad prior. The formulation of (6) results
in an interpretable kernel in the spectral domain, where the weights ?q correspond to the relative
contribution of each component, the means ?q represent spectral peaks, and the variances ?q play a
role similar to an inverse length-scale.
Through a realization of this Gaussian process, an analytical representation is obtained for the
marginal likelihood of the observed data y given the parameters {?, ?}, and the observation locations t, p(y|?, ?, t). The optimal set of kernel parameters {?, ?} can then be chosen as the set that
maximizes the marginal likelihood. Further discussions on the inference for the Gaussian process
parameters is presented in Section 3.
2.3.2
Generating observed data
To combine the clustering model with our SM kernel, each cluster ` is associated with a distinct
(ar)
(ar)
set of kernel parameters ?` . To generate the observations {yw }, where each yw ? RN has
observation times t = {t1 , . . . , tN } such that |ti ? tj | = |i ? j|? for all i and j, we consider a draw
from the multivariate normal distribution:
(ar)
yw
? N (0, ?z(ar) ),
(?` )ij = k(|ti ? tj |; ?` , ?),
w
(7)
(ar)
where each observation is generated from the cluster indicated by zw (described in Section 2.2),
and each cluster is represented uniquely by a covariance matrix, ?` , whose elements are defined
through the covariance kernel k(? ; ?` , ?). Therefore, the parameters ?z(ar) describe the autow
(ar)
correlation content associated with each yw .
We address two concerns with this formulation. First, this observation model ignores complex
cross-covariance functions between regions. Although LFP measurements exhibit coherence patterns across regions, the generative model in (7) only weakly couples the spectral densities of each
region through the brain state. In principle, the generative model could be extended to incorporate
this coherence information. Second, (7) does not model the time-series itself as a stochastic process,
but rather the preprocessed, ?independent? observation vectors. This shortcoming is not ideal, but
the windowing process allows for efficient computation via the mixture of Gaussian processes.
3
Inference
In the following, latent model variables are represented by ? = {Z, S, ?, ?, ?, ?, ?}, the kernel
parameters to be optimized are ? = {{?` }L
1 , ?}, and H and L are upper limit truncations on the
number of brain states and clusters, respectively. As described throughout this section, the proposed
algorithm adaptively adjusts the truncation levels on the number of brain states, H, and clusters, L,
4
through a series of split-merge moves. The joint probability of the proposed model is
p(Y ,?, ?) = p(Y |Z, ?)p(Z, S|?, ?, ?)p(?|?)p(?)p(?|?)p(?)p(?)p(?)
hY
ih
i
Y
(r)
(ar) (ar)
(ar) (a)
=
p(yw
|zw , ?)p(zw
|sw , ?) p(?|?1 )
p(?h |?, ?1 )
a,r,w
r,h
Y
h
i
Y
Y
W
(a)
(a)
(a)
p(s1 |? (a) )p(? (a) )
p(s(a)
) p(?|?0 )
p(?(a)
w |sw?1 , ?
g |?, ?0 )
a
w=2
a,g
YQ
p(?|e1 , f1 )
p(?q |e0 , f0 )p(?q |?min , ?max )p(?q |? max ) .
(8)
q=1
A variational inference scheme is developed to update ? and ?.
3.1
Variational inference
With variational inference, an approximate variational posterior distribution is sought that
is similar to the true posterior distribution, q(?, ?) ? p(?, ?|Y ).
This variational
posterior is assumed to have a factorization into simpler distributions, where q(?, ?) =
q(Z)q(S)q(?)q(?)q(?)q(?)q(?)q(?), with further factorization
Y
Y
(r)
(r)
(ar)
(ar)
Cat(zw
; ?w
),
q(?) =
Dir(?h ; ?h ),
q(?) = ??? (?),
q(Z) =
a,r,w
h,r
Y
Y
W
(a)
q(S) =
q({s(a)
q(?) =
Dir(?(a)
q(?) = ??? (?),
w }w=1 ),
g ; ?g ),
a
g,a
Y
Y
q(?) =
??(a)? (? (a) ),
q(?) =
???j (?j ),
(9)
a
j
(a)
where only necessary sufficient statistics of the latent factors q({sw }W
w=1 ) are required, and the
approximate posteriors of ?, ?, {? (a) } and {?j } are represented by point estimates at ? ? , ? ? ,
{? (a)? } and {??j }, respectively.
The degenerate distributions ??? (?) and ??? (?) are described in previous work on variational inference for HDPs [15, 16]. The idea is that the point estimates of the stick-breaking processes
simplify the derivation of the variational posterior, and the authors of [16] show that obtaining a
full posterior distribution on the stick-breaking weights has little impact on model fitting since the
variational lower bound is not heavily influenced by the terms dependent on ? and ?. Furthermore, the Dirichlet process is truncated for both the number of states and the number of clusters
(a)
(ar)
such that q(zw = `) = 0 for ` > L and q(sw = h) = 0 for h > H. This truncation method
(see [17] for details) is notably different than other common truncation methods of the DP (e.g., [18]
and [19]), and is primarily important for facilitating the split-merge inference techniques described
in Section 3.2.
In mean-field variational inference, the variational distribution q(?, ?) is chosen such that the
Kullback-Leibler divergence of p(?, ?|Y ) from q(?, ?), DKL (q(?, ?)||p(?, ?|Y )), is minimized. This is equivalent to maximizing the evidence lower bound (also known as the variational
free energy in the DCM literature), L(q) = Eq [log p(Y , ?, ?)] ? Eq [log q(?, ?)], where both
expectations are taken with respect to the variational distribution. The resulting lower bound is
L(q) = E[ln p(Y |Z, ?)] + E[ln p(Z, S|?, ?, ?)] + E[ln p(?|?)] + E[ln p(?)] + E[ln p(?|?)]
+ E[ln p(?)] + E[ln p(?)] + E[ln p(?)] + H[q(Z)] + H[q(S)] + H[q(?)] + H[q(?)], (10)
where all expectations are with respect to the variational distribution, the hyperparameters are excluded for notational simplicity, and we define H[q(?)] as the sum over the entropies of the individual
factors of q(?). Due to the degenerate approximations for q(?), q(?), q(?) and q(?), these full
posterior distributions are not obtained, and, therefore, the terms H[q(?)], H[q(?)], H[q(?)] and
H[q(?)] are set to zero in the lower bound.
(ar)
(r)
The updates for ?w and ?h are standard. Variational inference for the HDP-HMM is detailed
(a)
in other work (e.g., see [20, 21]); using these methods, updates for ?g , ? (a) and the necessary
expected sufficient statistics of the factors of q(S) are realized. Finally, updates for ? ? , ? ? and
{?j } are non-conjugate, so a gradient-ascent method is performed to optimize these values. We
use a simple resilient back-propagation (Rprop), though most line-search methods should suffice.
Details on all updates and taking the gradient of L(q) with respect to ?, ? and {?j } are found in
the Supplemental Material.
5
3.2
Split-merge moves
During inference, a series of split and merge operations are used to help the algorithm jump out of
local optima [22]. This work takes the viewpoint that two clusters (or states) should merge only if
the variational lower bound increases, and, when a split is proposed for a cluster (or state), it should
always be accepted, whether or not the split increases the variational lower bound. If the split is
not appropriate, a future merge step is expected to undo this operation. In this way, the opportunity
is provided for cluster and state assignments to jump out of local optima, allowing the inference
algorithm to readjust assignments as desired.
Merge states: To merge states h0 and h00 into a new state h, new parameters are initialized as:
(a)
(a)
(a)
(a)
(a)
(a)
(a)
(a)
(a)
?wh = ?wh0 + ?wh00 , ?gh = ?gh0 + ?gh00 , ?h? = ?h?0 + ?h?00 , and vh = vh0 + vh00 , such
that the model now has a truncation at H new = H ? 1 states. In order to account for problems
with merging two states in an HMM, a single restricted iteration is allowed, where only the statedependent variational parameters in ?new are updated, producing a new distribution q(?new ). The
merge is accepted (i.e., ? = ?new ) if L(q(?new )) > L(q(?)). Since these computations are not
excessive, all possible state merges are computed and a small number of merges are accepted per
iteration.
Merge clusters: To merge clusters `0 and `00 into a new cluster `, new parameters are initialized as:
(ar)
(ar)
(ar)
(r)
(r)
(r)
?w` = ?w`0 + ?w`00 , ?h` = ?h`0 + ?h`00 , ?`? = ?`?0 + ?`?00 , and ?`new = ? ? , such that there is a
new
truncation at L
= L ? 1 clusters. We set ? ? = ?`0 for simplicity, and allow a restricted iteration
new
of updates to ?
and ?`new . The merge is accepted (i.e., ? = ?new and ? = ?new ) if the lower
bound is improved, L(q(?new , ?new )) > L(q(?, ?)). Since the restricted iteration for ?`new is
expensive, only a few cluster merges may be proposed at a time. Therefore, merges are proposed
for clusters with the smallest earth mover?s distance [23] between their spectral densities.
Split step: When splitting states and clusters, the opposite process to the initialization of the merging
procedures described above is performed. For clusters, data points within a cluster ` are randomly
chosen to stay in cluster ` or split to a new cluster `0 . For splitting state h, the cluster assignment
(r)
vector ?h is replicated and windows within state h are randomly chosen to stay in state h or split
to a new cluster h0 . Regardless of how this effects the lower bound, a split step is always accepted.
For implementation details, we allow the model to accept 3 state merges every third iteration, propose 5 cluster merges every third iteration, and split one state and one cluster every third iteration.
Therefore, every iteration may affect the truncation level of either the number of states or clusters.
A ?burn-in? period is allowed before starting the proposing of splits/merges, and a ?burn-out? period
is employed in which split proposals cease. In this way, the algorithm has guarantees of improving
the lower bound only during iterations when a split is not proposed, and convergence tests are only
considered during the burn-out period.
4
Datasets
Three datasets are considered in this work, as follows:
Toy data: Data is generated for a single animal according to the proposed model in Section 2. The
purpose of this dataset is to ensure the inference scheme can recover known ground truth, since
ground truth information is not known for the real datasets. We set L = 5 and H = 3. For each
cluster, a spectral density was generated with Q = 4, ?q ? Unif(4, 50), ?q ? Unif(1, 50) and
(r)
1
1
, . . . , 10
). State
? ? Dir(1, . . . , 1). The cluster usage probability vector was drawn ?h ? Dir( 10
transition probabilities were drawn according to ?gh ? Unif(0, 1) + 10?(g=h) . States were assigned
to W = 1000 windows according to an HMM with transition matrix ?, and cluster assignments
were drawn conditioned on this state. Data with N = 200 was drawn for each window.
Sleep data: Twelve hours of LFP data from sixteen different brain regions were recorded from three
mice naturally transitioning through different levels of sleep arousal. Due to the high number of
brain regions, we present only three hours of sleep data from a single mouse for simplicity. The
multi-animal analysis is reserved for the novel environment dataset.
Novel environment data: Thirty minutes of LFP data from five brain regions was recorded from
five mice who were moved from their home cage to a novel environment approximately nine minutes
into the recording. Placing animals into novel environments has been shown to increase arousal, and
6
should therefore result in (at least one) network state change [3]. Data acquisition methods for the
latter two datasets are discussed in [24].
5
Results
For all results, we set Q = 10, H = 15, L = 25, stop the ?burn-in? period after iteration 6, and start
the subsequent computation period after iteration 25. Hyperparameters were set to ?0 = ?1 = .01,
?0 = ?1 = 1, ?min = 0, ?max = 50, ? max = 10, and e0 = f0 = 10?6 . In all results, the model
was seen to converge to a local optima after 30 iterations, and each iteration took on the order of 20
seconds using Matlab code on a PC with a 2.30GHz quad-core CPU and 8GB RAM.
Figure 2 shows results on the toy data. The model correctly recovers exactly 3 states and 5 clusters,
and, as seen in the figure, the state assignments and spectral densities of each cluster component
are recovered almost perfectly. The model was implemented for different values of the noise variance, ? ?1 , and, though not shown, in all cases the noise variance was recovered accurately during
inference, implying the spectral mixture kernels are not overfitting the noise. In this way, we confirm that the inference scheme recovers a ground truth. For further model verification, ten-fold
cross-validation was used to compute predictive probabilities for held-out data (reported in Table 1),
where we compare to two simpler versions of our model: 1) the HDP-HMM on brain states in (1) is
replaced with an HDP, and 2) a single brain state. For the HDP-HMM, the hold-out data was considered as ?missing data? in the training data and the window index was used to assign time-dependent
probabilities over clusters, whereas in the HDP and Single State models it was simply withheld from
the training data. We see large predictive performance gains when considering multiple brain states,
and even more improvement on average (though modest) when considering an HDP-HMM.
Cluster Usage for each Brain State
Log Spectral Density
Simulated Brain Activity over Time
?2
1
Clust 5
?Region 1?
Clust 5
Clust 4
?Region 2?
Clust 4
Clust 3
?Region 3?
Clust 3
Clust 2
?Region 4?
Clust 2
Clust 1
?Region 5?
?3
0.8
?4
0.6
?5
0.4
?6
0.2
10
20
30
Frequency, Hz
40
50
State Assignment Comparison
State 3
State 2
State 1
True State
Inferred State
5
10
Clust 1
5
1
2
3
Brain State, (5 Channels per State)
Log Spectral Density
0
?7
0
15
Minutes
10
15
Minutes
?2
?4
?6
?8
0
25
50
Freq, Hz
0
25
50
Freq, Hz
0
25
50
Freq, Hz
0
25
50
Freq, Hz
0
25
50
Freq, Hz
Figure 2: Toy data results. Top row shows the generated toy data. From left to right: the five spectral functions,
each associated with a component in the mixture model; the probability of each of these five components
occurring for all five regions in each brain state; the generated brain state assignments from a 3-state HMM
along with the generated cluster assignments for the five simulated regions. The bottom row shows the results
of our model. On the left, a comparison of the recovered state vs. the true state for all time; on the right, an
alignment of the five recovered kernels to the spectral density ground truth.
Brain Activity over Time
State 8
State 7
State 6
State 5
State 4
State 3
State 2
State 1
Dzirasa et al.
Tensor Method
Our Method
20
30
Minutes
DLS
DMS
FrA
M1
M_OFC_Cx
OFC
Basal_Amy
D_Hipp
L_Hb
NAc_Core
NAc_Shell
MD_Thal
PrL_Cx
SubNigra
V1
VTA
0.1
0.05
4
6
8
10
Frequency, Hz
35
40
12
14
Clust 7
Clust 6
Clust 5
Clust 4
Clust 3
Clust 2
45
Cluster Usage given State/Region
0.15
0
2
25
Brain Activity over Time
D Hipp
NAc core
OFC
VTA
15
D Hipp
NAc core
OFC
VTA
10
D Hipp
NAc core
OFC
VTA
5
Spectral Density
0.2
50
55
State Usage Given (Dzirasa et al.)
Clust 7
Clust 5
State 7
0.8
State 6
Clust 4
0.6
Clust 3
0.4
15
20
Minutes
25
30
State 5
State 4
State 3
Clust 2
State 2
0.2
State 1
0
10
State 8
1
Clust 6
Clust 1
Clust 1
5
60
1
3
2
Brain State (Showing 4/16 Regions)
Tensor
Our
Figure 3: Sleep data results. Top: A comparison of brain state assignments from our method to two other
methods. Bottom Left: Spectral density of the 7 inferred clusters. Middle Left: Cluster assignments over
time for 16 different brain regions, sorted by similarity. Middle Right: Given brain states 1, 2 and 3, we show
cluster assignment probabilities for 4 different brain regions: the hippocampus (D Hipp), nucleus accumbens
core (NAc core), orbitofrontal cortex (OFC) and ventral tegmental area (VTA) from left to right, respectively.
Right: State assignments of our method and the tensor method conditioned on the method of [6].
7
Cluster Usage for each Brain State
Brain Activity over Time
Spectral Density
0.35
Animal 1
0.3
Animal 2
0.25
Animal 3
State 7
1
Clust 6
0.8
Clust 5
State 6
State 5
Animal 4
0.2
Clust 4
0.6
Animal 5
State 4
0.15
Clust 3
Animal 6
0.1
Animal 7
0.05
Animal 8
State 3
Clust 2
State 2
0
2
6
8
10
Frequency, Hz
12
14
0.2
Clust 1
State 1
Animal 9
4
0.4
5
10
15
Minutes
20
25
0
1
2
3
4
5
6
7
Brain State, (5 Channels per State)
Figure 4: Novel environment data results. Left: The log spectral density of the 6 inferred clusters. Middle:
State assignments for all 9 animals over a 30 minute period. There are 7 inferred states, and each state has a
distribution over clusters for each region, as seen on the right.
Dataset
5
Toy (?10 )
Sleep (?106 )
Novel (?105 )
HDP-HMM
HDP
Single State
?1.686 (?0.053)
?1.677 (?0.030)
?5.932 (?0.040)
?1.688 (?0.053)
?1.682 (?0.020)
?5.973 (?0.034)
?1.718 (?0.054)
?1.874 (?0.019)
?6.962 (?0.063)
Table 1: Average held-out log predictive probability for different priors on brain states: HDP-HMM, HDP,
and a single state. The data consists of W time-series windows for R regions of A animals; at random, 10% of
these time-series windows were held-out, and the predictive distribution was used to determine their likelihood.
The sleep and novel environment results are presented in Figures 3 and 4, respectively. With the
sleep dataset, our results are compared with the two methods discussed in the Introduction: that
of [6, 7], and the tensor method of [8]. We refer to the Supplemental Material for exact specifications
of the tensor method.
For each of these datasets, we infer the intended arousal states. In the novel environment data,
we observe broad arousal changes at 9?minutes for all animals, as expected. In the sleep data, we
successfully uncover at least as many states as the simple approach of [6, 7], to include SWS, REM
and WK states. Thus far neuroscientists have focused primarily on 2 stages of sleep (NREM and
REM), but as many as 5 have been discussed (4 different stages of NREM sleep, and 1 stage of
REM). Different stages of sleep affect memory and behavior in different ways (e.g., see [25]), as
does the number of times animals transition between these states [26]. Our results suggest that there
may be even more levels of sleep that should be considered (e.g., transition states and sub states).
This is very interesting and important for neuroscientists to know, because it is possible that each
of our newly observed states could affect memory and behavior in different ways. There is no other
published method that has provided evidence of these other states.
In addition to brain states, we infer spectral information for each brain region through cluster assignments. Though not the primary focus of this work, it is interesting that groups of brain regions tend
to share similar attributes. In Figure 3, we have sorted brain regions into groups based on cluster
assignment similarity, essentially recovering a ?network? of the brain. This underscores the power of
the proposed method: not only do we develop unsupervised methods to classify whole-brain activity
into states, we infer the cross-region/animal relationships within these states.
6
Conclusion
The contributions of this paper are three-fold. First, we design an extension of the infinite tensor
mixture model, incorporating time dependency. Second, we develop variational inference for the
proposed generative model, including an efficient inference scheme using split-merge moves for two
general models: the ITM and iHMM. To the authors? knowledge, neither of these inference schemes
have been developed previously. Finally, with respect to neuroscience application, we model brain
states given multi-channel LFP data in a principled manner, showing significant advantages over
other potential approaches to modeling brain states. Using the proposed framework, we discover
distinct brain states directly from the raw, filtered data, defined by their spectral content and network
properties, and we can infer relationships between and share statistical strength across data from
multiple animals.
Acknowledgments
The research reported here was funded in part by ARO, DARPA, DOE, NGA and ONR.
8
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5,109 | 5,625 | Extracting Latent Structure From Multiple
Interacting Neural Populations
Jo?ao D. Semedo1,2,3 , Amin Zandvakili4 , Adam Kohn4 ,
?
Christian K. Machens3 , ? Byron M. Yu1,5
1
Department of Electrical and Computer Engineering, Carnegie Mellon University
2
Department of Electrical and Computer Engineering, Instituto Superior T?ecnico
3
Champalimaud Neuroscience Programme, Champalimaud Center for the Unknown
4
Dominick Purpura Department of Neuroscience, Albert Einstein College of Medicine
5
Department of Biomedical Engineering, Carnegie Mellon University
[email protected]
{amin.zandvakili,adam.kohn}@einstein.yu.edu
[email protected]
[email protected]
?
Denotes equal contribution.
Abstract
Developments in neural recording technology are rapidly enabling the recording of populations of neurons in multiple brain areas simultaneously, as well as
the identification of the types of neurons being recorded (e.g., excitatory vs. inhibitory). There is a growing need for statistical methods to study the interaction
among multiple, labeled populations of neurons. Rather than attempting to identify direct interactions between neurons (where the number of interactions grows
with the number of neurons squared), we propose to extract a smaller number of
latent variables from each population and study how these latent variables interact.
Specifically, we propose extensions to probabilistic canonical correlation analysis
(pCCA) to capture the temporal structure of the latent variables, as well as to distinguish within-population dynamics from across-population interactions (termed
Group Latent Auto-Regressive Analysis, gLARA). We then applied these methods to populations of neurons recorded simultaneously in visual areas V1 and V2,
and found that gLARA provides a better description of the recordings than pCCA.
This work provides a foundation for studying how multiple populations of neurons
interact and how this interaction supports brain function.
1
Introduction
In recent years, developments in neural recording technologies have enabled the recording of populations of neurons from multiple brain areas simultaneously [1?7]. In addition, it is rapidly becoming possible to identify the types of neurons being recorded (e.g., excitatory versus inhibitory [8]).
Enabled by these experimental advances, a major growing line of scientific inquiry is to ask how different populations of neurons interact, whether the populations correspond to different brain areas or
different neuron types. To address such questions, we need statistical methods that are well-suited
for assessing how different groups of neurons interact on a population level.
One way to characterize multi-population activity is to have the neurons interact directly [9?11],
then examine the properties of the interaction strengths. While this may be a reasonable approach
for small populations of neurons, the number of interactions grows with the square of the number
of recorded neurons, which may make it difficult to summarize how larger populations of neurons
interact [12]. Instead, it may be possible to obtain a more succinct account by extracting latent
variables for each population and asking how these latent variables interact.
1
(a) pCCA
(b) AR-pCCA
(c) gLARA
Figure 1: Directed graphical models for multi-population activity. (a) Probabilistic canonical
correlation analysis (pCCA). (b) pCCA with auto-regressive latent dynamics (AR-pCCA). (c) Group
latent auto-regressive analysis (gLARA). For clarity, we show only two populations in each panel
and auto-regressive dynamics of order 1 in panel (c).
Dimensionality reduction methods have been widely used to extract succinct representations of population activity [13?17] (see [18] for a review). Each observed dimension corresponds to the spike
count (or firing rate) of a neuron, and the goal is to extract latent variables that describe how the population activity varies across experimental conditions, experimental trials, and/or across time. These
previous studies use dimensionality reduction methods that do not explicitly account for multiple
populations of neurons. In other words, these methods are invariant to permutations of the ordering
of the neurons (i.e., the observed dimensions).
This work focuses on latent variable methods designed explicitly for studying the interaction between labelled populations of neurons. To motivate the need for these methods, consider applying
a standard dimensionality reduction method, such as factor analysis (FA) [19], to all neurons together by ignoring the population labels. The extracted latent variables would capture all modes
of covariability across the neurons, without distinguishing between-population interaction (i.e., the
quantity of interest) from within-population interaction. Alternatively, one might first apply a standard dimensionality reduction method to each population of neurons individually, then examine
how the latent variables extracted from each population interact. However, important features of the
between-population interaction may be eliminated by the dimensionality reduction step, whose sole
objective is to preserve the within-population interaction.
We begin by considering canonical correlations analysis (CCA) and its probabilistic formulation
(pCCA) [20], which identify a single set of latent variables that explicitly captures the betweenpopulation covariability. To understand how the different neural populations interact on different
timescales, we propose extensions of pCCA that introduce a separate set of latent variables for each
neural population, as well as dynamics on the latent variables to describe their interaction over time.
We then apply the proposed methods to populations of neurons recorded simultaneously in visual
areas V1 and V2 to demonstrate their utility.
2
Methods
We consider the setting where many neurons are recorded simultaneously, and the neurons belong
to distinct populations (either by brain area or by neuron type). Let yti ? Rqi represent the observed
activity vector of population i ? {1, ..., M } at time t ? {1, ..., T }, where qi denotes the number of
neurons in population i. Below, we consider three different ways to study the interaction between
the neural populations. To keep the notation simple, we?ll only consider two populations (M = 2);
the extension to more than two populations is straightforward.
2.1
Factor analysis and probabilistic canonical correlation analysis
Consider the following latent variable model, that defines a linear-Gaussian relationship between the
observed variables, yt1 and yt2 , and the latent state, xt ? Rp :
xt ? N (0, I)
2
(1)
yt1
yt2
| xt ? N
C1
C2
xt +
d1
d2
R11
,
T
R12
R12
R22
(2)
where C i ? Rqi ?p , di ? Rqi and:
R11
T
R12
R12
R22
? Sq++
with q = q1 + q2 . According to this model, the covariance of the observed variables is given by:
1 1 h
i R11 R12
C
yt
1T
2T
=
+
cov
(3)
C
C
T
yt2
C2
R12
R22
Factor analysis (FA) and probabilistic canonical correlation analysis (pCCA) can be seen as two
special cases of the general model presented above. FA assumes the noise covariance to be diagonal,
i.e., R11 = diag(r11 , ..., rq11 ), R22 = diag(r12 , ..., rq22 ) and R12 = 0. This noise covariance captures
only the independent variance of each neuron, and not the covariance between neurons. As a result,
the covariance between neurons is explained by the latent state through the observation matrices C 1
and C 2 . pCCA, on the other hand, considers a block diagonal noise covariance, i.e., R12 = 0. This
noise covariance accounts for the covariance observed between neurons in the same population. The
latent state is therefore only used to explain the covariance between neurons in different populations.
The directed graphical model for pCCA is shown in Fig.1a.
2.2
Auto-regressive probabilistic canonical correlation analysis (AR-pCCA)
While pCCA offers a succinct picture of the covariance structure between populations of neurons,
it does not capture any temporal structure. There are two main reasons as to why this time structure
may be interesting. First, pCCA is modelling the covariance structure at zero time lag, which may
not capture all of the interactions of interest. If the two populations of neurons correspond to two
different brain areas, there may be important interactions at non-zero time lags due to physical delays
in information transmission. Second, the two populations of neurons may interact at more than one
time delay, for example if multiple pathways exist between the neurons in these populations. To
take the temporal structure into account we will first extend pCCA by defining an auto-regressive
linear-Gaussian model on the latent state:
xt ? N (0, I),
if 1 ? t ? ?
?
X
xt | xt?1 , xt?2 , ..., xt?? ? N
(4)
!
Ak xt?k , Q ,
if t > ?
(5)
k=1
where Ak ? Rp?p , ?k, Q ? Sp++ and ? denotes the order of the autoregressive model. We term this
model AR-pCCA, which is defined by the state model in Eq.(4)-(5) and the observation model in
Eq.(2) with R12 = 0. Although the observation model is the same as that for pCCA, the latent state
here accounts for temporal dynamics, as well as the covariation structure between the populations.
The corresponding directed graphical model is shown in Fig.1b.
2.3
Group latent auto-regressive analysis (gLARA)
According to AR-pCCA, a single latent state drives the observed activity in both areas. As a result,
it?s not possible to distinguish the within-population dynamics from the between-population interactions. To allow for this, we propose using two separate latent states, one per population, that interact
over time. We refer to the proposed model as group latent auto-regressive analysis (gLARA):
xt ? N (0, I),
xit | xt?1 , xt?2 , ..., xt??
if 1 ? t ? ?
?
?
2 X
?
X
j
i?
?N?
Aij
,
k xt?k , Q
j=1 k=1
3
(6)
if t > ?
(7)
1
1 1 1
d
R
0
C
0
xt
yt1
+
,
(8)
|
x
?
N
t
0 R2
x2t
d2
yt2
0 C2
where xt is obtained by stacking x1t ? Rp1 and x2t ? Rp2 , the latent states for each population,
i
pi ?pj
, ?k and i ? {1, 2}. Note that the covariance structure
C i ? Rqi ?pi , Aij
and Qi ? Sp++
k ? R
observed on a population level now has to be completely reflected by the latent states (there are no
shared latent variables in this model) and is therefore defined by the dynamics matrices Aij
k , allowing
22
for the separation of the within-population dynamics (A11
and
A
)
and
the
between-population
k
k
21
interactions (A12
k and Ak ). Furthermore, the interaction between the populations is asymmetrically
21
defined by A12
k and Ak , allowing for a more in depth study of the way in each the two areas interact
by comparing these across the various time delays considered. Note that gLARA represents a special
case of the AR-pCCA model.
2.4
Parameter estimation for gLARA
The parameters of gLARA can be fit to the training data using the expectation-maximization (EM)
? t ? Rp? , with p = p1 + p2 :
algorithm. To do so, we start by defining the augmented latent state x
1 h
iT
?t
x
1T
1 T
2T
2 T
?t =
=
(9)
x
x
.
.
.
x
x
.
.
.
x
t
t??
t
t??
? 2t
x
? t ? Rq , with q = q1 + q2 :
and the augmented observation vector y
h
iT
? t = yt1 T yt2 T
y
(10)
? , the dynamics equation (Eq.(6) and (7)) can
for t ? {?, ..., T }. Using the augmented latent state x
be rewritten as:
? t ? N (0, I), if t = ?
x
(11)
?xt?1 , Q
? , if t > ?
?t | x
? t?1 ? N A?
x
(12)
p?
p? ?p?
?
?
for appropriately structured A ? R
and Q ? S++ . The observation model (Eq.(8)) can be
rewritten as:
?t
x
?
?t | x
? t ? N C?
,R
(13)
y
1
? ? Sq . Due to space constraints, we will not
for appropriately structured C? ? Rq?(p? +1) and R
++
? R,
? A,
? Q}.
? It is straightforward
explicitly show the structure of the augmented parameters ?? = {C,
to derive them by inspection of Eq.(9)-(13).
We fit the model parameters using the EM algorithm. In the E-step, because the latent and observed
? 1 , ..., y
? T ) is also Gaussian and can be computed exactly by
variables are jointly Gaussian, P (?
xt | y
applying the forward-backward recursion of the Kalman smoother [21] on the augmented vectors. In
the M-step, we directly estimate the original parameters ? = {C i , di , Ri , Aij
k }, as opposed to esti? R,
? A}
? (without loss of generality,
mating the structured form of the augmented parameters ?? = {C,
we set Qi = I):
! T "
#!?1
T
h
i
X
X E(xi xi T ) E(xi )
i
T
i
i
t
t
t
C d
=
(14)
yt E(xit ) 1
T
E(xit )
1
t=1
t=1
Ri =
T
1X
T
{(yti ? di )(yti ? di ) ? C i E(xit )(yti ? di )T
T t=1
T
T
T
T
?(yti ? di )E(xit )C i + C i E(xit xit )C i }
A11
1
A21
1
...
...
A11
k
A21
k
A12
1
A22
1
...
...
A12
k
A22
k
=
T
X
t=2
!
E
?tx
? Tt?1
x
T
X
(15)
!?1
E
? t?1 x
? Tt?1
x
(16)
t=2
To initialize the EM algorithm, we start by applying FA to each population individually, and use
the estimated observation matrices C 1 and C 2 , as well as the mean vectors d1 and d2 and the
observation covariance matrices R11 and R22 . The Aij
k matrices are initialized at 0.
4
cross?validated log?likelihood
?4.77
x 10
5
FA
pCCA
pCCA shuffled
?4.795
10
20
30
40
latent dimensionality
50
60
Figure 2: Comparing the optimal dimensionality for FA and pCCA. (a) Cross-validated loglikelihood plotted as a function of the dimensionality of the latent state for FA (black) and pCCA
(blue). pCCA was also applied to the same data after randomly shuffling the population labels
(green). Note that maximum possible dimensionality for pCCA is 31, which is the size of the
smaller of the two populations (in this case, V2).
2.5
Neural recordings
The methods described above were applied to multi-electrode recordings performed simultaneously
in visual area 1 (V1) and visual area 2 (V2) of an anaesthetised monkey, while the monkey was
shown a set of oriented gratings with 8 different orientations. Each of the 8 orientations was shown
400 times for a period of 1.28s, providing a total of 3200 trials. We used 1.23s of data in each trial,
from 50ms after stimulus onset until the end of the trial, and proceeded to bin the observed spikes
with a 5ms window. The recordings include a total of 97 units in V1 and 31 units in V2 (single- and
multi-units). For model comparison, we performed 4-fold cross-validation, splitting the data into
four non-overlapping test folds with 250 trials each. We chose to analyze a subset of the trials for
rapid iteration of the analyses, as the cross-validation procedure is computationally expensive for
the full dataset. Given that 1000 trials provides a total of 246,000 timepoints (at 5 ms resolution),
this provides a reasonable amount of data to fit any of the models with the 128 observed neurons.
In this study, we sought to investigate how trial-to-trial population variability in V1 relates to the
trial-to-trial population variability in V2. For these gratings stimuli (which are relatively simple
compared to naturalistic stimuli [22]), there is likely richer structure in the V1-V2 interaction for
the trial-to-trial variability than for the stimulus drive. To this end, we preprocessed the neural
activity by computing the peristimulus time histogram (PSTH), representing the trial-averaged firing
rate timecourse, for each neuron and experimental condition (grating orientation). For each spike
train, we then subtracted the appropriate PSTH from the binned spike counts to obtain a single-trial
?residual?. The residuals across all neurons and conditions were considered together in the analyses
shown in Section 3. Note that the methods considered in this study could also be applied to the
PSTHs of sequentially recorded neurons in multiple areas.
3
Results
We started by asking how many dimensions are needed to describe the between-population covariance, relative to the number of dimensions needed to describe the within-population covariance.
This was assessed by applying pCCA to the labeled V1 and V2 populations, as well as FA to the two
populations together (which ignores the V1 and V2 labels). In this analysis, pCCA captures only
the between-population covariance, whereas FA captures both the between-population and withinpopulation covariance. By comparing cross-validated data likelihoods for different dimensionalities,
we found that pCCA required three latent dimensions, whereas FA required 40 latent dimensions
(Fig.2). This indicates that the zero time lag interaction between V1 and V2 is confined to a small
number of dimensions (three) relative to the number of dimensions (40) needed to describe all covariance among the neurons. The difference of these two dimensionalities (37) describes covariance
that is ?private? to each population (i.e., within-population covariance). The FA and pCCA curves
peak at similar cross-validated likelihoods in Fig.2 because the observation model for pCCA Eq.(2)
accounts for the within-population covariance (which is not captured by the pCCA latents).
5
(b)
cross?validated log?likelihood
(a)
?4.52
x 10
(c)
5
?=3
?=3
?=5
?=5
?=1
gLARA
?4.62
?=1
AR?pCCA
20
40
60
latent dimensionality
80
10
20
30
40
latent dimensionality p1
50
5
10
15
20
latent dimensionality p2
25
Figure 3: Model selection for AR-pCCA and gLARA. (a) Comparing AR-pCCA and gLARA as
a function of the latent dimensionality (defined as p1 + p2 for gLARA, where p2 was fixed at 15),
for ? = 3. (b) gLARA?s cross-validated log-likelihood plotted as a function of the dimensionality
of V1?s latent state, p1 (for p2 = 15), for different choices of ? . (c) gLARA?s cross-validated loglikelihood plotted as a function of the dimensionality of V2?s latent state, p2 (for p1 = 50), for
different choices of ? .
The distinction between within-population covariance and between-population covariance is further
supported by re-applying pCCA, but now randomly shuffling the population labels. The crossvalidated log-likelihood curve for these mixed populations now peaks at a larger dimensionality
than three. The reason is that the shuffling procedure removes the distinction between the two types
of covariance, such that the pCCA latents now capture both types of covariance (of the original
unmixed populations). The peak for mixed pCCA occurs at a lower dimensionality than for FA for
two reasons: i) because the mixed populations have the same number of neurons as the original
populations (97 and 31), the maximum number of dimensions that can be identified by pCCA is 31,
and ii) for the same latent dimensionality, pCCA has a larger number of parameters than FA, which
makes pCCA more prone to overfitting.
Together, the analyses in Fig.2 demonstrate two key points. First, if the focus of the analysis lies
in the interaction between populations, then pCCA provides a more parsimonious description, as it
focuses exclusively on the covariance between populations. In contrast, FA is unable to distinguish
within-population covariance from between-population covariance. Second, the neuron groupings
for V1 and V2 are meaningful, as the number of dimensions needed to describe the covariance
between V1 and V2 is small relative to that within each population.
We then analysed the performance of the models with latent dynamics (AR-pCCA and gLARA).
The cross-validated log-likelihood for these models depends jointly on the dimensionality of the
latent state, p, and the order of the auto-regressive model, ? . For gLARA, p is the sum of the dimensionalities of each population?s latent state, p1 + p2 , and we therefore want to jointly maximize
the cross-validated log-likelihood with respect to both p1 and p2 . AR-pCCA required a latent dimensionality of p = 70, while gLARA peaked for a joint latent dimensionality of 65 (p1 = 50 and
p2 = 15) (Fig.3a). When computing the performance of AR-pCCA we considered models with
p ? {5, 10, ..., 75} and ? ? {1, 3, ..., 7} (Fig.3a shows the ? = 3 case). To access how gLARA?s
cross-validated log-likelihood varied with the latent dimensionalities and the model order, we plotted it in Fig.3b, for p2 = 15 and p1 ? {5, 10, ..., 50}, for different choices of ? . This showed that
the performance is greater for an order 3 model, and that it saturates by the time p1 reaches 50.
In Fig.3c, we did a similar analysis for the dimensionality of V2?s latent state, where p1 was held
constant at 50 and p2 ? {5, 10, ..., 25}. The cross-validated log-likelihood shows a clear peak at
p2 = 15 regardless of ? . We found that, for both models, the cross-validated log-likelihood peaks
for ? = 3 (see Fig.3b and 3c for gLARA, results not shown for AR-pCCA).
Finally, we asked which model, AR-pCCA or gLARA, better describes the data. Note that gLARA
is a special case of AR-pCCA, where the observation matrix in Eq.(8) is constrained to have a block
diagonal structure (with blocks C 1 and C 2 ). The key difference between the two models is that
gLARA assigns a non-overlapping set of latent variables to each population. We found that gLARA
outperforms AR-pCCA (Fig.3a). This suggests that the extra flexibility of the AR-pCCA model
6
(b)
(a)
V1
average activity (spikes/s)
40
V2
20
observed activity
observed activity
predicted activity
?40
200
400
600
800
time (ms)
1000
predicted activity
?20
1200
200
400
600
800
time (ms)
1000
1200
Figure 4: Leave-one-neuron-out prediction using gLARA. Observed activity (black) and the
leave-one-neuron-out prediction of gLARA (blue) for a representative held-out trial, averaged over
(a) the V1 population and (b) the V2 population. Note that the activity can be negative because we
are analyzing the single-trial residuals (cf. Section 2.5).
leads to overfitting and that the data are better explained by considering two separate sets of latent
variables that interact.
The optimal latent dimensionalities found for AR-pCCA and gLARA are substantially higher than
those found for pCCA, as the latent states now also capture non-zero time lag interactions between
the populations, and the dynamics within each population. For gLARA, the between-population
covariance must be accounted for by the interaction between the population-specific latents, x1t and
x2t , because there are no shared latents in this model. Thus, the interaction between V1 and V2 is
21
summarized by the A12
k and Ak matrices. Also, both AR-pCCA and gLARA outperform FA and
pCCA (comparing vertical axes in Fig.2 and 3), showing that there is meaningful temporal structure
in how V1 and V2 interact that can be captured by these models.
Having performed a systematic, relative comparison between AR-pCCA and gLARA models of different complexities, we asked how well the best gLARA model fit the data in an absolute sense. To
do so, we used 3/4 of the data to fit the model parameters and performed leave-one-neuron-out pre-
1
diction [15] on the remaining 1/4. This is done by estimating the latent states E x11,...,T | y1,...,T
2
2
and E x1,...,T | y1,...,T using all but one neuron. This estimate of the latent state is then used to
predict the activity of the neuron that was left out (the same procedure was repeated for each neuron). For visualization purposes, we averaged the predicted activity across neurons for a given trial
and compared it to the recorded activity averaged across neurons for the same trial. We found that
they indeed tracked each other, as shown in Fig.4 for a representative trial.
Finally, we asked whether gLARA reveals differences in the time structures of the within-population
dynamics and the between-population interactions. We computed the Frobenius norm of both the
22
within-population dynamics matrices A11
k and Ak (Fig.5a) and the between-population interaction
12
21
matrices Ak and Ak (Fig.5b), for p1 = 50, p2 = 15 and ? = 3 (k ? {1, 2, 3}), which is the model
for which the cross-validated log-likelihood was the highest. The time structure of the withinpopulation dynamics appears to differ from that of the between-population interaction. In particular,
the latents for each area depend more strongly on its own previous latents as the time delay increases
up to 15 ms (Fig.5a). In contrast, the dependence between areas is stronger at time lags of 5 and
15 ms, compared to 10 ms (Fig.5b). Note that the peak of the cross-validated log-likelihood for
? = 3 (Fig.3) shows that delays longer than 15ms do not contribute to an increase in the accuracy
of the model and, therefore, the most significant interactions between these areas may occur within
this time window. The structure seen in Fig.5 is not present if the same analysis is performed on
data that are shuffled across time (results not shown). Because the latent states may have different
21
11
22
11
22
scales, it is not informative to compare the magnitude of A12
k and Ak or Ak and Ak (Ak and Ak
ij
also have different dimensions). Thus, we divided the norms for each Ak matrix by the respective
maximum across k.
7
(a)
(b)
Frobenius norm (a.u.)
1
0.4
5
V1
?
V1
V2
?
V2
10
time delay (ms)
1
0.75
15
5
V1
?
V2
V2
?
V1
10
time delay (ms)
15
Figure 5: Temporal structure of coupling matrices for gLARA. (a) Frobenius norm of the within22
population dynamics matrices A11
k and Ak , for k ? {1, 2, 3}. Each curve was divided by its
21
maximum value. (b) Same as (a) for the between-population interaction matrices A12
k and Ak .
4
Discussion
We started by applying standard methods, FA and pCCA, to neural activity recorded simultaneously
from visual areas V1 and V2. We found that the neuron groupings by brain area are meaningful,
as the covariance of the neurons across areas is lower dimensional than that within each area. We
then proposed an extension to pCCA that takes temporal dynamics into account and allows for
the separation of within-population dynamics from between-population interactions (gLARA). This
method was then shown to provide a better characterization of the two-population neural activity
than FA and pCCA.
In the context of studying the interaction between populations of neurons, capturing the information
flow is key to understanding how information is processed in the brain [3?7, 23]. To do so, one must
be able to characterize the directionality of these between-population interactions. Previous studies
have sought to identify the directionality of interactions directly between neurons, using measures
such as Granger causality [10] (and related extensions, such as directed transfer function (DTF)
[24]), and directed information [11]. Here, we proposed to study between-population interaction
on the level of latent variables, rather than of the neurons themselves. The advantage is that this
approach scales better with the number of recorded neurons and provides a more succinct picture
of the structure of these interactions. To detect fine timescale interactions, it may be necessary to
replace the linear-Gaussian model with a point process model on the spike trains [25].
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9
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5,110 | 5,626 | Learning convolution filters for inverse covariance
estimation of neural network connectivity
George O. Mohler?
Department of Mathematics and Computer Science
Santa Clara University University
Santa Clara, CA, USA
[email protected]
Abstract
We consider the problem of inferring direct neural network connections from Calcium imaging time series. Inverse covariance estimation has proven to be a fast
and accurate method for learning macro- and micro-scale network connectivity
in the brain and in a recent Kaggle Connectomics competition inverse covariance
was the main component of several top ten solutions, including our own and the
winning team?s algorithm. However, the accuracy of inverse covariance estimation is highly sensitive to signal preprocessing of the Calcium fluorescence time
series. Furthermore, brute force optimization methods such as grid search and
coordinate ascent over signal processing parameters is a time intensive process,
where learning may take several days and parameters that optimize one network
may not generalize to networks with different size and parameters. In this paper
we show how inverse covariance estimation can be dramatically improved using a
simple convolution filter prior to applying sample covariance. Furthermore, these
signal processing parameters can be learned quickly using a supervised optimization algorithm. In particular, we maximize a binomial log-likelihood loss function
with respect to a convolution filter of the time series and the inverse covariance
regularization parameter. Our proposed algorithm is relatively fast on networks
the size of those in the competition (1000 neurons), producing AUC scores with
similar accuracy to the winning solution in training time under 2 hours on a cpu.
Prediction on new networks of the same size is carried out in less than 15 minutes,
the time it takes to read in the data and write out the solution.
1
Introduction
Determining the topology of macro-scale functional networks in the brain and micro-scale neural
networks has important applications to disease diagnosis and is an important step in understanding
brain function in general [11, 19]. Modern neuroimaging techniques allow for the activity of hundreds of thousands of neurons to be simultaneously monitored [19] and recent algorithmic research
has focused on the inference of network connectivity from such neural imaging data. A number
of approaches to solve this problem have been proposed, including Granger causality [3], Bayesian
networks [6], generalized transfer entropy [19], partial coherence [5], and approaches that directly
model network dynamics [16, 18, 14, 22].
?
1
Several challenges must be overcome when reconstructing network connectivity from imaging data.
First, imaging data is noisy and low resolution. The rate of neuron firing may be faster than the
image sampling rate [19] and light scattering effects [13, 19] lead to signal correlations at short
distances irrespective of network connectivity. Second, causality must be inferred from observed
correlations in neural activity. Neuron spiking is highly correlated both with directly connected
neurons and those connected through intermediate neurons. Coupled with the low sampling rate
this poses a significant challenge, as it may be the case that neuron i triggers neuron j, which then
triggers neuron k, all within a time frame less than the sampling rate.
To solve the second challenge, sparse inverse covariance estimation has recently become a popular
technique for disentangling causation from correlation [11, 15, 23, 1, 9, 10]. While the sample
covariance matrix only provides information on variable correlations, zeros in the inverse covariance
matrix correspond to conditional independence of variables under normality assumptions on the
data. In the context of inferring network connectivity from leaky integrate and fire neural network
time-series, however, it is not clear what set of random variables one should use to compute sample
covariance (a necessary step for estimating inverse covariance). While the simplest choice is the raw
time-series signal, the presence of both Gaussian and jump-type noise make this significantly less
accurate than applying signal preprocessing aimed at filtering times at which neurons fire.
In a recent Kaggle competition focused on inferring neural network connectivity from Calcium
imaging time series, our approach used inverse covariance estimation to predict network connections. Instead of using the raw time series to compute sample covariance, we observed improved
Area Under the Curve (receiver operating characteristic [2]) scores by thresholding the time derivative of the time-series signal and then combining inverse covariance corresponding to several thresholds and time-lags in an ensemble. This is similar to the approach of the winning solution [21],
though they considered a significantly larger set of thresholds and nonlinear filters learned via coordinate ascent, the result of which produced a private leaderboard AUC score of .9416 compared to
our score of .9338. However, both of these approaches are computationally intensive, where prediction on a new network alone takes 10 hours in the case of the winning solution [21]. Furthermore,
parameters for signal processing were highly tuned for optimizing AUC of the competition networks
and don?t generalize to networks of different size or parameters [21]. Given that coordinate ascent
takes days for learning parameters of new networks, this makes such an approach impractical.
In this paper we show how inverse covariance estimation can be significantly improved by applying
a simple convolution filter to the raw time series signal. The filter can be learned quickly in a
supervised manner, requiring no time intensive grid search or coordinate ascent. In particular, we
optimize a smooth binomial log-likelihood loss function with respect to a time series convolution
kernel, along with the inverse covariance regularization parameter, using L-BFGS [17]. Training
the model is fast and accurate, running in under 2 hours on a CPU and producing AUC scores
that are competitive with the winning Kaggle solution. The outline of the paper is as follows. In
Section 2 we review inverse covariance estimation and introduce our convolution based method for
signal preprocessing. In Section 3 we provide the details of our supervised learning algorithm and
in Section 4 we present results of the algorithm applied to the Kaggle Connectomics dataset.
2
2.1
Modeling framework for inferring neural connectivity
Background on inverse covariance estimation
Let X ? Rn?p be a data set of n observations from a multivariate Gaussian distribution with p
variables, let ? denote the covariance matrix of the random variables, and S the sample covariance.
Variables i and j are conditionally independent given all other variables if the ijth component of
? = ??1 is zero. For this reason, a popular approach for inferring connectivity in sparse networks
is to estimate the inverse covariance matrix via l1 penalized maximum likelihood,
? = arg max log det(?) ? tr(S?) ? ?k?k1 ,
?
(1)
?
2
[11, 15, 23, 1, 9, 10], commonly referred to as GLASSO (graphical least absolute shrinkage and
selection operator). GLASSO has been used to infer brain connectivity for the purpose of diagnosing
Alzheimer?s disease [11] and determining brain architecture and pathologies [23].
While GLASSO is a useful method for imposing sparsity on network connections, in the Kaggle
Connectomics competition AUC was the metric used for evaluating competing models and on AUC
GLASSO only performs marginally better (AUC? .89) than the generalized transfer entropy Kaggle benchmark (AUC? .88). The reason for the poor performance of GLASSO on AUC is that
l1 penalization forces a large percentage of neuron connection scores to zero, whereas high AUC
performance requires ranking all possible connections.
We therefore use l2 penalized inverse covariance estimation [23, 12],
? =
?
?1
,
S + ?I
(2)
instead of optimizing Equation 1. While one advantage of Equation 2 is that all connections are
assigned a non-zero score, another benefit is derivatives with respect to model parameters are easy
to determine and compute using the standard formula for the derivative of an inverse matrix. In
particular, our model consists of parametrizing S using a convolution filter applied to the raw Calcium fluorescence time series and Equation 2 facilitates derivative based optimization. We return to
GLASSO in the discussion section at the end of the paper.
2.2
Signal processing
Next we introduce a model for the covariance matrix S taking as input observed imaging data from a
neural network. Let f be the Calcium fluorescence time series signal, where fti is the signal observed
at neuron i in the network at time t. The goal in this paper is to infer direct network connections
from the observed fluorescence time series (see Figure 1). While fti can be used directly to calculate
0.6
0.4
0.2
0
1200 1400 1600 1800 2000 2200
time (20 ms)
1
(C)
0.95
0.9
0.85
5
3
0.8
?k
(B)
(A)
Filtered fluorescence amplitude
Fluorescence amplitude
0.8
1
?1
?3
0.75
?5
0
2
4
6
8 10
k
1200 1400 1600 1800 2000 2200
time (20 ms)
Figure 1: (A) Fluorescence time series f i for neuron i = 1 (blue) of Kaggle Connectomics network
2 and time series for two neurons (red and green) connected to neuron 1. Synchronized firing of
all 1000 neurons occurs around time 1600. (B) Neuron locations (gray) in network 2 and direct
connections to neuron 1 (green and red connections correspond to time series in Fig 1A). The task is
to reconstruct network connectivity as in Fig 1B for all neurons given time series data as in Fig 1A.
(C) Filtered fluorescence time series ?(f i ? ? + ?bias ) using the convolution kernel ? (inset figure)
learned from our method detailed in Section 3.
covariance between fluorescence time series, significant improvements in model performance are
achieved by filtering the signal to obtain an estimate of nit , the number of times neuron i fired
between t and t + ?t. In the competition we used simple thresholding of the time series derivative
3
i
?fti = ft+?t
? fti to estimate neuron firing times,
nit = 1{?fti >?} .
(3)
The covariance matrix was then computed using a variety of threshold values ? and time-lags k. In
particular, the (i, j)th entry of S(?, k) was determined by,
sij =
T
1X i
(nt ? ni )(njt?k ? nj ),
T
(4)
t=k
where ni is the mean signal. The covariance matrices were then inverted using Equation 2 and
combined using LambdaMart [4] to optimize AUC, along with a restricted Boltzmann machine and
generalized linear model. In Figure 2, we illustrate the sensitivity of inverse covariance estimation
on the threshold parameter ?, regularization parameter ?, and time-lag parameter k. Using the raw
time series signal leads to AUC scores between 0.84 and 0.88, whereas for good choices of the
threshold and regularization parameter Equation 2 yields AUC scores above 0.92. Further gains are
achieved by using an ensemble over varying ?, ?, and k.
(A)
0.88
AUC
AUC
0.92
.916
?
.12
0.86
k=1
?=.01
?=.025
?=.05
.11
0.88
.86
k=0
.912
.1
(C)
(B)
AUC
.924
0.84
.13
.1
?=.01
?=.025
?=.05
?=.01
?=.025
?=.05
0.84
.11
?
.12
.13
2
1
k
0
Figure 2: (A) AUC scores for network 2 using Equations 2, 3, and 4 with a time lag of k = 0 and
varying threshold ? and regularization parameter ?. (B) AUC scores analogous to Figure 2A, but
for a time lag of k = 1. (C) AUC scores corresponding to inverse covariance estimation using raw
time series signal. For comparison, generalized transfer entropy [19] corresponds to AUC? .88 and
simple correlation corresponds to AUC? .66.
In this paper we take a different approach in order to jointly learn the processed fluorescence signal
and the inverse covariance estimate. In particular, we convolve the fluorescence time series f i with
a kernel ? and then pass the convolution through the logistic function ?(x),
yi = ?(f i ? ? + ?bias ).
(5)
Note for ?0 = ??1 (and ?k = 0 otherwise) this convolution filter approximates the threshold filter
in Equation 3. However, it turns out that the learned optimal filter is significantly different than time
derivative thresholding (see Figure 1C). Inverse covariance is then estimated via Equation 2, where
the sample covariance is given by,
sij =
T
1X i
(y ? y i )(ytj ? y j ).
T t=1 t
The time lags no longer appear in Equation 6, but instead are reflected in the convolution filter.
4
(6)
2.3
Supervised inverse covariance estimation
Given the sensitivity of model performance on signal processing illustrated in Figure 2, our goal is
now to learn the optimal filter ? by optimizing a smooth loss function. To do this we introduce a
model for the probability of neurons being connected as a function of inverse covariance.
Let zij = 1 if neuron i connects to neuron j and zero otherwise and let ?(?, ?) be the inverse
covariance matrix that depends on the smoothing parameter ? from Section 2.1 and the convolution
filter ? from Section 2.2. We model the probability of neuron i connecting to j as ?ij = ?(?ij ?0 +
?1 ) where ? is the logistic function and ?ij is the (i, j)th entry of ?. In summary, our model for
scoring the connection from i to j is detailed in Algorithm 1.
Algorithm 1: Inverse covariance scoring algorithm
Input: f ? ?bias ? ?0 ?1
\\ fluorescence signal and model parameters
yi = ?(f i ? ? + ?bias )
\\ apply convolution filter and logistic function to signal
for i ? 1 to N do
for j ? 1 to N do
PT
sij = T1 t=1 (yti ? y i )(ytj ? y j )
\\ compute sample covariance matrix
end
end
? = (S + ?I)?1
\\ compute inverse covariance matrix
Output: ?(??0 + ?1 )
\\ output connection probability matrix
The loss function we aim to optimize is the binomial log-likelihood, given by,
X
L(?, ?, ?0 , ?1 ) =
?zij log(?ij ) + (1 ? ?)(1 ? zij ) log(1 ? ?ij ),
(7)
i6=j
where the parameter ? is chosen to balance the dataset. The networks in the Kaggle dataset are
sparse, with approximately 1.2% connections, so we choose ? = .988. For ? values within 10% of
the true percentage of connections, AUC scores are above .935. Without data balancing, the model
achieves an AUC score of .925, so the introduction of ? is important. While smooth approximations
of AUC are possible, we find that optimizing Equation 7 instead still yields high AUC scores.
To use derivative based optimization methods that converge quickly, we need to calculate the derivatives of Equation 7. Defining,
?ij = ?zij (1 ? ?ij ) ? (1 ? ?)(1 ? zij )?ij ,
(8)
then the derivatives of the loss function with respect to the model parameters are specified by,
X
X
dL
dL
=
?ij ?ij ,
=
?ij ,
d?0
d?1
(9)
X
dL X
d?ij
dL
d?ij
=
?0 ?ij
,
=
?0 ?ij
.
d?
d? d?k
d?k
(10)
i6=j
i6=j
i6=j
i6=j
Using the inverse derivative formula, we have that the derivatives of the inverse covariance matrix
satisfy the following convenient equations,
2 d?
d?
dS
= ? (S(?) + ?I)?1 ,
= ?(S(?) + ?I)?1
(S(?) + ?I)?1 ,
d?
d?k
d?k
(11)
where S is the sample covariance matrix from Section 2.2. The derivatives of the sample covariance
dy i
i
are then found by substituting d?tk = yti (1 ? yti )ft?k
into Equation 6 and using the product rule.
5
3
Results
We test our methodology using data provided through the Kaggle Connectomics competition. In the
Kaggle competition, neural activity was modeled using a leaky integrate and fire model outlined in
[19]. Four 1000 neuron networks with 179,500 time series observations per network were provided
for training, a test network of the same size and parameters was provided without labels to determine
the public leaderboard, and final standings were computed using a 6th network for validation. The
goal of the competition was to infer the network connections from the observed Fluorescence time
series signal (see Figure 1) and the error metric for determining model performance was AUC.
There are two ways in which we determined the size of the convolution filter. The first is through
inspecting the decay of cross-correlation as a function of the time-lag. For the networks we consider
in the paper, this decay takes place over 10-15 time units. The second method is to add an additional
time unit one at a time until cross-validated AUC scores no longer improve. This happens for the
networks we consider at 10 time units. We therefore consider a convolution filter with k = 0...10.
We use the off-the-shelf optimization method L-BFGS [17] to optimize Equation 7. Prior to applying
the convolution filter, we attempt to remove light scattering effects simulated in the competition by
inverting the equation,
X j
Fti = fti + Asc
(12)
ft exp ? (dij /?sc )2 .
j6=i
Here Fti is the observed fluorescence provided for the competition with light scattering effects (see
[19]) and dij is the distance between neuron i and j. The parameter values Asc = .15 and ?sc =
.025 were determined such that the correlation between neuron distance and signal covariance was
approximately zero.
We learn the model parameters using network 2 and training time takes less than 2 hours in Matlab
on a laptop with a 2.3 GHz Intel Core i7 processor and 16GB of RAM. Whereas prediction alone
takes 10 hours on one network for the winning Kaggle entry [21], prediction using Algorithm 1
takes 15 minutes total and the algorithm itself runs in 20 seconds (the rest of the time is dedicated to
reading the competition csv files into and out of Matlab). In Figure 3 we display results for all four of
the training networks using 80 iterations of L-BFGS (we used four outer iterations with maxIter= 20
and TolX= 1e ? 5). The convolution filter is initialized to random values and at every 20 iterations
we plot the corresponding filtered signal for neuron 1 of network 2 over the first 1000 time series
observations. After 10 iterations all four networks have an AUC score above 0.9. After 80 iterations
the AUC private leaderboard score of the winning solution is within the range of the AUC scores of
networks 1, 3, and 4 (trained on network 2). We note that during training intermediate AUC scores
do not increase monotonically and also exhibit several plateaus. This is likely due to the fact that
AUC is a non-smooth loss function and we used the binomial likelihood in its stead.
4
Discussion
We introduced a model for inferring connectivity in neural networks along with a fast and easy to
implement optimization strategy. In this paper we focused on the application to leaky integrate and
fire models of neural activity, but our methodology may find application to other types of crossexciting point processes such as models of credit risk contagion [7] or contagion processes on social
networks [20].
It is worth noting that we used a Gaussian model for inverse covariance even though the data
was highly non-Gaussian. In particular, neural firing time series data is generated by a nonlinear,
mutually-exciting point process. We believe that it is the fact that the input data is non-Gaussian that
the signal processing is so crucial. In this case fti and fsj are highly dependent for 10 > t ? s > 0
6
(A)
.944
network1
network2
network3
network4
winning solution
on valid network
AUC
0.9
0.8
Filtered fluorescence
amplitude
0.7
0
10
20
30
40
L?BFGS iterations
(C)
(B)
50
60
70
(D)
0.94
.936
80
75 76 77 78 79 80
(F)
(E)
1
1
1
1
1
0.8
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.4
0.2
0
500
0.2
1000
0
500
0.2
1000
0
500
time (20 ms)
0.2
1000
0
500
1000
0.2
0
500
1000
Figure 3: (A) Networks 1-4 AUC values plotted against L-BFGS iterations where network 2 was
used to learn the convolution filter. The non-monotonic increase can be attributed to optimizing
the binomial log-likelihood rather than AUC directly. (B-F) Every 20 iterations we also plot a
subsection of the filtered signal of neuron 1 from network 2. The filter is initially given random
values but quickly produces impulse-like signals with high AUC scores. The AUC score of the
winning solution is within the range of the AUC scores of held-out networks 1, 3, 4 after 80 iterations
of L-BFGS.
and j ? i. Empirically, the learned convolution filter compensates for the model mis-specification
and allows for the ?wrong? model to still achieve a high degree of accuracy.
We also note that using directed network estimation did not improve our methods, nor the methods
of other top solutions in the competition. This may be due to the fact that the resolution of Calcium fluorescence imaging is coarser than the timescale of network dynamics, so that directionality
information is lost in the imaging process. That being said, it is possible to adapt our method for
estimation of directed networks. This can be accomplished by introducing two different filters ?i
and ?j into Equations 5 and 6 to allow for an asymmetric covariance matrix S in Equation 6. It
would be interesting to assess the performance of such a method on networks with higher resolution
imaging in future research.
While the focus here was on AUC maximization, other loss functions may be useful to consider. For
sparse networks where the average network degree is known, precision or discounted cumulative
gain may be reasonable alternatives to AUC. Here it is worth noting that l1 penalization is more
accurate for these types of loss functions that favor sparse solutions. In Table 1 we compare the accuracy of Equation 1 vs Equation 2 on both AUC and PREC@k (where k is chosen to be the known
number of network connections). For signal processing we return to time-derivative thresholding
and use the parameters that yielded the best single inverse covariance estimate during the competition. While l2 penalization is significantly more accurate for AUC, this is not the case for PREC@k
for which GLASSO achieves a higher precision.
It is clear that the sample covariance S in Equation 1 can be parameterized by a convolution kernel
?, but supervised learning is no longer as straightforward. Coordinate ascent can be used, but given
that Equation 1 is orders of magnitude slower to solve than Equation 2, such an approach may not be
practical. Letting G(?, S) be the penalized log-likelihood corresponding to GLASSO in Equation
1, another possibility is to jointly optimize
?G(?, S) + (1 ? ?)L(?, S)
7
(13)
Network1
Network2
Network3
?l1 = 5 ? 10?5
.894/.423
.894/.417
.894/.423
?l1 = 1 ? 10?4
.884/.420
.885/.416
.885/.425
?l1 = 5 ? 10?4
.882/.420
.885/.415
.884/.427
?l2 = 2 ? 10?2
.926/.394
.924/.385
.925/.397
Table 1: AUC/PREC@k for l1 vs. l2 penalized inverse covariance estimation (where k equals
the true number of connections). Time series preprocessed by a derivative threshold of .125 and
removing spikes when 800 or more neurons fire simultaneously. For l1 penalization AUC increases
as ?l1 decreases, though the Rglasso solver [8] becomes prohibitively slow for ?l1 on the order of
10?5 or smaller.
where L is the binomial log-likelihood in Equation 7. In this case both the convolution filter and the
inverse covariance estimate ? would need to be learned jointly and the parameter ? could be determined via cross validation on a held-out network. Extending the results in this paper to GLASSO
will be the focus of subsequent research.
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9
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correlated:1 |
5,111 | 5,627 | On Multiplicative Multitask Feature Learning
Xin Wang? , Jinbo Bi? , Shipeng Yu? , Jiangwen Sun?
?
Dept. of Computer Science & Engineering
Health Services Innovation Center
University of Connecticut
Siemens Healthcare
Storrs, CT 06269
Malvern, PA 19355
wangxin,jinbo,[email protected]
[email protected]
?
Abstract
We investigate a general framework of multiplicative multitask feature learning
which decomposes each task?s model parameters into a multiplication of two components. One of the components is used across all tasks and the other component
is task-specific. Several previous methods have been proposed as special cases of
our framework. We study the theoretical properties of this framework when different regularization conditions are applied to the two decomposed components.
We prove that this framework is mathematically equivalent to the widely used
multitask feature learning methods that are based on a joint regularization of all
model parameters, but with a more general form of regularizers. Further, an analytical formula is derived for the across-task component as related to the taskspecific component for all these regularizers, leading to a better understanding of
the shrinkage effect. Study of this framework motivates new multitask learning
algorithms. We propose two new learning formulations by varying the parameters
in the proposed framework. Empirical studies have revealed the relative advantages of the two new formulations by comparing with the state of the art, which
provides instructive insights into the feature learning problem with multiple tasks.
1
Introduction
Multitask learning (MTL) captures and exploits the relationship among multiple related tasks and
has been empirically and theoretically shown to be more effective than learning each task independently. Multitask feature learning (MTFL) investigates a basic assumption that different tasks may
share a common representation in the feature space. Either the task parameters can be projected to
explore the latent common substructure [18], or a shared low-dimensional representation of data can
be formed by feature learning [10]. Recent methods either explore the latent basis that is used to
develop the entire set of tasks, or learn how to group the tasks [16, 11], or identify if certain tasks
are outliers to other tasks [6].
A widely used MTFL strategy is to impose a blockwise joint regularization of all task parameters
to shrink the effects of features for the tasks. These methods employ a regularizer based on the
so-called `1,p matrix norm [12, 13, 15, 22, 24] that is the sum of the `p norms of the rows in a
matrix. Regularizers based on the `1,p norms encourage row sparsity. If rows represent features
and columns represent tasks, they shrink the entire rows of the matrix to have zero entries. Typical
choices for p are 2 [15, 4] and ? [20] which are used in the very early MTFL methods. Effective
algorithms have since then been developed for the `1,2 [13] and `1,? [17] regularization. Later,
the `1,p norm is generalized to include 1 < p ? ? with a probabilistic interpretation that the
resultant MTFL method solves a relaxed optimization problem with a generalized normal prior for
all tasks [22]. Recent research applies the capped `1,1 norm as a nonconvex joint regularizer [5].
The major limitation of joint regularized MTFL is that it either selects a feature as relevant to all
tasks or excludes it from all models, which is very restrictive in practice where tasks may share some
features but may have their own specific features as well.
1
To overcome this limitation, one of the most effective strategies is to decompose the model parameters into either summation [9, 3, 6] or multiplication [21, 2, 14] of two components with different
regularizers applied to the two components. One regularizer is used to take care of cross-task similarities and the other for cross-feature sparsity. Specifically, for the methods that decompose the
parameter matrix into summation of two matrices, the dirty model in [9] employs `1,1 and `1,?
regularizers to the two components. A robust MTFL method in [3] uses the trace norm on one component for mining a low-rank structure shared by tasks and a column-wise `1,2 -norm on the other
component for identifying task outliers. Another method applies the `1,2 -norm both row-wisely to
one component and column-wisely to the other [6].
For the methods that work with multiplicative decompositions, the parameter vector of each task
is decomposed into an element-wise product of two vectors where one is used across tasks and
the other is task-specific. These methods either use the `2 -norm penalty on both of the component
vectors [2], or the sparse `1 -norm on the two components (i.e., multi-level LASSO) [14]. The
multi-level LASSO method has been analytically compared to the dirty model [14], showing that
the multiplicative decomposition creates better shrinkage on the global and task-specific parameters.
The across-task component can screen out the features irrelevant to all tasks. To exclude a feature
from a task, the additive decomposition requires the corresponding entries in both components to
be zero whereas the multiplicative decomposition only requires one of the components to have a
zero entry. Although there are different ways to regularize the two components in the product, no
systematic work has been done to analyze the algorithmic and statistical properties of the different
regularizers. It is insightful to answer the questions such as how these learning formulations differ
from the early methods based on blockwise joint regularization, how the optimal solutions of the two
components look like, and how the resultant solutions are compared with those of other methods that
also learn both shared and task-specific features.
In this paper, we investigate a general framework of the multiplicative decomposition that enables
a variety of regularizers to be applied. This general form includes all early methods that represent
model parameters as a product of two components [2, 14]. Our theoretical analysis has revealed that
this family of methods is actually equivalent to the joint regularization based approach but with a
more general form of regularizers, including those that do not correspond to a matrix norm. The
optimal solution of the across-task component can be analytically computed by a formula of the
optimal task-specific parameters, showing the different shrinkage effects. Statistical justification
and efficient algorithms are derived for this family of formulations. Motivated by the analysis, we
propose two new MTFL formulations. Unlike the existing methods [2, 14] where the same kind of
vector norm is applied to both components, the shrinkage of the global and task-specific parameters
differ in the new formulations. Hence, one component is regularized by the `2 -norm and the other
is by the `1 -norm, which aims to reflect the degree of sparsity of the task-specific parameters relative to the sparsity of the across-task parameters. In empirical experiments, simulations have been
designed to examine the various feature sharing patterns where a specific choice of regularizer may
be preferred. Empirical results on benchmark data are also discussed.
2
The Proposed Multiplicative MTFL
Given T tasks in total, for each task t, t ? {1, ? ? ? , T }, we have sample set (Xt ? R`t ?d , yt ? R`t ).
The data set of Xt has `t examples, where the i-th row corresponds to the i-th example xti of task
t, i ? {1, ? ? ? , `t }, and each column represents a feature. The vector yt contains yit , the label of the
i-th example of task t. We consider functions of the linear form Xt ?t where ?t ? Rd . We define
the parameter matrix or weight matrix A = [?1 , ? ? ? , ?T ] and ?j are the rows, j ? {1, ? ? ? , d}.
A family of multiplicative MTFL methods can be derived by rewriting ?t = diag(c)? t where
diag(c) is a diagonal matrix with its diagonal elements composing a vector c. The c vector is used
across all tasks, indicating if a feature is useful for any of the tasks, and the vector ? t is only for task
t. Let j index the entries in these vectors. We have ?jt = cj ?jt . Typically c comprises binary entries
that are equal to 0 or 1, but the integer constraint is often relaxed to require just non-negativity. We
minimize a regularized loss function as follows for the best c and ? t:t=1,??? ,T :
min
? t ,c?0
T
P
L(c, ? t , Xt , yt ) + ?1
t=1
T
P
t=1
2
||? t ||pp + ?2 ||c||kk
(1)
where L(?) is a loss function, e.g., the least squares loss for regression problems or the logistic loss
Pd
Pd
for classification problems, ||? t ||pp = j=1 |?jt |p and ||c||kk = j=1 (cj )k , which are the `p -norm
of ? t to the power of p and the `k -norm of c to the power of k if p and k are positive integers.
The tuning parameters ?1 , ?2 are used to balance the empirical loss and regularizers. At optimality,
if cj = 0, the j-th variable is removed for all tasks, and the corresponding row vector ?j = 0;
otherwise the j-th variable is selected for use in at least one of the ??s. Then, a specific ? t can rule
out the j-th variable from task t if ?jt = 0.
In particular, if both p = k = 2, Problem (1) becomes the formulation in [2] and if p = k = 1,
Problem (1) becomes the formulation in [14]. Any other choices of p and k will derive into new
formulations for MTFL. We first examine the theoretical properties of this entire family of methods,
and then empirically study two new formulations by varying p and k.
3
Theoretical Analysis
PT
Pd
The joint `1,p regularized MTFL method minimizes t=1 L(?t , Xt , yt ) + ? j=1 ||?j ||p for the
best ?t:t=1,??? ,T where ? is a tuning parameter. We now extend this formulation to allow more
choices of regularizers. We introduce
a new notation that is an operator applied to a vector, such
qP
T
q
j
j p/q
t p
as ? . The operator ||? ||
=
t=1 |?j | , p, q ? 0, which corresponds to the `p norm if
p = q and both are positive integers. A joint regularized MTFL approach can solve the following
optimization problem with pre-specified values of p, q and ?, for the best parameters ?t:t=1,??? ,T :
min
?t
T
P
L(?t , Xt , yt ) + ?
t=1
d p
P
||?j ||p/q .
(2)
j=1
Our main result of this paper is (i) a theorem that establishes the equivalence between the models
derived from solving Problem (1) and Problem (2) for properly chosen values of ?, q, k, ?1 and ?2 ;
and (ii) an analytical solution of Problem (1) for c which shows how the sparsity of the across-task
component is relative to the sparsity of task-specific components.
?) be the optimal solution to
Theorem 1 Let ??t be the optimal solution to Problem
(2) and (??t , c
q
p
p
2? kq
p
Problem (1). Then ??t = diag(?
c)??t when ? = 2 ?1
?2kq and q = k+p
2k (or k = 2q?1 ).
Proof. The theorem holds by proving the following two Lemmas. The first lemma proves that the
solution ??t of Problem (2) also minimizes the following optimization problem:
PT
Pd
Pd
?1
j p/q
(3)
min?t ,??0
+ ?2 j=1 ?j ,
t=1 L(?t , Xt , yt ) + ?1
j=1 ?j ||? ||
and the optimal solution of Problem (3) also minimizes Problem (2) when proper values of ?, ?1
and ?2 are chosen. The second lemma connects Problem (3) to our formulation (1). We show that
? can be computed from the optimal ?.
?
the optimal ?
?j is equal to (?
cj )k , and then the optimal ?
?
Lemma 1 The solution sets of Problem (2) and Problem (3) are identical when ? = 2 ?1 ?2 .
?
Proof. First, we show that when ? = 2 ?1 ?2 , the optimal solution ?
? jt of Problem (2) minimizes
q
1
1
?
? j ||p/q . By the Cauchy-Schwarz inequality, the
Problem (3) and the optimal ?
?j = ?12 ?2 2 ||?
following inequality holds
?1
d
X
j=1
?j?1 ||?j ||p/q + ?2
d
X
d
X
?
?j ? 2 ?1 ?2
j=1
1
2
q
||?j ||p/q
j=1
? 12
p
j ||p/q . Since Problems (3) and (2) use
where the equality holds if and only if ?j = ?1 ?2
||?q
1
1
?
? j ||p/q , Problems (3) and (2) have
the exactly same loss function, when we set ?
?j = ?12 ?2 2 ||?
?
? = (?
? = (?
identical objective function if ? = 2 ?1 ?2 . Hence the pair (A
? jt )jt , ?
?j )j=1,??? ,d )
minimizes Problem (3) as it entails the objective function to reach its lower bound.
3
? ?)
? also minimizes
? minimizes Problem (3), then A
Second, it can be proved that if the pair (A,
?
Problem (2) by proof of contradiction. Suppose that A does not minimize Problem (2), which means
? j (6= ?
? j for some j) that is an q
that there exists ?
optimal solution to Problem (2) and achieves a lower
1
? 12
j
2
? ?)
? . We set ?
? j ||p/q . The pair (A,
? is an optimal solution
objective value than ?
?j = ? ?
||?
1
2
? ?)
? will bring the objective function of
of Problem (3) as proved in the first paragraph. Then (A,
?
? ?)
? contradicting to the assumption that (A,
? be
Problem (3) to a lower value than that of (A, ?),
optimal to Problem (3).
?
Hence, we have proved that Problems (3) and (2) have identical solutions when ? = 2 ?1 ?2 .
From the proof of Lemma (1), we also see that the optimal objective value of Problem (2) gives a
lower bound to the objective of Problem (3). Let ?j = (cj )k , k ? R, k 6= 0 and ?jt = cj ?jt , an
equivalent objective function of Problem (3) can be derived.
? ?)
? c
? of Problem (3) is equivalent to the optimal solution (B,
?) of
Lemma 2 The optimal solution (A,
kq
kq?p
p
2kq?p
2kq?p
k
t
t
?
?j = (?
cj ) when ?1 = ?1
Problem (1) where ?
? j = c?j ?j and ?
.
?2
, ?2 = ?2 , and k = 2q?1
Proof. First, by proof of contradiction, we show that if ?
? jt and ?
?j optimize Problem (3), then
p
?
?t
c?j = k ?
?j and ??t = j optimize Problem (1). Denote the objectives of (1) and (3) by J (1)
j
c?j
? , Xt , y ) +
and J . Substituting ??jt , c?j for ?
? jt , ?
?j in J (3) yields an objective function L(?
c, ?
tq
t
1
1
Pd
P
j p/q (p?kq)/q
?
d
j
k
2
2
? || c?
? ||p/q .
?1 j=1 ||?
+ ?2 j=1 (?
cj ) . By the proof of Lemma 1, ?
?j = ?1 ?2
||?
j
q
? j p/q 2kq?p . Applying the formula of c?j and substituting ?1 and ?2 by
Hence, c?j = ?1 ??1
2 ||? ||
?1 and ?2 yield an objective identical to J (1) . Suppose ?(??jt , c?j )(6= (??jt , c?j )) that minimize (1), and
J (1) (??jt , c?j ) < J (1) (??jt , c?j ). Let ?
? jt = c?j ??jt and substitute ??jt by ?
? jt /?
cj in J (1) . By Cauchy-Schwarz
P
1
T
inequality, we similarly have c?j = (?1 ?2?1 t=1 (?
?jt )p ) p+k . Thus, J (1) (?
?jt , c?j ) can be derived into
J (3) (?
?jt , c?j ). Let ?
?j = (?
cj )k , and we have J (3) (?
?jt , ?
?j ) < J (3) (?
?jt , ?
?j ), which contradicts with
t
t
the optimality of (?
?j , ?
?j ). Second, we similarly prove that if ??j and c?j optimize Problem (1), then
?
? jt = c?j ??jt and ?
?j = (?
cj )k optimize Problem (3).
q
p
2? p
Now, combining the results from the two Lemmas, we can derive that when ? = 2 ?1 kq ?2kq and
q = k+p
2k , the optimal solutions to Problems (1) and (2) are equivalent. Solving Problem (1) will
? to Problem (2) and vice versa.
yield an optimal solution ?
(3)
? ,??? ,?
? ]
? = [?
Theorem 2 Let ??t , t = 1, ? ? ? , T, be the optimal solutions of Problem (1), Let B
1
T
j
? denote the j-th row of the matrix B.
? Then,
and ?
1
? j || 2kq?p ,
c?j = (?1 /?2 ) k ||?
for all j = 1, ? ? ? , d, is optimal to Problem (1).
p
(4)
Proof. This analytical formula can be directly derived from Lemma 1 and Lemma 2. When we set
q
? j ||p/q 2kq?p . In the proof
?
?j = (?
cj )k and ?
? jt = c?j ??jt in Problem (3), we obtain c?j = ?1 ??1
||
?
2
2kq?p
kq
of Lemma 2, we obtain that ?1 = ?1
formula of c yields the formula (4).
p?kq
?2 kq and ?2 = ?2 . Substituting these formula into the
Based on the derivation, for each pair of {p, q} and ? in Problem (2), there exists an equivalent
problem (1) with determined values of k, ?1 and ?2 , and vice versa. Note that if q = p/2, the
regularization term on ?j in Problem (2) becomes the standard p-norm. In particular, if {p, q} =
{2, 1} in Problem (2) as used in the methods of [15] and [1], the `2 -norm regularizer
is applied
?
to ?j . Then, this problem is equivalent to Problem (1) when k = 2 and ? = 2 ?1 ?2 , the same
formulation in [2]. If {p, q} = {1, 1}, the square root of `1 -norm regularizer is applied to ?j . Our
theorem 1 shows that this problem is equivalent
? to the multi-level LASSO MTFL formulation [14]
which is Problem (1) with k = 1 and ? = 2 ?1 ?2 .
4
4
Probabilistic Interpretation
In this section we show the proposed multiplicative formalism is related to the maximum a posteriori (MAP) solution of a probabilistic model. Let p(A|?) be the prior distribution of the
weight matrix A = [?1 , . . . , ?T ] = [?1> , . . . , ?d> ]> ? Rd?T , where ? denote the parameter of the prior. Then the a posteriori distribution of A can be calculated via Bayes rule as
QT
p(A|X, y, ?) ? p(A|?) t=1 p(yt |Xt , ?t ). Denote z ? GN (?, ?, q) the univariate generalized
q
1
normal distribution, with the density function p(z) = 2??(1+1/q)
exp(? |z??|
?q ), in which ? > 0,
q > 0, and ?(?) is the Gamma function [7]. Now let each element of A, ?jt , follow a generalized
normal prior, ?jt ? GN (0, ?j , q). Then with the i.i.d. assumption, the prior takes the form (also
refer to [22] for a similar treatment)
p(A|?) ?
d Y
T
d
k?j kq
|?t |q Y
Y
1
1
q
j
exp
?
exp ? q
=
,
q
T
?
?
?
?
j
j
j
j
j=1 t=1
j=1
(5)
where k?kq denote vector q-norm. With an appropriately chosen likelihood function p(yt |Xt , ?t ) ?
exp(?L(?t , Xt , yt )), finding the MAP solution
is equivalent
to solving the following problem:
PT
Pd k?j kqq
+ T ln ?j . By setting the derivative of J with
minA,? J = t=1 L(?t , Xt , yt ) + j=1
?jq
respect to ?j to zero, we obtain:
XT
Xd
min J =
L(?t , Xt , yt ) + T
ln k?j kq .
(6)
t=1
A
j=1
Now let us look at the multiplicative nature of ?jt with different q ? [1, ?]. When q = 1, we have:
!
!
!
T
d
d
T
d
T
d
X
X
X
X
X
X
X
t
t
t
j
|?j | . (7)
ln |cj | + ln
ln
|cj ?j | =
ln
|?j | =
ln k? k1 =
j=1
j=1
t=1
j=1
t=1
j=1
t=1
Because of ln z ? z ? 1 for any z > 0, we can optimize an upper bound of J in (6). We then have
PT
Pd
Pd PT
minA J1 = t=1 L(?t , Xt , yt ) + T j=1 |cj | + T j=1 t=1 |?jt |, which is equivalent to the
multiplicative formulation (1) where {p, k} = {1, 1}. For q > 1, we have:
!
!
d
T
d
T
d
X
X
X
1X
1X
t q
q
t q
j
ln
|cj ?j |
?
ln max{|c1 |, . . . , |cd |} ?
|?j |
(8)
ln k? kq =
q j=1
q j=1
t=1
t=1
j=1
=
d
X
ln kck? +
j=1
T
d
T
1X
1X X tq
d
|?j | ? dkck? +
ln
k? kq ? (d + ).
q j=1 t=1
q t=1 t q
q
(9)
Since vector norms satisfy kzk? ? kzkk for any vector z and k ? 1, these inequalities lead to
PT
PT
an upper bound of J in (6), i.e., minA Jq,k = t=1 L(?t , Xt , yt ) + T dkckk + Tq t=1 k? t kqq ,
which is equivalent to the general multiplicative formulation in (1).
5
Optimization Algorithm
Alternating optimization algorithms have been used in both of the early methods [2, 14] to solve
Problem (1) which alternate between solving two subproblems: solve for ? t with fixed c; solve for
c with fixed ? t . The convergence property of such an alternating algorithm has been analyzed in [2]
that it converges to a local minimizer. However, both subproblems in the existing methods can only
be solved using iterative algorithms such as gradient descent, linear or quadratic program solvers.
We design a new alternating optimization algorithm that utilizes the property that both Problems (1)
and (2) are equivalent to Problem (3) used in our proof and we derive a closed-form solution for c
for the second subproblem. The following theorem characterizes this result.
Theorem 3 For any given values of ?t:t=1,??? ,T , the optimal ? of Problem
qP (3) when ?t:t=1,??? ,T are
p
p
1
1? 2kq
T
2 ? 2kp 2q
t p
?2
fixed to the given values can be computed by ?j = ?1
t=1 (?j ) , j = 1, ? ? ? , d.
5
Proof. By the Cauchy-Schwarz inequality and the same argument used in the proof of Lemma 1,
1
?1 p
we obtain that the best ? for a given set of ?t:t=1,??? ,T is ?j = ?12 ?2 2 ||?j ||p/q . We also know
kq
kq?p
that ?1 and ?2 are chosen in such a way that ?1 = ?12kq?p ?22kq?p and ?2 = ?2 . This is equivalent to
2kq?p
kq
have ?1 = ?1
p?kq
?2 kq and ?2 = ?2 . Substituting them into the formula of ? yields the result.
Now, in the algorithm to solve Problem (1), we solve the first subproblem to obtain a new iterate
? new
, then we use the current value of c, cold , to compute the value of ?new
= diag(cold )? new
,
t
t
t
which is then used to compute ?j according to the formula in Theorem 3. Then, c is computed as
?
cj = k ?j , j = 1, ? ? ? , d. The overall procedure is summarized in Algorithm 1.
Algorithm 1 Alternating optimization for multiplicative MTFL
Input: Xt , yt , t = 1, ? ? ? , T , as well as ?1 , ?2 , p and k
Initialize: cj = 1, ?j = 1, ? ? ? , d
repeat
? t , ? t = 1, ? ? ? , T
1. Convert Xt diag(cs?1 ) ? X
for t = 1, ? ? ? , T do
? t , yt ) + ?1 ||? t ||p for ? s
Solve min?t L(? t , X
t
p
end for
?
2. Compute ?st = diag(c(s?1) )? st , and compute cs as csj = k ?j where ?j is computed
according to the formula in Theorem 3.
until max(|(?jt )s ? (?jt )s?1 |) <
Output: ?t , c and ? t , t = 1, ? ? ? , T
Algorithm 1 can be used to solve the entire family of methods characterized by Problem (1). The
first subproblem involves convex optimization if a convex loss function is chosen and p ? 1, and
can be solved separately for individual tasks using single task learning. The second subproblem
is analytically solved by a formula that guarantees that Problem (1) reaches a lower bound for the
current ?t . In this paper, the least squares and logistic regression losses are used and both of them are
convex and differentiable loss functions. When convex and differentiable losses are used, theoretical
results in [19] can be used to prove the convergence of the proposed algorithm. We choose to monitor
the maximum norm of the A matrix to terminate the process, but it can be replaced by any other
suitable termination criterion. Initialization can be important for this algorithm, and we suggest
starting with c = 1, which considers all features initially in the learning process.
6
Two New Formulations
The two existing methods discussed in [2, 14] use p = k in their formulations, which renders ?jt and
cj the same amount of shrinkage. To explore other feature sharing patterns among tasks, we propose
two new formulations where p 6= k. For the common choices of p and k, the relation between the
optimal c and ? can be computed according to Theorem 2, and is summarized in Table 1.
1. When the majority of the features is not relevant to any of the tasks, it requires a sparsityinducing norm on c. However, within the relevant features, many features are shared between tasks.
In other words, the features used in each task are not sparse relative to all the features selected by
c, which requires a non-sparsity-inducing norm on ?. Hence, we use `1 norm on c and `2 norm
on all ??s in Formulation (1). This formulation
is equivalent to the joint regularization method of
1
2
PT
PT
Pd q
3
t 2
3
3
min?t t=1 L(?t , Xt , yt ) + ? j=1
t=1 (?j ) where ? = 2?1 ?2 .
2. When many or all features are relevant to the given tasks, it may prefer the `2 norm penalty on
c. However, only a limited number of features are shared between tasks, i.e., the features used by
individual tasks are sparse with respect to the features selected as useful across tasks by c. We can
impose the `1 norm penalty on ?. This formulation
is equivalent to the joint regularization method
r
2
2
1
PT
Pd 3 PT
t|
of min?t t=1 L(?t , Xt , yt ) + ? j=1
|?
where ? = 2?13 ?23 .
j
t=1
6
Table 1: The shrinkage effect of c with respect to ? for four common choices of p and k.
7
p
k
2
2
1
1
c
q
q
PT ?t 2
c?j = ?1 ?2?1
t=1 ?j
?1 PT
?t
c?j = ?1 ?2
t=1 |?j |
p
k
2
1
1
2
c
P
2
c?j = ?1 ?2?1 Tt=1 ??jt
qP
q
T
?t
c?j = ?1 ?2?1
t=1 |?j |
Experiments
In this section, we empirically evaluate the performance of the proposed multiplicative MTFL with
the four parameter settings listed in Table 1 on synthetic and real-world data for both classification
and regression problems. The first two settings (p, k) = (2, 2), (1, 1) give the same methods respectively in [2, 14], and the last two settings correspond to our new formulations. The least squares
and logistic regression losses are used, respectively, for regression and classification problems. We
focus on the understanding of the shrinkage effects created by the different choices of regularizers
in multiplicative MTFL. These methods are referred to as MMTFL and are compared with the dirty
model (DMTL) [9] and robust MTFL (rMTFL) [6] that use the additive decomposition.
The first subproblem of Algorithm 1 was solved using CPLEX solvers and single task learning in the
initial first subproblem served as baseline. We used respectively 25%, 33% and 50% of the available
data in each data set for training and the rest data for test. We repeated the random split 15 times
and reported the averaged performance. For each split, the regularization parameters of each method
were tuned by a 3-fold cross validation within the training data. The regression performance was
measured by the coefficient of determination, denoted as R2 , which was computed as 1 minus the
ratio of the sum of squared residuals and the total sum of squares. The classification performance
was measured by the F1 score, which was the harmonic mean of precision and recall.
Synthetic Data. We created two synthetic data sets which included 10 and 20 tasks, respectively.
For each task, we created 200 examples using 100 features with pre-defined combination weights
?. Each feature was generated following the N (0, 1) distribution. We added noise and computed
yt = Xt ?t + t for each task t where the noise followed a distribution N (0, 1). We put the
different tasks? ??s together as rows in Figure 1. The values of ??s were specified in such a way
for us to explore how the structure of feature sharing influences the multitask learning models when
various regularizers are used. In particular, we illustrate the cases where the two newly proposed
formulations outperformed other methods.
(a) Synthetic data D1
(b) Synthetic data D2
Figure 1: Parameter matrix learned by different methods (darker color indicates greater values.).
Synthetic Data 1 (D1). As shown in Figure 1a, 40% of components in all ??s were set to 0, and
these features were irrelevant to all tasks. The rest features were used in every task?s model and
hence these models were sparse with respect to all of the features, but not sparse with respect to the
selected features. This was the assumption for the early joint regularized methods to work. To learn
this feature sharing structure, however, we observed that the amount of shrinkage needed would be
different for c and ?. This case might be in favor of the `1 norm penalty on c.
Synthetic Data 2 (D2). The designed parameter matrix is shown in Figure 1b where tasks were split
into 6 groups. Five features were irrelevant to all tasks, 10 features were used by all tasks, and each
7
of the remaining 85 features was used by only 1 or 2 groups. The neighboring groups of tasks in
Figure 1b shared only 7 features besides those 10 common features. Non-neighboring tasks did not
share additional features. We expected c to be non-sparse. However, each task only used very few
features with respect to all available features, and hence each ? should be sparse.
Figure 1 shows the parameter matrices (with columns representing features for illustrative convenience) learned by different methods using 33% of the available examples in each data set. We can
clearly see that MMTFL(2,1) performs the best for Synthetic data D1. This result suggests that the
classic choices of using `2 or `1 penalty on both c and ? (corresponding to early joint regularized
methods) might not always be optimal. MMTFL(1,2) is superior for Synthetic data D2, where each
model shows strong feature sparsity but few features can be removed if all tasks are considered.
Table 2 summarizes the performance comparison where the best performance is highlighted in bold
font. Note that the feature sharing patterns may not be revealed by the recent methods on clustered
multitask learning that cluster tasks into groups [10, 8, 23] because no cluster structure is present in
Figure 1b, for instance. Rather, the sharing pattern in Figure 1b is in the shape of staircase.
Table 2: Comparison of the performance between various multitask learning models
Data set
Synthetic data
D1 (R2 )
25%
33%
50%
D2 (R2 )
25%
33%
50%
Real-world data
SARCOS
25%
(R2 )
33%
50%
USPS
25%
(F1 score)
33%
50%
STL
DMTL
rMTFL
MMTFL(2,2) MMTFL(1,1) MMTFL(2,1) MMTFL(1,2)
0.40?0.02
0.55?0.03
0.60?0.02
0.28?0.02
0.35?0.01
0.75?0.01
0.60?0.02
0.73?0.01
0.75?0.01
0.36?0.01
0.42?0.02
0.81?0.01
0.58?0.02
0.61?0.02
0.66?0.01
0.46?0.01
0.63?0.03
0.83?0.01
0.64?0.02
0.79?0.02
0.86?0.01
0.45?0.01
0.69?0.02
0.91?0
0.54?0.03
0.76?0.01
0.88?0.01
0.35?0.05
0.75?0.01
0.95?0
0.73?0.02
0.86?0.01
0.90?0.01
0.46?0.02
0.67?0.03
0.92?0.01
0.42?0.04
0.65?0.03
0.84?0.01
0.49?0.02
0.83?0.02
0.97?0
0.78?0.02
0.78?0.02
0.83?0.06
0.83?0.01
0.84?0.02
0.87?0.02
0.90? 0
0.88?0.11
0.87? 0.1
0.89?0.01
0.90?0.01
0.91?0.01
0.90?0
0.89?0.1
0.89?0.1
0.91?0.01
0.90?0.01
0.92?0.01
0.89? 0
0.90? 0
0.91? 0
0.90?0.01
0.89?0.01
0.92?0.01
0.89? 0
0.90? 0
0.90? 0.01
0.90?0.01
0.90?0.01
0.92?0.01
0.90?0.01
0.91?0.01
0.91?0.01
0.90?0.01
0.90?0.01
0.92?0.01
0.87?0.01
0.89?0.01
0.89?0.01
0.91?0.01
0.91?0.01
0.93?0.01
Real-world Data. Two benchmark data sets, the Sarcos [1] and the USPS data sets [10], were used
for regression and classification tests respectively. The Sarcos data set has 48,933 observations and
each observation (example) has 21 features. Each task is to map from the 21 features to one of the 7
consecutive torques of the Sarcos robot arm. We randomly selected 2000 examples for use in each
task. USPS handwritten digits data set has 2000 examples and 10 classes as the digits from 0 to 9.
We first used principle component analysis to reduce the feature dimension to 87. To create binary
classification tasks, we randomly chose images from the other 9 classes to be the negative examples.
Table 2 provides the performance of the different methods on these two data sets, which shows the
effectiveness of MMTFL(2,1) and MMTFL(1,2).
8
Conclusion
In this paper, we study a general framework of multiplicative multitask feature learning. By decomposing the model parameter of each task into a product of two components: the across-task feature
indicator and task-specific parameters, and applying different regularizers to the two components,
we can select features for individual tasks and also search for the shared features among tasks. We
have studied the theoretical properties of this framework when different regularizers are applied and
found that this family of methods creates models equivalent to those of the joint regularized MTL
methods but with a more general form of regularization. Further, an analytical formula is derived for
the across-task component as related to the task-specific component, which shed light on the different shrinkage effects in the various regularizers. An efficient algorithm is derived to solve the entire
family of methods and also tested in our experiments. Empirical results on synthetic data clearly
show that there may not be a particular choice of regularizers that is universally better than other
choices. We empirically show a few feature sharing patterns that are in favor of two newly-proposed
choices of regularizers, which is confirmed on both synthetic and real-world data sets.
Acknowledgements
Jinbo Bi and her students Xin Wang and Jiangwen Sun were supported by NSF grants IIS-1320586,
DBI-1356655, IIS-1407205, and IIS-1447711.
8
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9
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5,112 | 5,628 | Multitask learning meets tensor factorization: task
imputation via convex optimization
Kishan Wimalawarne
Tokyo Institute of Technology
Meguro-ku, Tokyo, Japan
[email protected]
Masashi Sugiyama
The University of Tokyo
Bunkyo-ku, Tokyo, Japan
[email protected]
Ryota Tomioka
TTI-C
Illinois, Chicago, USA
[email protected]
Abstract
We study a multitask learning problem in which each task is parametrized by a
weight vector and indexed by a pair of indices, which can be e.g, (consumer,
time). The weight vectors can be collected into a tensor and the (multilinear-)rank
of the tensor controls the amount of sharing of information among tasks. Two
types of convex relaxations have recently been proposed for the tensor multilinear rank. However, we argue that both of them are not optimal in the context of
multitask learning in which the dimensions or multilinear rank are typically heterogeneous. We propose a new norm, which we call the scaled latent trace norm
and analyze the excess risk of all the three norms. The results apply to various settings including matrix and tensor completion, multitask learning, and multilinear
multitask learning. Both the theory and experiments support the advantage of the
new norm when the tensor is not equal-sized and we do not a priori know which
mode is low rank.
1
Introduction
We consider supervised multitask learning problems [1, 6, 7] in which the tasks are indexed by a
pair of indices known as multilinear multitask learning (MLMTL) [17, 19]. For example, when we
would like to predict the ratings of different aspects (e.g., quality of service, food, etc) of restaurants
by different customers, the tasks would be indexed by aspects ? customers. When each task is
parametrized by a weight vector over features, the goal would be to learn a features ? aspects ?
customers tensor. Another possible task dimension would be time, since the ratings may change
over time.
This setting is interesting, because it would allow us to exploit the similarities across different customers as well as similarities across different aspects or time-points. Furthermore this would allow
us to perform task imputation, that is to learn weights for tasks for which we have no training examples. On the other hand, the conventional matrix-based multitask learning (MTL) [2, 3, 13, 16] may
fail to capture the higher order structure if we consider learning a flat features ? tasks matrix and
would require at least r samples, where r is the rank of the matrix to be learned, for each task.
Recently several norms that induce low-rank tensors in the sense of Tucker decomposition or multilinear singular value decomposition [8, 9, 14, 25] have been proposed. The mean squared error for
recovering a n1 ? ? ? ? ? nK tensor of multilinear rank (r1 , . . . , rK ) from its noisy version scale as
?K ? 2 1 ?K
?
1
O(( K
rk ) ( K k=1 1/ nk )2 ) for the overlapped trace norm [23]. On the other hand, the
k=1
error of the latent trace norm scales as O(mink rk / mink nk ) in the same setting [21]. Thus while
the latent trace norm has the better dependence in terms of the multilinear rank rk , it has the worse
dependence in terms of the dimensions nk .
Tensors that arise in multitask learning typically have heterogeneous dimensions. For example,
the number of aspects for a restaurant (quality of service, food, atmosphere, etc.) would be much
1
? ?
Table 1: Tensor denoising performance using different norms. The mean squared error |||W
? 2
W |||F /N is shown for the denoising algorithms (3) using different norms for tensors.
Op
((
1
K
Overlapped trace norm
K ? )2 (
K
?
?
? )2 )
1
rk
1/ nk
K
k=1
Latent trace norm
(
)
Op min rk / min nk
k
k=1
k
Scaled latent trace norm
(
)
Op min (rk /nk )
k
smaller than the number of customers or the number of features. In addition, it is a priori unclear
which mode (or dimension) would have the most redundancy or sharing that could be exploited by
multitask learning. Some of the modes may have full ranks if there is no sharing of information
along them. Therefore, both the latent trace norm and the overlapped trace norm would suffer either
from the heterogeneous multilinear rank or the heterogeneous dimensions in this context.
In this paper, we propose a modification to the latent trace norm whose mean squared error scales as
O(mink (rk /nk )) in the same setting, which is better than both the previously proposed extensions
of trace norm for tensors. We study the excess risk of the three norms through their Rademacher
complexities in various settings including matrix completion, multitask learning, and MLMTL. The
new analysis allows us to also study the tensor completion setting, which was only empirically
studied in [22, 23]. Our analysis consistently shows the advantage of the proposed scaled latent
trace norm in various settings in which the dimensions or ranks are heterogeneous. Experiments on
both synthetic and real data sets are also consistent with our theoretical findings.
2
Norms for tensors and their denoising performance
?K
Let W ? Rn1 ?????nK be a K-way tensor. We denote the total number of entries by N := k=1 nk .
A mode-k fiber of W is an nk dimensional vector we obtain by fixing all but the kth index. The
mode-k unfolding W (k) of W is the nk ?N/nk matrix formed by concatenating all the N/nk modek fibers along columns. We say that W has multilinear rank (r1 , . . . , rK ) if rk = rank(W (k) ).
2.1
Existing norms for tensors
First we review two norms proposed in literature in order to convexify tensor decomposition.
The overlapped trace norm (see [12, 15, 18, 22]) is defined as the sum of the trace norms of the
mode-k unfoldings as follows:
?K
|||W|||overlap =
?W (k) ?tr ,
(1)
k=1
where ? ? ?tr is the trace norm (also known as the nuclear norm) [10, 20], which is defined as the
absolute sum of singular values. Romera-Paredes et al. [17] has used the overlapped trace norm in
MLMTL.
The latent trace norm [21, 22] is defined as the infimum over K tensors as follows:
?K
(k)
|||W|||latent =
inf
?W (k) ?tr .
W (1) +???+W (K) =W
k=1
(2)
Table 1 summarizes the denoising performance in mean squared error analyzed in Tomioka and
Suzuki [21] for the above two norms. The setting is as follows: we observe a noisy version Y of a
tensor W ? with multilinear rank (r1 , . . . , rK ) and would like to recover W ? by solving
(
)
1
2
?
W = argmin
|||W ? Y|||F + ? |||W|||? ,
(3)
2
W
where |||?|||? is either the overlapped trace norm or the latent trace norm. We can see that while the
latent trace norm has the better dependence in terms of the multilinear rank, it has the worse dependence in terms of the dimensions. Intuitively, the latent trace norm recognizes the mode with
the lowest rank. However, it does not have a good control of the dimensions; in fact, the factor
2
1/ mink nk comes from the fact that for a random tensor X with i.i.d. ?
Gaussian entries, the expectation of the dual norm ?X ?latent? = maxk ?X (k) ?op behaves like Op ( maxk N/nk ), where ? ? ?op
is the operator norm.
2.2
A new norm
In order to correct the unfavorable behavior of the dual norm, we propose
the scaled latent trace
?
norm. It is defined similarly to the latent trace norm with weights 1/ nk as follows:
K
?
1
(k)
(4)
? ?W (k) ?tr .
nk
k=1
?
?
Now the expectation of the dual norm ?X ?scaled? = maxk nk ?X (k) ?op behaves like Op ( N )
for X with random i.i.d. Gaussian entries and combined with the following relation
?
rk
|||W|||F ,
(5)
|||W|||scaled ? min
k
nk
we obtain the scaling of the mean squared error in the last column of Table 1. We can see that the
scaled latent norm recognizes the mode with the lowest rank relative to its dimension.
|||W|||scaled =
3
inf
W (1) +???+W (K) =W
Theory for multilinear multitask learning
m
pq
We consider T = P Q supervised learning tasks. Training samples (xipq , yipq )i=1
((p, q) ? S)
are provided for a relatively small fraction of the task index pairs S ? [P ] ? [Q]. Each task is
parametrized by a weight vector wpq ? Rd , which can be collected into a 3-way tensor W =
(wpq ) ? Rd?P ?Q whose (p, q) fiber is wpq . We define the learning problem as follows:
? = argmin L(W),
?
W
subject to |||W||| ? B0 ,
(6)
?
W?Rd?P ?Q
where the norm |||?|||? is either the overlapped trace norm, latent trace norm, or the scaled latent trace
? is defined as follows:
norm, and the empirical risk L
mpq
1 ?
1 ?
?
L(W)
=
? (?xipq , wpq ? ? yipq ) .
|S|
mpq i=1
(p,q)?S
The true risk we are interested in minimizing is defined as follows:
1 ?
E(x,y)?Ppq ? (?x, wpq ? ? y) ,
L(W) =
P Q p,q
m
pq
where Ppq is the distribution from which the samples (xipq , yipq )i=1
are drawn from.
? ? L(W ? ) with the expected dual norm E |||D||| ?
The next lemma relates the excess risk L(W)
?
through Rademacher complexity.
Lemma 1. We assume that the output yipq is bounded as |yipq | ? b, and the number of samples
mpq ? m > 0 for the observed tasks. We also assume that the loss function ? is Lipschitz continuous
with the constant ?, bounded in [0, c] and ?(0) = 0. Let W ? be any tensor such that |||W ? |||? ? B0 .
Then with probability at least 1 ? ?, any minimizer of (6) satisfies the following bound:
?
(
? )
b
?
2B
log(2/?)
0
?
?
? ? L(W ) ? 2?
E |||D|||?? + ?
,
L(W)
+c
|S|
2|S|m
|S|m
?
mpq
1
d?P ?Q
where c? = c + 1, |||?|||?? is the dual norm of |||?|||? , ? := |S|
(p,q)?S m . The tensor D ? R
?
?mpq ipq
ipq
d?P ?Q
is defined as the sum D = (p,q)?S i=1 Z , where Z ? R
is defined as
{
1
?ipq xipq , if p = p? and q = q ? ,
(p? , q ? )th fiber of Z ipq = mpq
0,
otherwise.
Here ?ipq ? {?1, +1} are Rademacher random variables and the expectation in the above inequalmpq
ity is with respect to ?ipq , the random draw of tasks S, and the training samples (xipq , yipq )i=1
.
3
Proof. The proof is a standard one following the line of [5] and it is presented in Appendix A.
The next theorem computes the expected dual norm E |||D|||?? for the three norms for tensors (the
proof can be found in Appendix B).
Theorem 1. We assume that C pq := E[xipq xipq ? ] ? ?d I d and there is a constant R > 0 such that
?xipq ? ? R almost surely. Let us define
D1 := d + P Q,
D2 := P + dQ,
D3 := Q + dP.
In order to simplify the presentation, we assume that maxk Dk ? 3 and dP Q ? max(d2 , P 2 , Q2 ).
For the overlapped trace norm, the latent trace norm, and the scaled latent trace norm, the expectation E |||D|||?? can be bounded as follows:
(?
)
1
?
R
E |||D|||overlap? ? C min
Dk log Dk +
log Dk ,
(7)
k
|S|
m|S|dP Q
m|S|
)
(?
?
R
1
E |||D|||latent? ? C ?
max (Dk log Dk ) +
log(max Dk ) ,
(8)
k
|S|
m|S|dP Q k
m|S|
(?
)
?
R maxk nk
1
?
E |||D|||scaled? ? C ??
log(max Dk ) +
log(max Dk ) ,
(9)
k
k
|S|
m|S|
m|S|
where C, C ? , C ?? are constants, n1 = d, n2 = P , and n3 = Q. Furthermore, if m|S| ?
R2 (maxk nk ) log(maxk Dk )/?, the O(1/m|S|) terms in the above inequalities can be dropped.
Note that the assumption that the norm of xipq is bounded is natural because the target yipq is also
bounded. The parameter ? in the assumption C pq ? ?/dI d controls the amount of correlation in
the data. Since Tr(C) = E?xipq ?2 ? R2 , we have ? = O(1) when the features are uncorrelated;
on the other hand, we have ? = O(d), if they lie in a one dimensional subspace. The number of
?
samples m|S| = O(max
k nk ) is enough to drop the O(1/m|S|) term even if ? = O(1).
Now we state the consequences of Theorem 1 for the three norms for tensors. The common assumptions are the same as in Lemma 1 and Theorem 1. We also assume m|S| ?
R2 (maxk nk ) log(maxk Dk )/? to drop the O(1/m|S|) terms. Let W ? be any d ? P ? Q tensor
with multilinear-rank (r1 , r2 , r3 ) and bounded element-wise as |||W ? |||?? ? B.
Corollary 1 (Overlapped
trace norm). With probability at least 1 ? ?, any minimizer of (6) with
?
|||W|||overlap ? B ?r?1/2 dP Q satisfies the following inequality:
?
?
?
?
?
log(2/?)
?
? ? L(W ) ? c1 ?B
L(W)
?r?1/2 min (Dk log Dk ) + c2 ?b
+ c3
,
k
m|S|
m|S|
m|S|
?3 ?
where ?r?1/2 = ( k=1 rk /3)2 and c1 , c2 , c3 are constants.
?3 ?
Note that Tomioka et al. [23] obtained a bound that depends on ( k=1 Dk /3)2 instead of
min(Dk log Dk ). Although the minimum may look better than the average, our bound has the
worse constant K = 3 hidden in c1 . The log Dk factor allows us to apply the above result to the
setting of tensor completion as we show below.
Corollary 2 (Latent
trace norm). With probability at least 1 ? ?, any minimizer of (6) with
?
|||W|||latent ? B mink rk dP Q satisfies the following inequality:
?
?
?
?
?
log(2/?)
?
?
?
L(W) ? L(W ) ? c1 ?B
min rk max(Dk log Dk ) + c2 ?b
+ c3
,
k
m|S| k
m|S|
m|S|
where c?1 , c2 , c3 are constants.
Corollary 3 (Scaled
? latent trace norm). With probability at least 1 ? ?, any minimizer of (6) with
|||W|||scaled ? B mink (rk /nk )dP Q satisfies the following inequality:
?
?
( )
?
?
r
?
log(2/?)
k
? ? L(W ? ) ? c??1 ?B
L(W)
min
dP Q log(max Dk ) + c2 ?b
+ c3
,
k
m|S| k
nk
m|S|
m|S|
where n1 = d, n2 = P , n3 = Q, and c??1 , c2 , c3 are constants.
4
5
k
k
k
k
?r?1/2 min(Dk log Dk )
k
??r?1/2 P Q log(P Q)
(r1 , r2 , r3 )
(1, 1, |S|)
(?, 1, |S|)
n3!
(r, P, r? )
Tensor completion
n1! n
2!
Q!
MLMTL [17] (heterogeneous case)
d! P!
MLMTL [17] (homogeneous case)
min rk max(Dk log Dk )
?dP Q log(dQ)
k
d!
d! d!
min( nrkk )N log(max Dk )
? min(rP Q, dP r? ) log(dQ)
k
?(min rk )d2 log(d2 )
??r?1/2 d2 log(d2 )
(?, 1, |S|)
?r?1/2 log(P Q)
?
(d, 1/ d, P Q)
(r, P, r? )
PQ!
d!
(heterogeneous
MTL
case)
MTL [16] (homogeneous case)
(r1 , r2 , r3 )
?r?1/2 log(d )
2
(r, d, d)
(1, r, r)
d2!
d!
Matrix completion [11]
?(min rk )d2 log(d2 )
r log(dQ)
d log(dQ)
2
r log(d )
P Q log(P Q)
?r?1/2 (P + Q) log(P + Q)
(1, 1, |S|)
?
(d, 1/ d, d2 )
Overlap
(?, B, |S|)
(r1 , r2 , r3 )
Q!
P!
1!
(n1 , n2 , n3 )
r log(d2 )
r(P + Q) log(P Q)
Scaled
Sample complexities (per 1/?2 )
Latent
Table 2: Sample complexities of the overlapped trace norm, latent trace norm, and the scaled latent trace norm in various settings. The common factor 1/?2 is
omitted from the sample complexities. The sample complexities are defined with respect to |S| for matrix completion, m for multitask learning, and m|S| for tensor
?3 ?
completion and multilinear multitask learning. In the heterogeneous cases, we assume P ? r < r? . We define ?r?1/2 = ( k=1 rk /K)2 and N := n1 n2 n3 .
We summarize the implications of the
above corollaries for different settings
in Table 2. We almost recover the settings for matrix completion [11] and
multitask learning (MTL) [16]. Note
that these simpler problems sometimes
disguise themselves as the more general
tensor completion or multilinear multitask learning problems. Therefore it
is important that the new tensor based
norms adapts to the simplicity of the
problems in these cases.
Matrix completion is when d = ? =
m = r1 = 1, and we assume that
r2 = r3 = r < P, Q. The sample complexities are the number of samples |S|
that we need to make the leading term
in Corollaries 1, 2, and 3 equal ?. We
can see that the overlapped trace norm
and the scaled latent trace norm recover
the known result for matrix completion
[11]. The plain latent trace norm requires O(P Q) samples because it recognizes the first mode as the mode with
the lowest rank 1. Although the rank r
of the last two modes are low relative to
their dimensions, the latent trace norm
fails to recognize this.
In multitask learning (MTL), only the
first mode corresponding to features has
a low rank r and the other two modes
have full rank. Note that a tensor is
a matrix when its multilinear rank is
full except for one mode. We also assume that all the pairs (p, q) are observed (|S| = P Q) as in [16]. The sample complexities are defined the same
way as above with respect to the number
of samples m because |S| is fixed. The
homogeneous case is when d = P =
Q. The heterogeneous case is when
P ? r < d. Our bound for the overlapped trace norm is almost as good as
the one in [16] but has an multiplicative
log(P Q) factor (as oppose to their additive log(P Q) term) and ?r?1/2 ? r.
Also note that the results in [16] can be
applied when d is much larger than P
and Q. Turning back to our bounds,
both the latent trace norm and its scaled
version can perform as well as knowing
the mode with the lowest rank (the first
mode) (see also [21]) when d = P =
Q. However, when the dimensions are
heterogeneous, similarly to the matrix
completion case above, the plain latent
trace norm fails to recognize the lowrank-ness of the first mode and
requires O(d) samples, because the second mode has the lowest rank P .
In multilinear multitask learning (MLMTL) [17], any mode could possibly be low rank but it is
a priori unknown. The sample complexities are defined the same way as above with respect to
m|S|. The homogeneous case is when d = P = Q. The heterogeneous case is when the first
mode or the third mode is low rank but P ? r < d. Similarly to the above two settings, the
overlapped trace norm has a mild dependence on the dimensions but a higher dependence on the
rank ?r?1/2 ? r. The latent trace norm performs as well as knowing the mode that has the lowest
rank in the homogeneous case. However, it fails to recognize the mode with the lowest rank relative
to its dimension. The scaled latent trace norm does this and although it has a higher logarithmic
dependence, it is competitive in both cases.
Finally, our bounds also hold for tensor completion. Although Tomioka et al. [22, 23] studied
tensor completion algorithms, their analysis assumed that the inputs xipq are drawn from a Gaussian
distribution, which does not hold for tensor completion. Note that in our setting xipq can be an
indicator vector that has one in the jth position uniformly over 1, . . . , d. In this case, ? = 1. The
sample complexities of different norms with respect to m|S| is shown in the last row of Table 2. The
sample complexity for the overlapped trace norm is the same as the one in [23] with a logarithmic
factor. The sample complexities for the latent and scaled latent trace norms are new. Again we can
see that while the latent trace norm recognize the mode with the lowest rank, the scaled latent trace
norm is able to recognize the mode with the lowest rank relative to its dimension.
4
Experiments
We conducted several experiments to evaluate performances of tensor based multitask learning setting we have discussed in Section 3. In Section 4.1, we discuss simulation we conducted using
synthetic data sets. In Sections 4.2 and 4.3, we discuss experiments on two real world data sets,
namely the Restaurant data set [26] and School Effectiveness data set [3, 4]. Both of our real world
data sets have heterogeneous dimensions (see Figure 2) and it is a priori unclear across which mode
has the most amount of information sharing.
4.1
Synthetic data sets
The true d ? P ? Q tensor W ? was generated by first sampling a r1 ? r2 ? r3 core tensor and then
multiplying random orthonormal matrix to each of its modes. For each task (p, q) ? [P ] ? [Q], we
generated training set of m vectors (xipq , yipq )m
i=1 by first sampling xipq from the standard normal
distribution and then computing yipq = ?xipq , wpq ? + ?i , where ?i was drawn from a zero-mean
normal distribution with variance 0.1. We used the penalty formulation of (6) with the squared loss
and selected the regularization parameter ? using two-fold cross validation on the training set from
the range 0.01 to 10 with the interval 0.1.
In addition to the three norms for tensors we discussed in the previous section, we evaluated the
matrix-based multitask learning approaches that penalizes the trace norm of the unfolding of W at
specific modes. The conventional convex multitask learning [2, 3, 16] corresponds to one of these
approaches that penalizes the trace norm of the first unfolding ?W (1) ?tr . The convex MLMTL in
[17] corresponds to the overlapped trace norm.
In the first experiment, we chose d = P = Q = 10 and r1 = r2 = r3 = 3. Therefore, both
the dimensions and the multilinear rank are homogeneous. The result is shown in Figure 1(a). The
overlapped trace norm performed the best, the matrix-based approaches performed next, and the
latent trace norm and the scaled latent trace norm were the worst. The scaling of the latent trace
norm had no effect because the dimensions were homogeneous. Since the sample complexities for
all the methods were the same in this setting (see Table 2), the difference in the performances could
be explained by a constant factor K(= 3) that is not shown in the sample complexities.
In the second experiment, we chose the dimensions to be homogeneous as d = P = Q = 10, but
(r1 , r2 , r3 ) = (3, 6, 8). The result is shown in Figure 1(b). In this setting, the (scaled) latent trace
norm and the mode-1 regularization performed the best. The lower the rank of the corresponding
mode, the lower were the error of the matrix-based MTL approaches. The overlapped trace norm
was somewhat in the middle of the three matrix-based approaches.
6
0.016
0.022
Overlapped trace norm
Latent trace norm
Scaled Latent trace norm
Mode 1
Mode 2
Mode 3
0.015
0.014
0.024
Overlapped trace norm
Latent trace norm
Scaled Latent trace norm
Mode 1
Mode 2
Mode 3
0.02
0.018
Overlapped trace norm
Latent trace norm
Scaled Latent trace norm
Mode 1
Mode 2
Mode 3
0.022
0.02
0.013
MSE
MSE
MSE
0.018
0.016
0.016
0.012
0.014
0.011
0.012
0.014
0.01
10
20
30
40
50
60
70
80
90
100
110
Sample size
0.01
10
0.012
20
30
40
50
60
70
80
90
100
110
0.01
10
Sample size
(a) Synthetic experiment for the
case when both the dimensions and
the ranks are homogeneous. The
true tensor is 10?10?10 with multilinear rank (3, 3, 3).
(b) Synthetic experiment for the
case when the dimensions are homogeneous but the ranks are heterogeneous. The true tensor is
10 ? 10 ? 10 with multilinear rank
(3, 6, 8).
20
30
40
50
60
70
80
90
100
110
Sample size
(c) Synthetic experiment for the
case when both the dimensions and
the ranks are heterogeneous. The
true tensor is 10 ? 3 ? 10 with multilinear rank (3, 3, 8).
Figure 1: Results for the synthetic data sets.
In the last experiment, we chose both the dimensions and the multilinear rank to be heterogeneous
as (d, P, Q) = (10, 3, 10) and (r1 , r2 , r3 ) = (3, 3, 8). The result is shown in Figure 1(c). Clearly the
first mode had the lowest rank relative to its dimension. However, the latent trace norm recognizes
the second mode as the mode with the lowest rank and performed similarly to the mode-2 regularization. The overlapped trace norm performed better but it was worse than the mode-1 regularization.
The scaled latent trace norm performed comparably to the mode-1 regularization.
4.2
Restaurant data set
The Restaurant data set contains data for a recommendation system for restaurants where different
customers have given ratings to different aspects of each restaurant. Following the same approach
as in [17] we modelled the problem as a MLMTL problem with d = 45 features, P = 3 aspects,
and Q = 138 customers.
The total number of instances for all the tasks were 3483 and we randomly selected training set of
sizes 400, 800, 1200, 1600, 2000, 2400, and 2800. When the size was small many tasks contained
no training example. We also selected 250 instances as the validation set and the rest was used as the
test set. The regularization parameter for each norm was selected by minimizing the mean squared
error on the validation set from the candidate values in the interval [50, 1000] for the overlapped,
[0.5, 40] for the latent, [6000, 20000] for the scaled latent norms, respectively.
We also evaluated matrix-based MTL approaches on different modes and ridge regression (Frobenius norm regularization; abbreviated as RR) as baselines. The convex MLMTL in [17] corresponds
to the overlapped trace norm.
The result is shown in Figure 2(a). We found the multilinear rank of the solution obtained by the
overlapped trace norm to be typically (1, 3, 3). This was consistent with the fact that the performances of the mode-1 regularization and the ridge regression were equal. In other words, the effective dimension of the first mode (features) was one instead of 45. The latent trace norm recognized
the first mode as the mode with the lowest rank and it failed to take advantage of the low-rank-ness
of the second and the third modes. The scaled latent trace norm was able to perform the best matching with the performances of mode-2 and mode-3 regularization. When the number of samples was
above 2400, the latent trace norm caught up with other methods, probably because the effective
dimension became higher in this regime.
4.3
School data set
The data set comes from the inner London Education Authority (ILEA) consisting of examination
records from 15362 students at 139 schools in years 1985, 1986, and 1987. We followed [4] for
the preprocessing of categorical attributes and obtained 24 features. Previously Argyriou et al. [3]
modeled this data set as a 27 ? 139 matrix-based MTL problem in which the year was modeled as a
7
40
0.7
Overlapped trace norm
Latent trace norm
Scaled latent trace norm
Mode 1
Mode 2
Mode 3
RR
MSE
0.6
35
Explained variance
0.65
0.55
0.5
30
25
Overlapped trace norm
Latent trace norm
Scaled latent trace norm
Mode 1
Mode 2
Mode 3
RR
20
0.45
15
0.4
0.35
0
500
1000
1500
2000
Sample size
2500
3000
(a) Result for the 45 ? 3 ? 138 Restaurant data set.
10
0
2000
4000
6000
8000
Sample size
10000
12000
(b) Result for the 24 ? 139 ? 3 School data set.
Figure 2: Results for the real world data sets.
trinomial attribute. Instead here we model this data set as a 24 ? 139 ? 3 MLMTL problem in which
the third mode corresponds to the year. Following earlier papers, [3, 4], we used the percentage of
explained variance, defined as 100 ? (1 ? (test MSE)/(variance of y)), as the evaluation metric.
The results are shown in Figure 2(b). First, ridge regression performed the worst because it was
not able to take advantage of the low-rank-ness of any mode. Second, the plain latent trace norm
performed similarly to the mode-3 regularization probably because the dimension 3 was lower than
the rank of the other two modes. Clearly the scaled latent trace norm performed the best matching
with the performance of the mode-2 regularization; probably the second mode had the most redundancy. The performance of the overlapped trace norm was comparable or slightly better than the
mode-1 regularization. The percentage of the explained variance of the latent trace norm exceeds
30 % around sample size 4000 (around 30 samples per school), which is higher than the Hierarchical
Bayes [4] (around 29.5 %) and matrix-based MTL [3] (around 26.7 %) that used around 80 samples
per school.
5
Discussion
Using tensors for modeling multitask learning [17, 19] is a promising direction that allows us to
take advantage of similarity of tasks in multiple dimensions and even make prediction for a task
with no training example. However, having multiple modes, we would have to face with more
hyperparameters to choose in the conventional nonconvex tensor decomposition framework. Convex
relaxation of tensor multilinear rank allows us to side-step this issue. In fact, we have shown that the
sample complexity of the latent trace norm is as good as knowing the mode with the lowest rank.
This is consistent with the analysis of [21] in the tensor denoising setting (see Table 1).
In the setting of tensor-based MTL, however, the notion of mode with the lowest rank may be
vacuous because some modes may have very low dimension. In fact, the sample complexity of
the latent trace norm can be as bad as not using any low-rank-ness at all if there is a mode with
dimension lower than the rank of the other modes. The scaled latent trace norm we proposed in this
paper recognizes the mode with the lowest rank relative to its dimension and lead to the competitive
sample complexities in various settings we have shown in Table 2.
Acknowledgment: MS acknowledges support from the JST CREST program.
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5,113 | 5,629 | Learning Multiple Tasks in Parallel with
a Shared Annotator
Koby Crammer
Department of Electrical Engeneering
The Technion ? Israel Institute of Technology
Haifa, 32000 Israel
[email protected]
Haim Cohen
Department of Electrical Engeneering
The Technion ? Israel Institute of Technology
Haifa, 32000 Israel
[email protected]
Abstract
We introduce a new multi-task framework, in which K online learners are sharing
a single annotator with limited bandwidth. On each round, each of the K learners
receives an input, and makes a prediction about the label of that input. Then, a
shared (stochastic) mechanism decides which of the K inputs will be annotated.
The learner that receives the feedback (label) may update its prediction rule, and
then we proceed to the next round. We develop an online algorithm for multitask binary classification that learns in this setting, and bound its performance in
the worst-case setting. Additionally, we show that our algorithm can be used to
solve two bandits problems: contextual bandits, and dueling bandits with context,
both allow to decouple exploration and exploitation. Empirical study with OCR
data, vowel prediction (VJ project) and document classification, shows that our
algorithm outperforms other algorithms, one of which uses uniform allocation,
and essentially achieves more (accuracy) for the same labour of the annotator.
1
Introduction
A triumph of machine learning is the ability to predict many human aspects: is certain mail spam or
not, is a news-item of interest or not, does a movie meet one?s taste or not, and so on. The dominant
paradigm is supervised learning, in which the main bottleneck is the need to annotate data. A
common protocol is problem centric: first collect data or inputs automatically (with low cost), and
then pass it on to a user or an expert to be annotated. Annotation can be outsourced to the crowed by
a service like Mechanical Turk, or performed by experts as in the Linguistic data Consortium. Then,
this data may be used to build models, either for a single task or many tasks. This approach is not
making optimal use of the main resource - the annotator - as some tasks are harder than others, yet
we need to give the annotator the (amount of) data to be annotated for each task a-priori . Another
aspect of this problem is the need to adapt systems to individual users, to this end, such systems
may query the user for the label of some input, yet, if few systems will do so independently, the user
will be flooded with queries, and will avoid interaction with those systems. For example, sometimes
there is a need to annotate news items from few agencies. One person cannot handle all of them,
and only some items can be annotated, which ones? Our setting is designed to handle exactly this
problem, and specifically, how to make best usage of annotation time.
We propose a new framework of online multi-task learning with a shared annotator. Here, algorithms
are learning few tasks simultaneously, yet they receive feedback using a central mechanism that
trades off the amount of feedback (or labels) each task receives. We derive a specific algorithm based
on the good-old Perceptron algorithm, called SHAMPO (SHared Annotator for Multiple PrOblems)
for binary classification and analyze it in the mistake bound model, showing that our algorithm
may perform well compared with methods that observe all annotated data. We then show how to
reduce few contextual bandit problems into our framework, and provide specific bounds for such
1
settings. We evaluate our algorithm with four different datasets for OCR , vowel prediction (VJ) and
document classification, and show that it can improve performance either on average over all tasks,
or even if their output is combined towards a single shared task, such as multi-class prediction. We
conclude with discussion of related work, and few of the many routes to extend this work.
2
Problem Setting
We study online multi-task learning with a shared annotator. There are K tasks to be learned simultaneously. Learning is performed in rounds. On round t, there are K input-output pairs (xi,t , yi,t )
where inputs xi,t ? Rdi are vectors, and labels are binary yi,t ? {?1, +1}. In the general case, the
input-spaces for each task may be different. We simplify the notation and assume that di = d for all
tasks. Since the proposed algorithm uses the margin that is affected by the vector norm, there is a
need to scale all the vectors into a ball. Furthermore, no dependency between tasks is assumed.
On round t, the learning algorithm receives K inputs xi,t for i = 1, . . . , K, and outputs K binary-labels y?i,t , where y?i,t ? {?1, +1} is the label predicted for the input
xi,t of task i. The algorithm then chooses a task Jt ? {1, . . . , K} and receives from
an annotator the true-label yJt ,t for that task Jt . It does not observe any other label.
Then, the algorithm updates its models, and proceeds to the
next round (and inputs). For easing calculations below, we
(x , y )
w ,..., w
x
(x , y )
denote by K indicators Zt = (Z1,t , . . . , ZK,t ) the identity
x
of the task which was queried on round
P t, and set ZJt ,t = 1
(x , y )
x
and Zi,t = 0 for i 6= Jt . Clearly, i Zi,t = 1. Below, we
(a)
define the notation Et?1 [x] to be the conditional expectation
x
E [x|Z1 , ...Zt?1 ] given all previous choices.
w
x
1
1
2
2
1
1
Annotator
K
K
K
Alg.
2
K
1
Illustration of a single iteration of multi-task algorithms is
shown in Fig. 1. The top panel shows the standard setting
with shared annotator, that labels all inputs, which are fed to
the corresponding algorithms to update corresponding models. The bottom panel shows the SHAMPO algorithm, which
couples labeling annotation and learning process, and synchronizes a single annotation per round. At most one task
performs an update per round (the annotated one).
J
Alg.
2
xK
Annotator
( xJ , y J )
xJ
(b)
Figure 1: Illustration of a single iteration of multi-task algorithms (a)
standard setting (b) SHAMPO
We focus on linear-functions of the form f (x) = sign(p) for
a quantity p = w> x, w ? Rd , called the margin. Specifically, the algorithm maintains a set of K
>
weight vectors. On round t, the algorithm predicts y?i,t = sign(?
pi,t ) where p?i,t = wi,t?1
xi,t . On
rounds for which the label of some task Jt is queried, the algorithm, is not updating the models of
all other tasks, that is, we have wi,t = wi,t?1 for i 6= Jt .
We say that the algorithm has a prediction mistake in task i if yi,t 6= y?i,t , and denote this event by
Mi,t = 1, otherwise, if there is no mistake
P Pwe set Mi,t = 0. The goal of the algorithm is to minimize
the cumulative number of mistakes, t i Mi,t . Models are also evaluated using the Hinge-loss.
Specifically, let ui ? Rd be some vector associated with task i. We
denote the Hinge-loss of it,
with respect to some input-output by, `?,i,t (ui ) = ? ? yi,t u>
i xi,t + , where, (x)+ = max{x, 0},
and ? > 0 is some parameter. The cumulative loss over all tasks and a sequence of n inputs,
Pn PK
is, L?,n = L? ({ui }) =
t=1
i=1 `?,i,t (ui ). We also useh the following expected hinge-loss
i
? ?,n = L
? {u } = E Pn PK Mi,t Zi,t `?,i,t (ui ) . We
over the random choices of the algorithm, L
i
t
i=1
proceed by describing our algorithm and specifying how to choose a task to query its label, and how
to perform an update.
3
SHAMPO: SHared Annotator for Multiple Problems
We turn to describe an algorithm for multi-task learning with a shared annotator setting, that works
with linear models. Two steps are yet to be specified: how to pick a task to be labeled and how to
perform an update once the true label for that task is given.
To select a task, the algorithm uses the absolute margin |?
pi,t |. Intuitively, if |?
pi,t | is small, then there
is uncertainty about the labeling of xi,t , and vise-versa for large values of |?
pi,t |. Similar argument
2
was used by Tong and Koller [22] for picking an example to be labeled in batch active learning.
Yet, if the model wi,t?1 is not accurate enough, due to small number of observed examples, this
estimation may be rough, and may lead to a wrong conclusion. We thus perform an explorationexploitation strategy, and query tasks randomly, with a bias towards tasks with low |?
pi,t |. To the
best of our knowledge, exploration-exploitation usage in this context of choosing an examples to be
labeled (eg. in settings such as semi-supervised learning or selective sampling) is novel and new.
We introduce b ? 0 to be a tradeoff parameter between exploration and exploitation and ai ? 0 as a
prior for query distribution over tasks. Specifically, we induce a distribution over tasks,
Pr [Jt = j] =
?1
K
?1
X
aj b + |?
pj,t |?minK
pm,t |
m=1 |?
for Dt =
ai b + |?
pi,t |?min |?
pm,t |
. (1)
m
Dt
i=1
Parameters: b, ?, ai ? R+ for i = 1, . . . , K
Initialize: wi,0 = 0 for i = 1, . . . , K
for t = 1, 2, ..., n do
1. Observe K instance vectors, xi,t , (i = 1, . . . , K).
>
2. Compute margins p?i,t = wi,t?1
xi,t .
3. Predict K labels, y?i,t = sign(?
pi,t ).
4. Draw task Jt with the distribution:
?1
aj b + |?
pj,t | ? minK
pm,t |
m=1 |?
,
Pr [Jt = j] =
Dt
?1
X
K
Dt =
ai b + |?
pi,t | ? min |?
pm,t |
.
Clearly,
Pr [Jt = j] ? 0 and
P
Pr
[J
= j] = 1. For b = 0
t
j
we have Pr [Jt = j] = 1 for the
task with minimal margin, Jt =
arg minK
pi,t |, and for b ? ?
i=1 |?
the distribution is proportional to
the prior
P weights, Pr [Jt = j] =
aj /( i ai ). As noted above we
denote by Zi,t = 1 iff i = Jt .
Since the distribution is invariant
to a multiplicative factor of ai we
assume 1 ? ai ?i.
The update of the algorithm
is performed with the aggresi
sive perceptron rule, that is
5. Query the true label ,yJt ,t ? {?1, 1}.
wJt ,t = wJt ,t?1 + (AJt ,t +
6. Set indicator MJt ,t = 1 iff yJt ,t p?i,t ? 0 (Error)
MJt ,t ) yJt ,t xJt ,t and wi,t =
7. Set indicator AJt ,t = 1 iff 0 < yJt ,t p?i,t ? ? (Small wi,t?1 for i 6= Jt . we define
margin)
Ai,t , the aggressive update in8. Update with the perceptron rule:
dicator introducing and the agwJt ,t = wJt ,t?1 + (AJt ,t + MJt ,t ) yJt ,t xJt ,t (2) gressive update threshold, ? ?
R > 0 such that, Ai = 1 iff
wi,t = wi,t?1 for i 6= Jt
0 < yi,t p?i,t ? ?, i.e, there is no
mistake but the margin is small,
end for
and Ai,t = 0 otherwise. An upOutput: wi,n for i = 1, . . . , K.
date is performed if either there
Figure 2: SHAMPO: SHared Annotator for Multiple PrOblems. is a mistake (MJi ,t = 0) or the
margin is low (AJi ,t = 1). Note
that these events are mutually exclusive. For simplicity of presentation we write this update as,
wi,t = wi,t?1 + Zi,t (Ai,t + Mi,t )yi,t xi,t . Although this notation uses labels for all-tasks, only the
label of the task Jt is used in practice, as for other tasks Zi,t = 0.
m=1
We call this algorithm SHAMPO for SHared Annotator for Multiple PrOblems. The pseudo-code
appears in Fig. 2. We conclude this section by noting that the algorithm can be incorporated with
Mercer-kernels as all operations depend implicitly on inner-product between inputs.
4
Analysis
The following theorem states that the expected cumulative number of mistakes that the algorithm
makes, may not be higher than the algorithm that observes the labels of all inputs.
Theorem 1 If SHAMPO algorithm runs on K tasks with K parallel example pair sequences
(xi,1 , yi,1 ), ...(xi,n , yi,n ) ? Rd ? {?1, 1}, i = 1, ..., K with input parameters 0 ? b, 0 ? ? ? b/2,
and prior 1 ? ai ?i, denote by X = maxi,t kxi,t k, then, for all ? > 0, all ui ? Rd and all n ? 1
3
PK
there exists 0 < ? ? i=1 ai , such that,
"K n
"
#
#
#
2
"X
K X
n
XX
2b + X 2 U 2
X2 ?
?
?
E
1+
L?,n +
+ 2 ?1 E
ai Ai,t .
Mi,t ?
?
2b
8?b
b
i=1 t=1
i=1 t=1
where we denote U 2 =
PK
i=1
2
kui k . The expectation is over the random choices of the algorithm.
Due to lack of space, the proof appears in Appendix A.1 in the supplementary material. Few notes
on the mistake bound: First, the right term of the bound is equals zero either when ? = 0 (as
Ai,t = 0) or ? = b/2. Any value in between, may yield an strict negative value of this term, which
? ?,n is non-increasing with the number of
in turn, results in a lower bound. Second, the quantity L
P
tasks. The first terms depends on the number of tasks only via ? ? i ai . Thus, if ai = 1 (uniform
prior) the quantity ? ? K is bounded by the number of tasks. Yet, when the hardness of the tasks is
not equal or balanced, one may expect ? to be closer to 1 than K, which we found empirically to be
true. Additionally, the prior ai can be used to make the algorithm focus on the hard tasks, thereby
improving the bound. While ? multiplying the first term can be larger, the second term can be lower.
A task i which corresponds to a large value of ai will be updated more in early rounds than tasks
with low ai . If more of these updates are aggressive, the second term will be negative and far from
zero.
One can use the bound to tune the algorithm for a good value of b for the non aggressive case
? ?,n de(? = 0), by minimizing the bound over b. This may not be possible directly since L
1
?
pends implicitly on the value of b . Alternatively, we can take a loose estimate of L?,n , and replace it with
q L?,n (which is ? K times larger). The optimal value of b can now be calculated,
b =
X2
2
1+
4?L?,n
U 2X2 .
lowing bound, E
Substituting this value in the bound of Eq. (1) with L?,n leads to the fol
q
i
4?L?,n
K Pn
U 2X2
U2
?
1 + U 2 X 2 , which has the same
i=1
t=1 Mi,t ? ? L?,n + 2? + 2?
hP
dependency in the number of inputs n as algorithm that observes all of them.
We conclude this section by noting that the algorithm and analysis can be extended to the case that
more than single query is allowed per task. Analysis and proof appears in Appendix A.2 in the
supplementary material.
5
From Multi-task to Contextual Bandits
Although our algorithm is designed for many binary-classification tasks, it can also be applied in
two settings of contextual bandits, when decoupling exploration and exploitation is allowed [23, 3].
In this setting, the goal is to predict a label Y?t ? {1, . . . , C} given an input xt . As before, the
algorithm works in rounds. On round t the algorithm receives an input xt and gives as an output
multicalss label Y?t ? {1, . . . , C}. Then, it queries for some information about the label via a single
binary ?yes-no? question, and uses the feedback to update its model. We consider two forms of
questions. Note that our algorithm subsumes past methods since they also allow the introduction of
a bias (or prior knowledge) towards some tasks, which in turn, may improve performance.
5.1 One-vs-Rest
The first setting is termed one-vs-rest. The algorithm asks if the true label is some label Y?t ?
{1, . . . , C}, possibly not the predicted label, i.e. it may be the case that Y?t 6= Y?t . Given the response
whether Y?t is the true label Yt , the algorithm updates its models. The reduction we perform is by
introducing K tasks, one per class. The problem of the learning algorithm for task i is to decide
whether the true label is class i or not. Given the output of all (binary) classifiers, the algorithm
generates a single multi-class prediction to be the single label for which the output of the corresponding binary classifier is positive. If such class does not exist, or there are more than one classes
as such, a random prediction is used, i.e., given an input xt we define Y?t = arg maxi y?i,t , where ties
are broken arbitrarily. The label to be queried is Y?t = Jt , i.e. the problem index that SHAMPO is
querying. We analyze the performance of this reduction as a multiclass prediction algorithm.
1
Similar issue appears also after the discussion of Theorem 1 in a different context [7].
4
Exploit
Uniform
SHAMPO
12
8
6
16
14
8
6
12
10
8
4
2
Aggressive ? ?=b
Aggrssive ? ?=b +prior
plain
18
10
Error
Error
Exploit
Uniform
SHAMPO
14
Error [%]
10
4
6
2
4
2
0
1
2
3
4
Task #
(a) 4 text classification tasks
0
1
2
3
4
5
6
7
8
?5
0
10
10
b [log]
Task #
(b) 8 text classification tasks
5
10
(c) Error vs. b
Figure 3: Left and middle: Test error of aggressive SHAMPO on (a) four and (b) eight binary text
classification tasks. Three algorithms are evaluated: uniform, exploit, and aggressive SHAMPO.
(Right) Mean test error over USPS One-vs-One binary problems vs b of aggressive SHAMPO with
prior, aggressive with uniform prior, and non-aggressive with uniform prior.
Corollary 2 Assume the SHAMPO algorithm is executed as above with K = C one-vs-rest
problems, on a sequence (x1 , Y1 ), ...(xn , Yn ) ? Rd ? {1, ..., C}, and input parameter b > 0
PC
and prior 1 ? ai ?i. Then for all ? > 0 and all ui ? Rd , there exist 0 h< ? ?
i=1
i ai
P
?
such that the expected number of multi-class errors is bounded as follows E
t [[Yt 6= Yt ]] ?
hP
i
2 2 2
P
2
2b+X
U
)
K
n
?
? ?,n + (
1 + X2b L
+ 2 ?b ? 1 E
i=1
t=1 ai Ai,t , where [[I]] = 1 if the pred?
8?b
icate I is true, and zero otherwise.
P
The corollary follows directly from Thm. 1 by noting that, [[Yt 6= Y?t ]] ? i Mi,t . That is, there is a
multiclass mistake if there is at least one prediction mistake of one of the one-vs-rest problems. The
closest setting is contextual bandits, yet we allow decoupling of exploration and exploitation. Ignor2/3
ing this decoupling, the Banditron algorithm [17]
? is the closest to ours, with a regret of O(T ).
designed for the log loss, with coeffiHazan et al [16] proposed an algorithm with O( T ) regret but?
cient that may be very large, and another [9] algorithm has O( T ) regret with respect to prediction
mistakes, yet they assumed stochastic labeling, rather than adversarial.
5.2 One-vs-One
In the second setting, termed by one-vs-one, the algorithm picks two labels Y?t+ , Y?t? ? {1 . . . C},
possibly both not the predicted label. The feedback for the learner is three-fold: it is yJt ,t = +1 if
the first alternative is the correct label, Y?t+ = Yt , yJt ,t = ?1 if the second alternative is the correct
label, Y?t? = Yt , and it is yJt ,t = 0 otherwise (in this case there is no error and we set MJt ,t = 0).
The reduction we perform is by introducing K = C2 problems, one per pair of classes. The goal
of the learning algorithm for a problem indexed with two labels (y1 , y2 ) is to decide which is the
correct label, given it is one of the two. Given the output of all (binary) classifiers the algorithm
generates a single multi-class prediction using a tournament in a round-robin approach [15]. If there
is no clear winner, a random prediction is used. We now analyze the performance of this reduction
as a multiclass prediction algorithm.
Corollary 3 Assume the SHAMPO algorithm is executed as above, with K = C2 one-vs-one
problems, on a sequence (x1 , Y1 ), ...(xn , Yn ) ? Rd ? {1, ..., C}, and input parameter b > 0 and
P(C2 )
prior 1 ? ai ?i . Then for all ? > 0 and all ui ? Rd , there exist 0 < ?h ?
i=1 ai such
i
P
?
that the expected number of multi-class errors can be bounded as follows E
[[Y
t t 6= Yt ]] ?
i
2
hPK Pn
(2b+X 2 ) U 2
?
X2
?
2
?
1 + 2b L?,n +
+ 2b ?1 E
.
i=1
t=1 ai Ai,t
?
8?b
((C
2 )?1)/2+1
The corollary follows directly from Thm. 1 by noting that, [[Yt 6= Y?t ]] ?
2
((C
2 )?1)/2+1
P(C2 )
i=1
Mi,t .
Note, that the bound is essentially independent of C as the coefficient in the bound is upper bounded
by 6 for C ? 3.
5
We conclude this section with two algorithmic modifications, we employed in this setting. Currently, when the feedback is zero, there is no update of the weights, because there are no errors. This causes the algorithm to effectively ignore such examples, as in these cases the algorithm is not modifying any model, furthermore, if such example is repeated, a problem with possibly ?0? feedback may be queried again. We fix this issue with one of two modifications: In
the first one, if the feedback is zero, we modify the model to reduce the chance that the chosen problem, Jt , would be chosen again for the same input (i.e. not to make the same wrongchoice of choosing irrelevant problem again). To this end, we modify the weights a bit, to increase the confidence (absolute margin) of the model for the same input, and replace Eq. (2)
with, wJt ,t = wJt ,t?1 + [[yJt ,t 6= 0]] yJt ,t xJt ,t + [[yJt ,t = 0]]? y?Jt ,t xJt ,t , for some ? > 0.
In other words, if there is a possible error (i.e. yJt ,t 6= 0) the update follows the Perceptron?s rule. Otherwise, the weights are updated such that the absolute margin will increase, as
|wJ>t ,t xJt ,t | = |(wJt ,t?1 + ? y?Jt ,t xJt ,t )> xJt ,t | = |wJ>t ,t?1 xJt ,t + ?sign(wJ>t ,t?1 xJt ,t )kxJt ,t k2 | =
|wJ>t ,t?1 xJt ,t | + ?kxJt ,t k2 > |wJ>t ,t?1 xJt ,t |. We call this method one-vs-one-weak, as it performs
weak updates for zero feedback. The second alternative is not to allow 0 value feedback, and if this
is the case, to set the label to be either +1 or ?1, randomly. We call this method one-vs-one-random.
6
Experiments
We evaluated the SHAMPO algorithm
using four datasets: USPS, MNIST
(both OCR), Vocal Joystick (VJ, vowel
recognition) and document classification. The USPS dataset, contains 7, 291
training examples and 2, 007 test examples, each is a 16 ? 16 pixels gray-scale
images converted to a 256 dimensional
vector. The MNIST dataset with 28 ?
(a) Training mistakes vs b (b) Test error vs no.
of
28 gray-scale images, contains 60, 000
queries
(10, 000) training (test) examples. In
both cases there are 10 possible labels,
Figure 4: Left: mean of fraction no. of mistakes digits. The VJ tasks is to predict a vowel
SHAMPO made during training time on MNIST of all from eight possible vowels. Each examexamples and only queried. Right: test error vs no. of ple is a frame of spoken value described
queries is plotted for all MNIST one-vs-one problems.
with 13 MFCC coefficients transformed
into 27 features. There are 572, 911
training examples and 236, 680 test examples. We created binary tasks from these multi-class
datasets using two reductions: One-vs-Rest setting and One-vs-One setting. For example, in both
USPS and MNIST there are 10 binary one-vs-rest tasks and 45 binary one-vs-one tasks. The NLP
document classification include of spam filtering, news items and news-group classification, sentiment classification, and product domain categorization. A total of 31 binary prediction tasks over
all, with a total of 252, 609 examples, and input dimension varying between 8, 768 and 1, 447, 866.
Details of the individual binary tasks can be found elsewhere [8]. We created an eighth collection,
named MIXED, which consists of 40 tasks: 10 random tasks from each one of the four basic datasets
(one-vs-one versions). This yielded eight collections (USPS, MNIST and VJ; each as one-vs-rest or
one-vs-one), document classification and mixed. From each of these eight collections we generated
between 6 to 10 combinations (or problems), each problem was created by sampling between 2 and
8 tasks which yielded a total of 64 multi-task problems. We tried to diversify problems difficulty by
including both hard and easy binary classification problems. The hardness of a binary problem is
evaluated by the number of mistakes the Perceptron algorithm performs on the problem.
total
queried
35
Error [%]
30
25
20
15
10
?6
10
?4
10
?2
10
b [log]
0
10
2
10
We evaluated two baselines as well as our algorithm. Algorithm uniform picks a random task to
be queried and updated (corresponding to b ? ?), exploit which picks the tasks with the lowest
absolute margin (i.e. the ?hardest instance?), this combination corresponds to b ? 0 of SHAMPO.
We tried for SHAMPO 13 values for b, equally spaced on a logarithmic scale. All algorithms made
a single pass over the training data. We ran two version of the algorithm: plain version, without
aggressiveness (updates on mistakes only, ? = 0) and an Aggressive version ? = b/2 (we tried
lower values of ? as in the bound, but we found that ? = b/2 gives best results), both with uniform
prior (ai = 1). We used separate training set and a test set, to build a model and evaluate it.
6
Table 1: Test errors percentage . Scores are shown in parenthesis.
Dataset
VJ 1 vs 1
VJ 1 vs Rest
USPS 1 vs 1
USPS 1 vs Rest
MNIST 1 vs 1
MNIST 1 vs Rest
NLP documents
MIXED
Mean score
Aggressive ? = b/2
exploit
SHAMPO
uniform
5.22 (2.9)
4.57 (1.1)
5.67 (3.9)
13.26 (3.5) 11.73 (1.2) 12.43 (2.5)
3.31 (2.5)
2.73 (1.0) 19.29 (6.0)
5.45 (2.8)
4.93 (1.2) 10.12 (6.0)
1.08 (2.3)
0.75 (1.0)
5.9 (6.0)
4.74 (2.8)
3.88 (1.0) 10.01 (6.0)
19.43 (2.3)
16.5 (1.0) 23.21 (5.0)
2.75 (2.4)
2.06 (1.0) 13.59 (6.0)
(2.7)
(1.1)
(5.2)
exploit
5.21 (2.7)
13.11 (3.0)
3.37 (2.5)
5.31 (2.0)
1.2 (2.7)
4.44 (2.8)
19.46 (2.7)
2.78 (2.6)
(2.6)
Plain
SHAMPO
6.93 (4.6)
14.17 (5.0)
4.83 (4.0)
6.51 (4.0)
1.69 (4.1)
5.4 (3.8)
21.54 (4.7)
4.2 (4.3)
(4.3)
uniform
6.26 (5.8)
14.71 (5.8)
5.33 (5,0)
7.06 (5.0)
1.94 (4.9)
6.1 (5.0)
21.74 (5.3)
4.45 (4.7)
(5.2)
Results are evaluated using 2 quantities. First, the average test error (over all the dataset combinations) and the average score. For each combination we assigned a score of 1 to the algorithm with
the lowest test error, and a score of 2, to the second best, and all the way up to a score of 6 to the
algorithm with the highest test error.
Multi-task Binary Classification : Fig. 3(a) and Fig. 3(b) show the test error of the three algorithms on two of document classification combinations, with four and eight tasks. Clearly, not only
SHAMPO performs better, but it does so on each task individually. (Our analysis above bounds
the total number of mistakes over all tasks.) Fig. 3(c) shows the average test error vs b using the
one-vs-one binary USPS problems for the three variants of SHAMPO: non-aggressive (called plain),
aggressive and aggressive with prior.Clearly, the plain version does worse than both the aggressive
version and the non-uniform prior version. For other combinations the prior was not always improving results. We hypothesise that this is because our heuristic may yield a bad prior which is not
focusing the algorithm on the right (hard) tasks.
Results are summarized in Table 1. In general exploit is better than uniform and aggressive is
better than non-aggressive. Aggressive SHAMPO yields the best results both evaluated as average
(over tasks per combination and over combinations). Remarkably, even in the mixed dataset (where
tasks are of different nature: images, audio and documents), the aggressive SHAPO improves over
uniform (4.45% error) and the aggressive-exploit baseline (2.75%), and achieves a test error of
2.06%.
Next, we focus on the problems that the algorithm chooses to annotate on each iteration for various
values of b. Fig. 4(a) shows the total number of mistakes SHAMPO made during training time on
MNIST , we show two quantities: fraction of mistakes over all training examples (denoted by ?total?
- blue) and fraction of mistakes over only queried examples (denoted by ?queried? - dashed red).
In pure exploration (large b) both quantities are the same, as the choice of problem to be labeled
is independent of the problem and example, and essentially the fraction of mistakes in queried
examples is a good estimate of the fraction of mistakes over all examples. The other extreme is
when performing pure exploitation (low b), here the fraction of mistakes made on queried examples
went up, while the overall fraction of mistakes went down. This indicates that the algorithm indeed
focuses its queries on the harder inputs, which in turn, improves overall training mistake. There is a
sweet point b ? 0.01 for which SHAMPO is still focusing on the harder examples, yet reduces the
total fraction of training mistakes even more. The existence of such tradeoff is predicted by Thm. 1.
Another perspective of the phenomena is that for values of b ? SHAMPO focuses on the harder
examples, is illustrated in Fig. 4(b) where test error vs number of queries is plotted for each problem
for MNIST. We show three cases: uniform, exploit and a mid-value of b ? 0.01 which tradeoffs
exploration and exploitation. Few comments: First, when performing uniform querying, all problems have about the same number of queries (266), close to the number of examples per problem
(12, 000), divided by the number of problems (45). Second, when having a tradeoff between exploration and exploitation, harder problems (as indicated by test error) get more queries than easier
problems. For example, the four problems with test error greater than 6% get at least 400 queries,
which is about twice the number of queries received by each of the 12 problems with test error less
than 1%. Third, as a consequence, SHAMPO performs equalization, giving the harder problems
more labeled data, and as a consequence, reduces the error of these problems, however, is not increasing the error of the easier problems which gets less queries (in fact it reduces the test error of
all 45 problems!). The tradeoff mechanism of SHAMPO, reduces the test error of each problem
7
by more than 40% compared to full exploration. Fourth, exploits performs similar equalization, yet
in some hard tasks it performs worse than SHAMPO. This could be because it overfits the training
data, by focusing on hard-examples too much, as SHAMPO has a randomness mechanism.
Indeed, Table 1 shows that aggressive SHAMPO outperforms better alternatives. Yet, we claim
that a good prior may improve results. We compute prior over the 45 USPS tasks, by running the
perceptron algorithm on 1000 examples and computing the number of mistakes. We set the prior
to be proportional to this number. We then reran aggressive SHAMPO with prior, comparing it to
aggressive SHAMPO with no prior (i.e. ai = 1). Aggressive SHAMO with prior achieves average
error of 1.47 (vs. 2.73 with no prior) on 1-vs-1 USPS and 4.97 (vs 4.93) on one-vs-rest USPS, with
score rank of 1.0 (vs 2.9) and 1.7 (vs 2.0) respectively. Fig. 3(c) shows the test error for a all values
of b we evaluated. A good prior is shown to outperform the case ai = 1 for all values of b.
Reduction of Multi-task to Contextual Bandits Next, we evaluated SHAMPO as a contextual
bandit algorithm, by breaking a multi-class problem into few binary tasks, and integrating their
output into a single multi-class problem. We focus on the VJ data, as there are many examples,
and linear models perform relatively well [18]. We implemented all three reductions mentioned in
Sec. 5.2, namely, one-vs-rest, one-vs-one-random which picks a random label if the feedback is zero,
one-vs-one-weak (which performs updates to increase confidence when the feedback is zero), where
we set ? = 0.2, and the Banditron algorithm [17]. The one-vs-rest reduction and the Banditron
have a test error of about 43.5%, and the one-vs-one-random of about 42.5%. Finally, one-vs-oneweak achieves an error of 39.4%. This is slightly worst than PLM [18] with test error of 38.4%
(and higher than MLP with 32.8%), yet all of these algorithms observe only one bit of feedback per
example, while both MLP and PLM observe 3 bits (as class identity can be coded with 3 bits for
8 classes). We claim that our setting can be easily used to adapt a system to individual user, as we
only need to assume the ability to recognise three words, such as three letters. Given an utterance of
the user, the system may ask: ?Did you say (a) ?a? like ?bad? (b) ?o? like in ?book?) (c) none?. The
user can communicate the correct answer with no need for a another person to key in the answer.
7
Related Work and Conclusion
In the past few years there is a large volume of work on multi-task learning, which clearly we can not
cover here. The reader is referred to a recent survey on the topic [20]. Most of this work is focused
on exploring relations between tasks, that is, find similarities dissimilarities between tasks, and use it
to share data directly (e.g. [10]) or model parameters [14, 11, 2]. In the online settings there are only
a handful of work on multi-task learning. Dekel et al [13] consider the setting where all algorithms
are evaluated using a global loss function, and all work towards the shared goal of minimizing it.
Logosi et al [19] assume that there are constraints on the predictions of all learners, and focus in
the expert setting. Agarwal et al [1] formalize the problem in the framework of stochastic convex
programming with few matrix regularization, each captures some assumption about the relation
between the models. Cavallanti et al [4] and Cesa-Bianci et al [6] assume a known relation between
tasks which is exploited during learning. Unlike these approaches, we assume the ability to share an
annotator rather than data or parameters, thus our methods can be applied to problems that do not
share a common input space.
Our analysis is similar to that of Cesa-Bianchi et al [7], yet they focus in selective sampling (see
also [5, 12]), that is, making individual binary decisions of whether to query, while our algorithm
always query, and needs to decide for which task. Finally, there have been recent work in contextual
bandits [17, 16, 9], each with slightly different assumptions. To the best of our knowledge, we are the
first to consider decoupled exploration and exploitation in this context. Finally, there is recent work
in learning with relative or preference feedback in various settings [24, 25, 26, 21]. Unlike this work,
our work allows again decoupled exploitation and exploration, and also non-relevant feedback.
To conclude, we proposed a new framework for online multi-task learning, where learners share a
single annotator. We presented an algorithm (SHAMPO) that works in this settings and analyzed it
in the mistake-bound model. We also showed how learning in such a model can be used to learn
in contextual-bandits setting with few types of feedback. Empirical results show that our algorithm
does better for the same price. It focuses the annotator on the harder instances, and is improving
performance in various tasks and settings. We plan to integrate other algorithms to our framework,
extend it to other settings, investigate ways to generate good priors, and reduce multi-class to binary
also via error-correcting output-codes.
8
References
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categorization. J. Mach. Learn. Res., 98888:1891?1926, June 2012.
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447. ACM, 2008.
[18] Hui Lin, Jeff Bilmes, and Koby Crammer. How to lose confidence: Probabilistic linear machines for
multiclass classification. In Tenth Annual Conference of the International Speech Communication Association, 2009.
[19] G?abor Lugosi, Omiros Papaspiliopoulos, and Gilles Stoltz. Online multi-task learning with hard constraints. In COLT, 2009.
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Data Engineering, 22(10):1345?1359, 2010.
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9
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5,114 | 563 | A Simple Weight Decay Can Improve
Generalization
Anders Krogh?
The Niels Bohr Institute
Blegdamsvej 17
DK-2100 Copenhagen, Denmark
[email protected]
CONNECT,
John A. Hertz
Nordita
Blegdamsvej 17
DK-2100 Copenhagen, Denmark
[email protected]
Abstract
It has been observed in numerical simulations that a weight decay can improve generalization in a feed-forward neural network. This paper explains
why. It is proven that a weight decay has two effects in a linear network.
First, it suppresses any irrelevant components of the weight vector by
choosing the smallest vector that solves the learning problem. Second, if
the size is chosen right, a weight decay can suppress some of the effects of
static noise on the targets, which improves generalization quite a lot. It
is then shown how to extend these results to networks with hidden layers
and non-linear units. Finally the theory is confirmed by some numerical
simulations using the data from NetTalk.
1
INTRODUCTION
Many recent studies have shown that the generalization ability of a neural network
(or any other 'learning machine') depends on a balance between the information in
the training examples and the complexity of the network, see for instance [1,2,3].
Bad generalization occurs if the information does not match the complexity, e.g.
if the network is very complex and there is little information in the training set.
In this last instance the network will be over-fitting the data, and the opposite
situation corresponds to under-fitting.
?Present address: Computer and Information Sciences, Univ. of California Santa Cruz,
Santa Cruz, CA 95064.
950
A Simple Weight Decay Can Improve Generalization
Often the number of free parameters, i. e. the number of weights and thresholds, is
used as a measure of the network complexity, and algorithms have been developed,
which minimizes the number of weights while still keeping the error on the training
examples small [4,5,6]. This minimization of the number of free parameters is not
always what is needed.
A different way to constrain a network, and thus decrease its complexity, is to limit
the growth of the weights through some kind of weight decay. It should prevent the
weights from growing too large unless it is really necessary. It can be realized by
adding a term to the cost function that penalizes large weights,
E(w) = Eo(w)
1 ""
2
+ 2A
L..J Wi'
(1)
i
where Eo is one's favorite error measure (usually the sum of squared errors), and
A is a parameter governing how strongly large weights are penalized. w is a vector
containing all free parameters of the network, it will be called the weight vector. If
gradient descend is used for learning, the last term in the cost function leads to a
new term -AWi in the weight update:
.
Wi
fJEo
\
ex: --fJ - ",Wi?
(2)
Wi
Here it is formulated in continuous time. If the gradient of Eo (the 'force term')
were not present this equation would lead to an exponential decay of the weights.
Obviously there are infinitely many possibilities for choosing other forms of the
additional term in (1), but here we will concentrate on this simple form.
It has been known for a long time that a weight decay of this form can improve
generalization [7], but until now not very widely recognized. The aim of this paper
is to analyze this effect both theoretically and experimentally. Weight decay as a
special kind of regularization is also discussed in [8,9] .
2
FEED-FORWARD NETWORKS
A feed-forward neural network implements a function of the inputs that depends
on the weight vector w, it is called fw. For simplicity it is assumed that there is
only one output unit. When the input is the output is fw (e) . Note that the input
vector is a vector in the N-dimensional input space, whereas the weight vector is a
vector in the weight space which has a different dimension W.
e
The aim of the learning is not only to learn the examples, but to learn the underlying
function that produces the targets for the learning process. First, we assume that
this target function can actually be implemented by the network . This means there
exists a weight vector u such that the target function is equal to fu . The network
with parameters u is often called the teacher, because from input vectors it can
produce the right targets . The sum of squared errors is
p
Eo(w) =
~ 2:[fu(e Jl ) JI=l
fw(eJl)]2,
(3)
951
952
Krogh and Hertz
where p is the number of training patterns. The learning equation (2) can then be
written
Wi
<X
2)fu(e Jl ) Jl
fw(eJl)]&~:~)
- AWj.
(4)
?
Now the idea is to expand this around the solution u, but first the linear case will
be analyzed in some detail.
3
THE LINEAR PERCEPTRON
The simplest kind of 'network' is the linear perceptron characterized by
(5)
where the N-l/2 is just a convenient normalization factor. Here the dimension of
the weight space (W) is the same as the dimension of the input space (N) .
The learning equation then takes the simple form
Wi
<X
(6)
L N- 1 L[Uj - Wj]ejer - AWi.
Jl
j
Defining
(7)
and
(8)
Aij = N- 1 L ere;
Jl
it becomes
Vj
<X -
L AijVj
+ A(Uj - Vi)'
(9)
j
Transforming this equation to the basis where A is diagonal yields
vr
<X
-(Ar
+ A)Vr + AUr,
(10)
where Ar are the eigenvalues of A, and a subscript r indicates transformation to this
basis. The generalization error is defined as the error averaged over the distribution
of input vectors
N- 1
L VjVj(eiej)?
ij
(11)
Here it is assumed that (eiej)? = 6ij . The generalization error F is thus proportional
to Iv1 2 , which is also quite natural.
The eigenvalues of the covariance matrix A are non-negative, and its rank can easily
be shown to be less than or equal to p. It is also easily seen that all eigenvectors
belonging to eigenvalues larger than 0 lies in the subspace of weight space spanned
A Simple Weight Decay Can Improve Generalization
e, ... ,e.
by the input patterns
This subspace, called the pattern subspace, will
be denoted Vp , and the orthogonal subspace is denoted by V.l. When there are
sufficiently many examples they span the whole space, and there will be no zero
eigenvalues. This can only happen for p 2:: N.
When A = 0 the solution to (10) inside Vp is just a simple exponential decay to
Vr = o. Outside the pattern subspace Ar = 0, and the corresponding part of Vr will
be constant. Any weight vector which has the same projection onto the pattern
subspace as u gives a learning error O. One can think of this as a 'valley' in the
error surface given by u +
V/.
The training set contains no information that can help us choose between all these
solutions to the learning problem. When learning with a weight decay A > 0, the
constant part in
will decay to zero asymptotically (as e->'t, where t is the time).
An infinitesimal weight decay will therefore choose the solution with the smallest
norm out of all the solutions in the valley described above. This solution can be
shown to be the optimal one on average.
V/
4
LEARNING WITH AN UNRELIABLE TEACHER
Random errors made by the teacher can be modeled by adding a random term 11 to
the targets:
(12)
The variance of TJ is called u 2 , and it is assumed to have zero mean. Note that these
targets are not exactly realizable by the network (for Q' > 0), and therefore this is
a simple model for studying learning of an unrealizable function.
With this noise the learning equation (2) becomes
Wi ex L:(N- 1
/J
L: Vjf.j + N-
1 / 2 11/J)f.f
-
AWi?
(13)
j
Transforming it to the basis where A is diagonal as before,
vr
ex
-(Ar
+ A)Vr + AUr -
N- 1/ 2
L 11/Jf.~?
(14)
/J
The asymptotic solution to this equation is
AU r - N-l/ 2 L/J TJ/Jf.~
Vr =
A + Ar
.
(15)
The contribution to the generalization error is the square of this summed over all
r. If averaged over the noise (shown by the bar) it becomes for each r
(16)
The last expression has a minimum in A, which can be found by putting the derivative with respect to A equal to zero, A~Ptimal u 2 /u;. Remarkably it depends only
=
953
954
Krogh and Hertz
Figure 1: Generalization error as a
function of Q' = pIN. The full line is
for A = u 2 = 0.2, and the dashed line
for A = O. The dotted line is the generalization error with no noise and A = O.
LI..
o~
__________., __________
~
o
1
pIN
~
2
on u and the variance of the noise, and not on A. If it is assumed that u is random
(16) can be averaged over u. This yields an optimal A independent of r,
u2
Aoptlmai
(17)
= ---;;-,
u~
where u 2 is the average of N- 1 IuI 2 .
In this case the weight decay to some extent prevents the network from fitting the
nOIse.
From equation (14) one can see that the noise is projected onto the pattern subspace.
Therefore the contribution to the generalization error from
is the same as before,
and this contribution is on average minimized by a weight decay of any size.
V/
Equation (17) was derived in [10] in the context of a particular eigenvalue spectrum.
Figure fig. 1 shows the dramatic improvement in generalization error when the
optimal weight decay is used in this case, The present treatment shows that (17)
is independent of the spectrum of A.
We conclude that a weight decay has two positive effects on generalization in a
linear network: 1) It suppresses any irrelevant components of the weight vector by
choosing the smallest vector that solves the learning problem. 2) If the size is chosen
right, it can suppress some of the effect of static noise on the targets .
5
NON-LINEAR NETWORKS
It is not possible to analyze a general non-linear network exactly, as done above
for the linear case. By a local linearization, it is however, possible to draw some
interesting conclusions from the results in the previous section.
Assume the function is realizable, f = fu. Then learning corresponds to solving the
p equations
(18)
A Simple Weight Decay Can Improve Generalization
in W variables, where W is the number of weights. For p < W these equations
define a manifold in weight space of dimension at least W - p. Any point W on this
manifold gives a learning error of zero, and therefore (4) can be expanded around
w. Putting v W - w, expanding fw in v, and using it in (4) yields
=
Vi
ex
-
L
(8f;~:Jj?) v/9f;~:Jj) + A(Wi - vd
Jj ,1
- LAij(W)Vj - AVi
+ AWj
(19)
j
(The derivatives in this equation should be taken at iV.)
The analogue of A is defined as
A'1??( w-) -=
L
Jj
8fw(eJj)
8fw(eJj)
~
uW',
:::l
uW'
1
?
(20)
Since it is of outer product form (like A) its rank R( in) ~ min{p, W}. Thus when
p < W, A is never of full rank. The rank of A is of course equal to W minus the
dimension of the manifold mentioned above.
From these simple observations one can argue that good generalization should not
be expected for p < W. This is in accordance with other results (cf. [3]), and with
current 'folk-lore'. The difference from the linear case is that the 'rain gutter' need
not be (and most probably is not) linear, but curved in this case. There may in fact
be other valleys or rain gutters disconnected from the one containing u. One can
also see that if A has full rank, all points in the immediate neighborhood of W = u
give a learning error larger than 0, i.e. there is a simple minimum at u.
Assume that the learning finds one of these valleys. A small weight decay will
pick out the point in the valley with the smallest norm among all the points in the
valley. In general it can not be proven that picking that solution is the best strategy.
But, at least from a philosophical point of view, it seems sensible, because it is (in a
loose sense) the solution with the smallest complexity-the one that Ockham would
probably have chosen.
The value of a weight decay is more evident if there are small errors in the targets.
In that case one can go through exactly the same line of arguments as for the linear
case to show that a weight decay can improve generalization, and even with the
same optimal choice (17) of A. This is strictly true only for small errors (where the
linear approximation is valid).
6
NUMERICAL EXPERIMENTS
A weight decay has been tested on the NetTalk problem [11]. In the simulations
back-propagation derived from the 'entropic error measure' [12] with a momentum
term fixed at 0.8 was used. The network had 7 x 26 input units, 40 hidden units and
26 output units. In all about 8400 weights. It was trained on 400 to 5000 random
words from the data base of around 20.000 words, and tested on a different set of
1000 random words. The training set and test set were independent from run to
run .
955
956
Krogh and Hertz
1.2
0.26
0.24
0.22
....0
.... 0.20
....
f/)
1.0
w
0.18
0.16
lL.
0.14
0
0.8
2 104
P
0
4 104
0
..
0.6
. --
o
Figure 2: The top full line corresponds to the generalization error after 300 epochs
(300 cycles through the training set) without a weight decay. The lower full line is
with a weight decay. The top dotted line is the lowest error seen during learning
without a weight decay, and the lower dotted with a weight decay. The size of the
weight decay was .A = 0.00008.
Insert : Same figure except that the error rate is shown instead of the squared error.
The error rate is the fraction of wrong phonemes when the phoneme vector with
the smallest angle to the actual output is chosen, see [11].
Results are shown in fig. 2. There is a clear improvement in generalization error
when weight decay is used. There is also an improvement in error rate (insert of
fig. 2), but it is less pronounced in terms of relative improvement. Results shown
here are for a weight decay of .A = 0.00008. The values 0.00005 and 0.0001 was also
tried and gave basically the same curves.
7
CONCLUSION
It was shown how a weight decay can improve generalization in two ways: 1) It
suppresses any irrelevant components of the weight vector by choosing the smallest
vector that solves the learning problem. 2) If the size is chosen right, a weight decay
can suppress some of the effect of static noise on the targets. Static noise on the
targets can be viewed as a model of learning an unrealizable function. The analysis
assumed that the network could be expanded around an optimal weight vector, and
A Simple Weight Decay Can Improve Generalization
therefore it is strictly only valid in a little neighborhood around that vector.
The improvement from a weight decay was also tested by simulations. For the
NetTalk data it was shown that a weight decay can decrease the generalization
error (squared error) and also, although less significantly, the actual mistake rate
of the network when the phoneme closest to the output is chosen.
Acknowledgements
AK acknowledges support from the Danish Natural Science Council and the Danish
Technical Research Council through the Computational Neural Network Center
(CONNECT).
References
[1] D.B. Schwartz, V.K. Samalam, S.A. Solla, and J.S. Denker. Exhaustive learning. Neural Computation, 2:371-382, 1990.
[2] N. Tishby, E. Levin, and S.A. Solla. Consistent inference of probabilities in
layered networks: predictions and generalization. In International Joint Conference on Neural Networks, pages 403-410, (Washington 1989), IEEE, New
York, 1989.
[3] E.B. Baum and D. Haussler. What size net gives valid generalization? Neural
Computation, 1:151-160, 1989.
[4] Y. Le Cun, J .S. Denker, and S.A. Solla. Optimal brain damage. In D.S. Touretzky, editor, Advances in Neural Information Processing Systems, pages 598605, (Denver 1989), Morgan Kaufmann, San Mateo, 1990.
[5] H.H. Thodberg. Improving generalization of neural networks through pruning.
International Journal of Neural Systems, 1:317-326, 1990.
[6] D.H. Weigend, D.E. Rumelhart, and B.A. Huberman. Generalization by
weight-elimination with application to forecasting. In R.P. Lippmann et ai,
editors, Advances in Neural Information Processing Systems, page 875-882,
(Denver 1989), Morgan Kaufmann, San Mateo, 1991.
[7] G.E. Hinton. Learning translation invariant recognition in a massively parallel
network. In G. Goos and J. Hartmanis, editors, PARLE: Parallel Architectures and Languages Europe. Lecture Notes in Computer Science, pages 1-13,
Springer-Verlag, Berlin, 1987.
[8] J .Moody. Generalization, weight decay, and architecture selection for nonlinear learning systems. These proceedings.
[9] D. MacKay. A practical bayesian framework for backprop networks. These
proceedings.
[10] A. Krogh and J .A. Hertz. Generalization in a Linear Perceptron in the Presence
of Noise. To appear in Journal of Physics A 1992.
[11] T.J. Sejnowski and C.R. Rosenberg. Parallel networks that learn to pronounce
english text . Complex Systems, 1:145-168,1987.
[12] J .A. Hertz, A. Krogh, and R .G. Palmer. Introduction to the Theory of Neural
Computation. Addison-Wesley, Redwood City, 1991.
957
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5,115 | 5,630 | Multivariate Regression with Calibration?
Han Liu
Department of Operations Research and Financial Engineering
Princeton University
Lie Wang
Department of Mathematics
Massachusetts Institute of Technology
Tuo Zhao?
Department of Computer Science
Johns Hopkins University
Abstract
We propose a new method named calibrated multivariate regression (CMR) for
fitting high dimensional multivariate regression models. Compared to existing
methods, CMR calibrates the regularization for each regression task with respect
to its noise level so that it is simultaneously tuning insensitive and achieves an
improved finite-sample performance. Computationally, we develop an efficient
smoothed proximal gradient algorithm which has a worst-case iteration complexity O(1/?), where ? is a pre-specified numerical accuracy. Theoretically, we prove
that CMR achieves the optimal rate of convergence in parameter estimation. We
illustrate the usefulness of CMR by thorough numerical simulations and show
that CMR consistently outperforms other high dimensional multivariate regression methods. We also apply CMR on a brain activity prediction problem and find
that CMR is as competitive as the handcrafted model created by human experts.
1
Introduction
Given a design matrix X 2 Rn?d and a response matrix Y 2 Rn?m , we consider a multivariate
linear model Y = XB0 + Z, where B0 2 Rd?m is an unknown regression coefficient matrix and
Z 2 Rn?m is a noise matrix [1]. For a matrix A = [Ajk ] 2 Rd?m , we denote Aj? = (Aj1 , ...,
Ajm ) 2 Rm and A?k = (A1k , ..., Adk )T 2 Rd to be its j th row and k th column respectively. We
assume that all Zi? ?s are independently sampled from an m-dimensional Gaussian distribution with
mean 0 and covariance matrix ? 2 Rm?m .
We can represent the multivariate linear model as an ensemble of univariate linear regression models:
Y?k = XB0?k +Z?k , k = 1, ..., m. Then we get a multi-task learning problem [3, 2, 26]. Multi-task
learning exploits shared common structure across tasks to obtain improved estimation performance.
In the past decade, significant progress has been made towards designing a variety of modeling
assumptions for multivariate regression.
A popular assumption is that all the regression tasks share a common sparsity pattern, i.e., many
B0j? ?s are zero vectors. Such a joint sparsity assumption is a natural extension of that for univariate
linear regressions. Similar to the L1 -regularization used in Lasso [23], we can adopt group regularization to obtain a good estimator of B0 [25, 24, 19, 13]. Besides the aforementioned approaches,
there are other methods that aim to exploit the covariance structure of the noise matrix Z [7, 22]. For
?
The authors are listed in alphabetical order. This work is partially supported by the grants NSF
IIS1408910, NSF IIS1332109, NSF Grant DMS-1005539, NIH R01MH102339, NIH R01GM083084, and
NIH R01HG06841.
?
Tuo Zhao is also affiliated with Department of Operations Research and Financial Engineering at Princeton
University.
1
instance, [22] assume that all Zi? ?s follow a multivariate Gaussian distribution with a sparse inverse
covariance matrix ? = ? 1 . They propose an iterative algorithm to estimate sparse B0 and ? by
maximizing the penalized Gaussian log-likelihood. Such an iterative procedure is effective in many
applications, but the theoretical analysis is difficult due to its nonconvex formulation.
In this paper, we assume an uncorrelated structure for the noise matrix Z, i.e., ? =
2
2
diag( 12 , 22 , . . . , m
1 , m ). Under this setting, we can efficiently solve the resulting estimation
problem with a convex program as follows
b = argmin p1 ||Y XB||2 + ||B||1,p ,
B
(1.1)
F
n
B
qP
2
where > 0 is a tuning parameter, and ||A||F =
j,k Ajk is the Frobenius norm of a maP d qP m
2
trix A. Popular choices of p include p = 2 and p = 1: ||B||1,2 =
j=1
k=1 Bjk and
Pd
||B||1,1 =
j=1 max1?k?m |Bjk |. Computationally, the optimization problem in (1.1) can be
efficiently solved by some first order algorithms [11, 12, 4].
The problem with the uncorrelated noise structure is amenable to statistical analysis. Under suitable conditions on the noise and design matrices, let max = maxk k , if we choose
=
p
b in (1.1) achieves the opti2c ? max
log d + m1 1/p , for some c > 1, then the estimator B
mal rates of convergence1 [13], i.e., there exists some universal constant C such that with high
probability, we have
!
r
r
1 b
s log d
sm1 2/p
0
p ||B B ||F ? C ? max
+
,
nm
n
m
where s is the number of rows with non-zero entries in B0 . However, the estimator in (1.1) has two
drawbacks: (1) All the tasks are regularized by the same tuning parameter , even though different
tasks may have different k ?s. Thus more estimation bias is introduced to the tasks with smaller k ?s
to compensate the tasks with larger k ?s. In another word, these tasks are not calibrated. (2) The
tuning parameter selection involves the unknown parameter max . This requires tuning the regularization parameter over a wide range of potential values to get a good finite-sample performance.
To overcome the above two drawbacks , we formulate a new convex program named calibrated
multivariate regression (CMR). The CMR estimator is defined to be the solution of the following
convex program:
b = argmin ||Y XB||2,1 + ||B||1,p ,
B
(1.2)
P qP
B
where ||A||2,1 = k
is the nonsmooth L2,1 norm of a matrix A = [Ajk ] 2 Rd?m .
This is a multivariate extension of the square-root Lasso [5]. Similar to the square-root Lasso, the
tuning parameter selection of CMR does not involve max . Moreover, the L2,1 loss function can
be viewed as a special example of the weighted least square loss, which calibrates each regression
task (See more details in ?2). Thus CMR adapts to different k ?s and achieves better finite-sample
performance than the ordinary multivariate regression estimator (OMR) defined in (1.1).
2
j Ajk
Since both the loss and penalty functions in (1.2) are nonsmooth, CMR is computationally more
challenging than OMR. To efficiently solve CMR, we propose a smoothed proximal gradient (SPG)
algorithm with an iteration complexity O(1/?), where ? is the pre-specified accuracy of the objective value [18, 4]. Theoretically, we provide sufficient conditions under which CMR achieves the
optimal rates of convergence in parameter estimation. Numerical experiments on both synthetic and
real data show that CMR universally outperforms existing multivariate regression methods. For a
brain activity prediction task, prediction based on the features selected by CMR significantly outperforms that based on the features selected by OMR, and is even competitive with that based on the
handcrafted features selected by human experts.
Notations: Given a vector v = (v1 , . . . , vd )T 2 Rd , for 1 ? p ? 1, we define the Lp -vector
?P
?1/p
d
p
norm of v as ||v||p =
if 1 ? p < 1 and ||v||p = max1?j?d |vj | if p = 1.
j=1 |vj |
1
The rate of convergence is optimal when p = 2, i.e., the regularization function is ||B||1,p
2
Given two matrices A = [Ajk ] and C = [Cjk ] 2 Rd?m , we define the inner product of A
Pd Pm
and C as hA, Ci = j=1 k=1 Ajk Cjk = tr(AT C), where tr(A) is the trace of a matrix A.
We use A?k = (A1k , ..., Adk )T and Aj? = (Aj1 , ..., Ajm ) to denote the k th column and j th
row of A. Let S be some subspace of Rd?m , we use AS to denote the projection of A onto S:
2
AS = argmin
p C2S ||C A||F . Moreover, we define the Frobenius and spectral norms of A as
||A||F = hA, Ai and ||A||2 = 1 (A), 1 (A)
the largest singular value of A. In addition,
Pis
m
we define the matrix block norms as ||A||2,1 = k=1 ||A?k ||2 , ||A||2,1 = max1?k?m ||A?k ||2 ,
Pd
||A||1,p = j=1 ||Aj? ||p , and ||A||1,q = max1?j?d ||Aj? ||q , where 1 ? p ? 1 and 1 ? q ? 1.
It is easy to verify that ||A||2,1 is the dual norm of ||A||2,1 . Let 1/1 = 0, then if 1/p + 1/q = 1,
||A||1,q and ||A||1,p are also dual norms of each other.
2
Method
We solve the multivariate regression problem by the following convex program,
b = argmin ||Y XB||2,1 + ||B||1,p .
B
(2.1)
B
The only difference between (2.1) and (1.1) is that we replace the L2 -loss function by the nonsmooth
L2,1 -loss function. The L2,1 -loss function can be viewed as a special example of the weighted square
loss function. More specifically, we consider the following optimization problem,
m
X
1
b = argmin
p ||Y?k XB?k ||22 + ||B||1,p ,
B
(2.2)
k n
B
k=1
where k n is a weight assigned to calibrate the k th regression task. Without prior knowledge on
k ?s, we use the following replacement of k ?s,
1
ek = p ||Y?k XB?k ||2 , k = 1, ..., m.
(2.3)
n
By plugging (2.3) into the objective function in (2.2), we get (2.1). In another word, CMR calibrates
different tasks by solving a penalized weighted least square program with weights defined in (2.3).
1p
The optimization problem in (2.1) can be solved by the alternating direction method of multipliers
(ADMM) with a global convergence guarantee [20]. However, ADMM does not take full advantage
of the problem structure in (2.1). For example, even though the L2,1 norm is nonsmooth, it is
nondifferentiable only when a task achieves exact zero residual, which is unlikely in applications.
In this paper, we apply the dual smoothing technique proposed by [18] to obtain a smooth surrogate
function so that we can avoid directly evaluating the subgradient of the L2,1 loss function. Thus we
gain computational efficiency like other smooth loss functions.
We consider the Fenchel?s dual representation of the L2,1 loss:
||Y
XB||2,1 =
max
||U||2,1 ?1
hU, Y
XBi.
(2.4)
Let ? > 0 be a smoothing parameter. The smooth approximation of the L2,1 loss can be obtained
by solving the following optimization problem
?
||Y XB||? = max hU, Y XBi
||U||2F ,
(2.5)
2
||U||2,1 ?1
where ||U||2F is the proximity function. Due to the fact that ||U||2F ? m||U||22,1 , we obtain the
following uniform bound by combing (2.4) and (2.5),
m?
||Y XB||2,1
? ||Y XB||? ? ||Y XB||2,1 .
(2.6)
2
From (2.6), we see that the approximation error introduced by the smoothing procedure can be
controlled by a suitable ?. Figure 2.1 shows several two-dimensional examples of the L2 norm
b B with
smoothed by different ??s. The optimization problem in (2.5) has a closed form solution U
B
b
U?k = (Y?k XB?k )/ max {||Y?k XB?k ||2 , ?}.
The next lemma shows that ||Y
XB||? is smooth in B with a simple form of gradient.
3
(a) ? = 0
(b) ? = 0.1
(c) ? = 0.25
(d) ? = 0.5
Figure 2.1: The L2 norm (? = 0) and its smooth surrogates with ? = 0.1, 0.25, 0.5. A larger ?
makes the approximation more smooth, but introduces a larger approximation error.
Lemma 2.1. For any ? > 0, ||Y XB||? is a convex and continuously differentiable function in
B. In addition, G? (B)?the gradient of ||Y XB||? w.r.t. B?has the form
?
?
b B , Y XBi + ?||U
b B ||2 /2
@ hU
F
b B.
G? (B) =
= XT U
(2.7)
@B
Moreover, let = ||X||22 , then we have that G? (B) is Lipschitz continuous in B with the Lipschitz
constant /?, i.e., for any B0 , B00 2 Rd?m ,
b B0 U
b B00 i||F ? 1 ||XT X(B0 B00 )||F ? ||B0 B00 ||F .
||G? (B0 ) G? (B00 )||F = ||hX, U
?
?
Lemma 2.1 is a direct result of Theorem 1 in [18] and implies that ||Y XB||? has good computational structure. Therefore we apply the smooth proximal gradient algorithm to solve the smoothed
version of the optimization problem as follows,
e = argmin ||Y XB||? + ||B||1,p .
B
(2.8)
B
We then adopt the fast proximal gradient algorithm to solve (2.8) [4]. To derive the algorithm,
we first define three sequences of auxiliary variables {A(t) }, {V(t) }, and {H(t) } with A(0) =
H(0) = V(0) = B(0) , a sequence of weights {?t = 2/(t + 1)}, and a nonincreasing sequence of
step-sizes {?t > 0}. For simplicity, we can set ?t = ?/ . In practice, we use the backtracking
line search to dynamically adjust ?t to boost the performance. At the tth iteration, we first take
V(t) = (1 ?t )B(t 1) + ?t A(t 1) . We then consider a quadratic approximation of ||Y XH||?
as
?
?
1
Q H, V(t) , ?t = ||Y XV(t) ||? + hG? (V(t) ), H V(t) i +
||H V(t) ||2F .
2?t
e (t) = V(t) ?t G? (V(t) ), we take
Consequently, let H
?
?
1
e (t) ||2 + ||H||1,p . (2.9)
H(t) = argmin Q H, V(t) , ?t + ||H||1,p = argmin
||H H
F
2?t
H
H
n
o
(t)
e j? ? max 1 ?t /||H
e j? ||2 , 0 . More
When p = 2, (2.9) has a closed form solution Hj? = H
details about other choices of p in the L1,p norm can be found in [11] and [12]. To ensure that the
objective value is nonincreasing, we choose
B(t) =
argmin
B2{H(t) , B(t
1) }
At last, we take A(t) = B(t 1) + ?1t (H(t) B(t
where " is the stopping precision.
||Y
1)
XB||? + ||B||1,p .
(2.10)
). The algorithm stops when ||H(t) V(t) ||F ? ",
The numerical rate of convergence of the proposed algorithm with respect to the original optimization problem (2.1) is presented in the following theorem.
p
b F /?
Theorem 2.2. Given a pre-specified accuracy ? and let ? = ?/m, after t = 2 m ||B(0) B||
(t)
(t)
b 2,1 + ||B||
b 1,p + ?.
1 = O (1/?) iterations, we have ||Y XB ||2,1 + ||B ||1,p ? ||Y XB||
The proof of Theorem 2.2 is provided in Appendix A.1. This result achieves the minimax optimal
rate of convergence over all first order algorithms [18].
4
3
Statistical Properties
For notational simplicity, we define a re-scaled noise matrix W = [Wik ] 2 Rn?m with Wik =
Zik / k , where EZ2ik = k2 . Thus W is a random matrix with all entries having mean 0 and variance
1. We define G0 to be the gradient of ||Y XB||2,1 at B = B0 . It is easy to see that
XT Z?k
XT W?k k
XT W?k
=
=
||Z?k ||2
||W?k k ||2
||W?k ||2
does not depend on the unknown quantities k for all k = 1, ..., m. G0?k works as an important
pivotal in our analysis. Moreover, our analysis exploits the decomposability of the L1,p norm [17].
More specifically, we assume that B0 has s rows with all zero entries and define
G0?k =
S = C 2 Rd?m | Cj? = 0 for all j such that B0j? = 0 ,
N = C2R
d?m
| Cj? = 0 for all j such that
B0j?
6= 0 .
(3.1)
(3.2)
Note that we have B 2 S and the L1,p norm is decomposable with respect to the pair (S, N ), i.e.,
||A||1,p = ||AS ||1,p + ||AN ||1,p .
The next lemma shows that when is suitably chosen, the solution to the optimization problem in
(2.1) lies in a restricted set.
b be the optimum to (2.1), and 1/p + 1/q = 1. We denote the
Lemma 3.1. Let B0 2 S and B
0
b
b
estimation error as = B B . If
c||G0 ||1,q for some c > 1, we have
?
c+1
b 2 Mc :=
2 Rd?m | || N ||1,p ?
|| S ||1,p .
(3.3)
c 1
0
The proof of Lemma 3.1 is provided in Appendix B.1. To prove the main result, we also need to
assume that the design matrix X satisfies the following condition.
Assumption 3.1. Let B0 2 S, then there exist positive constants ? and c > 1 such that
||X ||F
p
??
min
.
n|| ||F
2Mc \{0}
Assumption 3.1 is the generalization of the restricted eigenvalue conditions for analyzing univariate
sparse linear models [17, 15, 6], Many common examples of random design satisfy this assumption
[13, 21].
Note that Lemma 3.1 is a deterministic result of the CMR estimator for a fixed . Since G is
essentially a random matrix, we need to show that
cR? (G0 ) holds with high probability to
deliver a concrete rate of convergence for the CMR estimator in the next theorem.
p
Theorem 3.2. We assume that each column of X is normalized as m1/2 1/p kX?j k2 = n for all
j = 1, ..., d. Then for some universal constant c0 and large enough n, taking
p
2c(m1 1/p + log d)
p
=
,
(3.4)
1 c0
with probability at least 1 2 exp( 2 log d) 2 exp nc20 /8 + log m , we have
!
r
r
r
1 b
16c max
1 + c0
sm1 2/p
s log d
0
p ||B B ||F ? 2
+
.
? (c 1) 1 c0
n
nm
m
The proof of Theorem 3.2 is provided in Appendix B.2.pNote that when we choose p = 2, the
column normalization condition is reduced to kX?j k2 = n. Meanwhile, the corresponding error
bound is further reduced to
!
r
r
1 b
s
s log d
0
p ||B B ||F = OP
+
,
n
nm
m
which achieves the minimax optimal rate of convergence presented in [13]. See Theorem 6.1 in [13]
for more technical details. From Theorem 3.2, we see that CMR achieves the same rates of convergence as the noncalibrated counterpart, but the tuning parameter in (3.4) does not involve k ?s.
Therefore CMR not only calibrates all the regression tasks, but also makes the tuning parameter
selection insensitive to max .
5
4
Numerical Simulations
To compare the finite-sample performance between the calibrated multivariate regression (CMR)
and ordinary multivariate regression (OMR), we generate a training dataset of 200 samples. More
specifically, we use the following data generation scheme: (1) Generate each row of the design
matrix Xi? , i = 1, ..., 200, independently from a 800-dimensional normal distribution N (0, ?)
where ?jj = 1 and ?j` = 0.5 for all ` 6= j.(2) Let k = 1, . . . , 13, set the regression coefficient
matrix B0 2 R800?13 as B01k = 3, B02k = 2, B04k = 1.5, and B0jk = 0 for all j 6= 1, 2, 4. (3)
Generate the random noise matrix Z = WD, where W 2 R200?13 with all entries of W are
independently generated from N (0, 1), and D is either of the following matrices
?
?
DI = max ? diag 20/4 , 2 1/4 , ? ? ? , 2 11/4 , 2 12/4 2 R13?13
DH = max ? I 2 R13?13 .
We generate a validation set of 200 samples for the regularization parameter selection and a testing
set of 10,000 samples to evaluate the prediction accuracy.
In numerical experiments, we set max = 1, 2, and 4 to illustrate the tuning insensitivity
of CMR. The regularization parameter of both CMR and
p OMR ispchosen over a grid ? =
240/4 0 , 239/4 0 , ? ? ? , 2 17/4 0 , 2 18/4 0 , where 0 = log d + m. The optimal regulare X
eB
b ||2 , where
ization parameter b is determined by the prediction error as b = argmin 2? ||Y
F
b denotes the obtained estimate using the regularization parameter , and X
e and Y
e denote the
B
design and response matrices of the validation set.
Since the noise level k ?s are different in regression tasks, we adopt the following three crite1
b F , Adj. Pre. Err. =
ria to evaluate the empirical performance: Pre. Err. = 10000
||Y XB||
1
1
0
2
1
2
b
b B || , where X and Y denotes the design
XB)D
||F , and Est. Err. = m ||B
F
10000m ||(Y
and response matrices of the testing set.
All simulations are implemented by MATLAB using a PC with Intel Core i5 3.3GHz CPU and 16GB
memory. CMR is solved by the proposed smoothing proximal gradient algorithm, where we set the
stopping precision " = 10 4 , the smoothing parameter ? = 10 4 . OMR is solved by the monotone
fast proximal gradient algorithm, where we set the stopping precision " = 10 4 . We set p = 2, but
the extension to arbitrary p > 2 is straightforward.
We first compare the smoothed proximal gradient (SPG) algorithm with the ADMM algorithm (the
detailed derivation of ADMM can be found in Appendix A.2). We adopt the backtracking line search
to accelerate both algorithms with a shrinkage parameter ? = 0.8. We set max = 2 for the adopted
multivariate linear models. We conduct 200 simulations. The results are presented in Table 4.1. The
SPG and ADMM algorithms attain similar objective values, but SPG is up to 4 times faster than
ADMM. Both algorithms also achieve similar estimation errors.
We then compare the statistical performance between CMR and OMR. Tables 4.2 and 4.3 summarize the results averaged over 200 replicates. In addition, we also present the results of the oracle
estimator, which is obtained by solving (2.2), since we know the true values of k ?s. Note that the
oracle estimator is only for comparison purpose, and it is not a practical estimator. Since CMR
calibrates the regularization for each task with respect to k , CMR universally outperforms OMR,
and achieves almost the same performance as the oracle estimator when we adopt the scale matrix
DI to generate the random noise. Meanwhile, when we adopt the scale matrix DH , where all k ?s
are the same, CMR and OMR achieve similar performance. This further implies that CMR can be a
safe replacement of OMR for multivariate regressions.
In addition, we also examine the optimal regularization parameters for CMR and OMR over all
replicates. We visualize the distribution of all 200 selected b?s using the kernel density estimator.
In particular, we adopt the Gaussian kernel, and the kernel bandwidth is selected based on the 10fold cross validation. Figure 4.1 illustrates the estimated density
functions.
The horizontal axis
?
?
b
corresponds to the rescaled regularization parameter as log plog d+pm . We see that the optimal
regularization parameters of OMR significantly vary with different max . In contrast, the optimal
regularization parameters of CMR are more concentrated. This is inconsistent with our claimed
tuning insensitivity.
6
Table 4.1: Quantitive comparison of the computational performance between SPG and ADMM with
the noise matrices generated using DI . The results are averaged over 200 replicates with standard
errors in parentheses. SPG and ADMM attain similar objective values, but SPG is up to about 4
times faster than ADMM.
2
0
0
0.5
0
Algorithm
Timing (second)
Obj. Val.
Num. Ite.
Est. Err.
SPG
ADMM
2.8789(0.3141)
8.4731(0.8387)
508.21(3.8498)
508.22(3.7059)
493.26(52.268)
437.7(37.4532)
0.1213(0.0286)
0.1215(0.0291)
SPG
ADMM
3.2633(0.3200)
11.976(1.460)
370.53(3.6144)
370.53(3.4231)
565.80(54.919)
600.94(74.629)
0.0819(0.0205)
0.0822(0.0233)
SPG
ADMM
3.7868(0.4551)
18.360(1.9678)
297.24(3.6125)
297.25(3.3863)
652.53(78.140)
1134.0(136.08)
0.1399(0.0284)
0.1409(0.0317)
Table 4.2: Quantitive comparison of the statistical performance between CMR and OMR with the
noise matrices generated using DI . The results are averaged over 200 simulations with the standard
errors in parentheses. CMR universally outperforms OMR, and achieves almost the same performance as the oracle estimator.
Method
Pre. Err.
Adj. Pre.Err
Est. Err.
1
Oracle
CMR
OMR
5.8759(0.0834)
5.8761(0.0673)
5.9012(0.0701)
1.0454(0.0149)
1.0459(0.0123)
1.0581(0.0162)
0.0245(0.0086)
0.0249(0.0071)
0.0290(0.0091)
2
Oracle
CMR
OMR
23.464(0.3237)
23.465(0.2598)
23.580(0.2832)
1.0441(0.0148)
1.0446(0.0121)
1.0573(0.0170)
0.0926(0.0342)
0.0928(0.0279)
0.1115(0.0365)
4
Oracle
CMR
OMR
93.532(0.8843)
93.542(0.9794)
94.094(1.0978)
1.0418(0.0962)
1.0421(0.0118)
1.0550(0.0166)
0.3342(0.1255)
0.3346(0.1063)
0.4125(0.1417)
max
Table 4.3: Quantitive comparison of the statistical performance between CMR and OMR with the
noise matrices generated using DH . The results are averaged over 200 simulations with the standard
errors in parentheses. CMR and OMR achieve similar performance.
Method
Pre. Err.
Adj. Pre.Err
Est. Err.
1
CMR
OMR
13.565(0.1408)
13.697(0.1554)
1.0435(0.0108)
1.0486(0.0142)
0.0599(0.0164)
0.0607(0.0128)
2
CMR
OMR
54.171(0.5771)
54.221(0.6173)
1.0418(0.0110)
1.0427(0.0118)
0.2252(0.0649)
0.2359(0.0821)
4
CMR
OMR
215.98(2.104)
216.19(2.391)
1.0384(0.0101)
1.0394(0.0114)
0.80821(0.25078)
0.81957(0.31806)
max
1.4
Oracle(1)
Oracle(2)
Oracle(4)
CMR(1)
CMR(2)
CMR(4)
OMR(1)
OMR(2)
OMR(4)
1.2
1
0.8
0.6
1.4
1
0.8
0.6
0.4
0.4
0.2
0.2
0
?2
?1.5
?1
?0.5
0
0.5
1
1.5
2
CMR(1)
CMR(2)
CMR(4)
OMR(1)
OMR(2)
OMR(4)
1.2
0
?2
2.5
(a) The noise matrices are generated using DI
?1.5
?1
?0.5
0
0.5
1
1.5
2
2.5
(b) The noise matrices are generated using DH
Figure 4.1: The distributions of the selected regularization parameters using the kernel density estimator. The numbers in the parentheses are max ?s. The optimal regularization parameters of OMR
are spreader with different max than those of CMR and the oracle estimator.
7
5
Real Data Experiment
We apply CMR on a brain activity prediction problem which aims to build a parsimonious model to
predict a person?s neural activity when seeing a stimulus word. As is illustrated in Figure 5.1, for
a given stimulus word, we first encode it into an intermediate semantic feature vector using some
corpus statistics. We then model the brain?s neural activity pattern using CMR. Creating such a
predictive model not only enables us to explore new analytical tools for the fMRI data, but also
helps us to gain deeper understanding on how human brain represents knowledge [16].
Predict fMRI brain activity patterns in response to text stimulus
89/:4:2%,-.&2
!"#$%)/01'2
!"#$%&'()*'
%0334'&
(Mitchell et al., Science,2008)
!"#$%0334'&
stimulus word
!"#$%50//'.&
%50//'.&
predicted
activities
for "apple"
"apple"
!"#$%6)*7*4'&
%6)*7*4'&
+',%,-.&
?
Model
intermediate semantic features
Standard solution
Our solution
Linear models
(More restrictive)
(Less restrictive)
(a) illustration of the data collection procedure
Nonlinear models
.
mapping learned from fMRI data
(b) model for predicting fMRI brain activity pattern
Figure
5.1: An illustration of the fMRI brain activity prediction problem [16]. (a) To collect the
;5'%'<3'.)/'+=2%0.'%*-+&:*='&%)+%>")=*5'44%'=%0?%8*)'+*'%@AB
data, a human participant sees a sequence of English words and their images. The corresponding
fMRI images are recorded to represent the brain activity patterns; (b) To build a predictive model,
each stimulus word is encoded into intermediate semantic features (e.g. the co-occurrence statistics
of this stimulus word in a large text corpus). These intermediate features can then be used to predict
the brain activity pattern.
Our experiments involves 9 participants, and Table 5.1 summarizes the prediction performance of
different methods on these participants. We see that the prediction based on the features selected by
CMR significantly outperforms that based on the features selected by OMR, and is as competitive
as that based on the handcrafted features selected by human experts. But due to the space limit, we
present the details of the real data experiment in the technical report version.
Table 5.1: Prediction accuracies of different methods (higher is better). CMR outperforms OMR for
8 out of 9 participants, and outperforms the handcrafted basis words for 6 out of 9 participants
Method
P. 1
CMR
0.840
OMR 0.803
Handcraft 0.822
6
P. 2
P. 3
P. 4
P. 5
P. 6
P. 7
P. 8
P. 9
0.794
0.789
0.776
0.861
0.801
0.773
0.651
0.602
0.727
0.823
0.766
0.782
0.722
0.623
0.865
0.738
0.726
0.734
0.720
0.749
0.685
0.780
0.765
0.819
Discussions
A related method is the square-root sparse multivariate regression [8]. They solve the convex program with the Frobenius loss function and L1,p regularization function
b = argmin ||Y
B
B
XB||F + ||B||1,p .
(6.1)
The Frobenius loss function in (6.1) makes the regularization parameter selection independent of
max , but it does not calibrate different regression tasks. Note that we can rewrite (6.1) as
1
b b) = argmin p 1 ||Y XB||2 + ||B||1,p s. t.
(B,
=p
||Y XB||F .
(6.2)
F
nm
nm
B,
Since in (6.2) is not specific to any individual task, it cannot calibrate the regularization. Thus it
is fundamentally different from CMR.
8
References
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9
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5,116 | 5,631 | Exclusive Feature Learning on Arbitrary Structures
via `1,2-norm
Deguang Kong1 , Ryohei Fujimaki2 , Ji Liu3 , Feiping Nie1 , Chris Ding1
1
Dept. of Computer Science, University of Texas Arlington, TX, 76019;
2
NEC Laboratories America, Cupertino, CA, 95014;
3
Dept. of Computer Science, University of Rochester, Rochester, NY, 14627
Email: [email protected], [email protected],
[email protected], [email protected], [email protected]
Abstract
Group LASSO is widely used to enforce the structural sparsity, which achieves
the sparsity at the inter-group level. In this paper, we propose a new formulation
called ?exclusive group LASSO?, which brings out sparsity at intra-group level in
the context of feature selection. The proposed exclusive group LASSO is applicable on any feature structures, regardless of their overlapping or non-overlapping
structures. We provide analysis on the properties of exclusive group LASSO, and
propose an effective iteratively re-weighted algorithm to solve the corresponding
optimization problem with rigorous convergence analysis. We show applications
of exclusive group LASSO for uncorrelated feature selection. Extensive experiments on both synthetic and real-world datasets validate the proposed method.
1
Introduction
Structure sparsity induced regularization terms [1, 8] have been widely used recently for feature
learning purpose, due to the inherent sparse structures of the real world data. Both theoretical and
empirical studies have suggested the powerfulness of structure sparsity for feature learning, e.g.,
Lasso [24], group LASSO [29], exclusive LASSO [31], fused LASSO [25], and generalized LASSO [22]. To make a compromise between the regularization term and the loss function, the sparseinduced optimization problem is expected to fit the data with better statistical properties. Moreover,
the results obtained from sparse learning are easier for interpretation, which give insights for many
practical applications, such as gene-expression analysis [9], human activity recognition [14], electronic medical records analysis [30], etc.
Motivation Of all the above sparse learning methods, group LASSO [29] is known to enforce the
sparsity on variables at an inter-group level, where variables from different groups are competing to
survive. Our work is motivated from a simple observation: in practice, not only features from different groups are competing to survive (i.e., group LASSO), but also features in a seemingly cohesive
group are competing to each other. The winner features in a group are set to large values, while the
loser features are set to zeros. Therefore, it leads to sparsity at the intra-group level. In order to make
a distinction with standard LASSO and group LASSO, we called it ?exclusive group LASSO? regularizer. In ?exclusive group LASSO? regularizer, intra-group sparsity is achieved via `1 norm, while
inter-group non-sparsity is achieved via `2 norm. Essentially, standard group LASSO achieves sparsity via `2,1 norm, while the proposed exclusive group LASSO achieves sparsity via `1,2 norm. An
example of exclusive group LASSO is shown in Fig.(1) via Eq.(2). The significant difference from
the standard LASSO is to encourage similar features in different groups to co-exist (Lasso usually
allows only one of them surviving). Overall, the exclusive group LASSO regularization encourages
intra-group competition but discourages inter-group competition.
1
zero value
w1
w2
w3
w4
w5
w6
w7
w1
w2
w3
w4
w5
w6
w7
G1
G2
G3
(a) group lasso
G1
G2
G3
(b) exclusive lasso
Figure 1: Explanation of differences between group LASSO and exclusive group LASSO. Group setting:
G1 = {1, 2}, G2 = {3, 4}, G3 = {5, 6, 7}. Group LASSO solution of Eq.(3) at ? = 2 using least square loss
is: w = [0.0337; 0.0891; 0; 0; ?0.2532; 0.043; 0.015]. exclusive group LASSO solution of Eq.(2) at ? = 10
is: w = [0.0749; 0; 0; ?0.0713; ?0.1888; 0; 0]. Clearly, group LASSO introduces sparsity at an inter-group
level, whereas exclusive LASSO enforces sparsity at an intra-group level.
We note that ?exclusive LASSO? was first used in [31] for multi-task learning. Our ?exclusive group
LASSO? work, however, has clear difference from [31]: (1) we give a clear physical intuition of
?exclusive group LASSO?, which leads to sparsity at an intra-group level (Eq.2), whereas [31] focuses on ?Exclusive LASSO? problem in a multi-task setting; (2) we target a general ?group? setting
which allows arbitrary group structure, which can be easily extended to multi-task/multi-label learning. The main contributions of this paper include: (1) we propose a new formulation of ?exclusive
group LASSO? with clear physical meaning, which allow any arbitrary structure on feature space;
(2) we propose an effective iteratively re-weighted algorithm to tackle non-smooth ?exclusive group
LASSO? term with rigorous convergence guarantees. Moreover, an effective algorithm is proposed
to handle both non-smooth `1 and exclusive group LASSO term (Lemma 4.1); (3) The proposed approach is validated via experiments on both synthetic and real data sets, specifically for uncorrelated
feature selection problems.
Notation Throughout the paper, matrices are written as boldface uppercase, vectors are written
as boldface lowercase, and scalars are denoted by lower-case letters (a, b). n is the number of
data points, p is the dimension of data, K is the number of class in a dataset. For any vector
1
q
Pp
q
w ? <p , `q norm of w is kwkq =
|w
|
for q ? (0, ?). A group of variables is
j
i=1
a subset g ? {1, 2, ? ? ? , p}. Thus, the set of possible groups is the power set of {1, 2, ? ? ? , p}:
P({1, 2, ? ? ? , p}). Gg ? P({1, 2, ? ? ? , p}) denotes a set of group g, which is known in advance
depending on applications. If two groups have one or more overlapped variables, we say that they are
overlapped. For any group variable wGg ? <p , only the entries in the group g are preserved which
are the same as those in w, while the other entriesp
are set to zeros. For example, if Gg = {1, 2, 4},
wGg = [w1 , w2 , 0, w4 , 0, ? ? ? , 0], then kwGg k2 = w12 + w22 + w42 . Let supp(w) ? {1, 2, ? ? ? , p}
be a set which wi 6= 0, and zero(w) ? {1, 2, ? ? ? , p} be a set which wi = 0. Clearly, zero(w) =
{1, 2, ? ? ? , p} \ supp(w). Let 5f (w) be gradient of f at w ? <p , for any differentiable function f :
<p ? <.
2
Exclusive group LASSO
Let G be a group set, the exclusive group LASSO penalty is defined as:
X
?w ? <p , ?GEg (w) =
kwGg k21 .
(1)
g?G
When the groups of g form different partitions of the set of variables, ?GEg is a `1 /`2 norm penalty.
A `2 norm is enforced on different groups, while in each group, `1 norm is used to make a sum
over each intra-group variable. Minimizing such a convex risk function often leads to a solution that
some entries in a group are zeros. For example, for a group Gg = {1, 2, 4}, there exists a solution
w, such that w1 = 0, w2 6= 0, w4 6= 0. A concrete example is shown in Fig.1, in which we solve:
min J1 (w), J1 (w) = f (w) + ??GEg (w).
w?<p
(2)
using least square loss function f (w) = ky ?XT wk22 . As compared to standard group LASSO [29]
solution of Eq.(3),
2
1
0.9
2
1
1
0.5
0.5
0.8
4
0.7
6
0
0
?0.5
?0.5
0.6
0.5
8
?1
1
0.4
?1
1
1
0
0
?1
1
0
10
0.3
12
0.2
0
?1
?1
0.1
14
?1
2
(a) non-overlap
4
6
8
10
12
14
(b) overlap
(c) feature correlation matrix
3
Figure 2: (a-b): Geometric shape of ?(w) ? 1 in R . (a) non-overlap exclusive group LASSO: ?(w) =
(|w1 | + |w2 |)2 + (|w3 |)2 ; (b) overlap exclusive group LASSO: ?(w) = (|w1 | + |w2 |)2 + (|w2 | + |w3 |)2 ;
(c) feature correlation matrix R on dataset House (506 data points, 14 variables). Rij indicates the feature
correlation between feature i and j. Red colors indicate large values, while blue colors indicate small values.
f (w) + ?
X
kwGg k2 .
(3)
g
We observe that group LASSO introduces sparsity at an inter-group level, whereas exclusive LASSO
enforces sparsity at an intra-group level.
Analysis of exclusive group LASSO For each group g, feature index u ? supp(g) will be non-zero.
Let vg ? <p be a variable which preserves the values of non-zero index for group g. Consider all
groups, for optimization goal w, we have supp(w) = ? supp(vg ). (1) For non-overlapping case,
g
different P
groups form a partition of feature set {1, 2, ? ? ? , p}, and there exists a unique decomposition
of w = g vg . Since there is not any common elements for any two different groups Gu and Gv ,
i.e., supp(wGu ) ? supp(wGv ) = ?. thus it is easy to see: vg = wGg , ?g ? G. (2) However, for
overlapping groups, there could be element sets I ? (Gu ? Gv ), and therefore, different groups Gu
and Gv may have opposite effects to optimize the features in set I. For feature i ? I, it is prone to
give different values if optimized separately, i.e., (wGu )i 6= (wGv )i . For example, Gu = [1, 2], Gv =
[2, 3], whereas group u may require w2 = 0 and group v may require w2 6= 0. P
Thus, there will be
2
many possible combinations of feature values, and it leads to: ?GEg = P inf
g kvg k1 . Further,
g
vg =w
if some groups are overlapped, the final zeros sets will be a subset of unions of all different groups.
zero(w) ? ? zero(vg ).
g
Illustration of Geometric shape of exclusive LASSO Figure 2 shows the geometric shape for
both norms in R3 with different group settings, where in (a): G1 = [1, 2], G2 = [3]; and in (b):
G1 = [1, 2], G2 = [2, 3]. For the non-overlapping case, variables w1 , w2 usually can not be zero
simultaneously. In contrast, for the overlapping case, variable w2 cannot be zero unless both groups
G1 and G2 require w2 = 0.
Properties of exclusive LASSO The
q regularization term of Eq.(1) is a convex formulation. If
?g?G = {1, 2, ? ? ? , p}, then ?GE := ?GEg is a norm. See Appendix for proofs.
3
An effective algorithm for solving ?GEg regularizer
The challenge of solving Eq. (1) is to tackle the exclusive group LASSO term, where f (w) can be
any convex loss function w.r.t w. It is generally felt that exclusive group LASSO term is much more
difficult to solve than the standard LASSO term (shrinkage thresholding). Existing algorithm can
formulate it as a quadratic programming problem [19], which can be solved by interior point method
or active set method. However, the computational cost is expensive, which limits its use in practice.
Recently, a primal-dual algorithm [27] is proposed to solve the similar problem, which casts the
non-smooth problem into a min-max problem. However, the algorithm is a gradient descent type
method and converges slowly. Moreover, the algorithm is designed for multi-task learning problem,
and cannot be applied directly for exclusive group LASSO problem with arbitrary structures.
In the following, we first derive a very efficient yet simple algorithm. Moreover, the proposed
algorithm is a generic algorithm, which allows arbitrary structure on feature space, irrespective of
specific feature structures [10], e.g., linear structure [28], tree structure [15], graph structure [7], etc.
3
Theoretical analysis guarantees the convergence of algorithm. Moreover, the algorithm is easy to
implement and ready to use in practice.
Key idea The idea of the proposed algorithm is to find an auxiliary function for Eq.(1) which can be
easily solved. Then the updating rules for w is derived. Finally, we prove the solution is exactly the
optimal solution we are seeking for the original problem. Since it is a convex problem, the optimal
solution is the global optimal solution.
Procedure Instead of directly optimizing Eq. (1), we propose to optimize the following objective
(the reasons will be seen immediately below), i.e.,
J2 (w) = f (w) + ?wT Fw,
(4)
p?p
where F ? <
is a diagonal matrices which encodes the exclusive group information, and its
diagonal element is given by1
X (I ) kw k
Gg i
Gg 1
.
(5)
Fii =
|wi |
g
Let IGg ? {0, 1}p?1 be group index indicator for group g ? G. For example, group G1 is {1, 2},
then IG1 = [1, 1, 0, ? ? ? , 0]. Thus the group variable wGg can be explicitly expressed as wGg =
diag(IGg ) ? w.
Note computation of F depends on w, thus minimization of w depends on both F. In the following,
we propose an efficient iteratively re-weighted algorithm to find out the optimal global solution for
w, where in each iteration, w is updated along the gradient descent direction. This process is iterated
2
until the algorithm converges. Taking the derivative of Eq.(4) w.r.t w and set ?J
?w = 0. We have
?w f (w) + 2?Fw = 0.
(6)
Then the complete algorithm is:
(1) Updating wt via Eq.(6);
(2) Updating Ft via Eq.(5).
The above two steps are iterated until the algorithm converges. We can prove the obtained optimal
solution is exactly the global optimal solution for Eq.(1).
3.1 Convergence Analysis
In the following, we prove the convergence of algorithm.
Theorem 3.1. Under the updating rule of Eq. (6), J1 (wt+1 ) ? J1 (wt ) ? 0.
The proof is provided in Appendix.
Discussion We note reweighted strategy [26] was also used in solving problems like zero-norm
of the parameters of linear models. However, it cannot be directly used to solve ?exclusive group
LASSO? problem proposed in this paper, and cannot handle arbitrary structures on feature space.
4
Uncorrelated feature learning via exclusive group LASSO
Motivation It is known that in Lasso-type (including elastic net) [24, 32] variable selection, variable
correlations are not taken into account. Therefore, some strongly correlated variables tend to be in
or out of the model together. However, in practice, feature variables are often correlated. See an
example shown on housing dataset [4] with 506 samples and 14 attributes. Although there are only
14 attributes, feature 5 is highly correlated with feature 6, 7, 11, 12, etc. Moveover, the strongly correlated variables may share similar properties, with overlapped or redundant information. Especially
1
when wi = 0, then Fii is related to subgradient of w w.r.t to wi . However, we can not set
F
ii
= 0, otherwise the derived
algorithm cannot be guaranteed to converge. We can regularize Fii =
p
P
2
(IGg )i kwGg k1 / wi + , then the derived algorithm can be proved to minimize the regularized
Pg
2
g k(w + )Gg k1 . It is easy to see the regularized exclusive `1 norm of w approximates exclusive `1 norm of
+
w when ? 0 .
4
Table 1: Characteristics of datasets
Dataset
isolet
ionosphere
mnist(0,1)
Leuml
# data
1560
351
3125
72
#dimension
617
34
784
3571
#domain
UCI
UCI
image
biology
when the number of selected variables are limited, more discriminant information with minimum
correlations are desirable for prediction or classification purpose. Therefore, it is natural to eliminate
the correlations in the feature learning process.
Formulation The above observations motivate our work of uncorrelated feature learning via exclusive group LASSO. We consider the variable selection problem based on the LASSO-type optimization, where we can make the selected variables uncorrelated as much as possible. To be exact, we
propose to optimize the following objective:
X
minp f (w) + ?kwk1 + ?
kwGg k21 ,
(7)
w?<
g
where f (w) is loss function involving class predictor y ? <n and data matrix X =
[x1 , x2 , ? ? ? , xn ] ? <p?n , kwGg k21 is the exclusive group LASSO term involving feature correlation information ? and ? are tuning parameters, which can make balances between plain LASSO
term and the exclusive group LASSO term.
The core part of Eq.(7) is to use exclusive group LASSO regularizer to eliminate the correlated
features, which cannot be done by plain LASSO. Let the feature correlation matrix be R = (Rst ) ?
<p?p , clearly, R = RT , Rst represents the correlations between features s and t, i.e.,
P
| i Xsi Xti |
Rst = pP 2 pP 2 , Rst > ?
(8)
i Xsi
i Xti
To let the selected features uncorrelated as much as possible, for any two features s, t, if their
correlations Rst > ?, then we put them in an exclusive group. Therefore, only one feature can
survive. For example, on the example shown in Fig.2(c), if we use ? = 0.93 as a threshold, we will
generate the following exclusive group LASSO term:
X
kwGg k21
= (|w3 | + |w10 |)2 + (|w5 | + |w6 |)2 + (|w5 | + |w7 |)2 + (|w5 | + |w11 |)2 + (|w6 | + |w11 |)2
g
+(|w6 | + |w12 |)2 + (|w6 | + |w14 |)2 + (|w7 | + |w11 |)2 .
(9)
Algorithm Solving Eq.(7) is to solve a convex optimization problem, because all the three involved
terms are convex. This also indicates that there exists a unique global solution. Eq.(7) can be
efficiently solved via accelerated proximal gradient (FISTA) method [17, 2], irrespective of what
kind of loss function used in minimization of empirical risk. Thus solving Eq.(7) is transformed into
solving:
minp
w?<
X
1
kw ? ak22 + ?kwk1 + ?
kwGg k21 ,
2
g
(10)
where a = wt ? L1t ?f (wt ) which involves the current wt value and step size Lt . The challenge
of solving Eq.(10) is that, it involves two non-smooth terms. Fortunately, we have the following
lemma to establish the relations between the optimal solution of Eq.(10) to Eq.(11), the solution of
which has been well discussed in ?3.
minp
w?<
X
1
kw ? uk22 + ?
kwGg k21 .
2
g
(11)
Lemma 4.1. The optimal solution to Eq.(10) is the optimal solution to Eq.(11), where
1
u = arg min kx ? ak22 + ?kxk1 = sgn(a)(a ? ?)+ ,
2
x
(12)
and sgn(.), SGN(.) are the operators defined in the component fashion: if v > 0, sgn(v) = 1, SGN(v) =
{1}; else if v = 0, sgn(v) = 0, SGN(v) = [?1, 1]; else if v < 0, sgn(v) = ?1, SGN(v) = {?1}.
The proof is provided in Appendix.
5
?0.54
?1.8
?2
L1
Exclusive lasso+L1
optimal solution
?2.2
?2.4
?2.6
?2.8
120
140
160
180
200
# of features
220
?1.8
?2
?2.2
L1
Exclusive lasso+L1
optimal solution
?2.4
?2.6
?2.8
?3
120
240
140
160
180
200
# of features
220
?0.55
?0.56
?0.57
?0.58
?0.59
?0.6
200
240
?0.71
L1
Exclusive lasso+L1
optimal solution
log Generalization: MAE error
?1.6
log Generalization: RMSE error
?1.6
log Generalization: MAE error
log Generalization: RMSE error
?1.4
210
220
230
# of features
240
L1
Exclusive lasso+L1
optimal solution
?0.72
?0.73
?0.74
?0.75
?0.76
?0.77
?0.78
?0.79
200
250
210
220
230
# of features
240
250
86
82
80
78
F?statistic
ReliefF
LASSO
Exclusive
76
74
0
100
200
300
400
# of features
500
600
84
82
80
78
F?statistic
ReliefF
LASSO
Exclusive
76
74
72
0
(e) isolet
5
10
15
20
# of features
25
30
94
99
92
98
90
88
86
(f) ionosphere
F?statistic
ReliefF
LASSO
Exclusive
84
82
80
0
35
Feature Selection Accuracy
88
84
Feature Selection Accuracy
86
Feature Selection Accuracy
Feature Selection Accuracy
(a) RMSE on linear struc- (b) MAE on linear struc- (c) RMSE on hub structure (d) MAE on hub structure
ture
ture
20
40
60
# of features
80
(g) mnist (0,1)
100
97
96
95
94
F?statistic
ReliefF
LASSO
Exclusive
93
92
91
0
20
40
60
# of features
80
100
(h) leuml
Figure 3: (a-d): Feature selection results on synthetic dataset using (a, b) linear structure; (c, d) hub structure.
Evaluation metrics: RMSE, MAE. x-axis: number of selected features. y-axis: RMSE or MAE error in log
scale. (g-j): Classification accuracy using SVM (linear kernel) with different number of selected features on
four datasets. Compared methods: Exclusive LASSO of Eq.(7), LASSO, ReliefF [21], F-statistics [3]. x-axis:
number of selected features; y-axis: classification accuracy.
5
Experiment Results
To validate the effectiveness of our method, we first conduct experiment using Eq.(7) on two synthetic datasets, and then show experiments on real-world datasets.
5.1
Synthetic datasets
(1) Linear-correlated features. Let data X1 = [x11 , x12 , ? ? ? , x1n ] ? <p?n , X2 = [x21 , x22 , ? ? ? , x2n ] ?
<p?n , where each data x1i ? N [0p?1 , Ip?p ], x2i ? N [0p?1 , Ip?p ], I is identity matrix. We generate
a group of p-features, which is a linear combination of features in X1 and X2 , i.e., X3 = 0.5(X1 +
X2 ) + , ? N (?0.1e, 0.1Ip?p ). Construct data matrix X = [X1 ; X2 ; X3 ], clearly, X ? <3p?n .
Features in dimension [2p + 1, 3p] are highly correlated with features in dimension [1, p] and [p +
1, 2p]. Let w1 ? <p , where each wi1 ? Uniform(?0.5, 0.5), and w2 ? <p , where each wi2 ?
? = [w1 ; w2 ; 0p?1 ]. We generate predicator y ? <n , and y = w
?TX +
Uniform(?0.5, 0.5). Let w
y , where (y )i ? N (0, 0.1).
We solve Eq.(7) using current y and X with least square loss. The group settings are: (i, p+i, 2p+i),
? and plain LASSO
for 1 ? i ? p. We compare the computed w? against ground truth solution w
solution (i.e., ? = 0 in Eq.7). We use the root mean square error (RMSE) and mean absolute
error (MAE) error to evaluate the differences of values predicted by a model and the values actually
observed. We generate n = 1000 data, with p = [120, 140, ? ? ? , 220, 240] and do 5-fold cross
validation. Generalization error of RMSE and MAE are shown in Figures 3(a) and 3(b). Clearly,
our approach outperforms standard LASSO solution and exactly recovers the true features.
(2) Correlated features on Hub structure. Let data X = [X1 ; X2 ; ? ? ? , XB ] ? <q?n , where
b
b
b
each block Xb = [X1:
; X2:
; ? ? ? ; Xp:
] ? <p?n , 1 ? b ? B, q = p ? B. In each block, for
P
b
b
b
+ B1 zi + bi , where Xji
? N (0, 1), zi ?
each data point 1 ? i ? n, X1i = B1 2?j?p Xji
b
1
2
B
p
b
N (0, 1) and i ? Uniform(?0.1, 0.1). Let w , w , ? ? ? , w ? < , where w = [w1b 0]T , where
? = [w1 ; w2 ; ? ? ? ; wB ], we generate predicator y ? <n , and
w1b ? Uniform(?0.5, 0.5). Let w
T
?
y = w X + y , where (y )i ? N (0, 0.1).
The group settings are: ((b ? 1) ? p + 1, ? ? ? , b ? p), for 1 ? b ? B. We generate n = 1000 data,
B = 10, with varied p = [20, 21, ? ? ? , 24, 25] and do 5-fold cross validation. Generalization error of
RMSE and MAE are shown in Figs.3(c),3(d). Clearly, our approach outperforms standard LASSO
solution, and recovers the exact features.
6
5.2
Real-world datasets
To validate the effectiveness of proposed method, we perform feature selection via proposed uncorrelated feature learning framework of Eq.(7) on 4 datasets (shown in Table.1), including 2 UCI
datasets: isolet [6], ionosphere [5], 1 image dataset: mnist with only figure ?0? and ?1? [16], and 1
biology dataset: Leuml [13].
We perform classification tasks on these different datasets. The compared methods include: proposed method of Eq.(7) (shown as Exclusive), plain LASSO, ReliefF [21], F-statistics [3]. We use
logistic regression as the loss function in our method and plain LASSO method. In our method,
parameter ?, ? are tuned to select different numbers of features. Exclusive LASSO groups are set
according to feature correlations (i.e., threshold ? is set to 0.90 in Eq.8). After the specific number
of features are selected, we feed them into support vector machine (SVM) with linear kernel, and
classification results with different number of selected features are shown in Fig.(3).
A first glance at the experimental results indicates the better performance of our method as compared to plain LASSO. Moreover, our method is also generally better than the other two popularly
used feature selection methods, such as ReliefF and F-statistics. The experiment result also further
confirms our intuition: elimination of correlated features is really helpful for feature selection and
thus improves the classification performance. Because `1,? [20], `2,1 [12, 18], or non-convex feature learning via `p,? [11](0 < p < 1) are designed for multi-task or multi-label feature learning,
thus we do not compare against these methods.
Further, we list the mean and variance of classification accuracy of different algorithms in the following table, using 50% of all the features. Compared methods include (1) Lasso (L1); (2) Plain
exclusive group LASSO (? = 0 in Eq. (7)); (3) Exclusive group LASSO (? > 0 in Eq. (7)).
dataset
isolet
ionosphere
mnist(0,1)
leuml
# of features
308
17
392
1785
LASSO
81.75 ? 0.49
85.10 ? 0.27
92.35 ? 0.13
95.10 ? 0.31
plain exclusive
82.05 ? 0.50
85.21 ? 0.31
93.07 ? 0.20
95.67? 0.24
exclusive group LASSO
83.24 ? 0.23
87.28 ? 0.42
94.51 ? 0.19
97.70 ? 0.27
The above experiment results indicate that the advantage of our method (exclusive group LASSO)
over plain LASSO comes from the exclusive LASSO term. The experiment results also suggest that
the plain exclusive LASSO performs very similar to LASSO. However, the exclusive group LASSO
(? > 0 in Eq.7) performs definitely better than both standard LASSO and plain exclusive LASSO
(1%-4% performance improvement). The exclusive LASSO regularizer eliminates the correlated
and redundant features.
We show the running time of plain exclusive LASSO and exclusive group LASSO (? > 0 in Eq.7)
in the following table. We run different algorithms on a Intel i5-3317 CPU, 1.70GHz, 8GRAM
desktop.
dataset
isolet
ionosphere
mnist(0,1)
leuml
plain exclusive (running time: sec)
47.24
22.75
123.45
142.19
exclusive group LASSO (running time: sec)
51.93
24.18
126.51
144.08
The above experiment results indicate that the computational cost of exclusive group LASSO is
slightly higher than that of plain exclusive LASSO. The reason is that, the solution to exclusive
group LASSO is given by simple thresholding on the plain exclusive LASSO result. This further
confirms our theoretical analysis results shown in Lemma 4.1.
6
Conclusion
In this paper, we propose a new formulation called ?exclusive group LASSO? to enforce the sparsity
for features at an intra-group level. We investigate its properties and propose an effective algorithm
with rigorous convergence analysis. We show applications for uncorrelated feature selection, which
indicate the good performance of proposed method. Our work can be easily extended for multi-task
or multi-label learning.
Acknowledgement The majority of the work was done during the internship of the first author at NEC Laboratories America, Cupertino, CA.
7
Appendix
Proof of a valid norm of ?GE : Note if ?GE (w) = 0, then w = 0. For any scalar a, ?GE (aw) =
|a|?GE (w). This proves absolute homogeneity and zero property hold. Next we consider triangle
? ? <p . Letq
? g be optimal decomposition of w, w
? such that
inequality. qConsider w, w
vg and v
P
P
G
2 , and ?G (w)
2 . Since v + v
?
?
?
=
is
a
decomposition
of w + w,
?E (w) =
kv
k
k?
v
k
g
g
g 1
g 1
E
g
g
qP
qP
qP
2
? ?
? g k21 ?
thus we have: 1 ?GE (w + w)
vg k21 = ?GE (w) +
g kvg + v
g kvg k1 +
g k?
?
?GE (w).
u
?
To prove Theorem 3.1, we need two lemmas.
Lemma 6.1. Under the updating rule of Eq.(6), J2 (wt+1 ) < J2 (wt ).
Lemma 6.2. Under the updating rule of Eq.(6),
J1 (wt+1 ) ? J1 (wt ) ? J2 (wt+1 ) ? J2 (wt ) .
(13)
Proof of Theorem 3.1 From Lemma 6.1 and Lemma 6.2, it is easy to see J1 (w
t+1
t
) ? J1 (w ) ?
u
?
Proof of Lemma 6.1 Eq.(4) is a convex function, and optimal solution of Eq.(6) is obtained by
?
t+1
2
taking derivative ?J
) < J2 (wt ).
u
?
?w = 0, thus obtained w is global optimal solution, J2 (w
Before proof of Lemma 6.2, we need the following Proposition.
P
2
Proposition 6.3. wT Fw = G
g=1 (kwGg k1 ) .
0. This completes the proof.
Proof of Lemma 6.2 Let ? = LHS -RHS of Eq.(13). We have ?, where
?=
X
k21 ?
kwGt+1
g
g
=
X
g
X (IGg )i kwGt g k1 t+1 2 X (IGg )i kwGt g k1 t 2 X
(wi ) +
(wi ) ?
kwGt g k21
|wit |
|wit |
g
i,g
i,g
(14)
X
t
t+1
X
X
X
X
(IGg )i kwGg k1 t+1 2
(wi )2
t+1 2
t
2
(w
)
=
)
(
|w
|)
?
(
|w
|)(
k
?
kwGt+1
1
i
i
i
g
|wit |
|wit |
g
i,g
i?G
i?G
i?G
g
g
g
(15)
X X
X 2 X 2
bi ) ? 0,
=
(
ai bi )2 ? (
ai )(
g
i?Gg
t+1
|w
where ai = ?i
i?Gg
|
(16)
i?Gg
p
|wit |. Due to proposition 6.3, Eq.(14) is equivalent to Eq.(15). Eq.(16)
P
P
P
holds due to cauchy inequality [23]: for any scalar ai , bi , ( i ai bi )2 ? ( i a2i )( i b2i ).
u
?
P
Proof of Lemma 4.1 For notation simplicity, let ?GEg (w) = g kwGg k21 . Let w? be the optimal
solution to Eq.(11), then we have
|wit |
, bi =
0 ? w? ? u + ???GEg (w? ).
In order to prove that w? is also the global optimal solution to Eq.(10), i.e.,
(17)
0 ? w? ? a + ?SGN(w? ) + ???GEg (w? ).
(18)
First, from Eq.(12), we have 0 ? u ? a + ?SGN(u), and this leads to u ? a ? ?SGN(u). According
to the definition of ?GEg (w), from Eq.(11), it is easy to verify that (1) if ui = 0, then wi = 0; (2) if
ui 6= 0, then sign(wi ) = sign(ui ) and 0 ? |wi | ? |ui |. This indicates that SGN(u) ? SGN(w),
and thus
u ? a ? ?SGN(w).
(19)
Put Eqs.(17, 19) together, and this exactly recovers Eq.(18), which completes the proof.
1
(
Note the derivation needs Cauchy inequality [23], where for any scalar ai , bi , (
2 P
2
vg k1 , then we can get the inequality.
g ag )(
g bg ). Let ag = kvg k1 , bg = k?
P
8
P
g
ag bg )2
?
References
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9
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5,117 | 5,632 | Flexible Transfer Learning under Support and Model
Shift
Jeff Schneider
Robotics Institute
Carnegie Mellon University
[email protected]
Xuezhi Wang
Computer Science Department
Carnegie Mellon University
[email protected]
Abstract
Transfer learning algorithms are used when one has sufficient training data for
one supervised learning task (the source/training domain) but only very limited
training data for a second task (the target/test domain) that is similar but not identical to the first. Previous work on transfer learning has focused on relatively
restricted settings, where specific parts of the model are considered to be carried over between tasks. Recent work on covariate shift focuses on matching
the marginal distributions on observations X across domains. Similarly, work on
target/conditional shift focuses on matching marginal distributions on labels Y
and adjusting conditional distributions P (X|Y ), such that P (X) can be matched
across domains. However, covariate shift assumes that the support of test P (X)
is contained in the support of training P (X), i.e., the training set is richer than the
test set. Target/conditional shift makes a similar assumption for P (Y ). Moreover,
not much work on transfer learning has considered the case when a few labels in
the test domain are available. Also little work has been done when all marginal
and conditional distributions are allowed to change while the changes are smooth.
In this paper, we consider a general case where both the support and the model
change across domains. We transform both X and Y by a location-scale shift to
achieve transfer between tasks. Since we allow more flexible transformations, the
proposed method yields better results on both synthetic data and real-world data.
1
Introduction
In a classical transfer learning setting, we have sufficient fully labeled data from the source domain
(or the training domain) where we fully observe the data points X tr , and all corresponding labels
Y tr are known. On the other hand, we are given data points, X te , from the target domain (or the
test domain), but few or none of the corresponding labels, Y te , are given. The source and the target
domains are related but not identical, thus the joint distributions, P (X tr , Y tr ) and P (X te , Y te ), are
different across the two domains. Without any transfer learning, a statistical model learned from the
source domain does not directly apply to the target domain. The use of transfer learning algorithms
minimizes, or reduces the labeling work needed in the target domain. It learns and transfers a
model based on the labeled data from the source domain and the data with few or no labels from
the target domain, and should perform well on the unlabeled data in the target domain. Some realworld applications of transfer learning include adapting a classification model that is trained on
some products to help learn classification models for other products [17], and learning a model on
the medical data for one disease and transferring it to another disease.
The real-world application we consider is an autonomous agriculture application where we want to
manage the growth of grapes in a vineyard [3]. Recently, robots have been developed to take images
of the crop throughout the growing season. When the product is weighed at harvest at the end of
each season, the yield for each vine will be known. The measured yield can be used to learn a model
1
to predict yield from images. Farmers would like to know their yield early in the season so they
can make better decisions on selling the produce or nurturing the growth. Acquiring training labels
early in the season is very expensive because it requires a human to go out and manually estimate the
yield. Ideally, we can apply a transfer-learning model which learns from previous years and/or on
other grape varieties to minimize this manual yield estimation. Furthermore, if we decide that some
of the vines have to be assessed manually to learn the model shift, a simultaneously applied active
learning algorithm will tell us which vines should be measured manually such that the labeling cost
is minimized. Finally, there are two different objectives of interest. To better nurture the growth we
need an accurate estimate of the current yield of each vine. However, to make informed decisions
about pre-selling an appropriate amount of the crops, only an estimate of the sum of the vine yields
is needed. We call these problems active learning and active surveying respectively and they lead to
different selection criteria.
In this paper, we focus our attention on real-valued regression problems. We propose a transfer
learning algorithm that allows both the support on X and Y , and the model P (Y |X) to change
across the source and target domains. We assume only that the change is smooth as a function
of X. In this way, more flexible transformations are allowed than mean-centering and variancescaling. Specifically, we build a Gaussian Process to model the prediction on the transformed X,
then the prediction is matched with a few observed labels Y (also properly transformed) available
in the target domain such that both transformations on X and on Y can be learned. The GP-based
approach naturally lends itself to the active learning setting where we can sequentially choose query
points from the target dataset. Its final predictive covariance, which combines the uncertainty in the
transfer function and the uncertainty in the target label prediction, can be plugged into various GP
based active query selection criteria. In this paper we consider (1) Active Learning which reduces
total predictive covariance [18, 19]; and (2) Active Surveying [20, 21] which uses an estimation
objective that is the sum of all the labels in the test set.
As an illustration, we show a toy problem in Fig. 1. As we can see, the support of P (X) in the
training domain (red stars) and the support of P (X) in the test domain (blue line) do not overlap,
neither do the support of Y across the two domains. The goal is to learn a model on the training
data, with a few labeled test data (the filled blue circles), such that we can successfully recover the
target function (the blue line). In Fig. 3, we show two real-world grape image datasets. The goal
is to transfer the model learned from one kind of grape dataset to another. In Fig. 2, we show the
labels (the yield) of each grape image dataset, along with the 3rd dimension of its feature space.
We can see that the real-world problem is quite similar to the toy problem, which indicates that the
algorithm we propose in this paper will be both useful and practical for real applications.
Synthetic Data
3
6000
source data
underlying P(source)
underlying P(target)
selected test x
2
5000
Riesling
Traminette
4000
Y
labels
1
0
3000
2000
?1
1000
?2
?4
?2
0
X
2
4
Figure 1: Toy problem
6
0
?1.5
?1
?0.5
0
0.5
3rd dimension of the real grape data
Figure 2: Real grape data
1
Figure 3: A part of one image
from each grape dataset
We evaluate our methods on synthetic data and real-world grape image data. The experimental
results show that our transfer learning algorithms significantly outperform existing methods with
few labeled target data points.
2
Related Work
Transfer learning is applied when joint distributions differ across source and target domains. Traditional methods for transfer learning use Markov logic networks [4], parameter learning [5, 6], and
Bayesian Network structure learning [7], where specific parts of the model are considered to be
carried over between tasks.
2
Recently, a large part of transfer learning work has focused on the problem of covariate shift [8,
9, 10]. They consider the case where only P (X) differs across domains, while the conditional
distribution P (Y |X) stays the same. The kernel mean matching (KMM) method [9, 10], is one of
the algorithms that deal with covariate shift. It minimizes ||?(Pte ) ? Ex?Ptr (x) [?(x)?(x)]|| over a
re-weighting vector ? on training data points such that P (X) are matched across domains. However,
this work suffers two major problems. First, the conditional distribution P (Y |X) is assumed to be
the same, which might not be true under many real-world cases. The algorithm we propose will
allow more than just the marginal on X to shift. Second, the KMM method requires that the support
of P (X te ) is contained in the support of P (X tr ), i.e., the training set is richer than the test set. This
is not necessarily true in many real cases either. Consider the task of transferring yield prediction
using images taken from different vineyards. If the images are taken from different grape varieties
or during different times of the year, the texture/color could be very different across transferring
tasks. In these cases one might mean-center (and possibly also variance-scale) the data to ensure
that the support of P (X te ) is contained in (or at least largely overlapped with) P (X tr ). In this
paper, we provide an alternative way to solve the support shift problem that allows more flexible
transformations than mean-centering and variance-scaling.
Some more recent research [12] has focused on modeling target shift (P (Y ) changes), conditional
shift (P (X|Y ) changes), and a combination of both. The assumption on target shift is that X depends causally on Y , thus P (Y ) can be re-weighted to match the distributions on X across domains.
In conditional shift, the authors apply a location-scale transformation on P (X|Y ) to match P (X).
However, the authors still assume that the support of P (Y te ) is contained in the support of P (Y tr ).
In addition, they do not assume they can obtain additional labels, Y te , from the target domain, and
thus make no use of the labels Y te , even if some are available.
There also have been a few papers handling differences in P (Y |X). [13] designed specific methods
(change of representation, adaptation through prior, and instance pruning) to solve the label adaptation problem. [14] relaxed the requirement that the training and testing examples be drawn from
the same source distribution in the context of logistic regression. Similar to work on covariate shift,
[15] weighted the samples from the source domain to deal with domain adaptation. These settings
are relatively restricted while we consider a more general case that both the data points X and the
corresponding labels Y can be transformed smoothly across domains. Hence all data will be used
without any pruning or weighting, with the advantage that the part of source data which does not
help prediction in the target domain will automatically be corrected via the transformation model.
The idea of combining transfer learning and active learning has also been studied recently. Both
[22] and [23] perform transfer and active learning in multiple stages. The first work uses the source
data without any domain adaptation. The second work performs domain adaptation at the beginning
without further refinement. [24] and [25] consider active learning under covariate shift and still
assume P (Y |X) stays the same. In [16], the authors propose a combined active transfer learning
algorithm to handle the general case where P (Y |X) changes smoothly across domains. However,
the authors still apply covariate shift algorithms to solve the problem that P (X) might differ across
domains, which follows the assumption covariate shift made on the support of P (X). In this paper,
we propose an algorithm that allows more flexible transformations (location-scale transform on both
X and Y ). Our experiments on real-data shows this additional flexibility pays off in real applications.
3
3.1
Approach
Problem Formulation
We are given a set of n labeled training data points, (X tr , Y tr ), from the source domain where
each Xitr ? <dx and each Yitr ? <dy . We are also given a set of m test data points, X te , from
the target domain. Some of these will have corresponding labels, Y teL . When necessary we will
separately denote the subset of X te that has labels as X teL , and the subset that does not as X teU .
For simplicity we restrict Y to be univariate in this paper, but the algorithm we proposed easily
extends to the multivariate case.
For static transfer learning, the goal is to learn a predictive model using all the given data that
te 2
? te
?
minimizes the squared prediction error on the test data, ?m
i=1 (Yi ? Yi ) where Yi and Yi are the
predicted and true labels for the ith test data point. We will evaluate the transfer learning algorithms
3
by including a subset of labeled test data chosen uniformly at random. For active transfer learning
the performance metric is the same. The difference is that the active learning algorithm chooses the
test points for labeling rather than being given a randomly chosen set.
3.2
Transfer Learning
Our strategy is to simultaneously learn a nonlinear mapping X te ? X new and Y te ? Y ? . This
allows flexible transformations on both X and Y , and our smoothness assumption using GP prior
makes the estimation stable. We call this method Support and Model Shift (SMS).
We apply the following steps (K in the following represents the Gaussian kernel, and KXY represents the kernel between matrices X and Y , ? ensures invertible kernel matrix):
? Transform X teL to X new(L) by a location-scale shift: X new(L) = WteL X teL + BteL ,
such that the support of P (X new(L) ) is contained in the support of P (X tr );
? Build a Gaussian Process on (X tr , Y tr ) and predict on X new(L) to get Y new(L) ;
? Transform Y teL to Y ? by a location-scale shift: Y ? = wteL Y teL + bteL , then we
optimize the following empirical loss:
arg
min
WteL ,BteL ,wteL ,bteL ,wte
||Y ? ? Y new(L) ||2 + ?reg ||wte ? 1||2 ,
(1)
where WteL , BteL are matrices with the same size as X teL . wteL , bteL are vectors with the same
size as Y teL (l by 1, where l is the number of labeled samples in the target domain), and wte is an
m by 1 scale vector on all Y te . ?reg is a regularization parameter.
To ensure the smoothness of the transformation w.r.t. X, we parameterize WteL , BteL , wteL , bteL
using: WteL = RteL G, BteL = RteL H, wteL = RteL g, bteL = RteL h, where RteL =
LteL (LteL + ?I)?1 , LteL = KX teL X teL . Following the same smoothness constraint we also have:
wte = Rte g, where Rte = KX te X teL (LteL + ?I)?1 . This parametrization results in the new
objective function:
arg min ||(RteL g Y teL + RteL h) ? Y new(L) ||2 + ?reg ||Rte g ? 1||2 .
G,H,g,h
(2)
In the objective function, although we minimize the discrepancy between the transformed labels and
the predicted labels for only the labeled points in the test domain, we put a regularization term on
the transformation for all X te to ensure overall smoothness in the test domain. Note that the nonlinearity of the transformation makes the SMS approach capable of recovering a fairly wide set of
changes, including non-monotonic ones. However, because of the smoothness constraint imposed
on the location-scale transformation, it might not recover some extreme cases where the scale or
location change is non-smooth/discontinuous. However, under these cases the learning problem by
itself would be very challenging.
We use a Metropolis-Hasting algorithm to optimize the objective (Eq. 2) which is multi-modal due
to the use of the Gaussian kernel. The proposal distribution is given by ?t ? N (?t?1 , ?), where ?
is a diagonal matrix with diagonal elements determined by the magnitude of ? ? {G, H, g, h}. In
addition, the transformation on X requires that the support of P (X new ) is contained in the support
of P (X tr ), which might be hard to achieve on real data, especially when X has a high-dimensional
feature space. To ensure that the training data can be better utilized, we relax the support-containing
condition by enforcing an overlapping ratio between the transformed X new and X tr , i.e., we reject
those proposal distributions which do not lead to a transformation that exceeds this ratio.
After obtaining G, H, g, h, we make predictions on X teU by:
? Transform X teU to X new(U ) with the optimized G, H: X new(U ) = WteU X teU +
BteU = RteU G X teU + RteU H;
? Build a Gaussian Process on (X tr , Y tr ) and predict on X new(U ) to get Y new(U ) ;
? Predict using optimized g, h: Y? teU = (Y new(U ) ? bteU )./wteU = (Y new(U ) ?
RteU h)./RteU g,
4
where RteU = KX teU X teL (LteL + ?I)?1 .
With the use of W = RG, B = RH, w = Rg, b = Rh, we allow more flexible transformations
than mean-centering and variance-scaling while assuming that the transformations are smooth w.r.t
X. We will illustrate the advantage of the proposed method in the experimental section.
3.3
A Kernel Mean Embedding Point of View
After the transformation from X teL to X new(L) , we build a Gaussian Process on (X tr , Y tr ) and
predict on X new(L) to get Y new(L) . This is equivalent to estimating ?
?[PY new(L) ] using conditional
? Y tr |X tr ]?
?[PX new(L) ] =
distribution embeddings [11] with a linear kernel on Y : ?
?[PY new(L) ] = U[P
?(ytr )(?(xtr )> ?(xtr ) + ?I)?1 ?> (xtr )?(xnew(L) ) = (KX new(L) X tr (KX tr X tr + ?I)?1 Y tr )> .
Finally we want to find the optimal G, H, g, h such that the distributions on Y are matched across
domains, i.e., PY ? = PY new(L) . The objective function Eq. 2 is effectively minimizing the maximum
? Y tr |X tr ]?
mean discrepancy: ||?
?[PY ? ] ? ?
?[PY new(L) ]||2 = ||?
?[PY ? ] ? U[P
?[PX new(L) ]||2 , with a
Gaussian kernel on X and a linear kernel on Y .
The transformation {W, B, w, b} are smooth w.r.t X. Take w for example, ?
?[Pw ] =
? w|X teL ]?
?[PX teL ] = ?(g)(?> (xteL )?(xteL ) + ?I)?1 ?> (xteL )?(xteL ) = ?(g)(LteL +
U[P
?I)?1 LteL = (RteL g)> .
3.4
Active Learning
We consider two active learning goals and apply a myopic selection criteria to each:
(1) Active Learning which reduces total predictive covariance [18, 19]. An optimal myopic selection
is achieved by choosing the point which minimizes the trace of the predictive covariance matrix
conditioned on that selection.
(2) Active Surveying [20, 21] which uses an estimation objective that is the sum of all the labels in
the test set. An optimal myopic selection is achieved by choosing the point which minimizes the
sum over all elements of the predictive covariance conditioned on that selection.
Now we derive the predictive covariance of the SMS approach. Note the transformation between
Y? teU and Y new(U ) is given by: Y? teU = (Y new(U ) ? bteU )./wteU . Hence we have Cov[Y? teU ] =
diag{1./wteU } ? Cov(Y new(U ) ) ? diag{1./wteU }.
As for Y new(U ) , since we build on Gaussian Processes for the prediction from X new(U ) to Y new(U ) ,
it follows: Y new(U ) |X new(U ) ? N (?, ?), where ? = KX new(U ) X tr (KX tr X tr + ?I)?1 Y tr , and
? = KX new(U ) X new(U ) ? KX new(U ) X tr (KX tr X tr + ?I)?1 KX tr X new(U ) .
Note the transformation between X new(U ) and X teU is given by:
X new(U )
=
teU
teU
teU
new(U )
new(U
)
teU
?
W
X
+
B
.
Integrating
over
X
,
i.e.,
P
(
Y
|X
,
D)
=
R
teU
new(U )
new(U )
teU
new(U )
tr
tr
teL
teL
?
P
(
Y
|X
,
D)P
(X
|X
)dX
,
with
D
=
{X
,
Y
,
X
,
Y
}.
new(U )
X
Using the empirical form of P (X new(U ) |X teU ) which has probability 1/|X teU | for each sample,
we get: Cov[Y? new(U ) |X teU , X tr , Y tr , X teL , Y teL ] = ?. Plugging the covariance of Y new(U ) into
Cov[Y? teU ] we can get the final predictive covariance:
Cov(Y? teU ) = diag{1./wteU } ? ? ? diag{1./wteU }
4
(3)
Experiments
4.1
4.1.1
Synthetic Dataset
Data Description
We generate the synthetic data with (using matlab notation): X tr = randn(80, 1), Y tr =
sin(2X tr +1)+0.1?randn(80, 1); X te = [w ?min(X tr )+b : 0.03 : w ?max(X tr )/3+b], Y te =
sin(2(revw ? X te + revb ) + 1) + 2. In words, X tr is drawn from a standard normal distribution,
and Y tr is a sine function with Gaussian noise. X te is drawn from a uniform distribution with a
5
location-scale transform on a subset of X tr . Y te is the same sine function plus a constant offset.
The synthetic dataset used is with w = 0.5; b = 5; revw = 2; revb = ?10, as shown in Fig. 1.
4.1.2
Results
We compare the SMS approach with the following approaches:
(1) Only test x: prediction using labeled test data only;
(2) Both x: prediction using both the training data and labeled test data without transformation;
(3) Offset: the offset approach [16];
(4) DM: the distribution matching approach [16];
(5) KMM: Kernel mean matching [9];
(6) T/C shift: Target/Conditional shift [12], code is from http://people.tuebingen.mpg.de/kzhang/
Code-TarS.zip.
To ensure the fairness of comparison, we apply (3) to (6) using: the original data, the meancentered data, and the mean-centered+variance-scaled (mean-var-centered) data.
A detailed comparison with different number of labeled test points are shown in Fig. 4, averaged
over 10 experiments. The selection of which test points to label is done uniformly at random for each
experiment. The parameters are chosen by cross-validation. Since KMM and T/C shift do not utilize
the labeled test points, the MSE of these two approaches are constants as shown in the text box. As
we can see from the results, our proposed approach performs better than all other approaches.
As an example, the results for transfer learning with 5 labeled test points on the synthetic dataset
are shown in Fig. 5. The 5 labeled test points are shown as filled blue circles. First, our proposed
model, SMS, can successfully learn both the transformation on X and the transformation on Y , thus
resulting in almost a perfect fit on unlabeled test points. Using only labeled test points results in a
poor fit towards the right part of the function because there are no observed test labels in that part.
Using both training and labeled test points results in a similar fit as using the labeled test points
only, because the support of training and test domain do not overlap. The offset approach with
mean-centered+variance-scaled data, also results in a poor fit because the training model is not true
any more. It would have performed well if the variances are similar across domains. The support
of the test data we generated, however, only consists of part of the support of the training data and
hence simple variance-scaling does not yield a good match on P (Y |X). The distribution matching
approach suffers the same problem. The KMM approach, as mentioned before, applies the same
conditional model P (Y |X) across domains, hence it does not perform well. The Target/Conditional
Shift approach does not perform well either since it does not utilize any of the labeled test points.
Its predicted support of P (Y te ), is constrained in the support of P (Y tr ), which results in a poor
prediction of Y te once there exists an offset between the Y ?s.
Mean Sqaured Error
3
0.8
SMS
use only test x
use both x
offset (original)
offset (mean?centered)
offset (mean?var?centered)
DM (original)
DM (mean?centered)
DM (mean?var?centered)
0.06
2.5
0.6
2
0.04
1.5
0.4
1
0.02
0.2
0.5
0
0
2
0
5
10
original mean mean?var
KMM 4.46 2.25 4.63
T/C
1.97 3.51 4.71
Figure 4: Comparison of MSE on the synthetic dataset with {2, 5, 10} labeled test points
4.2
4.2.1
Real-world Dataset
Data Description
We have two datasets with grape images taken from vineyards and the number of grapes on them as
labels, one is riesling (128 labeled images), another is traminette (96 labeled images), as shown in
Figure 3. The goal is to transfer the model learned from one kind of grape dataset to another. The
total number of grapes for these two datasets are 19, 253 and 30, 360, respectively.
6
SMS
4
use only labeled test x
6
source data
target
selected test x
prediction
3
2
4
2
Y
Y
1
source data
target
selected test x
prediction
0
0
?1
?2
?4
?2
0
2
4
X
offset (mean?var?centered data)
?2
?4
6
?2
0
X
2
4
6
DM (mean?var?centered data)
3
3
2
Y
1
1
0
0
?1
?1
?2
?2
?1
0
X
1
2
?2
?2
3
KMM/TC Shift (mean?centered data)
3
?1
X
1
2
2
3
source data
target
selected test x
prediction (KMM)
prediction (T/C shift)
1
?1
0
1
2
0
?1
X
3
0
?2
?2
0
Y
Y
1
?1
KMM/TC Shift (mean?var?centered data)
source data
target
selected test x
prediction (KMM)
prediction (T/C shift)
2
source data
target
selected test x
prediction (p=1e?3)
prediction (p=0.1)
2
Y
source data
target
selected test x
prediction (w=1)
prediction (w=5)
?2
?2
3
?1
0
X
1
2
3
Figure 5: Comparison of results on the synthetic dataset: An example
We extract raw-pixel features from the images, and use Random Kitchen Sinks [1] to get the coefficients as feature vectors [2], resulting in 2177 features. On the traminette dataset we have achieved
a cross-validated R-squared correlation of 0.754. Previously specifically designed image processing
methods have achieved an R-squared correlation 0.73 [3]. This grape-detection method takes lots
of manual labeling work and cannot be directly applied across different varieties of grapes (due to
difference in size and color). Our proposed approach for transfer learning, however, can be directly
used for different varieties of grapes or even different kinds of crops.
4.2.2
Results
The results for transfer learning are shown in Table 1. We compare the SMS approach with the
same baselines as in the synthetic experiments. For {DM, offset, KMM, T/C shift}, we only
show their best results after applying them on the original data, the mean-centered data, and the
mean-centered+variance-scaled data. In each row the result in bold indicates the result with the
best RMSE. The result with a star mark indicates that the best result is statistically significant at a
p = 0.05 level with unpaired t-tests. We can see that our proposed algorithm yields better results under most cases, especially when the number of labeled test points is small. This means our proposed
algorithm can better utilize the source data and will be particularly useful in the early stage of learning model transfer, when only a small number of labels in the target domain is available/required.
The Active Learning/Active Surveying results are as shown in Fig. 6. We compare the SMS approach (covariance matrix in Eq. 3 for test point selection, and SMS for prediction) with:
(1) combined+SMS: combined covariance [16] for selection, and SMS for prediction;
(2) random+SMS: random selection, and SMS for prediction;
(3) combined+offset: the Active Learning/Surveying algorithm proposed in [16], using combined
covariance for selection, and the corresponding offset approach for prediction.
7
From the results we can see that SMS is the best model overall. SMS is better than the Active
Learning/Surveying approach proposed in [16] (combined+offset), especially in the Active Surveying result. Moreover, the combined+SMS result is better than combined+offset, which also indicates
that the SMS model is better for prediction than the offset approach in [16]. Also, given the better
model that SMS has, there is not much difference in which active learning algorithm we use. However, SMS with active selection is better than SMS with random selection, especially in the Active
Learning result.
Table 1: RMSE for transfer learning on real data
#X
5
10
15
20
25
30
40
50
70
90
teL
SMS
1197?23?
1046?35?
993?28
985?13
982?14
960?19
890?26
893?16
860?40
791?98
DM
1359?54
1196?59
1055?27
1056?54
1030?29
921?29
898?30
925?59
805?38
838?102
Offset
1303?39
1234?53
1063?30
1024?20
1040 ?27
961?30
938?30
935?59
819?40
863?99
Only test x
1479?69
1323?91
1104?46
1086?74
1039?31
937?29
901?31
926?64
804?37
838?104
Active Learning
1400
RMSE
1200
4
x 10
SMS
combined+SMS
random+SMS
combined+offset
2.5
Absolute Error
1600
1000
800
600
KMM
2127
2127
2127
2127
2127
2127
2127
2127
2127
2127
T/C Shift
2330
2330
2330
2330
2330
2330
2330
2330
2330
2330
Active Surveying
SMS
combined+SMS
random+SMS
combined+offset
1.5
1
0.5
400
200
0
2
Both x
2094?60
1939?41
1916?36
1832?46
1839?41
1663?31
1621?34
1558?51
1399?63
1288?117
5
10
15
Number of labeled test points
0
0
20
5
10
15
Number of labeled test points
20
Figure 6: Active Learning/Surveying results on the real dataset (legend: selection+prediction).
5
Discussion and Conclusion
Solving objective Eq. 2 is relatively involved. Gradient methods can be a faster alternative but the
non-convex property of the objective makes it harder to find the global optimum using gradient
methods. In practice we find it is relatively efficient to solve Eq. 2 with proper initializations (like
using the ratio of scale on the support for w, and the offset between the scaled-means for b). In our
real-world dataset with 2177 features, it takes about 2.54 minutes on average in a single-threaded
MATLAB process on a 3.1 GHz CPU with 8 GB RAM to solve the objective and recover the transformation. As part of the future work we are working on faster ways to solve the proposed objective.
In this paper, we proposed a transfer learning algorithm that handles both support and model shift
across domains. The algorithm transforms both X and Y by a location-scale shift across domains,
then the labels in these two domains are matched such that both transformations can be learned.
Since we allow more flexible transformations than mean-centering and variance-scaling, the proposed method yields better results than traditional methods. Results on both synthetic dataset and
real-world dataset show the advantage of our proposed method.
Acknowledgments
This work is supported in part by the US Department of Agriculture under grant number
20126702119958.
8
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9
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5,118 | 5,633 | Texture Synthesis Using Convolutional Neural
Networks
Leon A. Gatys
Centre for Integrative Neuroscience, University of T?ubingen, Germany
Bernstein Center for Computational Neuroscience, T?ubingen, Germany
Graduate School of Neural Information Processing, University of T?ubingen, Germany
[email protected]
Alexander S. Ecker
Centre for Integrative Neuroscience, University of T?ubingen, Germany
Bernstein Center for Computational Neuroscience, T?ubingen, Germany
Max Planck Institute for Biological Cybernetics, T?ubingen, Germany
Baylor College of Medicine, Houston, TX, USA
Matthias Bethge
Centre for Integrative Neuroscience, University of T?ubingen, Germany
Bernstein Center for Computational Neuroscience, T?ubingen, Germany
Max Planck Institute for Biological Cybernetics, T?ubingen, Germany
Abstract
Here we introduce a new model of natural textures based on the feature spaces
of convolutional neural networks optimised for object recognition. Samples from
the model are of high perceptual quality demonstrating the generative power of
neural networks trained in a purely discriminative fashion. Within the model, textures are represented by the correlations between feature maps in several layers of
the network. We show that across layers the texture representations increasingly
capture the statistical properties of natural images while making object information more and more explicit. The model provides a new tool to generate stimuli
for neuroscience and might offer insights into the deep representations learned by
convolutional neural networks.
1
Introduction
The goal of visual texture synthesis is to infer a generating process from an example texture, which
then allows to produce arbitrarily many new samples of that texture. The evaluation criterion for the
quality of the synthesised texture is usually human inspection and textures are successfully synthesised if a human observer cannot tell the original texture from a synthesised one.
In general, there are two main approaches to find a texture generating process. The first approach is
to generate a new texture by resampling either pixels [5, 28] or whole patches [6, 16] of the original
texture. These non-parametric resampling techniques and their numerous extensions and improvements (see [27] for review) are capable of producing high quality natural textures very efficiently.
However, they do not define an actual model for natural textures but rather give a mechanistic procedure for how one can randomise a source texture without changing its perceptual properties.
In contrast, the second approach to texture synthesis is to explicitly define a parametric texture
model. The model usually consists of a set of statistical measurements that are taken over the
1
Figure 1: Synthesis method. Texture analysis (left). The original texture is passed through the CNN
and the Gram matrices Gl on the feature responses of a number of layers are computed. Texture
? is passed through the CNN and a loss function El is
synthesis (right). A white noise image ~x
computed on every layer included in the texture model. The total loss function L is a weighted sum
of the contributions El from each layer. Using gradient descent on the total loss with respect to the
? l as the original texture.
pixel values, a new image is found that produces the same Gram matrices G
spatial extent of the image. In the model a texture is uniquely defined by the outcome of those
measurements and every image that produces the same outcome should be perceived as the same
texture. Therefore new samples of a texture can be generated by finding an image that produces the
same measurement outcomes as the original texture. Conceptually this idea was first proposed by
Julesz [13] who conjectured that a visual texture can be uniquely described by the Nth-order joint
histograms of its pixels. Later on, texture models were inspired by the linear response properties
of the mammalian early visual system, which resemble those of oriented band-pass (Gabor) filters
[10, 21]. These texture models are based on statistical measurements taken on the filter responses
rather than directly on the image pixels. So far the best parametric model for texture synthesis
is probably that proposed by Portilla and Simoncelli [21], which is based on a set of carefully
handcrafted summary statistics computed on the responses of a linear filter bank called Steerable
Pyramid [24]. However, although their model shows very good performance in synthesising a wide
range of textures, it still fails to capture the full scope of natural textures.
In this work, we propose a new parametric texture model to tackle this problem (Fig. 1). Instead
of describing textures on the basis of a model for the early visual system [21, 10], we use a convolutional neural network ? a functional model for the entire ventral stream ? as the foundation for
our texture model. We combine the conceptual framework of spatial summary statistics on feature
responses with the powerful feature space of a convolutional neural network that has been trained on
object recognition. In that way we obtain a texture model that is parameterised by spatially invariant
representations built on the hierarchical processing architecture of the convolutional neural network.
2
2
Convolutional neural network
We use the VGG-19 network, a convolutional neural network trained on object recognition that was
introduced and extensively described previously [25]. Here we give only a brief summary of its
architecture.
We used the feature space provided by the 16 convolutional and 5 pooling layers of the VGG-19
network. We did not use any of the fully connected layers. The network?s architecture is based on
two fundamental computations:
1. Linearly rectified convolution with filters of size 3 ? 3 ? k where k is the number of input
feature maps. Stride and padding of the convolution is equal to one such that the output
feature map has the same spatial dimensions as the input feature maps.
2. Maximum pooling in non-overlapping 2?2 regions, which down-samples the feature maps
by a factor of two.
These two computations are applied in an alternating manner (see Fig. 1). A number of convolutional
layers is followed by a max-pooling layer. After each of the first three pooling layers the number of
feature maps is doubled. Together with the spatial down-sampling, this transformation results in a
reduction of the total number of feature responses by a factor of two. Fig. 1 provides a schematic
overview over the network architecture and the number of feature maps in each layer. Since we
use only the convolutional layers, the input images can be arbitrarily large. The first convolutional
layer has the same size as the image and for the following layers the ratio between the feature map
sizes remains fixed. Generally each layer in the network defines a non-linear filter bank, whose
complexity increases with the position of the layer in the network.
The trained convolutional network is publicly available and its usability for new applications is
supported by the caffe-framework [12]. For texture generation we found that replacing the maxpooling operation by average pooling improved the gradient flow and one obtains slightly cleaner
results, which is why the images shown below were generated with average pooling. Finally, for
practical reasons, we rescaled the weights in the network such that the mean activation of each filter
over images and positions is equal to one. Such re-scaling can always be done without changing the
output of a neural network as long as the network is fully piece-wise linear 1 .
3
Texture model
The texture model we describe in the following is much in the spirit of that proposed by Portilla
and Simoncelli [21]. To generate a texture from a given source image, we first extract features of
different sizes homogeneously from this image. Next we compute a spatial summary statistic on the
feature responses to obtain a stationary description of the source image (Fig. 1A). Finally we find a
new image with the same stationary description by performing gradient descent on a random image
that has been initialised with white noise (Fig. 1B).
The main difference to Portilla and Simoncelli?s work is that instead of using a linear filter bank
and a set of carefully chosen summary statistics, we use the feature space provided by a highperforming deep neural network and only one spatial summary statistic: the correlations between
feature responses in each layer of the network.
To characterise a given vectorised texture ~x in our model, we first pass ~x through the convolutional
neural network and compute the activations for each layer l in the network. Since each layer in the
network can be understood as a non-linear filter bank, its activations in response to an image form a
set of filtered images (so-called feature maps). A layer with Nl distinct filters has Nl feature maps
each of size Ml when vectorised. These feature maps can be stored in a matrix F l ? RNl ?Ml , where
l
Fjk
is the activation of the j th filter at position k in layer l. Textures are per definition stationary,
so a texture model needs to be agnostic to spatial information. A summary statistic that discards
the spatial information in the feature maps is given by the correlations between the responses of
1
Source code to generate textures with CNNs as well as the rescaled VGG-19 network can be found at
http://github.com/leongatys/DeepTextures
3
different features. These feature correlations are, up to a constant of proportionality, given by the
Gram matrix Gl ? RNl ?Nl , where Glij is the inner product between feature map i and j in layer l:
X
l
l
Glij =
Fik
Fjk
.
(1)
k
1
2
L
A set of Gram matrices {G , G , ..., G } from some layers 1, . . . , L in the network in response to
a given texture provides a stationary description of the texture, which fully specifies a texture in our
model (Fig. 1A).
4
Texture generation
To generate a new texture on the basis of a given image, we use gradient descent from a white noise
image to find another image that matches the Gram-matrix representation of the original image.
This optimisation is done by minimising the mean-squared distance between the entries of the Gram
matrix of the original image and the Gram matrix of the image being generated (Fig. 1B).
? be the original image and the image that is generated, and Gl and G
? l their respective
Let ~x and ~x
Gram-matrix representations in layer l (Eq. 1). The contribution of layer l to the total loss is then
2
X
1
? lij
El =
Glij ? G
(2)
2
2
4Nl Ml i,j
and the total loss is
?) =
L(~x, ~x
L
X
wl El
(3)
l=0
where wl are weighting factors of the contribution of each layer to the total loss. The derivative of
El with respect to the activations in layer l can be computed analytically:
(
1
? l )T Gl ? G
?l
(
F
if F?ijl > 0
?El
2
2
Nl Ml
ji
=
(4)
? F? l
0
if F? l < 0 .
ij
ij
?), with respect to the pixels ~x
? can be readily
The gradients of El , and thus the gradient of L(~x, ~x
?L
computed using standard error back-propagation [18]. The gradient ? ~x? can be used as input for
some numerical optimisation strategy. In our work we use L-BFGS [30], which seemed a reasonable
choice for the high-dimensional optimisation problem at hand. The entire procedure relies mainly
on the standard forward-backward pass that is used to train the convolutional network. Therefore, in
spite of the large complexity of the model, texture generation can be done in reasonable time using
GPUs and performance-optimised toolboxes for training deep neural networks [12].
5
Results
We show textures generated by our model from four different source images (Fig. 2). Each row of
images was generated using an increasing number of layers in the texture model to constrain the
gradient descent (the labels in the figure indicate the top-most layer included). In other words, for
the loss terms above a certain layer we set the weights wl = 0, while for the loss terms below
and including that layer, we set wl = 1. For example the images in the first row (?conv1 1?) were
generated only from the texture representation of the first layer (?conv1 1?) of the VGG network. The
images in the second row (?pool1?) where generated by jointly matching the texture representations
on top of layer ?conv1 1?, ?conv1 2? and ?pool1?. In this way we obtain textures that show what
structure of natural textures are captured by certain computational processing stages of the texture
model.
The first three columns show images generated from natural textures. We find that constraining all
layers up to layer ?pool4? generates complex natural textures that are almost indistinguishable from
the original texture (Fig. 2, fifth row). In contrast, when constraining only the feature correlations
on the lowest layer, the textures contain little structure and are not far from spectrally matched noise
4
Figure 2: Generated stimuli. Each row corresponds to a different processing stage in the network.
When only constraining the texture representation on the lowest layer, the synthesised textures have
little structure, similarly to spectrally matched noise (first row). With increasing number of layers on
which we match the texture representation we find that we generate images with increasing degree of
naturalness (rows 2?5; labels on the left indicate the top-most layer included). The source textures in
the first three columns were previously used by Portilla and Simoncelli [21]. For better comparison
we also show their results (last row). The last column shows textures generated from a non-texture
image to give a better intuition about how the texture model represents image information.
5
Figure 3: A, Number of parameters in the texture model. We explore several ways to reduce the
number of parameters in the texture model (see main text) and compare the results. B, Textures
generated from the different layers of the caffe reference network [12, 15]. The textures are of
lesser quality than those generated with the VGG network. C, Textures generated with the VGG
architecture but random weights. Texture synthesis fails in this case, indicating that learned filters
are crucial for texture generation.
(Fig. 2, first row). We can interpolate between these two extremes by using only the constraints
from all layers up to some intermediate layer. We find that the statistical structure of natural images
is matched on an increasing scale as the number of layers we use for texture generation increases.
We did not include any layers above layer ?pool4? since this did not improve the quality of the
synthesised textures. For comparability we used source textures that were previously used by Portilla
and Simoncelli [21] and also show the results of their texture model (Fig. 2, last row). 2
To give a better intuition for how the texture synthesis works, we also show textures generated from
a non-texture image taken from the ImageNet validation set [23] (Fig. 2, last column). Our algorithm
produces a texturised version of the image that preserves local spatial information but discards the
global spatial arrangement of the image. The size of the regions in which spatial information is
preserved increases with the number of layers used for texture generation. This property can be
explained by the increasing receptive field sizes of the units over the layers of the deep convolutional
neural network.
When using summary statistics from all layers of the convolutional neural network, the number
of parameters of the model is very large. For each layer with Nl feature maps, we match Nl ?
(Nl + 1)/2 parameters, so if we use all layers up to and including ?pool4?, our model has ? 852k
parameters (Fig. 3A, fourth column). However, we find that this texture model is heavily overparameterised. In fact, when using only one layer on each scale in the network (i.e. ?conv1 1?,
2
A curious finding is that the yellow box, which indicates the source of the original texture, is also placed
towards the bottom left corner in the textures generated by our model. As our texture model does not store
any spatial information about the feature responses, the only possible explanation for such behaviour is that
some features in the network explicitly encode the information at the image boundaries. This is exactly what
we find when inspecting feature maps in the VGG network: Some feature maps, at least from layer ?conv3 1?
onwards, only show high activations along their edges. This might originate from the zero-padding that is used
for the convolutions in the VGG network and it could be interesting to investigate the effect of such padding on
learning and object recognition performance.
6
Classification performance
1.0
0.8
0.6
0.4
top1 Gram
top5 Gram
top1 VGG
top5 VGG
0.2
0
pool1
pool2
pool3
pool4
Decoding layer
pool5
Figure 4: Performance of a linear classifier on top of the texture representations in different layers in
classifying objects from the ImageNet dataset. High-level information is made increasingly explicit
along the hierarchy of our texture model.
and ?pool1-4?), the model contains ? 177k parameters while hardly loosing any quality (Fig. 3A,
third column). We can further reduce the number of parameters by doing PCA of the feature vector
in the different layers of the network and then constructing the Gram matrix only for the first k
principal components. By using the first 64 principal components for layers ?conv1 1?, and ?pool14? we can further reduce the model to ? 10k parameters (Fig. 3A, second column). Interestingly,
constraining only the feature map averages in layers ?conv1 1?, and ?pool1-4?, (1024 parameters),
already produces interesting textures (Fig. 3A, first column). These ad hoc methods for parameter
reduction show that the texture representation can be compressed greatly with little effect on the
perceptual quality of the synthesised textures. Finding minimal set of parameters that reproduces
the quality of the full model is an interesting topic of ongoing research and beyond the scope of the
present paper. A larger number of natural textures synthesised with the ? 177k parameter model
can be found in the Supplementary Material as well as on our website3 . There one can also observe
some failures of the model in case of very regular, man-made structures (e.g. brick walls).
In general, we find that the very deep architecture of the VGG network with small convolutional
filters seems to be particularly well suited for texture generation purposes. When performing the
same experiment with the caffe reference network [12], which is very similar to the AlexNet [15], the
quality of the generated textures decreases in two ways. First, the statistical structure of the source
texture is not fully matched even when using all constraints (Fig 3B, ?conv5?). Second, we observe
an artifactual grid that overlays the generated textures (Fig 3B). We believe that the artifactual grid
originates from the larger receptive field sizes and strides in the caffe reference network.
While the results from the caffe reference network show that the architecture of the network is
important, the learned feature spaces are equally crucial for texture generation. When synthesising
a texture with a network with the VGG architecture but random weights, texture generation fails
(Fig. 3C), underscoring the importance of using a trained network.
To understand our texture features better in the context of the original object recognition task of the
network, we evaluated how well object identity can be linearly decoded from the texture features
in different layers of the network. For each layer we computed the Gram-matrix representation of
each image in the ImageNet training set [23] and trained a linear soft-max classifier to predict object
identity. As we were not interested in optimising prediction performance, we did not use any data
augmentation and trained and tested only on the 224 ? 224 centre crop of the images. We computed
the accuracy of these linear classifiers on the ImageNet validation set and compared them to the
performance of the original VGG-19 network also evaluated on the 224 ? 224 centre crops of the
validation images.
The analysis suggests that our texture representation continuously disentangles object identity information (Fig. 4). Object identity can be decoded increasingly well over the layers. In fact, linear
decoding from the final pooling layer performs almost as well as the original network, suggesting
that our texture representation preserves almost all high-level information. At first sight this might
appear surprising since the texture representation does not necessarily preserve the global structure
of objects in non-texture images (Fig. 2, last column). However, we believe that this ?inconsis3
www.bethgelab.org/deeptextures
7
tency? is in fact to be expected and might provide an insight into how CNNs encode object identity.
The convolutional representations in the network are shift-equivariant and the network?s task (object
recognition) is agnostic to spatial information, thus we expect that object information can be read
out independently from the spatial information in the feature maps. We show that this is indeed the
case: a linear classifier on the Gram matrix of layer ?pool5? comes close to the performance of the
full network (87.7% vs. 88.6% top 5 accuracy, Fig. 4).
6
Discussion
We introduced a new parametric texture model based on a high-performing convolutional neural
network. Our texture model exceeds previous work as the quality of the textures synthesised using
our model shows a substantial improvement compared to the current state of the art in parametric
texture synthesis (Fig. 2, fourth row compared to last row).
While our model is capable of producing natural textures of comparable quality to non-parametric
texture synthesis methods, our synthesis procedure is computationally more expensive. Nevertheless, both in industry and academia, there is currently much effort taken in order to make the evaluation of deep neural networks more efficient [11, 4, 17]. Since our texture synthesis procedure
builds exactly on the same operations, any progress made in the general field of deep convolutional
networks is likely to be transferable to our texture synthesis method. Thus we expect considerable
improvements in the practical applicability of our texture model in the near future.
By computing the Gram matrices on feature maps, our texture model transforms the representations
from the convolutional neural network into a stationary feature space. This general strategy has
recently been employed to improve performance in object recognition and detection [9] or texture
recognition and segmentation [3]. In particular Cimpoi et al. report impressive performance in
material recognition and scene segmentation by using a stationary Fisher-Vector representation built
on the highest convolutional layer of readily trained neural networks [3]. In agreement with our
results, they show that performance in natural texture recognition continuously improves when using
higher convolutional layers as the input to their Fisher-Vector representation. As our main aim is
to synthesise textures, we have not evaluated the Gram matrix representation on texture recognition
benchmarks, but would expect that it also provides a good feature space for those tasks.
In recent years, texture models inspired by biological vision have provided a fruitful new analysis
tool for studying visual perception. In particular the parametric texture model proposed by Portilla and Simoncelli [21] has sparked a great number of studies in neuroscience and psychophysics
[8, 7, 1, 22, 20]. Our texture model is based on deep convolutional neural networks that are the
first artificial systems that rival biology in terms of difficult perceptual inference tasks such as object recognition [15, 25, 26]. At the same time, their hierarchical architecture and basic computational properties admit a fundamental similarity to real neural systems. Together with the increasing
amount of evidence for the similarity of the representations in convolutional networks and those in
the ventral visual pathway [29, 2, 14], these properties make them compelling candidate models for
studying visual information processing in the brain. In fact, it was recently suggested that textures
generated from the representations of performance-optimised convolutional networks ?may therefore prove useful as stimuli in perceptual or physiological investigations? [19]. We feel that our
texture model is the first step in that direction and envision it to provide an exciting new tool in the
study of visual information processing in biological systems.
Acknowledgments
This work was funded by the German National Academic Foundation (L.A.G.), the Bernstein Center
for Computational Neuroscience (FKZ 01GQ1002) and the German Excellency Initiative through
the Centre for Integrative Neuroscience T?ubingen (EXC307)(M.B., A.S.E, L.A.G.)
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9
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5,119 | 5,634 | Convolutional Neural Networks with Intra-layer
Recurrent Connections for Scene Labeling
Ming Liang
Xiaolin Hu
Bo Zhang
Tsinghua National Laboratory for Information Science and Technology (TNList)
Department of Computer Science and Technology
Center for Brain-Inspired Computing Research (CBICR)
Tsinghua University, Beijing 100084, China
[email protected], {xlhu,dcszb}@tsinghua.edu.cn
Abstract
Scene labeling is a challenging computer vision task. It requires the use of both
local discriminative features and global context information. We adopt a deep
recurrent convolutional neural network (RCNN) for this task, which is originally
proposed for object recognition. Different from traditional convolutional neural
networks (CNN), this model has intra-layer recurrent connections in the convolutional layers. Therefore each convolutional layer becomes a two-dimensional
recurrent neural network. The units receive constant feed-forward inputs from the
previous layer and recurrent inputs from their neighborhoods. While recurrent
iterations proceed, the region of context captured by each unit expands. In this
way, feature extraction and context modulation are seamlessly integrated, which
is different from typical methods that entail separate modules for the two steps.
To further utilize the context, a multi-scale RCNN is proposed. Over two benchmark datasets, Standford Background and Sift Flow, the model outperforms many
state-of-the-art models in accuracy and efficiency.
1
Introduction
Scene labeling (or scene parsing) is an important step towards high-level image interpretation. It
aims at fully parsing the input image by labeling the semantic category of each pixel. Compared
with image classification, scene labeling is more challenging as it simultaneously solves both segmentation and recognition. The typical approach for scene labeling consists of two steps. First,
extract local handcrafted features [6, 15, 26, 23, 27]. Second, integrate context information using
probabilistic graphical models [6, 5, 18] or other techniques [24, 21]. In recent years, motivated by
the success of deep neural networks in learning visual representations, CNN [12] is incorporated into this framework for feature extraction. However, since CNN does not have an explicit mechanism
to modulate its features with context, to achieve better results, other methods such as conditional
random field (CRF) [5] and recursive parsing tree [21] are still needed to integrate the context information. It would be interesting to have a neural network capable of performing scene labeling in an
end-to-end manner.
A natural way to incorporate context modulation in neural networks is to introduce recurrent connections. This has been extensively studied in sequence learning tasks such as online handwriting
recognition [8], speech recognition [9] and machine translation [25]. The sequential data has strong
correlations along the time axis. Recurrent neural networks (RNN) are suitable for these tasks because the long-range context information can be captured by a fixed number of recurrent weights.
Treating scene labeling as a two-dimensional variant of sequence learning, RNN can also be applied,
but the studies are relatively scarce. Recently, a recurrent CNN (RCNN) in which the output of the
top layer of a CNN is integrated with the input in the bottom is successfully applied to scene labeling
1
Patch-wise training
Extract patch and resize
Valid convolutions
concatenate
?
classify
? Softmax
??
??
?boat?
Cross entropy loss
label
Image-wise test
Same convolutions
Concatenate
Upsample
Classify
{? ? }
Downsample
Upsample
Figure 1: Training and testing processes of multi-scale RCNN for scene labeling. Solid lines denote
feed-forward connections and dotted lines denote recurrent connections.
[19]. Without the aid of extra preprocessing or post-processing techniques, it achieves competitive
results. This type of recurrent connections captures both local and global information for labeling
a pixel, but it achieves this goal indirectly as it does not model the relationship between pixels (or
the corresponding units in the hidden layers of CNN) in the 2D space explicitly. To achieve the goal
directly, recurrent connections are required to be between units within layers. This type of RCNN
has been proposed in [14], but there it is used for object recognition. It is unknown if it is useful for
scene labeling, a more challenging task. This motivates the present work.
A prominent structural property of RCNN is that feed-forward and recurrent connections co-exist
in multiple layers. This property enables the seamless integration of feature extraction and context
modulation in multiple levels of representation. In other words, an RCNN can be seen as a deep
RNN which is able to encode the multi-level context dependency. Therefore we expect RCNN to be
competent for scene labeling.
Multi-scale is another technique for capturing both local and global information for scene labeling
[5]. Therefore we adopt a multi-scale RCNN [14]. An RCNN is used for each scale. See Figure 1 for
its overall architecture. The networks in different scales have exactly the same structure and weights.
The outputs of all networks are concatenated and input to a softmax layer. The model operates in an
end-to-end fashion, and does not need any preprocessing or post-processing techniques.
2
Related Work
Many models, either non-parametric [15, 27, 3, 23, 26] or parametric [6, 13, 18], have been proposed
for scene labeling. A comprehensive review is beyond the scope of this paper. Below we briefly
review the neural network models for scene labeling.
In [5], a multi-scale CNN is used to extract local features for scene labeling. The weights are shared
among the CNNs for all scales to keep the number of parameters small. However, the multi-scale
scheme alone has no explicit mechanism to ensure the consistency of neighboring pixels? labels.
Some post-processing techniques, such as superpixels and CRF, are shown to significantly improve
the performance of multi-scale CNN. In [1], CNN features are combined with a fully connected
CRF for more accurate segmentations. In both models [5, 1] CNN and CRF are trained in separated
stages. In [29] CRF is reformulated and implemented as an RNN, which can be jointly trained with
CNN by back-propagation (BP) algorithm.
In [24], a recursive neural network is used to learn a mapping from visual features to the semantic
space, which is then used to determine the labels of pixels. In [21], a recursive context propagation
2
network (rCPN) is proposed to better make use of the global context information. The rCPN is fed a
superpixel representation of CNN features. Through a parsing tree, the rCPN recursively aggregates
context information from all superpixels and then disseminates it to each superpixel. Although
recursive neural network is related to RNN as they both use weight sharing between different layers,
they have significant structural difference. The former has a single path from the input layer to the
output layer while the latter has multiple paths [14]. As will be shown in Section 4, this difference
has great influence on the performance in scene labeling.
To the best of our knowledge, the first end-to-end neural network model for scene labeling refers
to the deep CNN proposed in [7]. The model is trained by a supervised greedy learning strategy.
In [19], another end-to-end model is proposed. Top-down recurrent connections are incorporated
into a CNN to capture context information. In the first recurrent iteration, the CNN receives a raw
patch and outputs a predicted label map (downsampled due to pooling). In other iterations, the CNN
receives both a downsampled patch and the label map predicted in the previous iteration and then
outputs a new predicted label map. Compared with the models in [5, 21], this approach is simple
and elegant but its performance is not the best on some benchmark datasets. It is noted that both
models in [14] and [19] are called RCNN. For convenience, in what follows, if not specified, RCNN
refers to the model in [14].
3
3.1
Model
RCNN
The key module of the RCNN is the RCL. A generic RNN with feed-forward input u(t), internal
state x(t) and parameters ? can be described by:
x(t) = F(u(t), x(t ? 1), ?)
(1)
where F is the function describing the dynamic behavior of RNN.
The RCL introduces recurrent connections into a convolutional layer (see Figure 2A for an illustration). It can be regarded as a special two-dimensional RNN, whose feed-forward and recurrent
computations both take the form of convolution.
xijk (t) = ? (wkf )> u(i,j) (t) + (wkr )> x(i,j) (t ? 1) + bk
(2)
where u(i,j) and x(i,j) are vectorized square patches centered at (i, j) of the feature maps of the
previous layer and the current layer, wkf and wkr are the weights of feed-forward and recurrent
connections for the kth feature map, and bk is the kth element of the bias. ? used in this paper
is composed of two functions ?(zijk ) = h(g(zijk )), where g is the widely used rectified linear
function g(zijk ) = max (zijk , 0), and h is the local response normalization (LRN) [11]:
g(zijk )
h(g(zijk )) = ?
?1 +
??
min(K,k+L/2)
X
?
L
(3)
(g(zijk0 ))2 ?
k0 =max(0,k?L/2)
where K is the number of feature maps, ? and ? are constants controlling the amplitude of normalization. The LRN forces the units in the same location to compete for high activities, which mimics
the lateral inhibition in the cortex. In our experiments, LRN is found to consistently improve the
accuracy, though slightly. Following [11], ? and ? are set to 0.001 and 0.75, respectively. L is set
to K/8 + 1.
During the training or testing phase, an RCL is unfolded for T time steps into a multi-layer subnetwork. T is a predetermined hyper-parameter. See Figure 2B for an example with T = 3. The
receptive field (RF) of each unit expands with larger T , so that more context information is captured. The depth of the subnetwork also increases with larger T . In the meantime, the number of
parameters is kept constant due to weight sharing.
Let u0 denote the static input (e.g., an image). The input to the RCL, denoted by u(t), can take this
constant u0 for all t. But here we adopt a more general form:
u(t) = ?u0
3
(4)
An RCL unit (red)
Unfold a RCL
Multiplicatively unfold two RCLs
Additively unfold two RCLs
RCNN
128
pooling
64
pooling
32
A
B
C
D
E
Figure 2: Illustration of the RCL and RCNN used in this paper. Sold arrows denote feed-forward
connections and dotted arrows denote recurrent connections.
where ? ? [0, 1] is a discount factor, which determines the tradeoff between the feed-forward component and the recurrent component. When ? = 0, the feed-forward component is totally discarded
after the first iteration. In this case the network behaves like the so-called recursive convolutional
network [4], in which several convolutional layers have tied weights. There is only one path from
input to output. When ? > 0, the network is a typical RNN. There are multiple paths from input to
output (see Figure 2B).
RCNN is composed of a stack of RCLs. Between neighboring RCLs there are only feed-forward
connections. Max pooling layers are optionally interleaved between RCLs. The total number of
recurrent iterations is set to T for all N RCLs. There are two approaches to unfold an RCNN.
First, unfold the RCLs one by one, and each RCL is unfolded for T time steps before feeding to
the next RCL (see Figure 2C). This unfolding approach multiplicatively increases the depth of the
network. The largest depth of the network is proportional to N T . In the second approach, at each
time step the states of all RCLs are updated successively (see Figure 2D). The unfolded network
has a two-dimensional structure whose x axis is the time step and y axis is the level of layer. This
unfolding approach additively increases the depth of the network. The largest depth of the network
is proportional to N + T .
We adopt the first unfolding approach due to the following advantages. First, it leads to larger
effective RF and depth, which are important for the performance of the model. Second, the second
approach is more computationally intensive since the feed-forward inputs need to be updated at each
time step. However, in the first approach the feed-forward input of each RCL needs to be computed
for only once.
3.2
Multi-scale RCNN
In natural scenes objects appear in various sizes. To capture this variability, the model should be
scale invariant. In [5], a multi-scale CNN is proposed to extract features for scene labeling, in which
several CNNs with shared weights are used to process images of different scales. This approach is
adopted to construct the multi-scale RCNN (see Figure 1). The original image corresponds to the
finest scale. Images of coarser scales are obtained simply by max pooling the original image. The
outputs of all RCNNs are concatenated to form the final representation. For pixel p, its probability
falling into the cth semantic category is given by a softmax layer:
exp wc> f p
p
yc = P
(c = 1, 2, ..., C)
(5)
> p
c0 exp wc0 f
where f p denotes the concatenated feature vector of pixel p, and wc denotes the weight for the cth
category.
The loss function is the cross entropy between the predicted probability ycp and the true hard label
y?cp :
XX
L=?
y?cp log ycp
(6)
p
c
where y?cp = 1 if pixel p is labeld as c and y?cp = 0 otherwise. The model is trained by backpropagation
through time (BPTT) [28], that is, unfolding all the RCNNs to feed-forward networks and apply the
BP algorithm.
4
3.3
Patch-wise Training and Image-wise Testing
Most neural network models for scene labeling [5, 19, 21] are trained by the patch-wise approach.
The training samples are randomly cropped image patches whose labels correspond to the categories
of their center pixels. Valid convolutions are used in both feed-forward and recurrent computation.
The patch is set to a proper size so that the last feature map has exactly the size of 1 ? 1. In
image-wise training, an image is input to the model and the output has exactly the same size as the
image. The loss is the average of all pixels? loss. We have conducted experiments with both training
methods, and found that image-wise training seriously suffered from over-fitting. A possible reason
is that the pixels in an image have too strong correlations. So patch-wise training is used in all our
experiments. In [16], it is suggested that image-wise and patch-wise training are equally effective
and the former is faster to converge. But their model is obtained by finetuning the VGG [22] model
pretrained on ImageNet [2]. This conclusion may not hold for models trained from scratch.
In the testing phase, the patch-wise approach is time consuming because the patches corresponding
to all pixels need to be processed. We therefore use image-wise testing. There are two image-wise
testing approaches to obtain dense label maps. The first is the Shift-and-stitch approach [20, 19].
When the predicted label map is downsampled by a factor of s, the original image will be shifted
and processed for s2 times. At each time, the image is shifted by (x, y) pixels to the right and
down. Both x and y take their value from {0, 1, 2, . . . , s ? 1}, and the shifted image is padded
in their left and top borders with zero. The outputs for all shifted images are interleaved so that
each pixel has a corresponding prediction. Shift-and-stitch approach needs to process the image for
s2 times although it produces the exact prediction as the patch-wise testing. The second approach
inputs the entire image to the network and obtains downsampled label map, then simply upsample
the map to the same resolution as the input image, using bilinear or other interpolation methods (see
Figure 1, bottom). This approach may suffer from the loss of accuracy, but is very efficient. The
deconvolutional layer proposed in [16] is adopted for upsampling, which is the backpropagation
counterpart of the convolutional layer. The deconvolutional weights are set to simulates the bilinear
interpolation. Both of the image-wise testing methods are used in our experiments.
4
4.1
Experiments
Experimental Settings
Experiments are performed over two benchmark datasets for scene labeling, Sift Flow [15] and
Stanford Background [6]. The Sift Flow dataset contains 2688 color images, all of which have the
size of 256 ? 256 pixels. Among them 2488 images are training data, and the remaining 200 images
are testing data. There are 33 semantic categories, and the class frequency is highly unbalanced.
The Stanford background dataset contains 715 color images, most of them have the size of 320 ?
240 pixels. Following [6] 5-fold cross validation is used over this dataset. In each fold there are
572 training images and 143 testing images. The pixels have 8 semantic categories and the class
frequency is more balanced than the Sift Flow dataset.
In most of our experiments, RCNN has three parameterized layers (Figure 2E). The first parameterized layer is a convolutional layer followed by a 2 ? 2 non-overlapping max pooling layer. This
is to reduce the size of feature maps and thus save the computing cost and memory. The other two
parameterized layers are RCLs. Another 2 ? 2 max pooling layer is placed between the two RCLs.
The numbers of feature maps in these layers are 32, 64 and 128. The filter size in the first convolutional layer is 7 ? 7, and the feed-forward and recurrent filters in RCLs are all 3 ? 3. Three scales
of images are used and neighboring scales differed by a factor of 2 in each side of the image.
The models are implemented using Caffe [10]. They are trained using stochastic gradient descent
algorithm. For the Sift Flow dataset, the hyper-parameters are determined on a separate validation
set. The same set of hyper-parameters is then used for the Stanford Background dataset. Dropout
and weight decay are used to prevent over-fitting. Two dropout layers are used, one after the second
pooling layer and the other before the concatenation of different scales. The dropout ratio is 0.5 and
weight decay coefficient is 0.0001. The base learning rate is 0.001, which is reduced to 0.0001 when
the training error enters a plateau. Overall, about ten millions patches have been input to the model
during training.
5
Data augmentation is used in many models [5, 21] for scene labeling to prevent over-fitting. It is a
technique to distort the training data with a set of transformations, so that additional data is generated
to improve the generalization ability of the models. This technique is only used in Section 4.3 for
the sake of fairness in comparison with other models. Augmentation includes horizontal reflection
and resizing.
4.2
Model Analysis
We empirically analyze the performance of RCNN models for scene labeling on the Sift Flow
dataset. The results are shown in Table 1. Two metrics, the per-pixel accuracy (PA) and the average per-class accuracy (CA) are used. PA is the ratio of correctly classified pixels to the total
pixels in testing images. CA is the average of all category-wise accuracies. The following results are obtained using the shift-and-stitch testing and without any data augmentation. Note that all
models have a multi-scale architecture.
Model
RCNN, ? = 1, T = 3
RCNN, ? = 1, T = 4
RCNN, ? = 1, T = 5
RCNN-large, ? = 1, T = 3
RCNN, ? = 0, T = 3
RCNN, ? = 0, T = 4
RCNN, ? = 0, T = 5
RCNN-large, ? = 0, T = 3
RCNN, ? = 0.25, T = 5
RCNN, ? = 0.5, T = 5
RCNN, ? = 0.75, T = 5
RCNN, no share, ? = 1, T = 5
CNN1
CNN2
Patch size
232
256
256
256
232
256
256
256
256
256
256
256
88
136
No. Param.
0.28M
0.28M
0.28M
0.65M
0.28M
0.28M
0.28M
0.65M
0.28M
0.28M
0.28M
0.28M
0.33M
0.28M
PA (%)
80.3
81.6
82.3
83.4
80.5
79.9
80.4
78.1
82.4
81.8
82.8
81.3
74.9
78.5
CA (%)
31.9
33.2
34.3
38.9
34.2
31.4
31.7
29.4
35.4
34.7
35.8
33.3
24.1
28.8
Table 1: Model analysis over the Sift Flow dataset. We limit the maximum size of input patch to
256, which is the size of the image in the Sift Flow dataset. This is achieved by replacing the first
few valid convolutions by same convolutions.
First, the influence of ? in (4) is investigated. The patch sizes of images for different models are set
such that the size of the last feature map is 1 ? 1. We mainly investigate two specific values ? = 1
and ? = 0 with different iteration number T. Several other values of ? are tested with T=5. See
Table 1 for details. For RCNN with ? = 1, the performance monotonously increase with more time
steps. This is not the case for RCNN with ? = 0, with which the network tends to be over-fitting
with more iterations. To further investigate this issue, a larger model denoted as RCNN-large is
tested. It has four RCLs, and has more parameters and larger depth. With ? = 1 it achieves a better
performance than RCNN. However, the RCNN-large with ? = 0 obtains worse performance than
RCNN. When ? is set to other values, 0.25, 0.5 or 0.75, the performance seems better than ? = 1
but the difference is small.
Second, the influence of weight sharing in recurrent connections is investigated. Another RCNN
with ? = 1 and T = 5 is tested. Its recurrent weights in different iterations are not shared anymore,
which leads to more parameters than shared ones. But this setting leads to worse accuracy both for
PA and CA. A possible reason is that more parameters make the model more prone to over-fitting.
Third, two feed-forward CNNs are constructed for comparison. CNN1 is constructed by removing
all recurrent connections from RCNN, and then increasing the numbers of feature maps in each
layer from 32, 64 and 128 to 60, 120 and 240, respectively. CNN2 is constructed by removing
the recurrent connections and adding two extra convolutional layers. CNN2 had five convolutional
layers and the corresponding numbers of feature maps are 32, 64, 64, 128 and 128, respectively.
With these settings, the two models have approximately the same number of parameters as RCNN,
which is for the sake of fair comparison. The two CNNs are outperformed by the RCNNs by a
significant margin. Compared with the RCNN, the topmost units in these two CNNs cover much
smaller regions (see the patch size column in Table 1). Note that all convolutionas in these models
are performed in ?valid? mode. This mode decreases the size of feature maps and as a consequence
6
Figure 3: Examples of scene labeling results from the Stanford Background dataset. ?mntn? denotes
mountains, and ?object? denotes foreground objects.
(together with max pooling) increases the RF size of the top units. Since the CNNs have fewer
convolutional layers than the time-unfolded RCNNs, their RF sizes of the top units are smaller.
Model
Liu et al.[15]
Tighe and Lazebnik [27]
Eigen and Fergus [3]
Singh and Kosecka [23]
Tighe and Lazebnik [26]
Multi-scale CNN + cover [5]
Multi-scale CNN + cover (balanced) [5]
Top-down RCNN [19]
Multi-scale CNN + rCPN [21]
Multi-scale CNN + rCPN (balanced) [21]
RCNN
RCNN (balanced)
RCNN-small
RCNN-large
FCNN [16] (?finetuned from VGG model [22])
No. Param.
NA
NA
NA
NA
NA
0.43 M
0.43 M
0.09 M
0.80 M
0.80 M
0.28 M
0.28 M
0.07 M
0.65 M
134 M
PA (%)
76.7
77.0
77.1
79.2
78.6
78.5
72.3
77.7
79.6
75.5
83.5
79.3
81.7
84.3
85.1
CA (%)
NA
30.1
32.5
33.8
39.2
29.6
50.8
29.8
33.6
48.0
35.8
57.1
32.6
41.0
51.7
Time (s)
31 (CPU)
8.4 (CPU)
16.6 (CPU)
20 (CPU)
? 8.4 (CPU)
NA
NA
NA
0.37 (GPU)
0.37 (GPU)
0.03 (GPU)
0.03 (GPU)
0.02 (GPU)
0.04 (GPU)
? 0.33 (GPU)
Table 2: Comparison with the state-of-the-art models over the Sift Flow dataset.
4.3
Comparison with the State-of-the-art Models
Next, we compare the results of RCNN and the state-of-the-art models. The RCNN with ? = 1
and T = 5 is used for comparison. The results are obtained using the upsampling testing approach
for efficiency. Data augmentation is employed in training because it is used by many other models
[5, 21]. The images are only preprocessed by removing the average RGB values computed over
training images.
Model
Gould et al. [6]
Tighe and Lazebnik [27]
Socher et al. [24]
Eigen and Fergus [3]
Singh and Kosecka [23]
Lempitsky et al. [13]
Multiscale CNN + CRF [5]
Top-down RCNN [19]
Single-scale CNN + rCPN [21]
Multiscale CNN + rCPN [21]
Zoom-out [17]
RCNN
No. Param.
NA
NA
NA
NA
NA
NA
0.43M
0.09M
0.80M
0.80M
0.23 M
0.28M
PA (%)
76.4
77.5
78.1
75.3
74.1
81.9
81.4
80.2
81.9
81.0
82.1
83.1
CA (%)
NA
NA
NA
66.5
62.2
72.4
76.0
69.9
73.6
78.8
77.3
74.8
Time (s)
30 to 60 (CPU)
12 (CPU)
NA
16.6 (CPU)
20 (CPU)
? 60 (CPU)
60.5 (CPU)
10.6 (CPU)
0.5 (GPU)
0.37 (GPU)
NA
0.03 (GPU)
Table 3: Comparison with the state-of-the-art models over the Stanford Background dataset.
The results over the Sift Flow dataset are shown in Table 2. Besides the PA and CA, the time for
processing an image is also presented. For neural network models, the number of parameters are
7
shown. When extra training data from other datasets is not used, the RCNN outperforms all other
models in terms of the PA metric by a significant margin.
The RCNN has fewer parameters than most of the other neural network models except the top-down
RCNN [19]. A small RCNN (RCNN-small) is then constructed by reducing the numbers of feature
maps in RCNN to 16, 32 and 64, respectively, so that its total number of parameters is 0.07 million.
The PA and CA of the small RCNN are 81.7% and 32.6%, respectively, significantly higher than
those of the top-down RCNN.
Note that better result over this dataset has been achieved by the fully convolutional network (FCN)
[16]. However, FCN is finetuned from the VGG [22] net trained over the 1.2 million images of
ImageNet, and has approximately 134 million parameters. Being trained over 2488 images, RCNN
is only outperformed by 1.6 percent on PA. This gap can be further reduced by using larger RCNN
models. For example, the RCNN-large in Table 1 achieves PA of 84.3% with data augmentation.
The class distribution in the Sift Flow dataset is highly unbalanced, which is harmful to the CA
performance. In [5], frequency balance is used so that patches in different classes appear in the
same frequency. This operation greatly enhance the CA value. For better comparison, we also test
an RCNN with weighted sampling (balanced) so that the rarer classes apprear more frequently. In
this case, the RCNN achieves a much higher CA than other methods including FCN, while still
keeping a good PA.
The results over the Stanford Background dataset are shown in Table 3. The set of hyper-parameters
used for the Sift Flow dataset is adopted without further tuning. Frequency balance is not used. The
RCNN again achieves the best PA score, although CA is not the best. Some typical results of RCNN
are shown in Figure 3.
On a GTX Titan black GPU, it takes about 0.03 second for the RCNN and 0.02 second for the
RCNN-small to process an image. Compared with other models, the efficiency of RCNN is mainly
attributed to its end-to-end property. For example, the rCPN model takes much time in obtaining the
superpixels.
5
Conclusion
A multi-scale recurrent convolutional neural network is used for scene labeling. The model is able to
perform local feature extraction and context integration simultaneously in each parameterized layer,
therefore particularly fits this application because both local and global information are critical for
determining the label of a pixel in an image. This is an end-to-end approach and can be simply
trained by the BPTT algorithm. Experimental results over two benchmark datasets demonstrate the
effectiveness and efficiency of the model.
Acknowledgements
We are grateful to the anonymous reviewers for their valuable comments. This work was supported
in part by the National Basic Research Program (973 Program) of China under Grant 2012CB316301
and Grant 2013CB329403, in part by the National Natural Science Foundation of China under Grant
61273023, Grant 91420201, and Grant 61332007, in part by the Natural Science Foundation of
Beijing under Grant 4132046.
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5,120 | 5,635 | Grammar as a Foreign Language
Oriol Vinyals?
Google
[email protected]
Terry Koo
Google
[email protected]
Lukasz Kaiser?
Google
[email protected]
Slav Petrov
Google
[email protected]
Ilya Sutskever
Google
[email protected]
Geoffrey Hinton
Google
[email protected]
Abstract
Syntactic constituency parsing is a fundamental problem in natural language processing and has been the subject of intensive research and engineering for decades.
As a result, the most accurate parsers are domain specific, complex, and inefficient. In this paper we show that the domain agnostic attention-enhanced
sequence-to-sequence model achieves state-of-the-art results on the most widely
used syntactic constituency parsing dataset, when trained on a large synthetic corpus that was annotated using existing parsers. It also matches the performance
of standard parsers when trained only on a small human-annotated dataset, which
shows that this model is highly data-efficient, in contrast to sequence-to-sequence
models without the attention mechanism. Our parser is also fast, processing over
a hundred sentences per second with an unoptimized CPU implementation.
1
Introduction
Syntactic constituency parsing is a fundamental problem in linguistics and natural language processing that has a wide range of applications. This problem has been the subject of intense research
for decades, and as a result, there exist highly accurate domain-specific parsers. The computational
requirements of traditional parsers are cubic in sentence length, and while linear-time shift-reduce
constituency parsers improved in accuracy in recent years, they never matched state-of-the-art. Furthermore, standard parsers have been designed with parsing in mind; the concept of a parse tree is
deeply ingrained into these systems, which makes these methods inapplicable to other problems.
Recently, Sutskever et al. [1] introduced a neural network model for solving the general sequenceto-sequence problem, and Bahdanau et al. [2] proposed a related model with an attention mechanism
that makes it capable of handling long sequences well. Both models achieve state-of-the-art results
on large scale machine translation tasks (e.g., [3, 4]). Syntactic constituency parsing can be formulated as a sequence-to-sequence problem if we linearize the parse tree (cf. Figure 2), so we can
apply these models to parsing as well.
Our early experiments focused on the sequence-to-sequence model of Sutskever et al. [1]. We found
this model to work poorly when we trained it on standard human-annotated parsing datasets (1M
tokens), so we constructed an artificial dataset by labelling a large corpus with the BerkeleyParser.
?
Equal contribution
1
(S
.
(VP
XX
LSTM3in
LSTM3out
LSTM2in
LSTM2out
LSTM1in
LSTM1out
Go
(S
END
(VP
.
)S
END
)VP
.
)S
)VP
XX
Figure 1: A schematic outline of a run of our LSTM+A model on the sentence ?Go.?. See text for details.
To our surprise, the sequence-to-sequence model matched the BerkeleyParser that produced the
annotation, having achieved an F1 score of 90.5 on the test set (section 23 of the WSJ).
We suspected that the attention model of Bahdanau et al. [2] might be more data efficient and we
found that it is indeed the case. We trained a sequence-to-sequence model with attention on the small
human-annotated parsing dataset and were able to achieve an F1 score of 88.3 on section 23 of the
WSJ without the use of an ensemble and 90.5 with an ensemble, which matches the performance of
the BerkeleyParser (90.4) when trained on the same data.
Finally, we constructed a second artificial dataset consisting of only high-confidence parse trees, as
measured by the agreement of two parsers. We trained a sequence-to-sequence model with attention
on this data and achieved an F1 score of 92.1 on section 23 of the WSJ, matching the state-of-the-art.
This result did not require an ensemble, and as a result, the parser is also very fast.
2
LSTM+A Parsing Model
Let us first recall the sequence-to-sequence LSTM model. The Long Short-Term Memory model of
[5] is defined as follows. Let xt , ht , and mt be the input, control state, and memory state at timestep
t. Given a sequence of inputs (x1 , . . . , xT ), the LSTM computes the h-sequence (h1 , . . . , hT ) and
the m-sequence (m1 , . . . , mT ) as follows.
it
i0t
ft
ot
mt
ht
=
=
=
=
=
=
sigm(W1 xt + W2 ht?1 )
tanh(W3 xt + W4 ht?1 )
sigm(W5 xt + W6 ht?1 )
sigm(W7 xt + W8 ht?1 )
mt?1 ft + it i0t
m t ot
The operator denotes element-wise multiplication, the matrices W1 , . . . , W8 and the vector h0 are
the parameters of the model, and all the nonlinearities are computed element-wise.
In a deep LSTM, each subsequent layer uses the h-sequence of the previous layer for its input
sequence x. The deep LSTM defines a distribution over output sequences given an input sequence:
P (B|A)
=
TB
Y
P (Bt |A1 , . . . , ATA , B1 , . . . , Bt?1 )
t=1
?
TB
Y
softmax(Wo ? hTA +t )> ?Bt ,
t=1
The above equation assumes a deep LSTM whose input sequence is x
=
(A1 , . . . , ATA , B1 , . . . , BTB ), so ht denotes t-th element of the h-sequence of topmost LSTM.
The matrix Wo consists of the vector representations of each output symbol and the symbol ?b
2
S
?
John has a dog .
NP
NNP
.
VP
VBZ
NP
DT
?
John has a dog .
NN
(S (NP NNP )NP (VP VBZ (NP DT NN )NP )VP . )S
Figure 2: Example parsing task and its linearization.
is a Kronecker delta with a dimension for each output symbol, so softmax(Wo ? hTA +t )> ?Bt is
precisely the Bt ?th element of the distribution defined by the softmax. Every output sequence
terminates with a special end-of-sequence token which is necessary in order to define a distribution
over sequences of variable lengths. We use two different sets of LSTM parameters, one for the input
sequence and one for the output sequence, as shown in Figure 1. Stochastic gradient descent is used
to maximize the training objective which is the average over the training set of the log probability
of the correct output sequence given the input sequence.
2.1
Attention Mechanism
An important extension of the sequence-to-sequence model is by adding an attention mechanism.
We adapted the attention model from [2] which, to produce each output symbol Bt , uses an attention
mechanism over the encoder LSTM states. Similar to our sequence-to-sequence model described
in the previous section, we use two separate LSTMs (one to encode the sequence of input words
Ai , and another one to produce or decode the output symbols Bi ). Recall that the encoder hidden
states are denoted (h1 , . . . , hTA ) and we denote the hidden states of the decoder by (d1 , . . . , dTB ) :=
(hTA +1 , . . . , hTA +TB ).
To compute the attention vector at each output time t over the input words (1, . . . , TA ) we define:
uti
ati
= v T tanh(W10 hi + W20 dt )
= softmax(uti )
d0t
=
TA
X
ati hi
i=1
W10 , W20
The vector v and matrices
are learnable parameters of the model. The vector ut has length
TA and its i-th item contains a score of how much attention should be put on the i-th hidden encoder
state hi . These scores are normalized by softmax to create the attention mask at over encoder hidden
states. In all our experiments, we use the same hidden dimensionality (256) at the encoder and the
decoder, so v is a vector and W10 and W20 are square matrices. Lastly, we concatenate d0t with dt ,
which becomes the new hidden state from which we make predictions, and which is fed to the next
time step in our recurrent model.
In Section 4 we provide an analysis of what the attention mechanism learned, and we visualize the
normalized attention vector at for all t in Figure 4.
2.2
Linearizing Parsing Trees
To apply the model described above to parsing, we need to design an invertible way of converting
the parse tree into a sequence (linearization). We do this in a very simple way following a depth-first
traversal order, as depicted in Figure 2.
We use the above model for parsing in the following way. First, the network consumes the sentence
in a left-to-right sweep, creating vectors in memory. Then, it outputs the linearized parse tree using
information in these vectors. As described below, we use 3 LSTM layers, reverse the input sentence
and normalize part-of-speech tags. An example run of our LSTM+A model on the sentence ?Go.?
is depicted in Figure 1 (top gray edges illustrate attention).
3
2.3
Parameters and Initialization
Sizes. In our experiments we used a model with 3 LSTM layers and 256 units in each layer, which
we call LSTM+A. Our input vocabulary size was 90K and we output 128 symbols.
Dropout. Training on a small dataset we additionally used 2 dropout layers, one between LSTM1
and LSTM2 , and one between LSTM2 and LSTM3 . We call this model LSTM+A+D.
POS-tag normalization. Since part-of-speech (POS) tags are not evaluated in the syntactic parsing F1 score, we replaced all of them by ?XX? in the training data. This improved our F1 score by
about 1 point, which is surprising: For standard parsers, including POS tags in training data helps
significantly. All experiments reported below are performed with normalized POS tags.
Input reversing. We also found it useful to reverse the input sentences but not their parse trees,
similarly to [1]. Not reversing the input had a small negative impact on the F1 score on our development set (about 0.2 absolute). All experiments reported below are performed with input reversing.
Pre-training word vectors. The embedding layer for our 90K vocabulary can be initialized randomly or using pre-trained word-vector embeddings. We pre-trained skip-gram embeddings of size
512 using word2vec [6] on a 10B-word corpus. These embeddings were used to initialize our network but not fixed, they were later modified during training. We discuss the impact of pre-training
in the experimental section.
We do not apply any other special preprocessing to the data. In particular, we do not binarize the
parse trees or handle unaries in any specific way. We also treat unknown words in a naive way: we
map all words beyond our 90K vocabulary to a single UNK token. This potentially underestimates
our final results, but keeps our framework task-independent.
3
3.1
Experiments
Training Data
We trained the model described above on 2 different datasets. For one, we trained on the standard
WSJ training dataset. This is a very small training set by neural network standards, as it contains
only 40K sentences (compared to 60K examples even in MNIST). Still, even training on this set, we
managed to get results that match those obtained by domain-specific parsers.
To match state-of-the-art, we created another, larger training set of ?11M parsed sentences (250M
tokens). First, we collected all publicly available treebanks. We used the OntoNotes corpus version
5 [7], the English Web Treebank [8] and the updated and corrected Question Treebank [9].1 Note
that the popular Wall Street Journal section of the Penn Treebank [10] is part of the OntoNotes
corpus. In total, these corpora give us ?90K training sentences (we held out certain sections for
evaluation, as described below).
In addition to this gold standard data, we use a corpus parsed with existing parsers using the
?tri-training? approach of [11]. In this approach, two parsers, our reimplementation of BerkeleyParser [12] and a reimplementation of ZPar [13], are used to process unlabeled sentences sampled
from news appearing on the web. We select only sentences for which both parsers produced the same
parse tree and re-sample to match the distribution of sentence lengths of the WSJ training corpus.
Re-sampling is useful because parsers agree much more often on short sentences. We call the set
of ?11 million sentences selected in this way, together with the ?90K golden sentences described
above, the high-confidence corpus.
After creating this corpus, we made sure that no sentence from the development or test set appears
in the corpus, also after replacing rare words with ?unknown? tokens. This operation guarantees that
we never see any test sentence during training, but it also lowers our F1 score by about 0.5 points.
We are not sure if such strict de-duplication was performed in previous works, but even with this,
we still match state-of-the-art.
1
All treebanks are available through the Linguistic Data Consortium (LDC): OntoNotes (LDC2013T19),
English Web Treebank (LDC2012T13) and Question Treebank (LDC2012R121).
4
Parser
baseline LSTM+D
LSTM+A+D
LSTM+A+D ensemble
baseline LSTM
LSTM+A
Petrov et al. (2006) [12]
Zhu et al. (2013) [13]
Petrov et al. (2010) ensemble [14]
Zhu et al. (2013) [13]
Huang & Harper (2009) [15]
McClosky et al. (2006) [16]
Training Set
WSJ only
WSJ only
WSJ only
BerkeleyParser corpus
high-confidence corpus
WSJ only
WSJ only
WSJ only
semi-supervised
semi-supervised
semi-supervised
WSJ 22
< 70
88.7
90.7
91.0
92.8
91.1
N/A
92.5
N/A
N/A
92.4
WSJ 23
< 70
88.3
90.5
90.5
92.1
90.4
90.4
91.8
91.3
91.3
92.1
Table 1: F1 scores of various parsers on the development and test set. See text for discussion.
In earlier experiments, we only used one parser, our reimplementation of BerkeleyParser, to create a
corpus of parsed sentences. In that case we just parsed ?7 million senteces from news appearing on
the web and combined these parsed sentences with the ?90K golden corpus described above. We
call this the BerkeleyParser corpus.
3.2
Evaluation
We use the standard EVALB tool2 for evaluation and report F1 scores on our developments set
(section 22 of the Penn Treebank) and the final test set (section 23) in Table 1.
First, let us remark that our training setup differs from those reported in previous works. To the best
of our knowledge, no standard parsers have ever been trained on datasets numbering in the hundreds
of millions of tokens, and it would be hard to do due to efficiency problems. We therefore cite the
semi-supervised results, which are analogous in spirit but use less data.
Table 1 shows performance of our models on the top and results from other papers at the bottom. We
compare to variants of the BerkeleyParser that use self-training on unlabeled data [15], or built an
ensemble of multiple parsers [14], or combine both techniques. We also include the best linear-time
parser in the literature, the transition-based parser of [13].
It can be seen that, when training on WSJ only, a baseline LSTM does not achieve any reasonable
score, even with dropout and early stopping. But a single attention model gets to 88.3 and an ensemble of 5 LSTM+A+D models achieves 90.5 matching a single-model BerkeleyParser on WSJ 23.
When trained on the large high-confidence corpus, a single LSTM+A model achieves 92.1 and so
matches the best previous single model result.
Generating well-formed trees. The LSTM+A model trained on WSJ dataset only produced malformed trees for 25 of the 1700 sentences in our development set (1.5% of all cases), and the model
trained on full high-confidence dataset did this for 14 sentences (0.8%). In these few cases where
LSTM+A outputs a malformed tree, we simply add brackets to either the beginning or the end of
the tree in order to make it balanced. It is worth noting that all 14 cases where LSTM+A produced
unbalanced trees were sentences or sentence fragments that did not end with proper punctuation.
There were very few such sentences in the training data, so it is not a surprise that our model cannot
deal with them very well.
Score by sentence length. An important concern with the sequence-to-sequence LSTM was that
it may not be able to handle long sentences well. We determine the extent of this problem by
partitioning the development set by length, and evaluating BerkeleyParser, a baseline LSTM model
without attention, and LSTM+A on sentences of each length. The results, presented in Figure 3,
are surprising. The difference between the F1 score on sentences of length upto 30 and that upto
70 is 1.3 for the BerkeleyParser, 1.7 for the baseline LSTM, and 0.7 for LSTM+A. So already the
baseline LSTM has similar performance to the BerkeleyParser, it degrades with length only slightly.
Surprisingly, LSTM+A shows less degradation with length than BerkeleyParser ? a full O(n3 ) chart
parser that uses a lot more memory.
2
http://nlp.cs.nyu.edu/evalb/
5
96
95
BerkeleyParser
baseline LSTM
LSTM+A
F1 score
94
93
92
91
90 10
20
30
40
Sentence length
50
60
70
Figure 3: Effect of sentence length on the F1 score on WSJ 22.
Beam size influence. Our decoder uses a beam of a fixed size to calculate the output sequence
of labels. We experimented with different settings for the beam size. It turns out that it is almost
irrelevant. We report report results that use beam size 10, but using beam size 2 only lowers the F1
score of LSTM+A on the development set by 0.2, and using beam size 1 lowers it by 0.5. Beam
sizes above 10 do not give any additional improvements.
Dropout influence. We only used dropout when training on the small WSJ dataset and its influence was significant. A single LSTM+A model only achieved an F1 score of 86.5 on our development set, that is over 2 points lower than the 88.7 of a LSTM+A+D model.
Pre-training influence. As described in the previous section, we initialized the word-vector embedding with pre-trained word vectors obtained from word2vec. To test the influence of this initialization, we trained a LSTM+A model on the high-confidence corpus, and a LSTM+A+D model
on the WSJ corpus, starting with randomly initialized word-vector embeddings. The F1 score on our
development set was 0.4 lower for the LSTM+A model and 0.3 lower for the LSTM+A+D model
(88.4 vs 88.7). So the effect of pre-training is consistent but small.
Performance on other datasets. The WSJ evaluation set has been in use for 20 years and is
commonly used to compare syntactic parsers. But it is not representative for text encountered on
the web [8]. Even though our model was trained on a news corpus, we wanted to check how well it
generalizes to other forms of text. To this end, we evaluated it on two additional datasets:
QTB 1000 held-out sentences from the Question Treebank [9];
WEB the first half of each domain from the English Web Treebank [8] (8310 sentences).
LSTM+A trained on the high-confidence corpus (which only includes text from news) achieved
an F1 score of 95.7 on QTB and 84.6 on WEB. Our score on WEB is higher both than the best
score reported in [8] (83.5) and the best score we achieved with an in-house reimplementation of
BerkeleyParser trained on human-annotated data (84.4). We managed to achieve a slightly higher
score (84.8) with the in-house BerkeleyParser trained on a large corpus. On QTB, the 95.7 score of
LSTM+A is also lower than the best score of our in-house BerkeleyParser (96.2). Still, taking into
account that there were only few questions in the training data, these scores show that LSTM+A
managed to generalize well beyond the news language it was trained on.
Parsing speed. Our LSTM+A model, running on a multi-core CPU using batches of 128 sentences
on a generic unoptimized decoder, can parse over 120 sentences from WSJ per second for sentences
of all lengths (using beam-size 1). This is better than the speed reported for this batch size in Figure
4 of [17] at 100 sentences per second, even though they run on a GPU and only on sentences of
under 40 words. Note that they achieve 89.7 F1 score on this subset of sentences of section 22,
while our model at beam-size 1 achieves a score of 93.2 on this subset.
6
Figure 4: Attention matrix. Shown on top is the attention matrix where each column is the attention
vector over the inputs. On the bottom, we show outputs for four consecutive time steps, where the
attention mask moves to the right. As can be seen, every time a terminal node is consumed, the
attention pointer moves to the right.
4
Analysis
As shown in this paper, the attention mechanism was a key component especially when learning
from a relatively small dataset. We found that the model did not overfit and learned the parsing
function from scratch much faster, which resulted in a model which generalized much better than
the plain LSTM without attention.
One of the most interesting aspects of attention is that it allows us to visualize to interpret what the
model has learned from the data. For example, in [2] it is shown that for translation, attention learns
an alignment function, which certainly should help translating from English to French.
Figure 4 shows an example of the attention model trained only on the WSJ dataset. From the
attention matrix, where each column is the attention vector over the inputs, it is clear that the model
focuses quite sharply on one word as it produces the parse tree. It is also clear that the focus moves
from the first word to the last monotonically, and steps to the right deterministically when a word is
consumed.
On the bottom of Figure 4 we see where the model attends (black arrow), and the current output
being decoded in the tree (black circle). This stack procedure is learned from data (as all the parameters are randomly initialized), but is not quite a simple stack decoding. Indeed, at the input side, if
the model focuses on position i, that state has information for all words after i (since we also reverse
the inputs). It is worth noting that, in some examples (not shown here), the model does skip words.
7
5
Related Work
The task of syntactic constituency parsing has received a tremendous amount of attention in the last
20 years. Traditional approaches to constituency parsing rely on probabilistic context-free grammars
(CFGs). The focus in these approaches is on devising appropriate smoothing techniques for highly
lexicalized and thus rare events [18] or carefully crafting the model structure [19]. [12] partially
alleviate the heavy reliance on manual modeling of linguistic structure by using latent variables
to learn a more articulated model. However, their model still depends on a CFG backbone and is
thereby potentially restricted in its capacity.
Early neural network approaches to parsing, for example by [20, 21] also relied on strong linguistic
insights. [22] introduced Incremental Sigmoid Belief Networks for syntactic parsing. By constructing the model structure incrementally, they are able to avoid making strong independence
assumptions but inference becomes intractable. To avoid complex inference methods, [23] propose
a recurrent neural network where parse trees are decomposed into a stack of independent levels.
Unfortunately, this decomposition breaks for long sentences and their accuracy on longer sentences
falls quite significantly behind the state-of-the-art. [24] used a tree-structured neural network to
score candidate parse trees. Their model however relies again on the CFG assumption and furthermore can only be used to score candidate trees rather than for full inference.
Our LSTM model significantly differs from all these models, as it makes no assumptions about
the task. As a sequence-to-sequence prediction model it is somewhat related to the incremental
parsing models, pioneered by [25] and extended by [26]. Such linear time parsers however typically
need some task-specific constraints and might build up the parse in multiple passes. Relatedly, [13]
present excellent parsing results with a single left-to-right pass, but require a stack to explicitly
delay making decisions and a parsing-specific transition strategy in order to achieve good parsing
accuracies. The LSTM in contrast uses its short term memory to model the complex underlying
structure that connects the input-output pairs.
Recently, researchers have developed a number of neural network models that can be applied to
general sequence-to-sequence problems. [27] was the first to propose a differentiable attention
mechanism for the general problem of handwritten text synthesis, although his approach assumed a
monotonic alignment between the input and output sequences. Later, [2] introduced a more general
attention model that does not assume a monotonic alignment, and applied it to machine translation,
and [28] applied the same model to speech recognition. [29] used a convolutional neural network to
encode a variable-sized input sentence into a vector of a fixed dimension and used a RNN to produce the output sentence. Essentially the same model has been used by [30] to successfully learn to
generate image captions. Finally, already in 1990 [31] experimented with applying recurrent neural
networks to the problem of syntactic parsing.
6
Conclusions
In this work, we have shown that generic sequence-to-sequence approaches can achieve excellent
results on syntactic constituency parsing with relatively little effort or tuning. In addition, while
we found the model of Sutskever et al. [1] to not be particularly data efficient, the attention model
of Bahdanau et al. [2] was found to be highly data efficient, as it has matched the performance of
the BerkeleyParser when trained on a small human-annotated parsing dataset. Finally, we showed
that synthetic datasets with imperfect labels can be highly useful, as our models have substantially
outperformed the models that have been used to create their training data. We suspect it is the
case due to the different natures of the teacher model and the student model: the student model
has likely viewed the teacher?s errors as noise which it has been able to ignore. This approach was
so successful that we obtained a new state-of-the-art result in syntactic constituency parsing with
a single attention model, which also means that the model is exceedingly fast. This work shows
that domain independent models with excellent learning algorithms can match and even outperform
domain specific models.
Acknowledgement. We would like to thank Amin Ahmad, Dan Bikel and Jonni Kanerva.
8
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9
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5,121 | 5,636 | Recursive Training of 2D-3D Convolutional Networks
for Neuronal Boundary Detection
Kisuk Lee, Aleksandar Zlateski
Massachusetts Institute of Technology
{kisuklee,zlateski}@mit.edu
Ashwin Vishwanathan, H. Sebastian Seung
Princeton University
{ashwinv,sseung}@princeton.edu
Abstract
Efforts to automate the reconstruction of neural circuits from 3D electron microscopic (EM) brain images are critical for the field of connectomics. An important
computation for reconstruction is the detection of neuronal boundaries. Images acquired by serial section EM, a leading 3D EM technique, are highly anisotropic,
with inferior quality along the third dimension. For such images, the 2D maxpooling convolutional network has set the standard for performance at boundary
detection. Here we achieve a substantial gain in accuracy through three innovations. Following the trend towards deeper networks for object recognition, we use
a much deeper network than previously employed for boundary detection. Second, we incorporate 3D as well as 2D filters, to enable computations that use 3D
context. Finally, we adopt a recursively trained architecture in which a first network generates a preliminary boundary map that is provided as input along with
the original image to a second network that generates a final boundary map. Backpropagation training is accelerated by ZNN, a new implementation of 3D convolutional networks that uses multicore CPU parallelism for speed. Our hybrid 2D3D architecture could be more generally applicable to other types of anisotropic
3D images, including video, and our recursive framework for any image labeling
problem.
1
Introduction
Neural circuits can be reconstructed by analyzing 3D brain images from electron microscopy (EM).
Image analysis has been accelerated by semiautomated systems that use computer vision to reduce
the amount of human labor required [1, 2, 3]. However, analysis of large image datasets is still
laborious [4], so it is critical to increase automation by improving the accuracy of computer vision
algorithms.
A variety of machine learning approaches have been explored for the 3D reconstruction of neurons,
a problem that can be formulated as image segmentation or boundary detection [5, 6]. This paper
focuses on neuronal boundary detection in images from serial section EM, the most widespread kind
of 3D EM [7]. The technique starts by cutting and collecting ultrathin (30 to 100 nm) sections of
brain tissue. A 2D image is acquired from each section, and then the 2D images are aligned. The
spatial resolution of the resulting 3D image stack along the z direction (perpendicular to the cutting
plane) is set by the thickness of the sections. This is generally much worse than the resolution that
EM yields in the xy plane. In addition, alignment errors may corrupt the image along the z direction.
Due to these issues with the z direction of the image stack [6, 8], most existing analysis pipelines
begin with 2D processing and only later transition to 3D. The stages are: (1) neuronal boundary
detection within each 2D image, (2) segmentation of neuron cross sections within each 2D image,
and (3) 3D reconstruction of individual neurons by linking across multiple 2D images [1, 9].
1
Boundary detection in serial section EM images is done by a variety of algorithms. Many algorithms
were compared in the ISBI?12 2D EM segmentation challenge, a publicly available dataset and
benchmark [10]. The winning submission was an ensemble of max-pooling convolutional networks
(ConvNets) created by IDSIA [11]. One of the ConvNet architectures shown in Figure 1 (N4) is the
largest architecture from [11], and serves as a performance baseline for the research reported here.
We improve upon N4 by adding several new elements (Fig. 1):
Increased depth Our VD2D architecture is deeper than N4 (Figure 1), and borrows other nowstandard practices from the literature, such as rectified linear units (ReLUs), small filter sizes, and
multiple convolution layers between pooling layers. VD2D already outperforms N4, without any
use of 3D context. VD2D is motivated by the principle ?the deeper, the better,? which has become
popular for ConvNets applied to object recognition [12, 13].
3D as well as 2D When human experts detect boundaries in EM images, they use 3D context
to disambiguate certain locations. VD2D3D is also able to use 3D context, because it contains
3D filters in its later layers. ConvNets with 3D filters were previously applied to block face EM
images [2, 3, 14]. Block face EM is another class of 3D EM techniques, and produces nearly
isotropic images, unlike serial section EM. VD2D3D also contains 2D filters in its earlier layers.
This novel hybrid use of 2D and 3D filters is suited for the highly anisotropic nature of serial
section EM images.
Recursive training of ConvNets VD2D and VD2D3D are concatenated to create an extremely
deep network. The output of VD2D is a preliminary boundary map, which is provided as input
to VD2D3D in addition to the original image (Fig. 1). Based on these two inputs, VD2D3D is
trained to compute the final boundary map. Such ?recursive? training has previously been applied
to neural networks for boundary detection [8, 15, 16], but not to ConvNets.
ZNN for 3D deep learning Very deep ConvNets with 3D filters are computationally expensive,
so an efficient software implementation is critical. We trained our networks with ZNN (https:
//github.com/seung-lab/znn-release, [17]), which uses multicore CPU parallelism
for speed. ZNN is one of the few deep learning implementations that is well-optimized for 3D.
2D
2DConvNet
ConvNet
VD2D
VD2D
1st input
193x193x5
boundary prediction
1x1x1
2D-3D ConvNet
VD2D3D
2nd input
85x85x5
initialize with
learned 2D representations
1st stage
tanh Pool1 tanh Pool2 tanh
48
2x2x1
48
2x2x1
48
Conv1
Conv2
Conv3
4x4x1
5x5x1
4x4x1
N4
Input
95x95x1
VD2D
Input
109x109x1
Conv1a
3x3x1
Conv1b
3x3x1
Conv1c
2x2x1
Input
85x85x5
Conv1a
3x3x1
Conv1b
3x3x1
Conv1c
2x2x1
VD2D3D
recursive input
85x85x5
ReLU
24
ReLU
24
tanh
24
Pool1
2x2x1
Pool3 tanh Pool4
2x2x1 48
2x2x1
Conv4
4x4x1
Conv2a
3x3x1
Conv2b
3x3x1
Conv2a
3x3x1
Conv2b
3x3x1
ReLU
36
tanh
36
Pool2
2x2x1
2nd stage
tanh
200
Conv5
3x3x1
Softmax
2
Conv3a
3x3x1
Conv3b
3x3x1
Pool3 ReLU
2x2x1
60
Conv4a
3x3x1
Conv3a
3x3x1
Conv3b
3x3x1
Conv4a
3x3x2
ReLU
48
Output
1x1x1
tanh
48
Pool3
2x2x2
ReLU
60
tanh Pool4 ReLU
2x2x1 200
60
Conv4b
Conv5
3x3x1
3x3x1
Conv4b Conv4c
3x3x2
3x3x2
ReLU
60
ReLU
100
Softmax
2
Output
1x1x1
Output
1x1x1
Softmax
2
Figure 1: An overview of our proposed framework (top) and model architectures (bottom). The
number of trainable parameters in each model is 220K (N4), 230K (VD2D), 310K (VD2D3D).
2
While we have applied the above elements to serial section EM images, they are likely to be generally useful for other types of images. The hybrid use of 2D and 3D filters may be useful for video,
which can also be viewed as an anisotropic 3D image. Previous 3D ConvNets applied to video
processing [18, 19] have used 3D filters exclusively.
Recursively trained ConvNets are potentially useful for any image labeling problem. The approach
is very similar to recurrent ConvNets [20], which iterate the same ConvNet. The recursive approach
uses different ConvNets for the successive iterations. The recursive approach has been justified in
several ways. In MRF/CRF image labeling, it is viewed as the sequential refinement of the posterior
probability of a pixel being assigned a label, given both an input image and recursive input from the
previous step [21]. Another viewpoint on recursive training is that statistical dependencies in label
(category) space can be directly modeled from the recursive input [15]. From the neurobiological
viewpoint, using a preliminary boundary map for an image to guide the computation of a better
boundary map for the image can be interpreted as employing a top-down or attentional mechanism.
We expect ZNN to have applications far beyond the one considered in this paper. ZNN can train very
large networks, because CPUs can access more memory than GPUs. Task parallelism, rather than
the SIMD parallelism of GPUs, allows for efficient training of ConvNets with arbitrary topology. A
self-tuning capability automatically optimizes each layer by choosing between direct and FFT-based
convolution. FFT convolution may be more efficient for wider layers or larger filter size [22, 23].
Finally, ZNN may incur less software development cost, owing to the relative ease of the generalpurpose CPU programming model.
Finally, we applied our ConvNets to images from a new serial section EM dataset from the mouse
piriform cortex. This dataset is important to us, because we are interested in conducting neuroscience research concerning this brain region. Even to those with no interest in piriform cortex,
the dataset could be useful for research on image segmentation algorithms. Therefore we make the
annotated dataset publicly available (http://seunglab.org/data/).
2
Dataset and evaluation
Images of mouse piriform cortex The datasets described here were acquired from the piriform cortex of an adult mouse prepared with aldehyde fixation and reduced osmium staining [24].
The tissue was sectioned using the automatic tape collecting ultramicrotome (ATUM) [25] and
sections were imaged on a Zeiss field emission scanning electron microscope [26]. The 2D images were assembled into 3D stacks using custom MATLAB routines and TrakEM2, and each
stack was manually annotated using VAST (https://software.rc.fas.harvard.edu/
lichtman/vast/, [25]) (Figure 2). Then each stack was checked and corrected by another annotator.
The properties of the four image stacks are detailed in Table 1. It should be noted that image
quality varies across the stacks, due to aging of the field emission source in the microscope. In all
experiments we used stack1 for testing, stack2 and stack3 for training, and stack4 as an
additional training data for recursive training.
Figure 2: Example dataset (stack1, Table 1) and results of each architecture on stack1.
3
Name
Resolution (nm3 )
Dimension (voxel3 )
# samples
Usage
Table 1: Piriform cortex datasets
stack1
stack2
stack3
7 ? 7 ? 40
7 ? 7 ? 40
7 ? 7 ? 40
255 ? 255 ? 168 512 ? 512 ? 170 512 ? 512 ? 169
10.9M
44.6 M
44.3 M
Test
Training
Training
stack4
10 ? 10 ? 40
256 ? 256 ? 121
7.9 M
Training (extra)
Pixel error We use softmax activation in the output layer of our networks to produce per-pixel
real-valued outputs between 0 and 1, each of which is interpreted as the probability of an output pixel
being boundary, or vice versa. This real-valued ?boundary map? can be thresholded to generate a
binary boundary map, from which the pixel-wise classification error is computed. We report the best
classification error obtained by optimizing the binarization threshold with line search.
Rand score We evaluate 2D segmentation performance with the Rand scoring system [27, 28].
Let nij denote the number of pixels simultaneously in the ith segment of the proposal segmentation
and the j th segment of the ground truth segmentation. The Rand merge score and the Rand split
score
P 2
P 2
ij nij
ij nij
Rand
Rand
Vmerge = P P
, Vsplit = P P
.
2
2
i(
j(
j nij )
i nij )
are closer to one when there are fewer merge and split errors, respectively. The Rand F-score is the
Rand
Rand
and Vsplit
.
harmonic mean of Vmerge
To compute the Rand scores, we need to first obtain 2D neuronal segmentation based on the realvalued boundary map. To this end, we apply two segmentation algorithms with different levels
of sophistication: (1) simple thresholding followed by computing 2D connected components, and
(2) modified graph-based watershed algorithm [29]. We report the best Rand F-score obtained by
optimizing parameters for each algorithm with line search, as well as the precision-recall curve for
the Rand scores.
3
Training with ZNN
ZNN [17] was built for 3D ConvNets. 2D convolution is regarded as a special case of 3D convolution, in which one of the three filter dimensions has size 1. For the details on how ZNN implements
task parallelism on multicore CPUs, we refer interested readers to [17]. Here we describe only aspects of ZNN that are helpful for understanding how it was used to implement the ConvNets of this
paper.
Dense output with maximum filtering In object recognition, a ConvNet is commonly applied
to produce a single output value for an entire input image. However, there are many applications in
which ?dense output? is required, i.e., the ConvNet should produce an output image with the same
resolution as the original input image. Such applications include boundary detection [11], image
labeling [30], and object localization [31].
ZNN was built from the ground up for dense output and also for dense feature maps.1 ZNN employs
max-filtering, which slides a window across the image and applies the maximum operation to the
window (Figure 3). Max-filtering is the dense variant of max-pooling. Consequently all feature
maps remain intact as dense 3D volumes during both forward and backward passes, making them
straightforward for visualization and manipulation.
On the other hand, all filtering operations are sparse, in the sense that the sliding window samples sparsely from a regularly spaced set of voxels in the image (Figure 3). ZNN can control the
spacing/sparsity of any filtering operation, either convolution or max-filtering.
ZNN can efficiently compute the dense output of a sliding window max-pooling ConvNet by making
filter sparsity depend on the number of prior max-filterings. More specifically, each max-filtering
1
Feature maps with the same resolution as the original input image. See Figure 5 for example. Note that the
feature maps shown in Figure 5 keep the original resolution even after a couple of max-pooling layers.
4
Convolution
Max-Pool
Convolution
Convolution
Max-Filter
Sparse Convolution
Figure 3: Sliding window max-pooling ConvNet (left) applied on three color-coded adjacent input
windows producing three outputs. Equivalent outputs produced by a max-filtering ConvNet with
sparse filters (right) applied on a larger window. Computation is minimized by reusing the intermediate values for computing multiple outputs (as color coded).
increases the sparsity of all subsequent filterings by a factor equal to the size of the max-pooling
window. This approach, which we employ for the paper, is also called ?skip-kernels? [31] or ?filter
rarefaction? [30], and is equivalent in its results to ?max-fragmentation-pooling? [32, 33]. Note
however that ZNN is more general, as the sparsity of filters need not depend on max-filtering, but
can be controlled independently.
Output patch training Training in ZNN is based on loss computed over a dense output patch of
arbitrary size. The patch can be arbitrarily large, limited only by memory. This includes the case of a
patch that spans the entire image [30, 33]. Although large patch sizes reduce the computational cost
per output pixel, neighboring pixels in the patch may provide redundant information. In practice,
we choose an intermediate output patch size.
4
Network architecture
N4 As a baseline for performance comparisons, we adopted the largest 2D ConvNet architecture
(named N4) from Cires?an et al. [11] (Figure 1).
VD2D The architecture of VD2D (?Very Deep 2D?) is shown in Figure 1. Multiple convolution
layers are between each max-pooling layer. All convolution filters are 3?3?1, except that Conv1c
uses a 2 ? 2 ? 1 filter to make the ?field of view? or ?receptive field? for a single output pixel have
an odd-numbered size and therefore centerable around the output pixel. Due to the use of smaller
filters, the number of trainable parameters in VD2D (230K) is roughly the same as in the shallower
N4 (220K).
VD2D3D The architecture of VD2D3D (?Very Deep 2D-3D?) is initially identical to VD2D (Figure 1), except that later convolution layers switch to 3 ? 3 ? 2 filters. This causes the number of
trainable parameters to increase, so we compensate by trimming the size of Conv4c to just 100
feature maps. The 3D filters in the later layers should enable the network to use 3D context to detect
neuronal boundaries. The use of 2D filters in the initial layers makes the network faster to run and
train.
Recursive training It is possible to apply VD2D3D by itself to boundary detection, giving the
raw image as the only input. However, we use a recursive approach in which VD2D3D receives an
extra input, the output of VD2D. As we will see below, this produces a significant improvement in
performance. It should be noted that instead of providing the recursive input directly to VD2D3D,
we added new layers 2 dedicated to processing it. This separate, parallel processing stream for
recursive input joins the main stream at Conv1c, allowing for more complex, highly nonlinear
interaction between the low-level features and the contextual information in the recursive input.
2
These layers are identical to Conv1a, Conv1b, and Conv1c.
5
5
Training procedures
Networks were trained using backpropagation with the cross-entropy loss function. We first trained
VD2D, and then trained VD2D3D. The 2D layers of VD2D3D were initialized using trained weights
from VD2D. This initialization meant that our recursive approach bore some similarity to recurrent
ConvNets, in which the first and second stage networks are constrained to be identical [20]. However, we did not enforce exact weight sharing, but fine-tuned the weights of VD2D3D.
Output patch As mentioned earlier, training with ZNN is done by dense output patch-based
gradient update with per-pixel loss. During training, an output patch of specified size is randomly
drawn from the training stacks at the beginning of each forward pass.
Class rebalancing In dense output patch-based training, imbalance between the number of training samples in different classes (e.g. boundary/non-boundary) can be handled by either sampling
a balanced number of pixels from an output patch, or by differentially weighting the per-pixel
loss [30]. In our experiment, we adopted the latter approach (loss weighting) to deal with the high
imbalance between boundary and non-boundary pixels.
Data augmentation We used the same data augmentation method used in [11], randomly rotating and flipping 2D image patches.
Hyperparameter We always used the fixed learning rate of 0.01 with the momentum of 0.9.
When updating weights we divided the gradient by the total number of pixels in an output patch,
similar to the typical minibatch averaging.
We first trained N4 with an output patch of size 200 ? 200 ? 1 for 90K gradient updates. Next, we
trained VD2D with 150 ? 150 ? 1 output patches, reflecting the increased size of model compared to
N4. After 60K updates, we evaluated the trained VD2D on the training stacks to obtain preliminary
boundary maps, and started training VD2D3D with 100?100?1 output patches, again reflecting the
increased model complexity. We trained VD2D3D for 90K updates. In this recursive training stage
we additionally used stack4 to prevent VD2D3D from being overly dependent on the good-quality
boundary maps for training stacks. It should be noted that stack4 has slightly lower xy-resolution
than other stacks (Table 1), which we think is helpful in terms of learning multi-scale representation.
Our proposed recursive framework is different from the training of recurrent ConvNets [20] in that
recursive input is not dynamically produced by the first ConvNet during training, but evaluated
before and being fixed throughout the recursive training stage. However, it is also possible to further
train the first ConvNet even after evaluating its preliminary output as recursive input to the second
ConvNet. We further trained VD2D for another 30K updates while VD2D3D is being trained. We
report the final performance of VD2D after a total of 90K updates. We also replaced the initial
VD2D boundary map with the final one when evaluating VD2D3D results. With ZNN, it took two
days to train both N4 and VD2D for 90K updates, and three days to train VD2D3D for 90K updates.
6
Results
In this section, we show both quantitative and qualitative results obtained by the three architectures
shown in Figure 1, namely N4, VD2D, and VD2D3D. The pixel-wise classification error of each
model on test set was 10.63% (N4), 9.77% (VD2D), and 8.76% (VD2D3D).
Quantitative comparison Figure 4 compares the result of each architecture on test set (stack1),
both quantitatively and qualitatively. The leftmost bar graph shows the best 2D Rand F-score of
each model obtained by 2D segmentation with (1) simpler connected component clustering and
(2) more sophisticated watershed-based segmentation. The middle and rightmost graphs show the
precision-recall curve of each model for the Rand scores obtained with the connected component and
watershed-based segmentation, respectively. We observe that VD2D performs significantly better
than N4, and also VD2D3D outperforms VD2D by a significant margin in terms of both best Rand
F-score and overall precision-recall curve.
Qualitative comparison Figure 2 shows the visualization of boundary detection results of each
model on test set, along with the original EM images and ground truth boundary map. We observe
that false detection of boundary on intracellular regions was significantly reduced in VD2D3D,
6
Rand score (connected component)
0.7
Connected
component
image data
Watershed
0.6
0.6
ground-truth
Rand score (watershed)
0.8
0.8
0.85
1
0.9
0.9
0.9
0.8
1
Rand merge score
Rand F-score
0.95
Best Rand F-score
N4
VD2D
VD2D3D
Rand merge score
1
0.7
N4
VD2D
VD2D3D
0.7
0.8
0.9
Rand split score
N4
1
0.6
0.6
VD2D
N4
VD2D
VD2D3D
0.7
0.8
0.9
Rand split score
1
VD2D3D
Figure 4: Quantitative (top) and qualitative (middle and bottom) evaluation of results.
which demonstrates the effectiveness of the proposed 2D-3D ConvNet combined with recursive
approach. The middle and bottom rows in Figure 4 show some example locations in test set where
both 2D models (N4 and VD2D) failed to correctly detect the boundary, or erroneously detected false
boundaries, whereas VD2D3D correctly predicted on those ambiguous locations. Visual analysis on
the boundary detection results of each model again demonstrates the superior performance of the
proposed recursively trained 2D-3D ConvNet over 2D models.
7
Discussion
Biologically-inspired recursive framework Our proposed recursive framework is greatly inspired by the work of Chen et al. [34]. In this work, they examined the close interplay between
neurons in the primary and higher visual cortical areas (V1 and V4, respectively) of monkeys performing contour detection tasks. In this task, monkeys were trained to detect a global contour pattern
that consists of multiple collinearly aligned bars in a cluttered background.
The main discovery of their work is as follows: initially, V4 neurons responded to the global contour
pattern. After a short time delay (?40 ms), the activity of V1 neurons responding to each bar composing the global contour pattern was greatly enhanced, whereas those responding to the background
was largely suppressed, despite the fact that those ?foreground? and ?background? V1 neurons have
similar response properties. They referred to it as ?push-pull response mode? of V1 neurons between foreground and background, which is attributable to the top-down influence from the higher
level V4 neurons. This process is also referred to as ?countercurrent disambiguating process? [34].
This experimental result readily suggests a mechanistic interpretation on the recursive training of
deep ConvNets for neuronal boundary detection. We can roughly think of V1 responses as lower
level feature maps in a deep ConvNet, and V4 responses as higher level feature maps or output activations. Once the overall ?contour? of neuonal boundaries is detected by the feedforward processing
of VD2D, this preliminary boundary map can then be recursively fed to VD2D3D. This process
7
VD2D
Conv2a
VD2D
Conv3b
VD2D3D
Conv2a
VD2D3D
Conv3b
Figure 5: Visualization of the effect of recursive training. Left: an example feature map from the
lower layer Conv2a in VD2D, and its corresponding feature map in VD2D3D. Right: an example feature map from the higher layer Conv3b in VD2D, and its corresponding feature map in
VD2D3D. Note that recursive training greatly enhances the signal-to-noise ratio of boundary representations.
can be thought of as corresponding to the initial detection of global contour patterns by V4 and its
top-down influence on V1.
During recursive training, VD2D3D will learn how to integrate the pixel-level contextual information in the recursive input with the low-level features, presumably in such a way that feature
activations on the boundary location are enhanced, whereas activations unrelated to the neuronal
boundary (intracellular space, mitochondria, etc.) are suppressed. Here the recursive input can also
be viewed as the modulatory ?gate? through which only the signals relevant to the given task of neuronal boundary detection can pass. This is convincingly demonstrated by visualizing and comparing
the feature maps of VD2D and VD2D3D.
In Figure 5, the noisy representations of oriented boundary segments in VD2D (first and third volumes) are greatly enhanced in VD2D3D (second and fourth volumes), with signals near boundary
being preserved or amplified, and noises in the background being largely suppressed. This is exactly
what we expected from the proposed interpretation of our recursive framework.
Potential of ZNN
We have shown that ZNN can serve as a viable alternative to the mainstream
GPU-based deep learning frameworks, especially when processing 3D volume data with 3D ConvNets. ZNN?s unique features including the large output patch-based training and the dense computation of feature maps can be further utilized for additional computations to better perform the given
task. In theory, we can perform any kind of computation on the dense output prediction between
each forward and backward passes. For instance, objective functions that consider topological constraints (e.g. MALIS [35]) or sampling of topologically relevant locations (e.g. LED weighting [15])
can be applied to the dense output patch, in addition to loss computation, before each backward pass.
Dense feature maps also enable the straighforward implementation of multi-level feature integration
for fine-grained segmentation. Long et al. [30] resorted to upsampling of the higher level features
with lower resolution in order to integrate them with the lower level features with higher resolution.
Since ZNN maintains every feature map at the original resolution of input, it is straighforward
enough to combine feature maps from any level, removing the need for upsampling.
Acknowledgments
We thank Juan C. Tapia, Gloria Choi and Dan Stettler for initial help with tissue handling and Jeff
Lichtman and Richard Schalek with help in setting up tape collection. Kisuk Lee was supported
by a Samsung Scholarship. The recursive approach proposed in this paper was partially motivated
by Matthew J. Greene?s preliminary experiments. We are grateful for funding from the Mathers
Foundation, Keating Fund for Innovation, Simons Center for the Social Brain, DARPA (HR001114-2-0004), and ARO (W911NF-12-1-0594).
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5,122 | 5,637 | Generative Image Modeling Using Spatial LSTMs
Matthias Bethge
University of T?ubingen
72076 T?ubingen, Germany
[email protected]
Lucas Theis
University of T?ubingen
72076 T?ubingen, Germany
[email protected]
Abstract
Modeling the distribution of natural images is challenging, partly because of
strong statistical dependencies which can extend over hundreds of pixels. Recurrent neural networks have been successful in capturing long-range dependencies in a number of problems but only recently have found their way into generative image models. We here introduce a recurrent image model based on multidimensional long short-term memory units which are particularly suited for image
modeling due to their spatial structure. Our model scales to images of arbitrary
size and its likelihood is computationally tractable. We find that it outperforms the
state of the art in quantitative comparisons on several image datasets and produces
promising results when used for texture synthesis and inpainting.
1
Introduction
The last few years have seen tremendous progress in learning useful image representations [6].
While early successes were often achieved through the use of generative models [e.g., 13, 23, 30],
recent breakthroughs were mainly driven by improvements in supervised techniques [e.g., 20, 34].
Yet unsupervised learning has the potential to tap into the much larger source of unlabeled data,
which may be important for training bigger systems capable of a more general scene understanding. For example, multimodal data is abundant but often unlabeled, yet can still greatly benefit
unsupervised approaches [36].
Generative models provide a principled approach to unsupervised learning. A perfect model of
natural images would be able to optimally predict parts of an image given other parts of an image and
thereby clearly demonstrate a form of scene understanding. When extended by labels, the Bayesian
framework can be used to perform semi-supervised learning in the generative model [19, 28] while it
is less clear how to combine other unsupervised approaches with discriminative learning. Generative
image models are also useful in more traditional applications such as image reconstruction [33, 35,
49] or compression [47].
Recently there has been a renewed strong interest in the development of generative image models
[e.g., 4, 8, 10, 11, 18, 24, 31, 35, 45, 47]. Most of this work has tried to bring to bear the flexibility of
deep neural networks on the problem of modeling the distribution of natural images. One challenge
in this endeavor is to find the right balance between tractability and flexibility. The present article
contributes to this line of research by introducing a fully tractable yet highly flexible image model.
Our model combines multi-dimensional recurrent neural networks [9] with mixtures of experts.
More specifically, the backbone of our model is formed by a spatial variant of long short-term
memory (LSTM) [14]. One-dimensional LSTMs have been particularly successful in modeling text
and speech [e.g., 38, 39], but have also been used to model the progression of frames in video [36]
and very recently to model single images [11]. In contrast to earlier work on modeling images,
here we use multi-dimensional LSTMs [9] which naturally lend themselves to the task of generative
image modeling due to their spatial structure and ability to capture long-range correlations.
1
RIDE
C
A
B
Pixels
MCGSM
xij
x<ij
xij
x<ij
SLSTM units
SLSTM units
Pixels
Figure 1: (A) We factorize the distribution of images such that the prediction of a pixel (black)
may depend on any pixel in the upper-left green region. (B) A graphical model representation of an
MCGSM with a causal neighborhood limited to a small region. (C) A visualization of our recurrent
image model with two layers of spatial LSTMs. The pixels of the image are represented twice and
some arrows are omitted for clarity. Through feedforward connections, the prediction of a pixel
depends directly on its neighborhood (green), but through recurrent connections it has access to the
information in a much larger region (red).
To model the distribution of pixels conditioned on the hidden states of the neural network, we use
mixtures of conditional Gaussian scale mixtures (MCGSMs) [41]. This class of models can be
viewed as a generalization of Gaussian mixture models, but their parametrization makes them much
more suitable for natural images. By treating images as instances of a stationary stochastic process,
this model allows us to sample and capture the correlations of arbitrarily large images.
2
A recurrent model of natural images
In the following, we first review and extend the MCGSM [41] and multi-dimensional LSTMs [9]
before explaining how to combine them into a recurrent image model. Section 3 will demonstrate
the validity of our approach by evaluating and comparing the model on a number of image datasets.
2.1
Factorized mixtures of conditional Gaussian scale mixtures
One successful approach to building flexible yet tractable generative models has been to use fullyvisible belief networks [21, 27]. To apply such a model to images, we have to give the pixels
an ordering and specify the distribution of each pixel conditioned on its parent pixels. Several
parametrizations have been suggested for the conditional distributions in the context of natural images [5, 15, 41, 44, 45]. We here review and extend the work of Theis et al. [41] who proposed to
use mixtures of conditional Gaussian scale mixtures (MCGSMs).
Let x be a grayscale image patch and xij be the intensity of the pixel at location ij. Further, let x<ij
designate the set of pixels xmn such that m < i or m = i and n < j (Figure 1A). Then
Q
p(x; ?) = i,j p(xij | x<ij ; ?)
(1)
for the distribution of any parametric model with parameters ?. Note that this factorization does
not make any independence assumptions but is simply an application of the probability chain rule.
Further note that the conditional distributions all share the same set of parameters. One way to
improve the representational power of a model is thus to endow each conditional distribution with
its own set of parameters,
Q
p(x; {?ij }) = i,j p(xij | x<ij ; ?ij ).
(2)
Applying this trick to mixtures of Gaussian scale mixtures (MoGSMs) yields the MCGSM [40].
Untying shared parameters can drastically increase the number of parameters. For images, it can
easily be reduced again by adding assumptions. For example, we can limit x<ij to a smaller neighborhood surrounding the pixel by making a Markov assumption. We will refer to the resulting set of
parents as the pixel?s causal neighborhood (Figure 1B). Another reasonable assumption is stationarity or shift invariance, in which case we only have to learn one set of parameters ?ij which can then
2
be used at every pixel location. Similar to convolutions in neural networks, this allows the model
to easily scale to images of arbitrary size. While this assumption reintroduces parameter sharing
constraints into the model, the constraints are different from the ones induced by the joint mixture
model.
The conditional distribution in an MCGSM takes the form of a mixture of experts,
X
p(xij | x<ij , ?ij ) =
p(c, s | x<ij , ?ij ) p(xij | x<ij , c, s, ?ij ),
|
{z
}|
{z
}
c,s
gate
(3)
expert
where the sum is over mixture component indices c corresponding to different covariances and scales
s corresponding to different variances. The gates and experts in an MCGSM are given by
p(c, s | x<ij ) ? exp ?cs ? 12 e?cs x>
(4)
<ij Kc x<ij ,
??cs
p(xij | x<ij , c, s) = N (xij ; a>
),
c x<ij , e
(5)
where Kc is positive definite. The number of parameters of an MCGSM still grows quadratically
with the dimensionality of the causal neighborhood. To further reduce the number of parameters, we
introduce
form of the MCGSM with additional parameter sharing by replacing Kc with
P 2 a factorized
>
n ?cn bn bn . This factorized MCGSM allows us to use larger neighborhoods and more mixture
components. A detailed derivation of a more general version which also allows for multivariate
pixels is given in Supplementary Section 1.
2.2
Spatial long short-term memory
In the following we briefly describe the spatial LSTM (SLSTM), a special case of the multidimensional LSTM first described by Graves & Schmidhuber [9]. At the core of the model are
memory units cij and hidden units hij . For each location ij on a two-dimensional grid, the
operations performed by the spatial LSTM are given by
? ? ?
?
gij
tanh
!
?oij ? ? ? ?
x<ij
c
r
cij = gij iij + ci,j?1 fij
+ ci?1,j fij
,
?i ? ?
?
? ij ? = ? ? ? TA,b hi,j?1 , (6)
?fr ? ? ? ?
hij = tanh (cij oij ) ,
hi?1,j
ij
c
fij
?
where ? is the logistic sigmoid function, indicates a pointwise product, and TA,b is an affine
transformation which depends on the only parameters of the network A and b. The gating units iij
and oij determine which memory units are affected by the inputs through gij , and which memory
states are written to the hidden units hij . In contrast to a regular LSTM defined over time, each
memory unit of a spatial LSTM has two preceding states ci,j?1 and ci?1,j and two corresponding
c
r
forget gates fij
and fij
.
2.3
Recurrent image density estimator
We use a grid of SLSTM units to sequentially read relatively small neighborhoods of pixels from
the image, producing a hidden vector at every pixel. The hidden states are then fed into a factorized
MCGSM to predict the state of the corresponding pixel, that is, p(xij | x<ij ) = p(xij | hij ). Importantly, the state of the hidden vector only depends on pixels in x<ij and does not violate the factorization given in Equation 1. Nevertheless, the recurrent network allows this recurrent image density
estimator (RIDE) to use pixels of a much larger region for prediction, and to nonlinearly transform
the pixels before applying the MCGSM. We can further increase the representational power of the
model by stacking spatial LSTMs to obtain a deep yet still completely tractable recurrent image
model (Figure 1C).
2.4
Related work
Larochelle & Murray [21] derived a tractable density estimator (NADE) in a manner similar to how
the MCGSM was derived [41], but using restricted Boltzmann machines (RBM) instead of mixture
models as a starting point. In contrast to the MCGSM, NADE tries to keep the weight sharing
3
constraints induced by the RBM (Equation 1). Uria et al. extended NADE to real values [44] and
introduced hidden layers to the model [45]. Gregor et al. [10] describe a related autoregressive
network for binary data which additionally allows for stochastic hidden units.
Gregor et al. [11] used one-dimensional LSTMs to generate images in a sequential manner (DRAW).
Because the model was defined over Bernoulli variables, normalized RGB values had to be treated
as probabilities, making a direct comparison with other image models difficult. In contrast to our
model, the presence of stochastic latent variables in DRAW means that its likelihood cannot be
evaluated but has to be approximated.
Ranzato et al. [31] and Srivastava et al. [37] use one-dimensional recurrent neural networks to model
videos, but recurrency is not used to describe the distribution over individual frames. Srivastava et
al. [37] optimize a squared error corresponding to a Gaussian assumption, while Ranzato et al. [31]
try to side-step having to model pixel intensities by quantizing image patches. In contrast, here we
also try to solve the problem of modeling pixel intensities by using an MCGSM, which is equipped
to model heavy-tailed as well as multi-modal distributions.
3
Experiments
RIDE was trained using stochastic gradient descent with a batch size of 50, momentum of 0.9, and
a decreasing learning rate varying between 1 and 10?4 . After each pass through the training set,
the MCGSM of RIDE was finetuned using L-BFGS for up to 500 iterations before decreasing the
learning rate. No regularization was used except for early stopping based on a validation set. Except
where indicated otherwise, the recurrent model used a 5 pixel wide neighborhood and an MCGSM
with 32 components and 32 quadratic features (bn in Section 2.1). Spatial LSTMs were implemented using the Caffe framework [17]. Where appropriate, we augmented the data by horizontal
or vertical flipping of images.
We found that conditionally whitening the data greatly sped up the training process of both models.
Letting y represent a pixel and x its causal neighborhood, conditional whitening replaces these with
?1
?1
1
> ?2
x
? = Cxx2 (x ? mx ) , y
? = W(y ? Cyx Cxx2 x
? ? my ), W = (Cyy ? Cyx C?1
, (7)
xx Cyx )
where Cyx is the covariance of y and x, and mx is the mean of x. In addition to speeding up training, this variance normalization step helps to make the learning rates less dependent on the training
data. When evaluating the conditional log-likelihood, we compensate for the change in variance by
adding the log-Jacobian log | det W|. Note that this preconditioning introduces a shortcut connection from the pixel neighborhood to the predicted pixel which is not shown in Figure 1C.
3.1
Ensembles
Uria et al. [45] found that forming ensembles of their autoregressive model over different pixel
orderings significantly improved performance. We here consider a simple trick to produce an ensemble without the need for training different models or to change training procedures. If Tk are
linear transformations leaving the targeted image distribution invariant (or approximately
invariant)
P
1
and if p is the distribution of a pretrained model, then we form the ensemble K
p(T
x)|
det Tk |.
k
k
Note that this is simply a mixture model over images x. We considered rotating as well as flipping
images along the horizontal and vertical axes (yielding an ensemble over 8 transformations). While
it could be argued that most of these transformations do not leave the distribution over natural images
invariant, we nevertheless observed a noticeable boost in performance.
3.2
Natural images
Several recent image models have been evaluated on small image patches sampled from the Berkeley
segmentation dataset (BSDS300) [25]. Although our model?s strength lies in its ability to scale to
large images and to capture long-range correlations, we include results on BSDS300 to make a
connection to this part of the literature. We followed the protocol of Uria et al. [44]. The RGB
images were turned to grayscale, uniform noise was added to account for the integer discretization,
and the resulting values were divided by 256. The training set of 200 images was split into 180
images for training and 20 images for validation, while the test set contained 100 images. We
4
63 dim.
[nat]
RNADE [44]
152.1
RNADE, 1 hl [45]
143.2
RNADE, 6 hl [45]
155.2
EoRNADE, 6 layers [45] 157.0
GMM, 200 comp. [47, 50] 153.7
STM, 200 comp. [46]
155.3
Deep GMM, 3 layers [47] 156.2
MCGSM, 16 comp.
155.1
MCGSM, 32 comp.
155.8
MCGSM, 64 comp.
156.2
MCGSM, 128 comp.
156.4
EoMCGSM, 128 comp.
158.1
RIDE, 1 layer
150.7
RIDE, 2 layers
152.1
EoRIDE, 2 layers
154.5
Model
64 dim. ? dim.
[bit/px] [bit/px]
3.346
3.146
3.416
3.457
3.360
3.418
3.439
3.413
3.688
3.430
3.706
3.439
3.716
3.443
3.717
3.481
3.748
3.293
3.802
3.869
3.346
3.400
3.899
256 dim. ? dim.
[bit/px] [bit/px]
GRBM [13]
0.992
1.072
ICA [1, 48]
GSM
1.349
ISA [7, 16]
1.441
MoGSM, 32 comp. [40]
1.526
MCGSM, 32 comp.
1.615
1.759
RIDE, 1 layer, 64 hid.
1.650
1.816
RIDE, 1 layer, 128 hid.
1.830
RIDE, 2 layers, 64 hid.
1.829
1.839
RIDE, 2 layers, 128 hid.
EoRIDE, 2 layers, 128 hid.
1.859
Model
Table 1: Average log-likelihoods and log-likelihood Table 2: Average log-likelihood rates for imrates for image patches (without/with DC comp.) and age patches and large images extracted from
large images extracted from BSDS300 [25].
van Hateren?s dataset [48].
extracted 8 by 8 image patches from each set and subtracted the average pixel intensity such that
each patch?s DC component was zero. Because the resulting image patches live on a 63 dimensional
subspace, the bottom-right pixel was discarded. We used 1.6 ? 106 patches for training, 1.8 ? 105
patches for validation, and 106 test patches for evaluation.
MCGSMs have not been evaluated on this dataset and so we first tested MCGSMs by training a
single factorized MCGSM for each pixel conditioned on all previous pixels in a fixed ordering.
We find that already an MCGSM (with 128 components and 48 quadratic features) outperforms all
single models including a deep Gaussian mixture model [46] (Table 1). Our ensemble of MCGSMs1
outperforms an ensemble of RNADEs with 6 hidden layers, which to our knowledge is currently the
best result reported on this dataset.
Training the recurrent image density estimator (RIDE) on the 63 dimensional dataset is more cumbersome. We tried padding image patches with zeros, which was necessary to be able to compute a
hidden state at every pixel. The bottom-right pixel was ignored during training and evaluation. This
simple approach led to a reduction in performance relative to the MCGSM (Table 1). A possible
explanation is that the model cannot distinguish between pixel intensities which are zero and zeros
in the padded region. Supplying the model with additional binary indicators as inputs (one for each
neighborhood pixel) did not solve the problem.
However, we found that RIDE outperforms the MCGSM by a large margin when images were
treated as instances of a stochastic process (that is, using infinitely large images). MCGSMs were
trained for up to 3000 iterations of L-BFGS on 106 pixels and corresponding causal neighborhoods
extracted from the training images. Causal neighborhoods were 9 pixels wide and 5 pixels high.
RIDE was trained for 8 epochs on image patches of increasing size ranging from 8 by 8 to 22 by
22 pixels (that is, gradients were approximated as in backpropagation through time [32]). The right
column in Table 1 shows average log-likelihood rates for both models. Analogously to the entropy
rate [3], we have for the expected log-likelihood rate:
lim E log p(x)/N 2 = E[log p(xij | x<ij )],
(8)
N ??
where x is an N by N image patch. An average log-likelihood rate can be directly computed for the
MCGSM, while for RIDE and ensembles we approximated it by splitting the test images into 64 by
64 patches and evaluating on those.
To make the two sets of numbers more comparable, we transformed nats as commonly reported on
the 63 dimensional data, `1:63 , into a bit per pixel log-likelihood rate using the formula (`1:63 +`DC +
ln | det A|)/64/ ln(2). This takes into account a log-likelihood for the missing DC component,
1
Details on how the ensemble of transformations can be applied despite the missing bottom-right pixel are
given in Supplementary Section 2.1.
5
Log-likelihood [bit/px]
Model
[bit/px]
MCGSM, 12 comp. [41]
1.244
MCGSM, 32 comp.
1.294
Diffusion [35]
1.489
RIDE, 64 hid., 1 layer
1.402
RIDE, 64 hid., 1 layer, ext. 1.416
RIDE, 64 hid., 2 layers
1.438
RIDE, 64 hid., 3 layers
1.454
RIDE, 128 hid., 3 layers
1.489
EoRIDE, 128 hid., 3 layers 1.501
1.5
1.4
1.3
1.2
MCGSM
RIDE
1.1
1
3
5
7
9
11 13
Neighborhood size
Table 3: Average log-likelihood rates on dead
leaf images. A deep recurrent image model is
on a par with a deep diffusion model [35]. Using ensembles we are able to further improve the
likelihood.
Figure 2: Model performance on dead leaves
as a function of the causal neighborhood width.
Simply increasing the neighborhood size of the
MCGSM is not sufficient to improve performance.
`DC = 0.5020, and the Jacobian of the transformations applied during preprocessing, ln | det A| =
?4.1589 (see Supplementary Section 2.2 for details). The two rates in Table 1 are comparable in the
sense that their differences express how much better one model would be at losslessly compressing
BSDS300 test images than another, where patch-based models would compress patches of an image
independently. We highlighted the best result achieved with each model in gray. Note that most
models in this list do not scale as well to large images as the MCGSM or RIDE (GMMs in particular)
and are therefore unlikely to benefit as much from increasing the patch size.
A comparison of the log-likelihood rates reveals that an MCGSM with 16 components applied to
large images already captures more correlations than any model applied to small image patches.
The difference is particularly striking given that the factorized MCGSM has approximately 3,000
parameters while a GMM with 200 components has approximately 400,000 parameters. Using an
ensemble of RIDEs, we are able to further improve this number significantly (Table 1).
Another dataset frequently used to test generative image models is the dataset published by van
Hateren and van der Schaaf [48]. Details of the preprocessing used in this paper are given in Supplementary Section 3. We reevaluated several models for which the likelihood has been reported
on this dataset [7, 40, 41, 42]. Likelihood rates as well as results on 16 by 16 patches are given in
Table 2. Because of the larger patch size, RIDE here already outperforms the MCGSM on patches.
3.3
Dead leaves
Dead leaf images are generated by superimposing disks of random intensity and size on top of each
other [22, 26]. This simple procedure leads to images which already share many of the statistical
properties and challenges of natural images, such as occlusions and long-range correlations, while
leaving out others such as non-stationary statistics. They therefore provide an interesting test case
for natural image models.
We used a set of 1,000 images, where each image is 256 by 256 pixels in size. We compare the
performance of RIDE to the MCGSM and a very recently introduced deep multiscale model based
on a diffusion process [35]. The same 100 images as in previous literature [35, 41] were used
for evaluation and we used the remaining images for training. We find that the introduction of an
SLSTM with 64 hidden units greatly improves the performance of the MCGSM. We also tried an
extended version of the SLSTM which included memory units as additional inputs (right-hand side
of Equation 6). This yielded a small improvement in performance (5th row in Table 3) while adding
layers or using more hidden units led to more drastic improvements. Using 3 layers with 128 hidden
units in each layer, we find that our recurrent image model is on a par with the deep diffusion model.
By using ensembles, we are able to beat all previously published results for this dataset (Table 3).
Figure 2 shows that the improved performance of RIDE is not simply due to an effectively larger
causal neighborhood but that the nonlinear transformations performed by the SLSTM units matter.
Simply increasing the neighborhood size of an MCGSM does not yield the same improvement.
Instead, the performance quickly saturates. We also find that the performance of RIDE slightly
deteriorates with larger neighborhoods, which is likely caused by optimization difficulties.
6
D106
D93
D104
D12
D34
D110
Figure 3: From top to bottom: A 256 by 256 pixel crop of a texture [2], a sample generated by
an MCGSM trained on the full texture [7], and a sample generated by RIDE. This illustrates that
our model can capture a variety of different statistical patterns. The addition of the recurrent neural
network seems particularly helpful where there are strong long-range correlations (D104, D34).
3.4
Texture synthesis and inpainting
To get an intuition for the kinds of correlations which RIDE can capture or fails to capture, we tried
to use it to synthesize textures. We used several 640 by 640 pixel textures published by Brodatz [2].
The textures were split into sixteen 160 by 160 pixel regions of which 15 were used for training and
one randomly selected region was kept for testing purposes. RIDE was trained for up to 6 epochs
on patches of increasing size ranging from 20 by 20 to 40 by 40 pixels.
Samples generated by an MCGSM and RIDE are shown in Figure 3. Both models are able to
capture a wide range of correlation structures. However, the MCGSM seems to struggle with textures having bimodal marginal distributions and periodic patterns (D104, D34, and D110). RIDE
clearly improves on these textures, although it also struggles to faithfully reproduce periodic structure. Possible explanations include that LSTMs are not well suited to capture periodicities, or that
these failures are not penalized strong enough by the likelihood. For some textures, RIDE produces
samples which are nearly indistinguishable from the real textures (D106 and D110).
One application of generative image models is inpainting [e.g., 12, 33, 35]. As a proof of concept,
we used our model to inpaint a large (here, 71 by 71 pixels) region in textures (Figure 4). Missing
pixels were replaced by sampling from the posterior of RIDE. Unlike the joint distribution, the
posterior distribution cannot be sampled directly and we had to resort to Markov chain Monte Carlo
methods. We found the following Metropolis within Gibbs [43] procedure to be efficient enough.
The missing pixels were initialized via ancestral sampling. Since ancestral sampling is cheap, we
generated 5 candidates and used the one with the largest posterior density. Following initialization,
we sequentially updated overlapping 5 by 5 pixel regions via Metropolis sampling. Proposals were
generated via ancestral sampling and accepted using the acceptance probability
n
o
0
) p(xij |x<ij )
? = min 1, p(x
(9)
0
p(x) p(x |x<ij ) ,
ij
where here xij represents a 5 by 5 pixel patch and x0ij its proposed replacement. Since evaluating the
joint and conditional densities on the entire image is costly, we approximated p using RIDE applied
to a 19 by 19 pixel patch surrounding ij. Randomly flipping images vertically or horizontally in
between the sampling further helped. Figure 4 shows results after 100 Gibbs sampling sweeps.
4
Conclusion
We have introduced RIDE, a deep but tractable recurrent image model based on spatial LSTMs.
The model exemplifies how recent insights in deep learning can be exploited for generative image
7
Figure 4: The center portion of a texture (left and center) was reconstructed by sampling from the
posterior distribution of RIDE (right).
modeling and shows superior performance in quantitative comparisons. RIDE is able to capture
many different statistical patterns, as demonstrated through its application to textures. This is an
important property considering that on an intermediate level of abstraction natural images can be
viewed as collections of textures.
We have furthermore introduced a factorized version of the MCGSM which allowed us to use more
experts and larger causal neighborhoods. This model has few parameters, is easy to train and already
on its own performs very well as an image model. It is therefore an ideal building block and may be
used to extend other models such as DRAW [11] or video models [31, 37].
Deep generative image models have come a long way since deep belief networks have first been
applied to natural images [29]. Unlike convolutional neural networks in object recognition, however,
no approach has as of yet proven to be a likely solution to the problem of generative image modeling.
Further conceptual work will be necessary to come up with a model which can handle both the more
abstract high-level as well as the low-level statistics of natural images.
Acknowledgments
The authors would like to thank A?aron van den Oord for insightful discussions and Wieland Brendel,
Christian Behrens, and Matthias K?ummerer for helpful input on this paper. This study was financially supported by the German Research Foundation (DFG; priority program 1527, BE 3848/2-1).
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9
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5,123 | 5,638 | Faster R-CNN: Towards Real-Time Object Detection
with Region Proposal Networks
Shaoqing Ren? Kaiming He
Ross Girshick
Jian Sun
Microsoft Research
{v-shren, kahe, rbg, jiansun}@microsoft.com
Abstract
State-of-the-art object detection networks depend on region proposal algorithms
to hypothesize object locations. Advances like SPPnet [7] and Fast R-CNN [5]
have reduced the running time of these detection networks, exposing region proposal computation as a bottleneck. In this work, we introduce a Region Proposal Network (RPN) that shares full-image convolutional features with the detection network, thus enabling nearly cost-free region proposals. An RPN is a
fully-convolutional network that simultaneously predicts object bounds and objectness scores at each position. RPNs are trained end-to-end to generate highquality region proposals, which are used by Fast R-CNN for detection. With a
simple alternating optimization, RPN and Fast R-CNN can be trained to share
convolutional features. For the very deep VGG-16 model [19], our detection
system has a frame rate of 5fps (including all steps) on a GPU, while achieving
state-of-the-art object detection accuracy on PASCAL VOC 2007 (73.2% mAP)
and 2012 (70.4% mAP) using 300 proposals per image. Code is available at
https://github.com/ShaoqingRen/faster_rcnn.
1
Introduction
Recent advances in object detection are driven by the success of region proposal methods (e.g., [22])
and region-based convolutional neural networks (R-CNNs) [6]. Although region-based CNNs were
computationally expensive as originally developed in [6], their cost has been drastically reduced
thanks to sharing convolutions across proposals [7, 5]. The latest incarnation, Fast R-CNN [5],
achieves near real-time rates using very deep networks [19], when ignoring the time spent on region
proposals. Now, proposals are the computational bottleneck in state-of-the-art detection systems.
Region proposal methods typically rely on inexpensive features and economical inference schemes.
Selective Search (SS) [22], one of the most popular methods, greedily merges superpixels based
on engineered low-level features. Yet when compared to efficient detection networks [5], Selective
Search is an order of magnitude slower, at 2s per image in a CPU implementation. EdgeBoxes
[24] currently provides the best tradeoff between proposal quality and speed, at 0.2s per image.
Nevertheless, the region proposal step still consumes as much running time as the detection network.
One may note that fast region-based CNNs take advantage of GPUs, while the region proposal methods used in research are implemented on the CPU, making such runtime comparisons inequitable.
An obvious way to accelerate proposal computation is to re-implement it for the GPU. This may be
an effective engineering solution, but re-implementation ignores the down-stream detection network
and therefore misses important opportunities for sharing computation.
In this paper, we show that an algorithmic change?computing proposals with a deep net?leads
to an elegant and effective solution, where proposal computation is nearly cost-free given the de?
Shaoqing Ren is with the University of Science and Technology of China. This work was done when he
was an intern at Microsoft Research.
1
tection network?s computation. To this end, we introduce novel Region Proposal Networks (RPNs)
that share convolutional layers with state-of-the-art object detection networks [7, 5]. By sharing
convolutions at test-time, the marginal cost for computing proposals is small (e.g., 10ms per image).
Our observation is that the convolutional (conv) feature maps used by region-based detectors, like
Fast R-CNN, can also be used for generating region proposals. On top of these conv features, we
construct RPNs by adding two additional conv layers: one that encodes each conv map position
into a short (e.g., 256-d) feature vector and a second that, at each conv map position, outputs an
objectness score and regressed bounds for k region proposals relative to various scales and aspect
ratios at that location (k = 9 is a typical value).
Our RPNs are thus a kind of fully-convolutional network (FCN) [14] and they can be trained end-toend specifically for the task for generating detection proposals. To unify RPNs with Fast R-CNN [5]
object detection networks, we propose a simple training scheme that alternates between fine-tuning
for the region proposal task and then fine-tuning for object detection, while keeping the proposals
fixed. This scheme converges quickly and produces a unified network with conv features that are
shared between both tasks.
We evaluate our method on the PASCAL VOC detection benchmarks [4], where RPNs with Fast
R-CNNs produce detection accuracy better than the strong baseline of Selective Search with Fast
R-CNNs. Meanwhile, our method waives nearly all computational burdens of SS at test-time?the
effective running time for proposals is just 10 milliseconds. Using the expensive very deep models
of [19], our detection method still has a frame rate of 5fps (including all steps) on a GPU, and
thus is a practical object detection system in terms of both speed and accuracy (73.2% mAP on
PASCAL VOC 2007 and 70.4% mAP on 2012). Code is available at https://github.com/
ShaoqingRen/faster_rcnn.
2
Related Work
Several recent papers have proposed ways of using deep networks for locating class-specific or classagnostic bounding boxes [21, 18, 3, 20]. In the OverFeat method [18], a fully-connected (fc) layer
is trained to predict the box coordinates for the localization task that assumes a single object. The
fc layer is then turned into a conv layer for detecting multiple class-specific objects. The MultiBox methods [3, 20] generate region proposals from a network whose last fc layer simultaneously
predicts multiple (e.g., 800) boxes, which are used for R-CNN [6] object detection. Their proposal
network is applied on a single image or multiple large image crops (e.g., 224?224) [20]. We discuss
OverFeat and MultiBox in more depth later in context with our method.
Shared computation of convolutions [18, 7, 2, 5] has been attracting increasing attention for efficient, yet accurate, visual recognition. The OverFeat paper [18] computes conv features from an
image pyramid for classification, localization, and detection. Adaptively-sized pooling (SPP) [7] on
shared conv feature maps is proposed for efficient region-based object detection [7, 16] and semantic
segmentation [2]. Fast R-CNN [5] enables end-to-end detector training on shared conv features and
shows compelling accuracy and speed.
3
Region Proposal Networks
A Region Proposal Network (RPN) takes an image (of any size) as input and outputs a set of
rectangular object proposals, each with an objectness score.1 We model this process with a fullyconvolutional network [14], which we describe in this section. Because our ultimate goal is to share
computation with a Fast R-CNN object detection network [5], we assume that both nets share a
common set of conv layers. In our experiments, we investigate the Zeiler and Fergus model [23]
(ZF), which has 5 shareable conv layers and the Simonyan and Zisserman model [19] (VGG), which
has 13 shareable conv layers.
To generate region proposals, we slide a small network over the conv feature map output by the last
shared conv layer. This network is fully connected to an n ? n spatial window of the input conv
1
?Region? is a generic term and in this paper we only consider rectangular regions, as is common for many
methods (e.g., [20, 22, 24]). ?Objectness? measures membership to a set of object classes vs. background.
2
2k scores
4k coordinates
cls layer
person : 0.992
k anchor boxes
dog : 0.994
horse : 0.993
reg layer
car : 1.000
cat : 0.982
dog : 0.997
person : 0.979
256-d
intermediate layer
bus : 0.996
person : 0.736
boat : 0.970
person : 0.983
person : 0.983
person : 0.925
person : 0.989
sliding window
conv feature map
Figure 1: Left: Region Proposal Network (RPN). Right: Example detections using RPN proposals
on PASCAL VOC 2007 test. Our method detects objects in a wide range of scales and aspect ratios.
feature map. Each sliding window is mapped to a lower-dimensional vector (256-d for ZF and 512-d
for VGG). This vector is fed into two sibling fully-connected layers?a box-regression layer (reg)
and a box-classification layer (cls). We use n = 3 in this paper, noting that the effective receptive
field on the input image is large (171 and 228 pixels for ZF and VGG, respectively). This mininetwork is illustrated at a single position in Fig. 1 (left). Note that because the mini-network operates
in a sliding-window fashion, the fully-connected layers are shared across all spatial locations. This
architecture is naturally implemented with an n ? n conv layer followed by two sibling 1 ? 1 conv
layers (for reg and cls, respectively). ReLUs [15] are applied to the output of the n ? n conv layer.
Translation-Invariant Anchors
At each sliding-window location, we simultaneously predict k region proposals, so the reg layer
has 4k outputs encoding the coordinates of k boxes. The cls layer outputs 2k scores that estimate
probability of object / not-object for each proposal.2 The k proposals are parameterized relative to
k reference boxes, called anchors. Each anchor is centered at the sliding window in question, and is
associated with a scale and aspect ratio. We use 3 scales and 3 aspect ratios, yielding k = 9 anchors
at each sliding position. For a conv feature map of a size W ?H (typically ?2,400), there are W Hk
anchors in total. An important property of our approach is that it is translation invariant, both in
terms of the anchors and the functions that compute proposals relative to the anchors.
As a comparison, the MultiBox method [20] uses k-means to generate 800 anchors, which are not
translation invariant. If one translates an object in an image, the proposal should translate and the
same function should be able to predict the proposal in either location. Moreover, because the
MultiBox anchors are not translation invariant, it requires a (4+1)?800-dimensional output layer,
whereas our method requires a (4+2)?9-dimensional output layer. Our proposal layers have an order
of magnitude fewer parameters (27 million for MultiBox using GoogLeNet [20] vs. 2.4 million for
RPN using VGG-16), and thus have less risk of overfitting on small datasets, like PASCAL VOC.
A Loss Function for Learning Region Proposals
For training RPNs, we assign a binary class label (of being an object or not) to each anchor. We
assign a positive label to two kinds of anchors: (i) the anchor/anchors with the highest Intersectionover-Union (IoU) overlap with a ground-truth box, or (ii) an anchor that has an IoU overlap higher
than 0.7 with any ground-truth box. Note that a single ground-truth box may assign positive labels
to multiple anchors. We assign a negative label to a non-positive anchor if its IoU ratio is lower than
0.3 for all ground-truth boxes. Anchors that are neither positive nor negative do not contribute to the
training objective.
With these definitions, we minimize an objective function following the multi-task loss in Fast RCNN [5]. Our loss function for an image is defined as:
1 X
1 X ?
L({pi }, {ti }) =
Lcls (pi , p?i ) + ?
p Lreg (ti , t?i ).
(1)
Ncls i
Nreg i i
2
For simplicity we implement the cls layer as a two-class softmax layer. Alternatively, one may use logistic
regression to produce k scores.
3
Here, i is the index of an anchor in a mini-batch and pi is the predicted probability of anchor i being
an object. The ground-truth label p?i is 1 if the anchor is positive, and is 0 if the anchor is negative. ti
is a vector representing the 4 parameterized coordinates of the predicted bounding box, and t?i is that
of the ground-truth box associated with a positive anchor. The classification loss Lcls is log loss over
two classes (object vs. not object). For the regression loss, we use Lreg (ti , t?i ) = R(ti ? t?i ) where
R is the robust loss function (smooth L1 ) defined in [5]. The term p?i Lreg means the regression loss
is activated only for positive anchors (p?i = 1) and is disabled otherwise (p?i = 0). The outputs of
the cls and reg layers consist of {pi } and {ti } respectively. The two terms are normalized with Ncls
and Nreg , and a balancing weight ?.3
For regression, we adopt the parameterizations of the 4 coordinates following [6]:
tx = (x ? xa )/wa , ty = (y ? ya )/ha , tw = log(w/wa ), th = log(h/ha ),
?
tx = (x? ? xa )/wa , t?y = (y ? ? ya )/ha , t?w = log(w? /wa ), t?h = log(h? /ha ),
where x, y, w, and h denote the two coordinates of the box center, width, and height. Variables
x, xa , and x? are for the predicted box, anchor box, and ground-truth box respectively (likewise
for y, w, h). This can be thought of as bounding-box regression from an anchor box to a nearby
ground-truth box.
Nevertheless, our method achieves bounding-box regression by a different manner from previous
feature-map-based methods [7, 5]. In [7, 5], bounding-box regression is performed on features
pooled from arbitrarily sized regions, and the regression weights are shared by all region sizes. In
our formulation, the features used for regression are of the same spatial size (n ? n) on the feature
maps. To account for varying sizes, a set of k bounding-box regressors are learned. Each regressor
is responsible for one scale and one aspect ratio, and the k regressors do not share weights. As such,
it is still possible to predict boxes of various sizes even though the features are of a fixed size/scale.
Optimization
The RPN, which is naturally implemented as a fully-convolutional network [14], can be trained
end-to-end by back-propagation and stochastic gradient descent (SGD) [12]. We follow the ?imagecentric? sampling strategy from [5] to train this network. Each mini-batch arises from a single image
that contains many positive and negative anchors. It is possible to optimize for the loss functions of
all anchors, but this will bias towards negative samples as they are dominate. Instead, we randomly
sample 256 anchors in an image to compute the loss function of a mini-batch, where the sampled
positive and negative anchors have a ratio of up to 1:1. If there are fewer than 128 positive samples
in an image, we pad the mini-batch with negative ones.
We randomly initialize all new layers by drawing weights from a zero-mean Gaussian distribution
with standard deviation 0.01. All other layers (i.e., the shared conv layers) are initialized by pretraining a model for ImageNet classification [17], as is standard practice [6]. We tune all layers of
the ZF net, and conv3 1 and up for the VGG net to conserve memory [5]. We use a learning rate
of 0.001 for 60k mini-batches, and 0.0001 for the next 20k mini-batches on the PASCAL dataset.
We also use a momentum of 0.9 and a weight decay of 0.0005 [11]. Our implementation uses Caffe
[10].
Sharing Convolutional Features for Region Proposal and Object Detection
Thus far we have described how to train a network for region proposal generation, without considering the region-based object detection CNN that will utilize these proposals. For the detection
network, we adopt Fast R-CNN [5]4 and now describe an algorithm that learns conv layers that are
shared between the RPN and Fast R-CNN.
Both RPN and Fast R-CNN, trained independently, will modify their conv layers in different ways.
We therefore need to develop a technique that allows for sharing conv layers between the two networks, rather than learning two separate networks. Note that this is not as easy as simply defining
a single network that includes both RPN and Fast R-CNN, and then optimizing it jointly with backpropagation. The reason is that Fast R-CNN training depends on fixed object proposals and it is
3
In our early implementation (as also in the released code), ? was set as 10, and the cls term in Eqn.(1) was
normalized by the mini-batch size (i.e., Ncls = 256) and the reg term was normalized by the number of anchor
locations (i.e., Nreg ? 2, 400). Both cls and reg terms are roughly equally weighted in this way.
4
https://github.com/rbgirshick/fast-rcnn
4
not clear a priori if learning Fast R-CNN while simultaneously changing the proposal mechanism
will converge. While this joint optimizing is an interesting question for future work, we develop a
pragmatic 4-step training algorithm to learn shared features via alternating optimization.
In the first step, we train the RPN as described above. This network is initialized with an ImageNetpre-trained model and fine-tuned end-to-end for the region proposal task. In the second step, we
train a separate detection network by Fast R-CNN using the proposals generated by the step-1 RPN.
This detection network is also initialized by the ImageNet-pre-trained model. At this point the two
networks do not share conv layers. In the third step, we use the detector network to initialize RPN
training, but we fix the shared conv layers and only fine-tune the layers unique to RPN. Now the two
networks share conv layers. Finally, keeping the shared conv layers fixed, we fine-tune the fc layers
of the Fast R-CNN. As such, both networks share the same conv layers and form a unified network.
Implementation Details
We train and test both region proposal and object detection networks on single-scale images [7,
5]. We re-scale the images such that their shorter side is s = 600 pixels [5]. Multi-scale feature
extraction may improve accuracy but does not exhibit a good speed-accuracy trade-off [5]. We also
note that for ZF and VGG nets, the total stride on the last conv layer is 16 pixels on the re-scaled
image, and thus is ?10 pixels on a typical PASCAL image (?500?375). Even such a large stride
provides good results, though accuracy may be further improved with a smaller stride.
For anchors, we use 3 scales with box areas of 1282 , 2562 , and 5122 pixels, and 3 aspect ratios of
1:1, 1:2, and 2:1. We note that our algorithm allows the use of anchor boxes that are larger than the
underlying receptive field when predicting large proposals. Such predictions are not impossible?
one may still roughly infer the extent of an object if only the middle of the object is visible. With
this design, our solution does not need multi-scale features or multi-scale sliding windows to predict
large regions, saving considerable running time. Fig. 1 (right) shows the capability of our method
for a wide range of scales and aspect ratios. The table below shows the learned average proposal
size for each anchor using the ZF net (numbers for s = 600).
anchor 1282 , 2:1 1282 , 1:1 1282 , 1:2 2562 , 2:1 2562 , 1:1 2562 , 1:2 5122 , 2:1 5122 , 1:1 5122 , 1:2
proposal 188?111 113?114 70?92 416?229 261?284 174?332 768?437 499?501 355?715
The anchor boxes that cross image boundaries need to be handled with care. During training, we
ignore all cross-boundary anchors so they do not contribute to the loss. For a typical 1000 ? 600
image, there will be roughly 20k (? 60 ? 40 ? 9) anchors in total. With the cross-boundary anchors
ignored, there are about 6k anchors per image for training. If the boundary-crossing outliers are not
ignored in training, they introduce large, difficult to correct error terms in the objective, and training
does not converge. During testing, however, we still apply the fully-convolutional RPN to the entire
image. This may generate cross-boundary proposal boxes, which we clip to the image boundary.
Some RPN proposals highly overlap with each other. To reduce redundancy, we adopt nonmaximum suppression (NMS) on the proposal regions based on their cls scores. We fix the IoU
threshold for NMS at 0.7, which leaves us about 2k proposal regions per image. As we will show,
NMS does not harm the ultimate detection accuracy, but substantially reduces the number of proposals. After NMS, we use the top-N ranked proposal regions for detection. In the following, we
train Fast R-CNN using 2k RPN proposals, but evaluate different numbers of proposals at test-time.
4
Experiments
We comprehensively evaluate our method on the PASCAL VOC 2007 detection benchmark [4].
This dataset consists of about 5k trainval images and 5k test images over 20 object categories. We
also provide results in the PASCAL VOC 2012 benchmark for a few models. For the ImageNet
pre-trained network, we use the ?fast? version of ZF net [23] that has 5 conv layers and 3 fc layers,
and the public VGG-16 model5 [19] that has 13 conv layers and 3 fc layers. We primarily evaluate detection mean Average Precision (mAP), because this is the actual metric for object detection
(rather than focusing on object proposal proxy metrics).
Table 1 (top) shows Fast R-CNN results when trained and tested using various region proposal
methods. These results use the ZF net. For Selective Search (SS) [22], we generate about 2k SS
5
www.robots.ox.ac.uk/?vgg/research/very_deep/
5
Table 1: Detection results on PASCAL VOC 2007 test set (trained on VOC 2007 trainval). The
detectors are Fast R-CNN with ZF, but using various proposal methods for training and testing.
train-time region proposals
method
# boxes
SS
EB
RPN+ZF, shared
2k
2k
2k
test-time region proposals
method
# proposals
mAP (%)
SS
EB
RPN+ZF, shared
2k
2k
300
58.7
58.6
59.9
RPN+ZF, unshared
RPN+ZF
RPN+ZF
RPN+ZF
RPN+ZF (no NMS)
RPN+ZF (no cls)
RPN+ZF (no cls)
RPN+ZF (no cls)
RPN+ZF (no reg)
RPN+ZF (no reg)
RPN+VGG
300
100
300
1k
6k
100
300
1k
300
1k
300
58.7
55.1
56.8
56.3
55.2
44.6
51.4
55.8
52.1
51.3
59.2
ablation experiments follow below
RPN+ZF, unshared
SS
SS
SS
SS
SS
SS
SS
SS
SS
SS
2k
2k
2k
2k
2k
2k
2k
2k
2k
2k
2k
proposals by the ?fast? mode. For EdgeBoxes (EB) [24], we generate the proposals by the default
EB setting tuned for 0.7 IoU. SS has an mAP of 58.7% and EB has an mAP of 58.6%. RPN with
Fast R-CNN achieves competitive results, with an mAP of 59.9% while using up to 300 proposals6 .
Using RPN yields a much faster detection system than using either SS or EB because of shared conv
computations; the fewer proposals also reduce the region-wise fc cost. Next, we consider several
ablations of RPN and then show that proposal quality improves when using the very deep network.
Ablation Experiments. To investigate the behavior of RPNs as a proposal method, we conducted
several ablation studies. First, we show the effect of sharing conv layers between the RPN and Fast
R-CNN detection network. To do this, we stop after the second step in the 4-step training process.
Using separate networks reduces the result slightly to 58.7% (RPN+ZF, unshared, Table 1). We
observe that this is because in the third step when the detector-tuned features are used to fine-tune
the RPN, the proposal quality is improved.
Next, we disentangle the RPN?s influence on training the Fast R-CNN detection network. For this
purpose, we train a Fast R-CNN model by using the 2k SS proposals and ZF net. We fix this detector
and evaluate the detection mAP by changing the proposal regions used at test-time. In these ablation
experiments, the RPN does not share features with the detector.
Replacing SS with 300 RPN proposals at test-time leads to an mAP of 56.8%. The loss in mAP
is because of the inconsistency between the training/testing proposals. This result serves as the
baseline for the following comparisons.
Somewhat surprisingly, the RPN still leads to a competitive result (55.1%) when using the topranked 100 proposals at test-time, indicating that the top-ranked RPN proposals are accurate. On
the other extreme, using the top-ranked 6k RPN proposals (without NMS) has a comparable mAP
(55.2%), suggesting NMS does not harm the detection mAP and may reduce false alarms.
Next, we separately investigate the roles of RPN?s cls and reg outputs by turning off either of them
at test-time. When the cls layer is removed at test-time (thus no NMS/ranking is used), we randomly
sample N proposals from the unscored regions. The mAP is nearly unchanged with N = 1k
(55.8%), but degrades considerably to 44.6% when N = 100. This shows that the cls scores account
for the accuracy of the highest ranked proposals.
On the other hand, when the reg layer is removed at test-time (so the proposals become anchor
boxes), the mAP drops to 52.1%. This suggests that the high-quality proposals are mainly due to
regressed positions. The anchor boxes alone are not sufficient for accurate detection.
6
For RPN, the number of proposals (e.g., 300) is the maximum number for an image. RPN may produce
fewer proposals after NMS, and thus the average number of proposals is smaller.
6
Table 2: Detection results on PASCAL VOC 2007 test set. The detector is Fast R-CNN and VGG16. Training data: ?07?: VOC 2007 trainval, ?07+12?: union set of VOC 2007 trainval and VOC
2012 trainval. For RPN, the train-time proposals for Fast R-CNN are 2k. ? : this was reported in [5];
using the repository provided by this paper, this number is higher (68.0?0.3 in six runs).
method
# proposals
SS
SS
RPN+VGG, unshared
RPN+VGG, shared
RPN+VGG, shared
data
2k
2k
300
300
300
mAP (%)
time (ms)
?
07
07+12
07
07
07+12
66.9
70.0
68.5
69.9
73.2
1830
1830
342
198
198
Table 3: Detection results on PASCAL VOC 2012 test set. The detector is Fast R-CNN and VGG16. Training data: ?07?: VOC 2007 trainval, ?07++12?: union set of VOC 2007 trainval+test
and VOC 2012 trainval. For RPN, the train-time proposals for Fast R-CNN are 2k. ? : http://
host.robots.ox.ac.uk:8080/anonymous/HZJTQA.html. ? : http://host.robots.ox.ac.uk:8080/
anonymous/YNPLXB.html
method
# proposals
data
mAP (%)
SS
SS
RPN+VGG, shared?
RPN+VGG, shared?
2k
2k
300
300
12
07++12
12
07++12
65.7
68.4
67.0
70.4
Table 4: Timing (ms) on a K40 GPU, except SS proposal is evaluated in a CPU. ?Region-wise?
includes NMS, pooling, fc, and softmax. See our released code for the profiling of running time.
model
system
conv
proposal
region-wise
total
rate
VGG
VGG
ZF
SS + Fast R-CNN
RPN + Fast R-CNN
RPN + Fast R-CNN
146
141
31
1510
10
3
174
47
25
1830
198
59
0.5 fps
5 fps
17 fps
We also evaluate the effects of more powerful networks on the proposal quality of RPN alone.
We use VGG-16 to train the RPN, and still use the above detector of SS+ZF. The mAP improves
from 56.8% (using RPN+ZF) to 59.2% (using RPN+VGG). This is a promising result, because it
suggests that the proposal quality of RPN+VGG is better than that of RPN+ZF. Because proposals of
RPN+ZF are competitive with SS (both are 58.7% when consistently used for training and testing),
we may expect RPN+VGG to be better than SS. The following experiments justify this hypothesis.
Detection Accuracy and Running Time of VGG-16. Table 2 shows the results of VGG-16 for both
proposal and detection. Using RPN+VGG, the Fast R-CNN result is 68.5% for unshared features,
slightly higher than the SS baseline. As shown above, this is because the proposals generated by
RPN+VGG are more accurate than SS. Unlike SS that is pre-defined, the RPN is actively trained
and benefits from better networks. For the feature-shared variant, the result is 69.9%?better than
the strong SS baseline, yet with nearly cost-free proposals. We further train the RPN and detection
network on the union set of PASCAL VOC 2007 trainval and 2012 trainval, following [5]. The mAP
is 73.2%. On the PASCAL VOC 2012 test set (Table 3), our method has an mAP of 70.4% trained
on the union set of VOC 2007 trainval+test and VOC 2012 trainval, following [5].
In Table 4 we summarize the running time of the entire object detection system. SS takes 1-2
seconds depending on content (on average 1.51s), and Fast R-CNN with VGG-16 takes 320ms on
2k SS proposals (or 223ms if using SVD on fc layers [5]). Our system with VGG-16 takes in total
198ms for both proposal and detection. With the conv features shared, the RPN alone only takes
10ms computing the additional layers. Our region-wise computation is also low, thanks to fewer
proposals (300). Our system has a frame-rate of 17 fps with the ZF net.
Analysis of Recall-to-IoU. Next we compute the recall of proposals at different IoU ratios with
ground-truth boxes. It is noteworthy that the Recall-to-IoU metric is just loosely [9, 8, 1] related to
the ultimate detection accuracy. It is more appropriate to use this metric to diagnose the proposal
method than to evaluate it.
7
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ZWE&
ZWEs''
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Figure 2: Recall vs. IoU overlap ratio on the PASCAL VOC 2007 test set.
Table 5: One-Stage Detection vs. Two-Stage Proposal + Detection. Detection results are on the
PASCAL VOC 2007 test set using the ZF model and Fast R-CNN. RPN uses unshared features.
regions
Two-Stage
One-Stage
One-Stage
RPN + ZF, unshared
dense, 3 scales, 3 asp. ratios
dense, 3 scales, 3 asp. ratios
300
20k
20k
detector
mAP (%)
Fast R-CNN + ZF, 1 scale
Fast R-CNN + ZF, 1 scale
Fast R-CNN + ZF, 5 scales
58.7
53.8
53.9
In Fig. 2, we show the results of using 300, 1k, and 2k proposals. We compare with SS and EB, and
the N proposals are the top-N ranked ones based on the confidence generated by these methods.
The plots show that the RPN method behaves gracefully when the number of proposals drops from
2k to 300. This explains why the RPN has a good ultimate detection mAP when using as few as 300
proposals. As we analyzed before, this property is mainly attributed to the cls term of the RPN. The
recall of SS and EB drops more quickly than RPN when the proposals are fewer.
One-Stage Detection vs. Two-Stage Proposal + Detection. The OverFeat paper [18] proposes a
detection method that uses regressors and classifiers on sliding windows over conv feature maps.
OverFeat is a one-stage, class-specific detection pipeline, and ours is a two-stage cascade consisting
of class-agnostic proposals and class-specific detections. In OverFeat, the region-wise features come
from a sliding window of one aspect ratio over a scale pyramid. These features are used to simultaneously determine the location and category of objects. In RPN, the features are from square (3?3)
sliding windows and predict proposals relative to anchors with different scales and aspect ratios.
Though both methods use sliding windows, the region proposal task is only the first stage of RPN
+ Fast R-CNN?the detector attends to the proposals to refine them. In the second stage of our cascade, the region-wise features are adaptively pooled [7, 5] from proposal boxes that more faithfully
cover the features of the regions. We believe these features lead to more accurate detections.
To compare the one-stage and two-stage systems, we emulate the OverFeat system (and thus also
circumvent other differences of implementation details) by one-stage Fast R-CNN. In this system,
the ?proposals? are dense sliding windows of 3 scales (128, 256, 512) and 3 aspect ratios (1:1, 1:2,
2:1). Fast R-CNN is trained to predict class-specific scores and regress box locations from these
sliding windows. Because the OverFeat system uses an image pyramid, we also evaluate using conv
features extracted from 5 scales. We use those 5 scales as in [7, 5].
Table 5 compares the two-stage system and two variants of the one-stage system. Using the ZF
model, the one-stage system has an mAP of 53.9%. This is lower than the two-stage system (58.7%)
by 4.8%. This experiment justifies the effectiveness of cascaded region proposals and object detection. Similar observations are reported in [5, 13], where replacing SS region proposals with sliding
windows leads to ?6% degradation in both papers. We also note that the one-stage system is slower
as it has considerably more proposals to process.
5
Conclusion
We have presented Region Proposal Networks (RPNs) for efficient and accurate region proposal
generation. By sharing convolutional features with the down-stream detection network, the region
proposal step is nearly cost-free. Our method enables a unified, deep-learning-based object detection
system to run at 5-17 fps. The learned RPN also improves region proposal quality and thus the
overall object detection accuracy.
8
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9
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5,124 | 5,639 | Weakly-supervised Disentangling with
Recurrent Transformations for 3D View Synthesis
Jimei Yang1
Scott Reed2
Ming-Hsuan Yang1
Honglak Lee2
1
University of California, Merced
{jyang44, mhyang}@ucmerced.edu
2
University of Michigan, Ann Arbor
{reedscot, honglak}@umich.edu
Abstract
An important problem for both graphics and vision is to synthesize novel views
of a 3D object from a single image. This is in particular challenging due to the
partial observability inherent in projecting a 3D object onto the image space, and
the ill-posedness of inferring object shape and pose. However, we can train a
neural network to address the problem if we restrict our attention to specific object
classes (in our case faces and chairs) for which we can gather ample training
data. In this paper, we propose a novel recurrent convolutional encoder-decoder
network that is trained end-to-end on the task of rendering rotated objects starting
from a single image. The recurrent structure allows our model to capture longterm dependencies along a sequence of transformations, and we demonstrate the
quality of its predictions for human faces on the Multi-PIE dataset and for a dataset
of 3D chair models, and also show its ability of disentangling latent data factors
without using object class labels.
1
Introduction
Numerous graphics algorithms have been established to synthesize photorealistic images from 3D
models and environmental variables (lighting and viewpoints), commonly known as rendering. At
the same time, recent advances in vision algorithms enable computers to gain some form of understanding of objects contained in images, such as classification [16], detection [10], segmentation [18], and caption generation [26], to name a few. These approaches typically aim to deduce
abstract representations from raw image pixels. However, it has been a long-standing problem for
both graphics and vision to automatically synthesize novel images of applying intrinsic transformations (e.g. 3D rotation and deformation) to the subject of an input image. From an artificial
intelligent perspective, this can be viewed as answering questions about object appearance when the
view angle or illumination is changed, or some action is taken. These synthesized images may then
be perceived by humans in photo editing [14], or evaluated by other machine vision systems, such
as the game playing agent with vision-based reinforcement learning [20].
In this paper, we consider the problem of predicting transformed appearances of an object when
it is rotated in 3D from a single image. In general this is an ill-posed problem due to the loss of
information inherent in projecting a 3D object into the image space. Classic geometry-based approaches either recover a 3D object model from multiple related images, i.e. multi-view stereo and
structure-from-motion, or register a single image of a known object class to its prior 3D model, e.g.
faces [5]. The resulting mesh can be used to re-render the scene from novel viewpoints. However,
having 3D meshes as intermediate representations, these methods are 1) limited to particular object
classes, 2) vulnerable to image alignment mistakes and 3) easy to generate artifacts during unseen
texture synthesis. To overcome these limitations, we propose a learning approach without explicit
3D model recovery. Having observed rotations of similar 3D objects (e.g. faces, chairs, household
objects), the trained model can both 1) better infer the true pose, shape and texture of the object,
and 2) make plausible assumptions about potentially ambiguous aspects of appearance in novel
1
viewpoints. Thus, the learning algorithm relies on mappings between Euclidean image space and
underlying nonlinear manifold. In particular, 3D view synthesis can be cast as pose manifold traversal where a desired rotation can be decomposed to a sequence of small steps. A major challenge
arises due to the long-term dependency among multiple rotation steps; the key identifying information (e.g. shape, texture) from the original input must be remembered along the entire trajectory.
Furthermore, the local rotation at each step must generate the correct result on the data manifold, or
subsequent steps will also fail.
Closely related to the image generation task considered in this paper is the problem of 3D invariant
recognition, which involves comparing object images from different viewpoints or poses with dramatic changes of appearance. Shepard and Metzler in their mental rotation experiments [22] found
that the time taken for humans to match 3D objects from two different views increased proportionally with the angular rotational difference between them. It was as if the humans were rotating their
mental images at a steady rate. Inspired by this mental rotation phenomenon, we propose a recurrent convolutional encoder-decoder network with action units to model the process of pose manifold
traversal. The network consists of four components: a deep convolutional encoder [16], shared
identity units, recurrent pose units with rotation action inputs, and a deep convolutional decoder [8].
Rather than training the network to model a specific rotation sequence, we provide control signals
at each time step instructing the model how to move locally along the pose manifold. The rotation
sequences can be of varying length. To improve the ease of training, we employed curriculum learning, similar to that used in other sequence prediction problems [27]. Intuitively, the model should
learn how to make a one-step 15? rotation before learning how to make a series of such rotations.
The main contributions of this work are summarized as follows. First, a novel recurrent convolutional encoder-decoder network is developed for learning to apply out-of-plane rotations to human
faces and 3D chair models. Second, the learned model can generate realistic rotation trajectories
with a control signal supplied at each step by the user. Third, despite only trained to synthesize
images, our model learns discriminative view-invariant features without using class labels. This
weakly-supervised disentangling is especially notable with longer-term prediction.
2
Related Work
The transforming autoencoder [12] introduces the notion of capsules in deep networks, which tracks
both the presence and position of visual features in the input image. These models are shown to
be capable of applying affine transformations and 3D rotations to images. We address a similar
task of rendering object appearance undergoing 3D rotations, but we use a convolutional network
architecture in lieu of capsules (albeit with stride-2 convolution instead of max-pooling), and incorporate action inputs and recurrent structure to handle repeated rotation steps. The Predictive Gating
Pyramid (PGP) [19] is developed for time series prediction, and is able to learn image transformations including shifts and rotation over multiple time steps. Our task is related to this time series
prediction, but our formulation includes a control signal, uses disentangled latent features, and uses
convolutional encoder and decoder networks to model detailed images. Another gating network is
proposed in [7] to directly model mental rotation by optimizing transforming distance. Instead of
extracting invariant recognition features in one shot, their model learns to perform recognition by
exploring a space of relevant transformations. Similarly, our model can explore the space of rotation
about an object image by setting the control signal input at each time step of our recurrent network.
The problem of training neural networks that generate images is studied in [25]. A convolutional
network mapping shape, pose and transformation labels to images is proposed in [8] for generating
chairs. They are able to control these factors of variation and generate high quality renderings. We
also generate chair renderings in this paper, but our model adds several additional features: a deep
encoder network (so that we can generalize to novel images, rather than only decode), distributed
representations for appearance and pose, and recurrent structure for long-term prediction.
Contemporary to our work, the Inverse Graphics Network (IGN) [17] also adds an encoding function to learn graphics codes of images, along with a decoder similar to that in the chair generating
network. As in our model, IGN uses a deep convolutional encoder to extract image representations,
apply modifications to these, and then re-render. Our model differs in that we train a recurrent
network to perform trajectories of multiple transformations, we add control signal input at each
step, and we use deterministic feed-forward training rather than the variational auto-encoder (VAE)
framework [15] (although our approach could be extended to a VAE version).
2
Figure 1: Deep convolutional encoder-decoder network for learning 3d rotation
A related line of work to ours is disentangling the latent factors of variation that generate natural
images. Bilinear models for separating style and content are developed in [24], and are shown to
be capable of separating handwriting style and character identity, and also separating face identity and pose. The disentangling Boltzmann Machine (disBM) [21] applies this idea to augment
the Restricted Boltzmann Machine by partitioning its hidden state into distinct factors of variation
and modeling their higher-order interaction. The multi-view perceptron [29] employs a stochastic
feedforward network to disentangle the identity and pose factors of face images in order to achieve
view-invariant recognition. The encoder network for IGN is also trained to learn a disentangled representation of images by extracting a graphics code for each factor. In [6], the (potentially unknown)
latent factors of variation are both discovered and disentangled using a novel hidden unit regularizer.
Our work is also loosely related to the ?DeepStereo? algorithm [9] that synthesizes novel views of
scenes from multiple images using deep convolutional networks.
3
Recurrent Convolutional Encoder-Decoder Network
In this section we describe our model formulation. Given an image of 3D object, our goal is to synthesize its rotated views. Inspired by recent success of convolutional networks (CNNs) in mapping
images to high-level abstract representations [16] and synthesizing images from graphics codes [8],
we base our model on deep convolutional encoder-decoder networks. One example network structure is shown in Figure 1. The encoder network used 5 ? 5 convolution-relu layers with stride 2 and
2-pixel padding so that the dimension is halved at each convolution layer, followed by two fullyconnected layers. In the bottleneck layer, we define a group of units to represent the pose (pose
units) where the desired transformations can be applied. The other group of units represent what
does not change during transformations, named as identity units. The decoder network is symmetric
to the encoder. To increase dimensionality we use fixed upsampling as in [8]. We found that fixed
stride-2 convolution and upsampling worked better than max-pooling and unpooling with switches,
because when applying transformations the encoder pooling switches would not in general match
the switches produced by the target image. The desired transformations are reflected by the action
units. We used a 1-of-3 encoding, in which [100] encoded a clockwise rotation, [010] encoded a noop, and [001] encoded a counter-clockwise rotation. The triangle indicates a tensor product taking
as input the pose units and action units, and producing the transformed pose units. Equivalently, the
action unit selects the matrix that transforms the input pose units to the output pose units.
The action units introduce a small linear increment to the pose units, which essentially model the
local transformations in the nonlinear pose manifold. However, in order to achieve longer rotation
trajectories, if we simply accumulate the linear increments from the action units (e.g. [2 0 0] for
two-step clockwise rotation, the pose units will fall off the manifold resulting in bad predictions.
To overcome this problem, we generalize the model to a recurrent neural network, which have been
shown to capture long-term dependencies for a wide variety of sequence modeling problems. In
essence, we turn the pose units to be recurrent to model the step-by-step pose manifold traversals
and the identity units are shared across all time steps, since we assume that all training sequences
preserve the identity while only changing the pose. Figure 2 shows the unrolled version of our RNN
model. We only perform encoding at the first time step, and all transformations are carried out in the
latent space; i.e. the model predictions at time step t are not fed into the next time step input. The
training objective is based on pixel-wise prediction over all time steps for training sequences:
Lrnn =
N X
T
X
||y (i,t) ? g(fpose (x(i) , a(i) , t), fid (x(i) ))||22
(1)
i=1 t=1
where a(i) is the sequence of T actions, fid (x(i) ) produces the identity features invariant to all the
time steps, fpose (x(i) , a(i) , t) produces the transformed pose features at time step t, g(?, ?) is the
3
Figure 2: Unrolled recurrent convolutional network for learning to rotate 3d objects. The convolutional encoder
and decoders have been abstracted out, represented here as vertical rectangles.
image decoder producing an image given the output of fid (?) and fpose (?, ?, ?), x(i) is the i-th image,
y (i,t) is the i-th training image target at step t.
3.1 Curriculum Training
We trained the network parameters using backpropagation through time and the ADAM solver [3].
To effectively train our recurrent network, we found it beneficial to use curriculum learning [4], in
which we gradually increase the difficulty of training by increasing the trajectory length. This appears to be useful for sequence prediction with recurrent networks in other domains as well, such
as learning to execute Python programs [27]. In section 4, we show that increasing the training sequence length improves both the model?s image prediction performance as well as the pose-invariant
recognition performance of identity features.
Also, longer training sequences force the identity units to better disentangle themselves from the
pose. If the same identity units need to be used to predict both a 15? -rotated and a 120? -rotated
image during training, these units can not pick up pose-related information. In this way, our model
can learn disentangled features (i.e. identity units can do invariant identity recognition but are not
informative of pose, and vice-versa) without explicitly regularizing to achieve this effect. We did
not find it necessary to use gradient clipping.
4
Experiments
We carry out experiments to achieve the following objectives. First, we examine the ability of our
model to synthesize high quality images of both face and complex 3D objects (chairs) in a wide
range of rotational angles. Second, we evaluate the discriminative performance of disentangled
identity units through cross-view object recognition. Third, we demonstrate the ability to generate
and rotate novel object classes by interpolating identity units of seen objects.
4.1 Datasets
Multi-PIE. The Multi-PIE [11] dataset consists of 754,204 face images from 337 people. The
images are captured from 15 viewpoints under 20 illumination conditions in different sessions. To
evaluate our model for rotating faces, we select a subset of Multi-PIE that covers 7 viewpoints
evenly from ?45? to 45? under neutral illumination. Each face image is aligned through manually
annotated landmarks on eyes, nose and mouth corners, and then cropped to 80 ? 60 ? 3 pixels. We
use the images of first 200 people for training and the remaining 137 people as the test set.
Chairs. This dataset contains 1393 chair CAD models made public by Aubry et al. [1]. Each chair
model is rendered from 31 azimuth angles (with steps of 11? ) and 2 elevation angles (20? and 30? )
at a fixed distance to the virtual camera. We use a subset of 809 chair models in our experiments,
which are selected out of 1393 by Dosovitskiy et al. [8] in order to remove near-duplicate models,
models differing only in color or low-quality models. We crop the rendered images to have a small
border and resize them to a common size of 64 ? 64 ? 3 pixels. We also prepare their binary masks
by subtracting the white background. We use the images of the first 500 models as the training set
and the remaining 409 models as the test set.
4.2 Network Architectures and Training Details
Multi-PIE. The encoder network for Multi-PIE used two convolution-relu layers with stride 2 and
2-pixel padding, followed by one fully-connected layers: 5?5?64 ? 5?5?128 ? 1024. The identity
4
?45? ?30? ?15?
0?
15?
30?
45?
45?
30?
15?
0?
?15? ?30? ?45?
Input
Input
Figure 3: 3D view synthesis on Multi-PIE. For each panel, the top row shows the ground truth images from
?45? to 45? , the bottom row shows the re-renderings of 6-step clockwise rotation from an input image of ?45?
and of 6-step counter-clockwise rotation from an input image of 45? .
45?
30?
15?
?15? ?30? ?45?
45?
30?
15?
?15? ?30? ?45?
Input
RNN
3D
model
Figure 4: Comparing face pose normalization results with 3D morphable model [28].
and pose units are 512 and 128, respectively. The decoder network is symmetric to the encoder. The
curriculum training procedure starts with the single-step rotation model which we named RNN1.
We prepare training samples by pairing face images of the same person captured in the same session
with adjacent camera viewpoints. For example, x(i) at ?30? is mapped to y (i) at ?15? with action
a(i) = [001]; x(i) at ?15? is mapped to y (i) at ?30? with action a(i) = [100]; and x(i) at ?30? is
mapped to y (i) at ?30? with action a(i) = [010]. For face images with ending viewpoints ?45?
and 45? , only one-way rotation is feasible. We train the network using the ADAM solver with fixed
learning rate 1e?4 for 400 epochs1 .
Since there are 7 viewpoints per person per session, we schedule the curriculum training with t = 2,
t = 4 and t = 6 stages, which we named RNN2, RNN4 and RNN6, respectively. To sample training
sequences with fixed length, we allow both clockwise and counter-clockwise rotations. For example,
when t = 4, one input image x(i) at 30? is mapped to (y (i,1) , y (i,2) , y (i,3) , y (i,4) ) with corresponding
angles (45? , 30? , 15? , 0? ) and action inputs ([001], [100], [100], [100]). In each stage, we initialize
the network parameters with the previous stage and fine-tune the network with fixed learning rate
1e?5 for 10 additional epochs.
Chairs. The encoder network for chairs used three convolution-relu layers with stride 2 and 2-pixel
padding, followed by two fully-connected layers: 5?5?64 ? 5?5?128 ? 5?5?256 ? 1024 ? 1024.
The decoder network is symmetric, except that after the fully-connected layers it branches into image
and mask prediction layers.The mask prediction indicates whether a pixel belongs to foreground or
background. We adopted this idea from the generative CNNs [8] and found it beneficial to training
efficiency and image synthesis quality. A tradeoff parameter ? = 0.1 is applied to the mask prediction
loss. We train the single-step network parameters with fixed learning rate 1e?4 for 500 epochs. We
schedule the curriculum training with t = 2, t = 4, t = 8 and t = 16, which we named RNN2, RNN4,
RNN8 and RNN16. Note that the curriculum training stops at t = 16 because we reached the limit
of GPU memory. Since the images of each chair model are rendered from 31 viewpoints evenly
sampled between 0? and 360? , we can easily prepare training sequences of clockwise or counterclockwise t-step rotations around the circle. Similarly, the network parameters of the current stage
is initialized with those of previous stage and fine-tuned with fixed learning rate 1e?5 for 50 epochs.
4.3 3D View Synthesis of Novel Objects
We first examine the re-rendering quality of our RNN models for novel objects instances that were
not seen during training. On the Multi-PIE dataset, given one input image from the test set with
possible views between ?45? to 45? , the encoder produces identity units and pose units and then
the decoder renders images progressively with fixed identity units and action-driven recurrent pose
units up to t-steps. Examples are shown in Figure 3 of the longest rotations, i.e. clockwise from
1
We carry out experiments using Caffe [13] on Nvidia k40c and Titan X GPUs.
5
?45? to 45? and counter-clockwise from 45? to ?45? with RNN6. High quality renderings are
generated with smooth transformations between adjacent views. The characteristics of faces, such
as gender, expression, eyes, nose and glasses are also preserved during rotation. We also compare
our RNN model with a state-of-the-art 3D morphable model for face pose normalization [28] in
Figure 4. It can be observed that our RNN model produces stable renderings while 3D morphable
model is sensitive to facial landmark localization. One of the advantages of 3D morphable model is
that it preserves facial textures well.
On the chair dataset, we use RNN16 to synthesize 16 rotated views of novel chairs in the test set.
Given a chair image of certain view, we define two action sequences; one for progressive clockwise
rotation and and another for counter-clockwise rotation. It is a more challenging task compared
to rotating faces due to the complex 3D shapes of chairs and the large rotation angles (more than
180? after 16-step rotations). Since no previous methods tackle the exact same chair re-rendering
problem, we use a k-nearest-neighbor (KNN) method for baseline comparisons. The KNN baseline
is implemented as follows. We first extract the CNN features ?fc7? from VGG-16 net [23] for all
the chair images. For each test chair image, we find its k nearest neighbors in the training set by
comparing their ?fc7? features. The retrieved top-K images are expected to be similar to the query
in terms of both style and pose [2]. Given a desired rotation angle, we synthesize rotated views of
the test image by averaging the corresponding rotated views of the retrieved top-K images in the
training set at the pixel level. We tune the K value in [1,3,5,7], namely KNN1, KNN3, KNN5 and
KNN7 to achieve its best performance. Two examples are shown in Figure 5. In our RNN model,
the 3D shapes are well preserved with clear boundaries for all the 16 rotated views from different
input, and the appearance changes smoothly between adjacent views with consistent style.
Input t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7 t = 8 t = 9 t = 10 t = 11 t = 12 t = 13 t = 14 t = 15 t = 16
Figure 5: 3D view synthesis of 16-step rotations on Chairs. In each panel, the first and second row demonstrate re-renderings of 16-step clockwise and counter-clockwise rotations from our RNN16 model and KNN5
baseline, respectively.
Note that conceptually the learned network parameters during different stages of curriculum training
can be used to process an arbitrary number of rotation steps. Unsurprisingly, the RNN1 model (the
first row in Figure 6) only works well in the first rotation step and produce degenerate results from
the second step. The RNN2 (second row in Figure 6), trained with two-step rotations, generates
reasonable results in the third step. Progressively, the RNN4 and RNN8 seem to generalize well
on chairs with longer predictions (t = 6 for RNN4 and t = 12 for RNN8). We measure the
quantitative performance of KNN and our RNN by the mean squared error (MSE) in (1) in Figure 7.
As a result, the best KNN with 5 retrievals (KNN5) obtains ?310 MSE, which is comparable to
our RNN4 model, but significantly outperformed by our RNN16 model (?179 MSE) with a 42%
improvement.
4.4 Cross-View Object Recognition
In this experiment, we examine and compare the discriminative performance of disentangled representations through cross-view object recognition.
6
RNN1
RNN2
RNN4
RNN8
RNN16
GT
Model t=1 t = 2 t = 3 t = 4 t = 6 t = 8 t = 12 t = 16
Figure 6: Comparing chair synthesis results from
RNN at different curriculum stages.
Figure 7: Comparing reconstruction mean
squared errors (MSE) on chairs with RNNs
and KNNs.
Multi-PIE. We create 7 gallery/probe splits from the test set. In each split, the face images of
same view, e.g. ?45? are collected as gallery and the rest of other views as probes. We extract
512-d features from the identity units of RNNs for all the test images so that the probes are matched
to the gallery by their cosine distance. It is considered as a success if the matched gallery image
has the same identity with one probe. We also categorize the probes in each split by measuring their
angle offsets from the gallery. In particular, the angle offsets range from 15? to 90? . The recognition
difficulties increase with angle offsets. To demonstrate the discriminative performance of our learned
representations, we also implement a convolutional network (CNN) classifier. The CNN architecture
is setup by connecting our encoder and identity units with a 200-way softmax output layer, and its
parameters are learned on the training set with ground truth class labels. The 512-d features extracted
from the layer before the softmax layer are used to perform cross-view object recognition as above.
Figure 8 (left) compares the average success rates of RNNs and CNN with their standard deviations
over 7 splits for each angle offset. The success rates of RNN1 drop more than 20% from angle
offset 15? to 90? . The success rates keep improving in general with curriculum training of RNNs,
and the best results are achieved as RNN6. As expected, the performance gap for RNN6 between
15? to 90? reduces to 10%. This phenomenon demonstrates that our RNN model gradually learns
pose/viewpoint-invariant representations for 3D face recognition. Without using any class labels,
our RNN model achieves competitive results against CNN.
Chairs. The experiment setup is similar to Multi-PIE. There are in total 31 azimuth views per
chair instance. For each view we create its gallery/probe split so that we have 31 splits. We extract
512-d features from identity units of RNN1, RNN2, RNN4, RNN8 and RNN16. The probes for each
split are categorized into 15 angle offsets ranging from 12? to 174? . Note that this experiment is
particularly challenging because chair matching is a fine-grained recognition task and chair appearances change significantly with 3D rotations. We also compare our model against CNN, but instead
of training CNN from scratch we use the pre-trained VGG-16 net [23] to extract the 4096-d ?fc7?
features for chair matching. The success rates are shown in Figure 8 (right). The performance drops
quickly when the angle offset is greater than 45? , but the RNN16 significantly improves the overall success rates especially for large angle offsets. We notice that the standard deviations are large
around the angle offsets 70? to 120? . This is because some views contain more information about
the chair 3D shapes than the other views so that we see performance variations. Interestingly, the
performance of VGG-16 net surpasses our RNN model when the angle offset is greater than 120? .
We hypothesize that this phenomenon results from the symmetric structures of most of chairs. The
VGG-16 net was trained with mirroring data augmentation to achieve certain symmetric invariance
while our RNN model does not explore this structure.
To further demonstrate the disentangling property of our RNN model, we use the pose units extracted
from the input images to repeat the above cross-view recognition experiments. The mean success
rates are shown in Table 1. It turns out that the better the identity units perform the worse the pose
units perform. When the identity units achieve near-perfect recognition on Multi-PIE, the pose units
only obtain a mean success rate 1.4%, which is close to the random guess 0.5% for 200 classes.
7
RNN1
RNN2
RNN4
RNN6
CNN
RNN1
RNN4
RNN8
RNN16
VGG-16
90
95
90
85
80
75
70
65
60
RNN2
100
Classification success rates (%)
Classification success rates (%)
100
80
70
60
50
40
30
20
10
20
30
40
50
60
70
80
0
90
20
40
Angle offset
60
80
100
120
140
160
180
Angle offset
Figure 8: Comparing cross-view recognition success rates for faces (left) and chairs (right).
Table 1: Comparing mean cross-view recognition success rates (%) with identity and pose units.
Models
Multi-PIE
Chairs
RNN: identity
93.3
56.8
RNN: pose
1.4
9.0
CNN
92.6
52.5 (VGG-16)
4.5 Class Interpolation and View Synthesis
In this experiment, we demonstrate the ability of our RNN model to generate novel chairs by interpolating between two existing ones. Given two chair images of same view from different instances, the
2
1
2
1
, respec, zpose
encoder network is used to compute their identity units zid
, zid
and pose units zpose
2
1
2
1
,
tively. The interpolation is computed by zid = ?zid + (1 ? ?)zid and zpose = ?zpose + (1 ? ?)zpose
where ? = [0.0, 0.2, 0.4, 0.6, 0.8, 1.0]. The interpolated zid and zpose are then fed into the recurrent
decoder network to render its rotated views. Example interpolations between four chair instances
are shown in Figure 9. The Interpolated chairs present smooth stylistic transformations between any
pair of input classes (each row in Figure 9), and their unique stylistic characteristics are also well
preserved among its rotated views (each column in Figure 9).
Input
t=1
t=5
t=9
t=13
?
0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
Figure 9: Chair style interpolation and view synthesis. Given four chair images of same view (first row) from
test set, each row presents renderings of style manifold traversal with fixed view while each column presents
the renderings of pose manifold traversal with fixed interpolated identity.
5
Conclusion
In this paper we develop a recurrent convolutional encoder-decoder network, and demonstrate its
effectiveness for synthesizing 3D views of unseen object instances. On the Multi-PIE dataset and
a database of 3D chair CAD models, the model predicts accurate renderings across trajectories of
repeated rotations. The proposed curriculum training by gradually increasing trajectory length of
training sequences yields both better image appearance and more discriminative features for poseinvariant recognition. We also show that a trained model could interpolate across the identity manifold of chairs at fixed pose, and traverse the pose manifold while fixing the identity. This generative
disentangling of chair identity and pose emerged from our recurrent rotation prediction objective,
even though we do not explicitly regularize the hidden units to be disentangled. Our future work
includes introducing more actions into the proposed model other than rotation, handling objects embedded in complex scenes, and handling one-to-many mappings for which a transformation yields a
multi-modal distribution over future states in the trajectory.
8
References
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9
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5,125 | 564 | Image Segmentation with Networks of Variable Scales
Hans P. Grar
Craig R. Nohl
AT&T Bell Laboratories
Crawfords Comer Road
Holmdel, NJ 07733, USA
Jan Ben
ABSTRACT
We developed a neural net architecture for segmenting complex
images, i.e., to localize two-dimensional geometrical shapes in a scene,
without prior knowledge of the objects' positions and sizes. A scale
variation is built into the network to deal with varying sizes. This algorithm has been applied to video images of railroad cars, to find their
identification numbers. Over 95% of the characlers were located
correctly in a data base of 300 images, despile a large variation in lighting conditions and often a poor quality of the characters. A part of the
network is executed on a processor board containing an analog neural
net chip (Graf et aI. 1991). while the rest is implemented as a software
model on a workstation or a digital signal processor.
1 INTRODUCTION
Neural nets have been applied successfully to the classification of shapes, such as characters. However, typically, these networks do not tolerate large variations of an object's
size. Rather, a normalization of the size has to be done before the network is able to perform a reliable classification. But in many machine vision applications an object's size is
not known in advance and may vary over a wide range. If the objects are part of a complex image, finding their positions plus their sizes becomes a very difficult problem.
Traditional techniques to locate objects of variable scale include the generalized Hough
transform (Ballard 1981) and constraint search techniques through a feature space (Grimson 1990), possibly with some relaxation mechanisms. These techniques stan with a
feature representation and then try to sort features into groups that may represent an
object. Searches through feature maps lend to be very time consuming, since the number
of comparisons that need to be made grows fast, typically exponentionally, with the
number of features. Therefore, practical techniques must focus on ways to minimize the
time required for this search.
480
Image Segmentation with Networks of Variable Scales
Our solution can be viewed as a large network. divided into two parts. The first layer of
the network provides a feature representation of the image. while the second layer locates
the objects. The key element for this network to be practical. is a neural net chip (Graf et
al. 1991) which executes the first layer. The high compute power of this chip makes it
possible to extract a large number of features. Hence features specific to the objects to be
found can be extracted, reducing drastically the amount of computation required in the
second layer.
The output of our network is not necessarily the final solution of a problem. Rather, its
intended use is as part of a modular system, combined with other functional elements.
Figure 1 shows an example of such a system that was used to read the identification
numbers on railroad cars. In this system the network's outputs are the positions and sizes
of characters. These are then classified in an additional network (LeCun et aI. 1990). specialized for reading characters.
The net described here is not limited to finding characters. It can be combined with other
classifiers and is applicable to a wide variety of object recognition tasks. Details of the
network, for example the types of features that are extracted, are task specific and have to
be optimized for the problem to be solved. But the overall architecture of the network
and the data flow remains the same for many problems. Beside the application described
here, we used this network for reading the license plates of cars, locating the address
blocks on mail pieces, and for page layout analysis of printed documents.
Camera
Preprocessing
-
Segmentation
Classification
Find characters
scale size
Digital Signal
Neural Net
Digital Signal
Processor
Processor
Processor
Figure 1: Schematic of the recognition system for reading the identification numbers on
railroad cars. The network described here performs the part in the middle box, segmenting the image into characters and background.
2 THE NETWORK
2.1
THE ARCHITECTURE
The network consists of two parts, the input layer extracting features and the second
layer, which locates the objects. The second layer is not rigidly coupled through connections to the first one. Before data move from the first layer to the second, the input fields
of the neurons in the second layer are scaled to an appropriate size. This size depends on
481
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Graf, Noh), and Ben
the data and is dynamically adjusted.
MODEL
FEATURE REPRESENTATION
OF THE MODEL
~-....-r.
????
??
?:
???
..... ..
??
It
MATCH MODEL Willi IMAGE
?
..
I
..
I
..
I
?
?
I
?
FEATURE MAPS
SIMPLE FEATURES
(EDGES. CORNERS)
INPUT IMAGE
Figure 2: Schematic of the network.
Figure 2 shows a schematic of this whole network. The input data pass through the first
layer of connections. From the other end of the net the model of the object is entered, and
in the middle model and image are matched by scaling the input fields of the neurons in
the second layer. In this way a network architecture is obtained that can handle a large
variation of sizes. In the present paper we consider only scale variations, but other
transformations, such as rotations can be integrated into this architecture as well.
And how can a model representation be scaled to the proper size before one knows an
object's size? With a proper feature representation of the image, this can be done in a
straight-forward and time-efficient way. Distances between pairs of features are measured and used to scale the input fields. In section 4 it is described in detail how the distances between corners provide a robust estimate of the sizes of characters. There is no
need to determine an object's size with absolute certainty here. The goal is to limit the
further search to just a few possible sizes, in order to reduce the amount of computation.
The time to evaluate the second layer of the network is reduced further by determining
"areas-of-interest" and searching only these. Areas without any features, or without
characteristic combinations of features, are excluded from the search. In this way, the
neurons of the second layer have to analyze only a small part of the whole image. The
key for the size estimates and the "area-of-interest" algorithm to work reliably, is a good
feature representation. Thanks to the neural net chip, we can search an image for a large
number of geometric features and have great freedom in choosing their shapes.
Image Segmentation with Networks of Variable Scales
2.2
KERNELS FOR EXTRACTING FEATURES
The features extracted in the first layer have to be detectable regardless of an objecCs
size. Many features, for example comers, are in principle independent of size. In practice
however, one uses two-dimensional detectors of a finite extent These detectors introduce
a scale and tend to work best for a certain range of sizes. Hence, it may be necessary to
use several detectors of different sizes for one feature. Simple features tend to be less
sensitive to scale than complex ones. In the application described below, a variation of a
factor of five in the characters' sizes is covered with just a single set of edge and comer
detectors. Figure 3 shows a few of the convolution kernels used to extract these features.
Figure 3: Examples of kernels for detecting edges and comers. Each of the kernels is
stored as the connection weights of a neuron. These are ternary kernels with a size of 16
x 16 pixels. The values of the pixels are: black = -1, white = 0, hatched =+1. A total of
32 kernels of this size can be scanned simultaneously over an image with the neural net
chip.
These kernels are scanned over an image with the neural net chip and wherever an edge
or a comer of the proper orientation is located, the neuron tied to this kernel turns on. In
this way, the neural net chip scans 32 kernels simultaneously over an image, creating 32
feature maps. The kernels containing the feature detectors have a size of 16 x 16 pixels.
With kernels of such a large size, it is possible to create highly selective detectors. Moreover, a high noise immunity is obtained.
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Graf, Nohl, and Ben
2.3
THE SECOND LAYER
The neurons of the second layer have a rectangular receptive field with 72 inputs, 3 x 3
inputs from eight feature maps. These neurons are trained with feature representations of
shapes. normalized in size. The 3 x 3 input field of a neuron does not mean that only an
area of 9 pixels in a feature map is used as input. Before a neuron is scanned over a part
of a feature map. its input field is scaled to the size indicated by the size estimator.
Therefore. each input corresponds to a rectangular area in a feature map. For finding
objects in an image. the input fields. scaled to the proper size. are then scanned over the
areas marked by the "area-of-interest" algorithm. If an output of a neuron is high. the
area is marked as position of an object and is passed along to the classifier.
The second layer of the network does require only relatively few computations. typically
a few hundred evaluations of neurons with 72 inputs. Therefore. this can be handled
easily by a workstation or a digital signal processor. The same is true for the area-ofinterest algorithm. The computationally expensive part is the feature extraction. On an
image with 512 x 512 pixels this requires over 2 billion connection updates. In fact. on a
workstation this takes typically about half an hour. Therefore. here a special purpose chip
is crucial to provide a speed-up to make this approach useful for practical applications.
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Figure 4: Schematic of the neural net board.
A schematic of the neural net board used for these applications is shown in Figure 4. The
board contains an analog neural net chip. combined with a digital signal processor (DSP)
and 256k of fast static memory. On the board. the DSP controls the data flow and the
operation of the neural net chip. This board is connected over a VME bus to the host
workstation. Signals. such as images. are sent from the host to the neural net board,
where a local program operates on the data. The results are then sent back to the host for
further processing and display. The time it takes to process an image of 512 x 512 pixels
is one second. where the transfer of the data from the workstation to the board and back
requires two thirds of this time.
Image Segmentation with Networks of Variable Scales
The chip does a part of the computation in analog form. But analog signals are used only
inside the chip, while all the input and the output data are digital. This chip works only
with a low dynamic range of the signals. Therefore, the input signals are typically binarized before they are transferred to the chip. In the case of gray level images, the pictures
are halftoned first and then the features are extracted form the binarized images. This is
possible, since the large kernel sizes suppress the noise introduced by the halftoning pr0cess.
4 APPLICATION
This network was integrated into a system to read the identification numbers on railroad
cars. Identifying a rail car by its number has to be done before a train enters the switching yard, where the cars are directed to different tracks. Today this is handled by human
operators reading the numbers from video screens. The present investigation is a study to
determine whether this process can be automated.
The pictures represent very difficult segmentation tasks, since the size of the characters
varies by more than a factor of five and they are often of poor quality with parts rusted
away or covered by dirt. Moreover, the positions of the characters can be almost anywhere in the picture, and they may be arranged in various ways, in single or in multiple
lines. Also, they are written in many different fonts, and the contrast between characters
and background varies substantially from one car to the next. Despite these difficulties,
we were able to locate the characters correctly in over 95% of the cases, on a database of
300 video images of railroad cars.
As mentioned in section 2, in the first layer feature maps are created from which areas of
interest are determined. Since the characters are arranged in horizontal lines, the first step
is to determine where lines of characters might be present in the image. For that purpose
the feature maps are projected onto a vertical line. Rows of characters produce strong
responses of the comer detectors and are therefore detected as maxima in the projected
densities. The orientation of a comer indicates whether it resulted from the lower end of
a character or from the upper end. The simultaneous presence of maxima in the densities
of lower and upper ends is therefore a strong indication for the presence of a row of characters. In this way, bands within the image are identified that may contain characters. A
band not only indicates the presence of a row of characters, but also provides a good
guess of their heights.
This simple heuristic proved to be very effective for the rail car images. It was made
more robust by taking into account also the outputs of the vertical edge detectors. Characters produce strong responses of vertical edge detectors, while detractors, such as dirt
create fewer and weaker responses.
At this stage we do not attempted to identify a character. All we need is a yes/no answer
whether a part of the image should be analyzed by the classifier or nOl The whole alphabet is grouped into five classes, and only one neuron to recognize any member within a
class is created. A high output of one of these neurons therefore means that any character
of its class may be present. Figures 5 and 6 show two examples produced by the segmentation network. The time required for the whole segmentation is less than three seconds,
of which one second is spent for the feature extraction and the rest for the "focus-of-
485
486
Graf. Noh!, and Ben
attention" algorithm and the second layer of the network.
Figure 5: Image of a tank car. The crosses mark where corner detectors gave a strong
response. The inset shows an enlarged section around the identification number. The
result of the segmentation network is indicated by the black lines.
5 CONCLUSION
The algorithm described combines neural nel techniques with heuristics to obtain a practical solution for segmenting complex images reliably and fast. Clearly. a "conventional"
neural net with a fixed architecture lacks the flexibility to handle the scale variations
required in many machine vision applications. To extend the use of neural nets. transformations have to be built into the architecture.
We demonstrated the network's use for locating characters. but the same strategy works
for a wide variety of other objects. Some details need to be adjusted to the objects to be
found. In particular, the features extracted by the first layer are task specific. Their
choice is critical, as they determine to a large extent the computational requirements for
finding the objects in the second layer.
The use of a neural net chip is crucial to make this approach feasible. since it provides the
computational power needed for the feature extraction. The extraction of geometrical
features for pattern recognition applications has been studied extensively. However, its
use is not wide spread, since it is computationally very demanding. The neural net chip
opens the possibility for extracting large numbers of features in a short time. The large
size of the convolution kernels, 16 x 16 pixels, provides a great flexibility in choosing the
feature detectors' shapes. Their large size is also the main reason for a good noise
Image Segmentation with Networks of Variable Scales
suppression and a high robustness of the described network.
Figure 6: The result of the network on an image of high complexity. The white horizontal lines indicate the result of the "area-of-interest" algorithm. The final result is shown
by the vertical white lines.
References
H.P. Graf, R. Janow, C.R. Nohl, and J. Ben, (1991), n A Neural-Net Board System for
Machine Vision Applications", Proc. Int. Joint Con/. Neural Networks, Vol. I, pp. 481 486.
D.H. Ballard, (1981), "Generalizing the Hough transform to detect arbitrary shapes", Pattern Recognition, Vol. 13, p. 111.
W.H. Grimson, (1990), "The Combinatorics of Object Recognition in Cluttered Environments Using Constraint Search", Artificial Intelligence, Vol. 44, p. 121.
Y. LeCun, B. Boser, J.S. Denker, D. Henderson, R.E. Howard, W. Hubbard, and L.D.
Jackel, (1990), "Handwritten Digit Recognition with a Back-Propagation Network", in:
Neural Information Processing Systems, Vol. 2, D. Touretzky (ed.), Morgan Kaufman,
pp. 396 - 404.
487
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5,126 | 5,640 | Exploring Models and Data for Image Question
Answering
Mengye Ren1 , Ryan Kiros1 , Richard S. Zemel1,2
University of Toronto1
Canadian Institute for Advanced Research2
{mren, rkiros, zemel}@cs.toronto.edu
Abstract
This work aims to address the problem of image-based question-answering (QA)
with new models and datasets. In our work, we propose to use neural networks
and visual semantic embeddings, without intermediate stages such as object detection and image segmentation, to predict answers to simple questions about images. Our model performs 1.8 times better than the only published results on an
existing image QA dataset. We also present a question generation algorithm that
converts image descriptions, which are widely available, into QA form. We used
this algorithm to produce an order-of-magnitude larger dataset, with more evenly
distributed answers. A suite of baseline results on this new dataset are also presented.
1
Introduction
Combining image understanding and natural language interaction is one of the grand dreams of
artificial intelligence. We are interested in the problem of jointly learning image and text through a
question-answering task. Recently, researchers studying image caption generation [1, 2, 3, 4, 5, 6,
7, 8, 9, 10] have developed powerful methods of jointly learning from image and text inputs to form
higher level representations from models such as convolutional neural networks (CNNs) trained on
object recognition, and word embeddings trained on large scale text corpora. Image QA involves
an extra layer of interaction between human and computers. Here the model needs to pay attention
to details of the image instead of describing it in a vague sense. The problem also combines many
computer vision sub-problems such as image labeling and object detection.
In this paper we present our contributions to the problem: a generic end-to-end QA model using
visual semantic embeddings to connect a CNN and a recurrent neural net (RNN), as well as comparisons to a suite of other models; an automatic question generation algorithm that converts description
sentences into questions; and a new QA dataset (COCO-QA) that was generated using the algorithm,
and a number of baseline results on this new dataset.
In this work we assume that the answers consist of only a single word, which allows us to treat the
problem as a classification problem. This also makes the evaluation of the models easier and more
robust, avoiding the thorny evaluation issues that plague multi-word generation problems.
2
Related Work
Malinowski and Fritz [11] released a dataset with images and question-answer pairs, the DAtaset
for QUestion Answering on Real-world images (DAQUAR). All images are from the NYU depth v2
dataset [12], and are taken from indoor scenes. Human segmentation, image depth values, and object
labeling are available in the dataset. The QA data has two sets of configurations, which differ by the
1
DAQUAR 1553
What is there in front of the
sofa?
Ground truth: table
IMG+BOW: table (0.74)
2-VIS+BLSTM: table (0.88)
LSTM: chair (0.47)
COCOQA 5078
How many leftover donuts is
the red bicycle holding?
Ground truth: three
IMG+BOW: two (0.51)
2-VIS+BLSTM: three (0.27)
BOW: one (0.29)
COCOQA 1238
What is the color of the teeshirt?
Ground truth: blue
IMG+BOW: blue (0.31)
2-VIS+BLSTM: orange (0.43)
BOW: green (0.38)
COCOQA 26088
Where is the gray cat sitting?
Ground truth: window
IMG+BOW: window (0.78)
2-VIS+BLSTM: window (0.68)
BOW: suitcase (0.31)
Figure 1: Sample questions and responses of a variety of models. Correct answers are in green and
incorrect in red. The numbers in parentheses are the probabilities assigned to the top-ranked answer
by the given model. The leftmost example is from the DAQUAR dataset, and the others are from
our new COCO-QA dataset.
number of object classes appearing in the questions (37-class and 894-class). There are mainly three
types of questions in this dataset: object type, object color, and number of objects. Some questions
are easy but many questions are very hard to answer even for humans. Since DAQUAR is the only
publicly available image-based QA dataset, it is one of our benchmarks to evaluate our models.
Together with the release of the DAQUAR dataset, Malinowski and Fritz presented an approach
which combines semantic parsing and image segmentation. Their approach is notable as one of the
first attempts at image QA, but it has a number of limitations. First, a human-defined possible set
of predicates are very dataset-specific. To obtain the predicates, their algorithm also depends on the
accuracy of the image segmentation algorithm and image depth information. Second, their model
needs to compute all possible spatial relations in the training images. Even though the model limits
this to the nearest neighbors of the test images, it could still be an expensive operation in larger
datasets. Lastly the accuracy of their model is not very strong. We show below that some simple
baselines perform better.
Very recently there has been a number of parallel efforts on both creating datasets and proposing
new models [13, 14, 15, 16]. Both Antol et al. [13] and Gao et al. [15] used MS-COCO [17] images
and created an open domain dataset with human generated questions and answers. In Anto et al.?s
work, the authors also included cartoon pictures besides real images. Some questions require logical
reasoning in order to answer correctly. Both Malinowski et al. [14] and Gao et al. [15] use recurrent
networks to encode the sentence and output the answer. Whereas Malinowski et al. use a single
network to handle both encoding and decoding, Gao et al. used two networks, a separate encoder
and decoder. Lastly, bilingual (Chinese and English) versions of the QA dataset are available in Gao
et al.?s work. Ma et al. [16] use CNNs to both extract image features and sentence features, and fuse
the features together with another multi-modal CNN.
Our approach is developed independently from the work above. Similar to the work of Malinowski
et al. and Gao et al., we also experimented with recurrent networks to consume the sequential
question input. Unlike Gao et al., we formulate the task as a classification problem, as there is no
single well- accepted metric to evaluate sentence-form answer accuracy [18]. Thus, we place more
focus on a limited domain of questions that can be answered with one word. We also formulate and
evaluate a range of other algorithms, that utilize various representations drawn from the question
and image, on these datasets.
3
Proposed Methodology
The methodology presented here is two-fold. On the model side we develop and apply various forms
of neural networks and visual-semantic embeddings on this task, and on the dataset side we propose
new ways of synthesizing QA pairs from currently available image description datasets.
2
.21 .56
One Two
...
...
.09 .01
Red Bird
...
Softmax
LSTM
Image
Word Embedding
Linear
CNN
?How?
t=1
?many?
t=2
?books?
t=T
Figure 2: VIS+LSTM Model
3.1
Models
In recent years, recurrent neural networks (RNNs) have enjoyed some successes in the field of natural language processing (NLP). Long short-term memory (LSTM) [19] is a form of RNN which
is easier to train than standard RNNs because of its linear error propagation and multiplicative gatings. Our model builds directly on top of the LSTM sentence model and is called the ?VIS+LSTM?
model. It treats the image as one word of the question. We borrowed this idea of treating the image
as a word from caption generation work done by Vinyals et al. [1]. We compare this newly proposed
model with a suite of simpler models in the Experimental Results section.
1. We use the last hidden layer of the 19-layer Oxford VGG Conv Net [20] trained on ImageNet 2014 Challenge [21] as our visual embeddings. The CNN part of our model is kept
frozen during training.
2. We experimented with several different word embedding models: randomly initialized embedding, dataset-specific skip-gram embedding and general-purpose skip-gram embedding
model [22]. The word embeddings are trained with the rest of the model.
3. We then treat the image as if it is the first word of the sentence. Similar to DeViSE [23],
we use a linear or affine transformation to map 4096 dimension image feature vectors to a
300 or 500 dimensional vector that matches the dimension of the word embeddings.
4. We can optionally treat the image as the last word of the question as well through a different
weight matrix and optionally add a reverse LSTM, which gets the same content but operates
in a backward sequential fashion.
5. The LSTM(s) outputs are fed into a softmax layer at the last timestep to generate answers.
3.2
Question-Answer Generation
The currently available DAQUAR dataset contains approximately 1500 images and 7000 questions
on 37 common object classes, which might be not enough for training large complex models. Another problem with the current dataset is that simply guessing the modes can yield very good accuracy.
We aim to create another dataset, to produce a much larger number of QA pairs and a more even
distribution of answers. While collecting human generated QA pairs is one possible approach, and
another is to synthesize questions based on image labeling, we instead propose to automatically
convert descriptions into QA form. In general, objects mentioned in image descriptions are easier to
detect than the ones in DAQUAR?s human generated questions, and than the ones in synthetic QAs
based on ground truth labeling. This allows the model to rely more on rough image understanding
without any logical reasoning. Lastly the conversion process preserves the language variability in
the original description, and results in more human-like questions than questions generated from
image labeling.
As a starting point we used the MS-COCO dataset [17], but the same method can be applied to any
other image description dataset, such as Flickr [24], SBU [25], or even the internet.
3
3.2.1
Pre-Processing & Common Strategies
We used the Stanford parser [26] to obtain the syntatic structure of the original image description.
We also utilized these strategies for forming the questions.
1. Compound sentences to simple sentences
Here we only consider a simple case, where two sentences are joined together with a conjunctive
word. We split the orginial sentences into two independent sentences.
2. Indefinite determiners ?a(n)? to definite determiners ?the?.
3. Wh-movement constraints
In English, questions tend to start with interrogative words such as ?what?. The algorithm needs
to move the verb as well as the ?wh-? constituent to the front of the sentence. For example:
?A man is riding a horse? becomes ?What is the man riding?? In this work we consider the
following two simple constraints: (1) A-over-A principle which restricts the movement of a whword inside a noun phrase (NP) [27]; (2) Our algorithm does not move any wh-word that is
contained in a clause constituent.
3.2.2
Question Generation
Question generation is still an open-ended topic. Overall, we adopt a conservative approach to
generating questions in an attempt to create high-quality questions. We consider generating four
types of questions below:
1. Object Questions: First, we consider asking about an object using ?what?. This involves replacing the actual object with a ?what? in the sentence, and then transforming the sentence structure
so that the ?what? appears in the front of the sentence. The entire algorithm has the following stages: (1) Split long sentences into simple sentences; (2) Change indefinite determiners
to definite determiners; (3) Traverse the sentence and identify potential answers and replace
with ?what?. During the traversal of object-type question generation, we currently ignore all the
prepositional phrase (PP) constituents; (4) Perform wh-movement. In order to identify a possible
answer word, we used WordNet [28] and the NLTK software package [29] to get noun categories.
2. Number Questions: We follow a similar procedure as the previous algorithm, except for a different way to identify potential answers: we extract numbers from original sentences. Splitting
compound sentences, changing determiners, and wh-movement parts remain the same.
3. Color Questions: Color questions are much easier to generate. This only requires locating the
color adjective and the noun to which the adjective attaches. Then it simply forms a sentence
?What is the color of the [object]? with the ?object? replaced by the actual noun.
4. Location Questions: These are similar to generating object questions, except that now the answer
traversal will only search within PP constituents that start with the preposition ?in?. We also
added rules to filter out clothing so that the answers will mostly be places, scenes, or large objects
that contain smaller objects.
3.2.3
Post-Processing
We rejected the answers that appear too rarely or too often in our generated dataset. After this QA
rejection process, the frequency of the most common answer words was reduced from 24.98% down
to 7.30% in the test set of COCO-QA.
4
4.1
Experimental Results
Datasets
Table 1 summarizes the statistics of COCO-QA. It should be noted that since we applied the QA
pair rejection process, mode-guessing performs very poorly on COCO-QA. However, COCO-QA
questions are actually easier to answer than DAQUAR from a human point of view. This encourages the model to exploit salient object relations instead of exhaustively searching all possible relations. COCO-QA dataset can be downloaded at http://www.cs.toronto.edu/?mren/
imageqa/data/cocoqa
4
Table 1: COCO-QA question type break-down
C ATEGORY
O BJECT
N UMBER
C OLOR
L OCATION
T OTAL
T RAIN
54992
5885
13059
4800
78736
%
69.84%
7.47%
16.59%
6.10%
100.00%
T EST
27206
2755
6509
2478
38948
%
69.85%
7.07%
16.71%
6.36%
100.00%
Here we provide some brief statistics of the new dataset. The maximum question length is 55, and
average is 9.65. The most common answers are ?two? (3116, 2.65%), ?white? (2851, 2.42%), and
?red? (2443, 2.08%). The least common are ?eagle? (25, 0.02%) ?tram? (25, 0.02%), and ?sofa?
(25, 0.02%). The median answer is ?bed? (867, 0.737%). Across the entire test set (38,948 QAs),
9072 (23.29%) overlap in training questions, and 7284 (18.70%) overlap in training question-answer
pairs.
4.2
Model Details
1. VIS+LSTM: The first model is the CNN and LSTM with a dimensionality-reduction weight
matrix in the middle; we call this ?VIS+LSTM? in our tables and figures.
2. 2-VIS+BLSTM: The second model has two image feature inputs, at the start and the end of the
sentence, with different learned linear transformations, and also has LSTMs going in both the
forward and backward directions. Both LSTMs output to the softmax layer at the last timestep.
We call the second model ?2-VIS+BLSTM?.
3. IMG+BOW: This simple model performs multinomial logistic regression based on the image
features without dimensionality reduction (4096 dimension), and a bag-of-word (BOW) vector
obtained by summing all the learned word vectors of the question.
4. FULL: Lastly, the ?FULL? model is a simple average of the three models above.
We release the complete details of the models at https://github.com/renmengye/
imageqa-public.
4.3
Baselines
To evaluate the effectiveness of our models, we designed a few baselines.
1. GUESS: One very simple baseline is to predict the mode based on the question type. For example, if the question contains ?how many? then the model will output ?two.? In DAQUAR, the
modes are ?table?, ?two?, and ?white? and in COCO-QA, the modes are ?cat?, ?two?, ?white?,
and ?room?.
2. BOW: We designed a set of ?blind? models which are given only the questions without the
images. One of the simplest blind models performs logistic regression on the BOW vector to
classify answers.
3. LSTM: Another ?blind? model we experimented with simply inputs the question words into the
LSTM alone.
4. IMG: We also trained a counterpart ?deaf? model. For each type of question, we train a separate
CNN classification layer (with all lower layers frozen during training). Note that this model
knows the type of question, in order to make its performance somewhat comparable to models
that can take into account the words to narrow down the answer space. However the model does
not know anything about the question except the type.
5. IMG+PRIOR: This baseline combines the prior knowledge of an object and the image understanding from the ?deaf model?. For example, a question asking the color of a white bird flying
in the blue sky may output white rather than blue simply because the prior probability of the bird
being blue is lower. We denote c as the color, o as the class of the object of interest, and x as the
5
image. Assuming o and x are conditionally independent given the color,
p(c, o|x)
p(o|c, x)p(c|x)
p(o|c)p(c|x)
=P
=P
c?C p(c, o|x)
c?C p(o|c, x)p(c|x)
c?C p(o|c)p(c|x)
p(c|o, x) = P
(1)
This can be computed if p(c|x) is the output of a logistic regression given the CNN features alone,
and we simply estimate p(o|c) empirically: p?(o|c) = count(o,c)
count(c) . We use Laplace smoothing on
this empirical distribution.
6. K-NN: In the task of image caption generation, Devlin et al. [30] showed that a nearest neighbors
baseline approach actually performs very well. To see whether our model memorizes the training
data for answering new question, we include a K-NN baseline in the results. Unlike image
caption generation, here the similarity measure includes both image and text. We use the bag-ofwords representation learned from IMG+BOW, and append it to the CNN image features. We use
Euclidean distance as the similarity metric; it is possible to improve the nearest neighbor result
by learning a similarity metric.
4.4
Performance Metrics
To evaluate model performance, we used the plain answer accuracy as well as the Wu-Palmer similarity (WUPS) measure [31, 32]. The WUPS calculates the similarity between two words based on
their longest common subsequence in the taxonomy tree. If the similarity between two words is less
than a threshold then a score of zero will be given to the candidate answer. Following Malinowski
and Fritz [32], we measure all models in terms of accuracy, WUPS 0.9, and WUPS 0.0.
4.5
Results and Analysis
Table 2 summarizes the learning results on DAQUAR and COCO-QA. For DAQUAR we compare
our results with [32] and [14]. It should be noted that our DAQUAR results are for the portion of the
dataset (98.3%) with single-word answers. After the release of our paper, Ma et al. [16] claimed to
achieve better results on both datasets.
Table 2: DAQUAR and COCO-QA results
MULTI-WORLD [32]
GUESS
BOW
LSTM
IMG
IMG+PRIOR
K-NN (K=31, 13)
IMG+BOW
VIS+LSTM
ASK-NEURON [14]
2-VIS+BLSTM
FULL
HUMAN
ACC .
0.1273
0.1824
0.3267
0.3273
0.3185
0.3417
0.3441
0.3468
0.3578
0.3694
0.6027
DAQUAR
WUPS 0.9
0.1810
0.2965
0.4319
0.4350
0.4242
0.4499
0.4605
0.4076
0.4683
0.4815
0.6104
WUPS 0.0
0.5147
0.7759
0.8130
0.8162
0.8063
0.8148
0.8223
0.7954
0.8215
0.8268
0.7896
ACC .
0.0730
0.3752
0.3676
0.4302
0.4466
0.4496
0.5592
0.5331
0.5509
0.5784
-
COCO-QA
WUPS 0.9
0.1837
0.4854
0.4758
0.5864
0.6020
0.5698
0.6678
0.6391
0.6534
0.6790
-
WUPS 0.0
0.7413
0.8278
0.8234
0.8585
0.8624
0.8557
0.8899
0.8825
0.8864
0.8952
-
From the above results we observe that our model outperforms the baselines and the existing approach in terms of answer accuracy and WUPS. Our VIS+LSTM and Malinkowski et al.?s recurrent
neural network model [14] achieved somewhat similar performance on DAQUAR. A simple average
of all three models further boosts the performance by 1-2%, outperforming other models.
It is surprising to see that the IMG+BOW model is very strong on both datasets. One limitation of
our VIS+LSTM model is that we are not able to consume image features as large as 4096 dimensions
at one time step, so the dimensionality reduction may lose some useful information. We tried to give
IMG+BOW a 500 dim. image vector, and it does worse than VIS+LSTM (?48%).
6
Table 3: COCO-QA accuracy per category
GUESS
BOW
LSTM
IMG
IMG+PRIOR
K-NN
IMG+BOW
VIS+LSTM
2-VIS+BLSTM
FULL
O BJECT
0.0239
0.3727
0.3587
0.4073
0.4799
0.5866
0.5653
0.5817
0.6108
N UMBER
0.3606
0.4356
0.4534
0.2926
0.3739
0.3699
0.4410
0.4610
0.4479
0.4766
C OLOR
0.1457
0.3475
0.3626
0.4268
0.4899
0.3723
0.5196
0.4587
0.4953
0.5148
L OCATION
0.0908
0.4084
0.3842
0.4419
0.4451
0.4080
0.4939
0.4552
0.4734
0.5028
By comparing the blind versions of the BOW and LSTM models, we hypothesize that in Image QA
tasks, and in particular on the simple questions studied here, sequential word interaction may not be
as important as in other natural language tasks.
It is also interesting that the blind model does not lose much on the DAQUAR dataset, We speculate
that it is likely that the ImageNet images are very different from the indoor scene images, which
are mostly composed of furniture. However, the non-blind models outperform the blind models
by a large margin on COCO-QA. There are three possible reasons: (1) the objects in MS-COCO
resemble the ones in ImageNet more; (2) MS-COCO images have fewer objects whereas the indoor
scenes have considerable clutter; and (3) COCO-QA has more data to train complex models.
There are many interesting examples but due to space limitations we can only show a few in Figure 1 and Figure 3; full results are available at http://www.cs.toronto.edu/?mren/
imageqa/results. For some of the images, we added some extra questions (the ones have
an ?a? in the question ID); these provide more insight into a model?s representation of the image and
question information, and help elucidate questions that our models may accidentally get correct. The
parentheses in the figures represent the confidence score given by the softmax layer of the respective
model.
Model Selection: We did not find that using different word embedding has a significant impact on
the final classification results. We observed that fine-tuning the word embedding results in better
performance and normalizing the CNN hidden image features into zero-mean and unit-variance
helps achieve faster training time. The bidirectional LSTM model can further boost the result by a
little.
Object Questions: As the original CNN was trained for the ImageNet challenge, the IMG+BOW
benefited significantly from its single object recognition ability. However, the challenging part is
to consider spatial relations between multiple objects and to focus on details of the image. Our
models only did a moderately acceptable job on this; see for instance the first picture of Figure 1 and
the fourth picture of Figure 3. Sometimes a model fails to make a correct decision but outputs the
most salient object, while sometimes the blind model can equally guess the most probable objects
based on the question alone (e.g., chairs should be around the dining table). Nonetheless, the FULL
model improves accuracy by 50% compared to IMG model, which shows the difference between
pure object classification and image question answering.
Counting: In DAQUAR, we could not observe any advantage in the counting ability of the
IMG+BOW and the VIS+LSTM model compared to the blind baselines. In COCO-QA there is
some observable counting ability in very clean images with a single object type. The models can
sometimes count up to five or six. However, as shown in the second picture of Figure 3, the ability
is fairly weak as they do not count correctly when different object types are present. There is a lot
of room for improvement in the counting task, and in fact this could be a separate computer vision
problem on its own.
Color: In COCO-QA there is a significant win for the IMG+BOW and the VIS+LSTM against
the blind ones on color-type questions. We further discovered that these models are not only able
to recognize the dominant color of the image but sometimes associate different colors to different
objects, as shown in the first picture of Figure 3. However, they still fail on a number of easy
7
COCOQA 33827
What is the color of the cat?
Ground truth: black
IMG+BOW: black (0.55)
2-VIS+LSTM: black (0.73)
BOW: gray (0.40)
DAQUAR 1522
How many chairs are there?
Ground truth: two
IMG+BOW: four (0.24)
2-VIS+BLSTM: one (0.29)
LSTM: four (0.19)
COCOQA 14855
Where are the ripe bananas sitting?
Ground truth: basket
IMG+BOW: basket (0.97)
2-VIS+BLSTM: basket (0.58)
BOW: bowl (0.48)
DAQUAR 585
What is the object on the chair?
Ground truth: pillow
IMG+BOW: clothes (0.37)
2-VIS+BLSTM: pillow (0.65)
LSTM: clothes (0.40)
COCOQA 33827a
What is the color of the couch?
Ground truth: red
IMG+BOW: red (0.65)
2-VIS+LSTM: black (0.44)
BOW: red (0.39)
DAQUAR 1520
How many shelves are there?
Ground truth: three
IMG+BOW: three (0.25)
2-VIS+BLSTM: two (0.48)
LSTM: two (0.21)
COCOQA 14855a
What are in the basket?
Ground truth: bananas
IMG+BOW: bananas (0.98)
2-VIS+BLSTM: bananas (0.68)
BOW: bananas (0.14)
DAQUAR 585a
Where is the pillow found?
Ground truth: chair
IMG+BOW: bed (0.13)
2-VIS+BLSTM: chair (0.17)
LSTM: cabinet (0.79)
Figure 3: Sample questions and responses of our system
examples. Adding prior knowledge provides an immediate gain on the IMG model in terms of
accuracy on Color and Number questions. The gap between the IMG+PRIOR and IMG+BOW
shows some localized color association ability in the CNN image representation.
5
Conclusion and Current Directions
In this paper, we consider the image QA problem and present our end-to-end neural network models.
Our model shows a reasonable understanding of the question and some coarse image understanding, but it is still very na??ve in many situations. While recurrent networks are becoming a popular
choice for learning image and text, we showed that a simple bag-of-words can perform equally well
compared to a recurrent network that is borrowed from an image caption generation framework [1].
We proposed a more complete set of baselines which can provide potential insight for developing
more sophisticated end-to-end image question answering systems. As the currently available dataset
is not large enough, we developed an algorithm that helps us collect large scale image QA dataset
from image descriptions. Our question generation algorithm is extensible to many image description
datasets and can be automated without requiring extensive human effort. We hope that the release
of the new dataset will encourage more data-driven approaches to this problem in the future.
Image question answering is a fairly new research topic, and the approach we present here has a
number of limitations. First, our models are just answer classifiers. Ideally we would like to permit
longer answers which will involve some sophisticated text generation model or structured output.
But this will require an automatic free-form answer evaluation metric. Second, we are only focusing
on a limited domain of questions. However, this limited range of questions allow us to study the
results more in depth. Lastly, it is also hard to interpret why the models output a certain answer.
By comparing our models with some baselines we can roughly infer whether they understood the
image. Visual attention is another future direction, which could both improve the results (based on
recent successes in image captioning [8]) as well as help explain the model prediction by examining
the attention output at every timestep.
Acknowledgments
We would like to thank Nitish Srivastava for the support of Toronto Conv Net, from which we
extracted the CNN image features. We would also like to thank anonymous reviewers for their
valuable and helpful comments.
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5,127 | 5,641 | Are You Talking to a Machine?
Dataset and Methods for Multilingual Image Question Answering
Haoyuan Gao1
1
Junhua Mao2
Baidu Research
Jie Zhou1 Zhiheng Huang1 Lei Wang1
2
University of California, Los Angeles
Wei Xu1
[email protected], [email protected], {zhoujie01,huangzhiheng,wanglei22,wei.xu}@baidu.com
Abstract
In this paper, we present the mQA model, which is able to answer questions about
the content of an image. The answer can be a sentence, a phrase or a single word.
Our model contains four components: a Long Short-Term Memory (LSTM) to
extract the question representation, a Convolutional Neural Network (CNN) to
extract the visual representation, an LSTM for storing the linguistic context in an
answer, and a fusing component to combine the information from the first three
components and generate the answer. We construct a Freestyle Multilingual Image Question Answering (FM-IQA) dataset to train and evaluate our mQA model.
It contains over 150,000 images and 310,000 freestyle Chinese question-answer
pairs and their English translations. The quality of the generated answers of our
mQA model on this dataset is evaluated by human judges through a Turing Test.
Specifically, we mix the answers provided by humans and our model. The human
judges need to distinguish our model from the human. They will also provide
a score (i.e. 0, 1, 2, the larger the better) indicating the quality of the answer.
We propose strategies to monitor the quality of this evaluation process. The experiments show that in 64.7% of cases, the human judges cannot distinguish our
model from humans. The average score is 1.454 (1.918 for human). The details
of this work, including the FM-IQA dataset, can be found on the project page:
http://idl.baidu.com/FM-IQA.html.
1
Introduction
Recently, there is increasing interest in the field of multimodal learning for both natural language
and vision. In particular, many studies have made rapid progress on the task of image captioning
[26, 15, 14, 40, 6, 8, 4, 19, 16, 42]. Most of them are built based on deep neural networks (e.g.
deep Convolutional Neural Networks (CNN [17]), Recurrent Neural Network (RNN [7]) or Long
Short-Term Memory (LSTM [12])). The large-scale image datasets with sentence annotations (e.g.,
[21, 43, 11]) play a crucial role in this progress. Despite the success of these methods, there are still
many issues to be discussed and explored. In particular, the task of image captioning only requires
generic sentence descriptions of an image. But in many cases, we only care about a particular part
or object of an image. The image captioning task lacks the interaction between the computer and
the user (as we cannot input our preference and interest).
In this paper, we focus on the task of visual question answering. In this task, the method needs
to provide an answer to a freestyle question about the content of an image. We propose the mQA
model to address this task. The inputs of the model are an image and a question. This model has four
components (see Figure 2). The first component is an LSTM network that encodes a natural language
sentence into a dense vector representation. The second component is a deep Convolutional Neural
Network [36] that extracted the image representation. This component was pre-trained on ImageNet
Classification Task [33] and is fixed during the training. The third component is another LSTM
network that encodes the information of the current word and previous words in the answer into
dense representations. The fourth component fuses the information from the first three components
to predict the next word in the answer. We jointly train the first, third and fourth components by
maximizing the probability of the groundtruth answers in the training set using a log-likelihood loss
1
Image
Question
???????????
What is the color of the bus?
???????
What is there in yellow?
??????????????? ??????
What is there on the grass, except
Where is the kitty?
the person?
????????????????
?? ?
Please look carefully and tell me what is
the name of the vegetables in the plate?
Answer
?????????
The bus is red.
???
Bananas.
??
Sheep.
??? ?
Broccoli.
???? ?
On the chair.
Figure 1: Sample answers to the visual question generated by our model on the newly proposed
Freestyle Multilingual Image Question Answering (FM-IQA) dataset.
function. To lower down the risk of overfitting, we allow the weight sharing of the word embedding
layer between the LSTMs in the first and third components. We also adopt the transposed weight
sharing scheme as proposed in [25], which allows the weight sharing between word embedding layer
and the fully connected Softmax layer.
To train our method, we construct a large-scale Freestyle Multilingual Image Question Answering
dataset1 (FM-IQA, see details in Section 4) based on the MS COCO dataset [21]. The current
version of the dataset contains 158,392 images with 316,193 Chinese question-answer pairs and
their corresponding English translations.2 To diversify the annotations, the annotators are allowed
to raise any question related to the content of the image. We propose strategies to monitor the
quality of the annotations. This dataset contains a wide range of AI related questions, such as action
recognition (e.g., ?Is the man trying to buy vegetables??), object recognition (e.g., ?What is there in
yellow??), positions and interactions among objects in the image (e.g. ?Where is the kitty??) and
reasoning based on commonsense and visual content (e.g. ?Why does the bus park here??, see last
column of Figure 3).
Because of the variability of the freestyle question-answer pairs, it is hard to accurately evaluate
the method with automatic metrics. We conduct a Visual Turing Test [38] using human judges.
Specifically, we mix the question-answer pairs generated by our model with the same set of questionanswer pairs labeled by annotators. The human judges need to determine whether the answer is
given by a model or a human. In addition, we also ask them to give a score of 0 (i.e. wrong), 1 (i.e.
partially correct), or 2 (i.e. correct). The results show that our mQA model passes 64.7% of this
test (treated as answers of a human) and the average score is 1.454. In the discussion, we analyze
the failure cases of our model and show that combined with the m-RNN [24] model, our model can
automatically ask a question about an image and answer that question.
2
Related Work
Recent work has made significant progress using deep neural network models in both the fields of
computer vision and natural language. For computer vision, methods based on Convolutional Neural
Network (CNN [20]) achieve the state-of-the-art performance in various tasks, such as object classification [17, 34, 17], detection [10, 44] and segmentation [3]. For natural language, the Recurrent
Neural Network (RNN [7, 27]) and the Long Short-Term Memory network (LSTM [12]) are also
widely used in machine translation [13, 5, 35] and speech recognition [28].
The structure of our mQA model is inspired by the m-RNN model [24] for the image captioning and
image-sentence retrieval tasks. It adopts a deep CNN for vision and a RNN for language. We extend
the model to handle the input of question and image pairs, and generate answers. In the experiments,
we find that we can learn how to ask a good question about an image using the m-RNN model and
this question can be answered by our mQA model.
There has been recent effort on the visual question answering task [9, 2, 22, 37]. However, most of
them use a pre-defined and restricted set of questions. Some of these questions are generated from a
template. In addition, our FM-IQA dataset is much larger than theirs (e.g., there are only 2591 and
1449 images for [9] and [22] respectively).
1
We are actively developing and expanding the dataset, please find the latest information on the project page
: http://idl.baidu.com/FM-IQA.html
2
The results reported in this paper are obtained from a model trained on the first version of the dataset (a
subset of the current version) which contains 120,360 images and 250,569 question-answer pairs.
2
What
is
the
cat
doing
?
<BOA> Sitting
on
the
umbrella
Shared
Embedding
Shared
LSTM
Fusing
CNN
Intermediate
Softmax
Sitting
on
the umbrella <EOA>
Figure 2: Illustration of the mQA model architecture. We input an image and a question about the
image (i.e. ?What is the cat doing??) to the model. The model is trained to generate the answer to
the question (i.e. ?Sitting on the umbrella?). The weight matrix in the word embedding layers of
the two LSTMs (one for the question and one for the answer) are shared. In addition, as in [25], this
weight matrix is also shared, in a transposed manner, with the weight matrix in the Softmax layer.
Different colors in the figure represent different components of the model. (Best viewed in color.)
There are some concurrent and independent works on this topic: [1, 23, 32]. [1] propose a largescale dataset also based on MS COCO. They also provide some simple baseline methods on this
dataset. Compared to them, we propose a stronger model for this task and evaluate our method using
human judges. Our dataset also contains two different kinds of language, which can be useful for
other tasks, such as machine translation. Because we use a different set of annotators and different
requirements of the annotation, our dataset and the [1] can be complementary to each other, and lead
to some interesting topics, such as dataset transferring for visual question answering.
Both [23] and [32] use a model containing a single LSTM and a CNN. They concatenate the question
and the answer (for [32], the answer is a single word. [23] also prefer a single word as the answer),
and then feed them to the LSTM. Different from them, we use two separate LSTMs for questions
and answers respectively in consideration of the different properties (e.g. grammar) of questions and
answers, while allow the sharing of the word-embeddings. For the dataset, [23] adopt the dataset
proposed in [22], which is much smaller than our FM-IQA dataset. [32] utilize the annotations in
MS COCO and synthesize a dataset with four pre-defined types of questions (i.e. object, number,
color, and location). They also synthesize the answer with a single word. Their dataset can also be
complementary to ours.
3
The Multimodal QA (mQA) Model
We show the architecture of our mQA model in Figure 2. The model has four components: (I). a
Long Short-Term Memory (LSTM [12]) for extracting semantic representation of a question, (II). a
deep Convolutional Neural Network (CNN) for extracting the image representation, (III). an LSTM
to extract representation of the current word in the answer and its linguistic context, and (IV). a
fusing component that incorporates the information from the first three parts together and generates
the next word in the answer. These four components can be jointly trained together 3 . The details
of the four model components are described in Section 3.1. The effectiveness of the important
components and strategies are analyzed in Section 5.3.
The inputs of the model are a question and the reference image. The model is trained to generate
the answer. The words in the question and answer are represented by one-hot vectors (i.e. binary
vectors with the length of the dictionary size N and have only one non-zero vector indicating its
index in the word dictionary). We add a hBOAi sign and a hEOAi sign, as two spatial words in
the word dictionary, at the beginning and the end of the training answers respectively. They will be
used for generating the answer to the question in the testing stage.
In the testing stage, we input an image and a question about the image into the model first. To
generate the answer, we start with the start sign hBOAi and use the model to calculate the probability
distribution of the next word. We then use a beam search scheme that keeps the best K candidates
3
In practice, we fix the CNN part because the gradient returned from LSTM is very noisy. Finetuning the
CNN takes a much longer time than just fixing it, and does not improve the performance significantly.
3
with the maximum probabilities according to the Softmax layer. We repeat the process until the
model generates the end sign of the answer hBOAi.
3.1 The Four Components of the mQA Model
(I). The semantic meaning of the question is extracted by the first component of the model. It
contains a 512 dimensional word embedding layer and an LSTM layer with 400 memory cells. The
function of the word embedding layer is to map the one-hot vector of the word into a dense semantic
space. We feed this dense word representation into the LSTM layer.
LSTM [12] is a Recurrent Neural Network [7] that is designed for solving the gradient explosion or
vanishing problem. The LSTM layer stores the context information in its memory cells and serves
as the bridge among the words in a sequence (e.g. a question). To model the long term dependency
in the data more effectively, LSTM add three gate nodes to the traditional RNN structure: the input
gate, the output gate and the forget gate. The input gate and output gate regulate the read and write
access to the LSTM memory cells. The forget gate resets the memory cells when their contents
are out of date. Different from [23, 32], the image representation does not feed into the LSTM in
this component. We believe this is reasonable because questions are just another input source for
the model, so we should not add images as the supervision for them. The information stored in the
LSTM memory cells of the last word in the question (i.e. the question mark) will be treated as the
representation of the sentence.
(II). The second component is a deep Convolutional Neural Network (CNN) that generates the representation of an image. In this paper, we use the GoogleNet [36]. Note that other CNN models,
such as AlexNet [17] and VggNet [34], can also be used as the component in our model. We remove
the final SoftMax layer of the deep CNN and connect the remaining top layer to our model.
(III). The third component also contains a word embedding layer and an LSTM. The structure is
similar to the first component. The activation of the memory cells for the words in the answer, as
well as the word embeddings, will be fed into the fusing component to generate the next words in
the answer.
In [23, 32], they concatenate the training question and answer, and use a single LSTM. Because of
the different properties (i.e. grammar) of question and answer, in this paper, we use two separate
LSTMs for questions and answers respectively. We denote the LSTMs for the question and the
answer as LSTM(Q) and LSTM(A) respectively in the rest of the paper. The weight matrix in
LSTM(Q) is not shared with the LSTM(A) in the first components. Note that the semantic meaning
of single words should be the same for questions and answers so that we share the parameters in the
word-embedding layer for the first and third component.
(IV). Finally, the fourth component fuses the information from the first three layers. Specifically,
the activation of the fusing layer f (t) for the tth word in the answer can be calculated as follows:
f (t) = g(VrQ rQ + VI I + VrA rA (t) + Vw w(t));
(1)
where ?+? denotes element-wise addition, rQ stands for the activation of the LSTM(Q) memory
cells of the last word in the question, I denotes the image representation, rA (t) and w(t) denotes
the activation of the LSTM(A) memory cells and the word embedding of the tth word in the answer
respectively. VrQ , VI , VrA , and Vw are the weight matrices that need to be learned. g(.) is an
element-wise non-linear function.
After the fusing layer, we build an intermediate layer that maps the dense multimodal representation
in the fusing layer back to the dense word representation. We then build a fully connected Softmax
layer to predict the probability distribution of the next word in the answer. This strategy allows the
weight sharing between word embedding layer and the fully connected Softmax layer as introduced
in [25] (see details in Section 3.2).
Similar to [25], we use the sigmoid function as the activation function of the three gates and adopt
ReLU [30] as the non-linear function for the LSTM memory cells. The non-linear activation function
for the word embedding layer, the fusing layer and the intermediate layer is the scaled hyperbolic
tangent function [20]: g(x) = 1.7159 ? tanh( 32 x).
3.2 The Weight Sharing Strategy
As mentioned in Section 2, our model adopts different LSTMs for the question and the answer
because of the different grammar properties of questions and answers. However, the meaning of
4
single words in both questions and answers should be the same. Therefore, we share the weight
matrix between the word-embedding layers of the first component and the third component.
In addition, this weight matrix for the word-embedding layers is shared with the weight matrix in
the fully connected Softmax layer in a transposed manner. Intuitively, the function of the weight
matrix in the word-embedding layer is to encode the one-hot word representation into a dense word
representation. The function of the weight matrix in the Softmax layer is to decode the dense word
representation into a pseudo one-word representation, which is the inverse operation of the wordembedding. This strategy will reduce nearly half of the parameters in the model and is shown to
have better performance in image captioning and novel visual concept learning tasks [25].
3.3 Training Details
The CNN we used is pre-trained on the ImageNet classification task [33]. This component is fixed
during the QA training. We adopt a log-likelihood loss defined on the word sequence of the answer.
Minimizing this loss function is equivalent to maximizing the probability of the model to generate
the groundtruth answers in the training set. We jointly train the first, second and the fourth components using stochastic gradient decent method. The initial learning rate is 1 and we decrease it
by a factor of 10 for every epoch of the data. We stop the training when the loss on the validation set does not decrease within three epochs. The hyperparameters of the model are selected by
cross-validation.
For the Chinese question answering task, we segment the sentences into several word phrases. These
phrases can be treated equivalently to the English words.
4
The Freestyle Multilingual Image Question Answering (FM-IQA) Dataset
Our method is trained and evaluated on a large-scale multilingual visual question answering dataset.
In Section 4.1, we will describe the process to collect the data, and the method to monitor the quality
of annotations. Some statistics and examples of the dataset will be given in Section 4.2. The latest
dataset is available on the project page: http://idl.baidu.com/FM-IQA.html
4.1 The Data Collection
We start with the 158,392 images from the newly released MS COCO [21] training, validation and
testing set as the initial image set. The annotations are collected using Baidu?s online crowdsourcing server4 . To make the labeled question-answer pairs diversified, the annotators are free to give
any type of questions, as long as these questions are related to the content of the image. The question should be answered by the visual content and commonsense (e.g., we are not expecting to get
questions such as ?What is the name of the person in the image??). The annotators need to give an
answer to the question themselves.
On the one hand, the freedom we give to the annotators is beneficial in order to get a freestyle,
interesting and diversified set of questions. On the other hand, it makes it harder to control the
quality of the annotation compared to a more detailed instruction. To monitor the annotation quality,
we conduct an initial quality filtering stage. Specifically, we randomly sampled 1,000 images as
a quality monitoring dataset from the MS COCO dataset as an initial set for the annotators (they
do not know this is a test). We then sample some annotations and rate their quality after each
annotator finishes some labeling on this quality monitoring dataset (about 20 question-answer pairs
per annotator). We only select a small number of annotators (195 individuals) whose annotations are
satisfactory (i.e. the questions are related to the content of the image and the answers are correct).
We also give preference to the annotators who provide interesting questions that require high level
reasoning to give the answer. Only the selected annotators are permitted to label the rest of the
images. We pick a set of good and bad examples of the annotated question-answer pairs from the
quality monitoring dataset, and show them to the selected annotators as references. We also provide
reasons for selecting these examples. After the annotation of all the images is finished, we further
refine the dataset and remove a small portion of the images with badly labeled questions and answers.
4.2 The Statistics of the Dataset
Currently there are 158,392 images with 316,193 Chinese question-answer pairs and their English
translations. Each image has at least two question-answer pairs as annotations. The average lengths
4
http://test.baidu.com
5
Image
GT Question
???????????
What is the boy in green cap doing?
GT Answer
??????
He is playing skateboard.
???????
Is there any person
in the image?
??
Yes.
?????????????
What is the texture of the sofa in
the room?
???
Cloth.
????????
Is the man trying to
buy vegetables?
???
Yes.
GT Question
GT Answer
????????????? ????????
Is the computer on the right hand What is the color of the frisbee?
or left hand side of the gentleman?
????
???
On the right hand side.
Yellow.
?????????
How many layers are
there for the cake?
???
Six.
????????
What are the people doing?
??????
Walking with umbrellas.
?????????
Why does the bus park there?
?????
Preparing for repair.
???????????????
What does it indicate when the phone,
mouse and laptop are placed together?
????????
Their owner is tired and sleeping.
Figure 3: Sample images in the FM-IQA dataset. This dataset contains 316,193 Chinese questionanswer pairs with corresponding English translations.
of the questions and answers are 7.38 and 3.82 respectively measured by Chinese words. Some
sample images are shown in Figure 3. We randomly sampled 1,000 question-answer pairs and their
corresponding images as the test set.
The questions in this dataset are diversified, which requires a vast set of AI capabilities in order
to answer them. They contain some relatively simple image understanding questions of, e.g., the
actions of objects (e.g., ?What is the boy in green cap doing??), the object class (e.g., ?Is there any
person in the image??), the relative positions and interactions among objects (e.g., ?Is the computer
on the right or left side of the gentleman??), and the attributes of the objects (e.g., ?What is the color
of the frisbee??). In addition, the dataset contains some questions that need a high-level reasoning
with clues from vision, language and commonsense. For example, to answer the question of ?Why
does the bus park there??, we should know that this question is about the parked bus in the image
with two men holding tools at the back. Based on our commonsense, we can guess that there might
be some problems with the bus and the two men in the image are trying to repair it. These questions
are hard to answer but we believe they are actually the most interesting part of the questions in the
dataset. We categorize the questions into 8 types and show the statistics of them on the project page.
The answers are also diversified. The annotators are allowed to give a single phrase or a single word
as the answer (e.g. ?Yellow?) or, they can give a complete sentence (e.g. ?The frisbee is yellow?).
5
Experiments
For the very recent works for visual question answering ([32, 23]), they test their method on the
datasets where the answer of the question is a single word or a short phrase. Under this setting,
it is plausible to use automatic evaluation metrics that measure the single word similarity, such
as Wu-Palmer similarity measure (WUPS) [41]. However, for our newly proposed dataset, the
answers in the dataset are freestyle and can be complete sentences. For most of the cases, there are
numerous choices of answers that are all correct. The possible alternatives are BLEU score [31],
METEOR [18], CIDEr [39] or other metrics that are widely used in the image captioning task [24].
The problem of these metrics is that there are only a few words in an answer that are semantically
critical. These metrics tend to give equal weights (e.g. BLEU and METEOR) or different weights
according to the tf-idf frequency term (e.g. CIDEr) of the words in a sentence, hence cannot fully
show the importance of the keywords. The evaluation of the image captioning task suffers from the
same problem (not as severe as question answering because it only needs a general description).
To avoid these problems, we conduct a real Visual Turing Test using human judges for our model,
which will be described in details in Section 5.1. In addition, we rate each generated sentences
with a score (the larger the better) in Section 5.2, which gives a more fine-grained evaluation of our
method. In Section 5.3, we provide the performance comparisons of different variants of our mQA
model on the validation set.
6
Human
blind-QA
mQA
5.1
Pass
948
340
647
Visual Turing Test
Fail Pass Rate (%)
52
94.8
660
34.0
353
64.7
2
927
628
Human Rated Scores
1
0
Avg. Score
64
9
1.918
198 174
1.454
Table 1: The results of our mQA model for our FM-IQA dataset.
The Visual Turing Test
In this Visual Turing Test, a human judge will be presented with an image, a question and the answer
to the question generated by the testing model or by human annotators. He or she need to determine,
based on the answer, whether the answer is given by a human (i.e. pass the test) or a machine (i.e.
fail the test).
In practice, we use the images and questions from the test set of our FM-IQA dataset. We use our
mQA model to generate the answer for each question. We also implement a baseline model of the
question answering without visual information. The structure of this baseline model is similar to
mQA, except that we do not feed the image information extracted by the CNN into the fusing layer.
We denote it as blind-QA. The answers generated by our mQA model, the blind-QA model and
the groundtruth answer are mixed together. This leads to 3000 question answering pairs with the
corresponding images, which will be randomly assigned to 12 human judges.
The results are shown in Table 1. It shows that 64.7% of the answers generated by our mQA model
are treated as answers provided by a human. The blind-QA performs very badly in this task. But
some of the generated answers pass the test. Because some of the questions are actually multi-choice
questions, it is possible to get a correct answer by random guess based on pure linguistic clues.
To study the variance of the VTT evaluation across different sets of human judges, we conduct
two additional evaluations with different groups of judges under the same setting. The standard
deviations of the passing rate are 0.013, 0.019 and 0.024 for human, the blind-mQA model and
mQA model respectively. It shows that VTT is a stable and reliable evaluation metric for this task.
5.2
The Score of the Generated Answer
The Visual Turing Test only gives a rough evaluation of the generated answers. We also conduct a
fine-grained evaluation with scores of ?0?, ?1?, or ?2?. ?0? and ?2? mean that the answer is totally
wrong and perfectly correct respectively. ?1? means that the answer is only partially correct (e.g.,
the general categories are right but the sub-categories are wrong) and makes sense to the human
judges. The human judges for this task are not necessarily the same people for the Visual Turing
Test. After collecting the results, we find that some human judges also rate an answer with ?1? if the
question is very hard to answer so that even a human, without carefully looking at the image, will
possibly make mistakes. We show randomly sampled images whose scores are ?1? in Figure 4.
The results are shown in Table 1. We show that among the answers that are not perfectly correct (i.e.
scores are not 2), over half of them are partially correct. Similar to the VTT evaluation process, we
also conducts two additional groups of this scoring evaluation. The standard deviations of human
and our mQA model are 0.020 and 0.041 respectively. In addition, for 88.3% and 83.9% of the
cases, the three groups give the same score for human and our mQA model respectively.
5.3
Performance Comparisons of the Different mQA Variants
In order to show the effectiveness of the different components and strategies of our mQA model, we
implement three variants of the mQA in Figure 2. For the first variant (i.e. ?mQA-avg-question?), we
replace the first LSTM component of the model (i.e. the LSTM to extract the question embedding)
Image
Question
Answer
???????
What is in the plate?
???
food.
?????
What is the dog doing?
????
Surfing in the sea.
??????
Where is the cat?
???
On the bed.
?????
What is there in the image?
?????
There is a clock.
??????
What is the type of the vehicle?
??
Train.
Figure 4: Random examples of the answers generated by the mQA model with score ?1? given by
the human judges.
7
Image
Generated
Question
Answer
????????
Where is this?
?????????
Is this guy playing tennis?
???????
What kind of food is this?
??????
Where is the computer?
??????
This is the kitchen room.
???
Yes.
???
Pizza.
?????
On the desk.
Figure 5: The sample generated questions by our model and their answers.
with the average embedding of the words in the quesWord Error Loss
tion using word2vec [29]. It is used to show the effecmQA-avg-question
0.442
2.17
mQA-same-LSTMs
0.439
2.09
tiveness of the LSTM as a question embedding learner
mQA-noTWS
0.438
2.14
and extractor. For the second variant (i.e. ?mQAmQA-complete
0.393
1.91
same-LSTMs?), we use two shared-weights LSTMs to
model question and answer. It is used to show the effectiveness of the decoupling strategy of the weights of Table 2: Performance comparisons of the
the LSTM(Q) and the LSTM(A) in our model. For the different mQA variants.
third variant (i.e. ?mQA-noTWS?), we do not adopt the Transposed Weight Sharing (TWS) strategy.
It is used to show the effectiveness of TWS.
The word error rates and losses of the three variants and the complete mQA model (i.e. mQAcomplete) are shown in Table 2. All of the three variants performs worse than our mQA model.
6
Discussion
In this paper, we present the mQA model, which is able to give a sentence or a phrase as the answer
to a freestyle question for an image. To validate the effectiveness of the method, we construct
a Freestyle Multilingual Image Question Answering (FM-IQA) dataset containing over 310,000
question-answer pairs. We evaluate our method using human judges through a real Turing Test. It
shows that 64.7% of the answers given by our mQA model are treated as the answers provided by a
human. The FM-IQA dataset can be used for other tasks, such as visual machine translation, where
the visual information can serve as context information that helps to remove ambiguity of the words
in a sentence.
We also modified the LSTM in the first component to the multimodal LSTM shown in [25]. This
modification allows us to generate a free-style question about the content of image, and provide an
answer to this question. We show some sample results in Figure 5.
We show some failure cases of our model in Figure 6. The model sometimes makes mistakes when
the commonsense reasoning through background scenes is incorrect (e.g., for the image in the first
column, our method says that the man is surfing but the small yellow frisbee in the image indicates
that he is actually trying to catch the frisbee. It also makes mistakes when the targeting object that the
question focuses on is too small or looks very similar to other objects (e.g. images in the second and
fourth column). Another interesting example is the image and question in the fifth column of Figure
6. Answering this question is very hard since it needs high level reasoning based on the experience
from everyday life. Our model outputs a hOOV i sign, which is a special word we use when the
model meets a word which it has not seen before (i.e. does not appear in its word dictionary).
In future work, we will try to address these issues by incorporating more visual and linguistic information (e.g. using object detection or using attention models).
Image
???????
Question
GT Answer
mQA Answer
???????? ?
??????
?????????
What is the handsome boy doing? What is there in the image?
?????
Which fruit is there in the plate?
What is the type of the vehicle?
Why does the bus park there?
?????
?????? ?
????? ?
?????
?????
Trying to catch the frisbee.
Horses on the grassland.
Apples and oranges.
Bus.
Preparing for repair.
?? ?
????
????? ?
???
<OOV>?
Surfing.
They are buffalos.
Bananas and oranges.
Train.
<OOV> (I do not know.)
Figure 6: Failure cases of our mQA model on the FM-IQA dataset.
8
References
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9
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5,128 | 5,642 | Parallel Multi-Dimensional LSTM, With Application
to Fast Biomedical Volumetric Image Segmentation
Marijn F. Stollenga*123 , Wonmin Byeon*1245 , Marcus Liwicki4 , and Juergen Schmidhuber123
*
Shared first authors, both Authors contribruted equally to this work. Corresponding authors:
[email protected], [email protected]
1
Istituto Dalle Molle di Studi sull?Intelligenza Artificiale (The Swiss AI Lab IDSIA)
2
Scuola universitaria professionale della Svizzera italiana (SUPSI), Switzerland
3
Universit?a della Svizzera italiana (USI), Switzerland
4
University of Kaiserslautern, Germany
5
German Research Center for Artificial Intelligence (DFKI), Germany
Abstract
Convolutional Neural Networks (CNNs) can be shifted across 2D images or 3D
videos to segment them. They have a fixed input size and typically perceive only
small local contexts of the pixels to be classified as foreground or background. In
contrast, Multi-Dimensional Recurrent NNs (MD-RNNs) can perceive the entire
spatio-temporal context of each pixel in a few sweeps through all pixels, especially
when the RNN is a Long Short-Term Memory (LSTM). Despite these theoretical
advantages, however, unlike CNNs, previous MD-LSTM variants were hard to parallelise on GPUs. Here we re-arrange the traditional cuboid order of computations
in MD-LSTM in pyramidal fashion. The resulting PyraMiD-LSTM is easy to parallelise, especially for 3D data such as stacks of brain slice images. PyraMiD-LSTM
achieved best known pixel-wise brain image segmentation results on MRBrainS13
(and competitive results on EM-ISBI12).
1
Introduction
Long Short-Term Memory (LSTM) networks [1, 2] are recurrent neural networks (RNNs) initially
designed for sequence processing. They achieved state-of-the-art results on challenging tasks such
as handwriting recognition [3], large vocabulary speech recognition [4] and machine translation [5].
Their architecture contains gates to store and read out information from linear units called error
carousels that retain information over long time intervals, which is hard for traditional RNNs.
Multi-Dimensional LSTM networks (MD-LSTM [6]) connect hidden LSTM units in grid-like
fashion1 . Two dimensional MD-LSTM is applicable to image segmentation [6, 7, 8] where each pixel
is assigned to a class such as background or foreground. Each LSTM unit sees a pixel and receives
input from predecessing LSTM units, thus recursively gathering information about all other pixels in
the image.
There are many biomedical 3D volumetric data sources, such as computed tomography (CT), magnetic resonance (MR), and electron microscopy (EM). Most previous approaches process each 2D
slice separately, using image segmentation algorithms such as snakes [9], random forests [10] and
Convolutional Neural Networks [11]. 3D-LSTM, however, can process the full context of each pixel
in such a volume through 8 sweeps over all pixels by 8 different LSTMs, each sweep in the general
direction of one of the 8 directed volume diagonals.
1
For example, in two dimensions this yields 4 directions; up, down, left and right.
1
Due to the sequential nature of RNNs, however, MD-LSTM parallelisation was difficult, especially
for volumetric data. The novel Pyramidal Multi-Dimensional LSTM (PyraMiD-LSTM) networks
introduced in this paper use a rather different topology and update strategy. They are easier to
parallelise, need fewer computations overall, and scale well on GPU architectures.
PyraMiD-LSTM is applied to two challenging tasks involving segmentation of biological volumetric
images. Competitive results are achieved on EM-ISBI12 [12]; best known results are achieved on
MRBrainS13 [13].
2
Method
We will first describe standard one-dimensional LSTM [2]. Then we introduce the MD-LSTM and
topology changes to construct the PyraMiD-LSTM, which is formally described and discussed.
A one-dimensional LSTM unit consists of an input gate (i), forget gate2 (f ), output gate (o), and
memory cell (c) which control what should be remembered or forgotten over potentially long periods
of time. The input x and all gates and activations are real-valued vectors: x, i, f, c?, c, o, h ? RT ,
where T is the length of the input. The gates and activations at discrete time t (t=1,2,...) are computed
as follows:
it = ?(xt ? ?xi + ht-1 ? ?hi + ?ibias ),
ft = ?(xt ? ?xf + ht-1 ? ?hf + ?fbias ),
c?t = tanh(xt ? ?x?c + ht-1 ? ?h?c + ?c?bias ),
ct = c?t it + ct-1 ft ,
ot = ?(xt ? ?xo + ht-1 ? ?ho + ?obias ),
ht = ot tanh(ct )
(1)
(2)
(3)
(4)
(5)
(6)
where (?) is a (matrix) multiplication, ( ) an element-wise multiplication, and ? denotes the weights.
c? is the input to the ?cell? c, which is gated by the input gate, and h is the output. The non-linear
functions ? and tanh are applied element-wise, where ?(x) = 1+e1?x . Equations (1, 2) determine
gate activations, Equation (3) cell inputs, Equation (4) the new cell states (here ?memories? are stored
or forgotten), Equation (5) output gate activations which appear in Equation (6), the final output.
2.1
Pyramidal Connection Topology
(a) Standard MD-LSTM
(b) 'Turned' MD-LSTM
(c) PyraMiD LSTM
Figure 1: The standard MD-LSTM topology (a) evaluates the context of each pixel recursively from neighbouring
pixel contexts along the axes, that is, pixels on a simplex can be processed in parallel. Turning this order by 45?
(b) causes the simplex to become a plane (a column vector in the 2D case here). The resulting gaps are filled by
adding extra connections, to process more than 2 elements of the context (c).
The multi-dimensional LSTM (MD-LSTM; [6]) aligns LSTM-units in a grid and connects them over
the axis. Multiple grids are needed to process information from all directions. A 2D-LSTM adds the
pixel-wise outputs of 4 LSTMs: one scanning the image pixel by pixel from north-west to south-east,
one from north-east to south-west, one from south-west to north-east, and one from south-east to
north-west. Figure 1?a shows one of these directions.
2
Although the forget gate output is inverted and actually ?remembers? when it is on, and forgets when it is
off, the traditional nomenclature is kept.
2
However, a small change in connections can greatly facilitate parallelisations: If the connections
are rotated by 45? , all inputs to all units come from either left, right, up, or down (left in case of
Figure 1?b), and all elements of a row in the grid row can be computed independently. However,
this introduces context gaps as in Figure 1?b. By adding an extra input, these gaps are filled as in
Figure 1?c. Extending this approach in 3 dimensions results in a Pyramidal Connection Topology,
meaning the context of a pixel is formed by a pyramid in each direction.
Figure 2: On the left we see the context scanned so far by one of the 8 LSTMs of a 3D-LSTM: a cube. In
general, given d dimensions, 2d LSTMs are needed. On the right we see the context scanned so far by one of
the 6 LSTMs of a 3D-PyraMiD-LSTM: a pyramid. In general, 2 ? d LSTMs are needed.
One of the striking differences between PyraMiD-LSTM and MD-LSTM is the shape of the scanned
contexts. Each LSTM of an MD-LSTM scans rectangle-like contexts in 2D or cuboids in 3D. Each
LSTM of a PyraMiD-LSTM scans triangles in 2D and pyramids in 3D (see Figure 2). An MD-LSTM
needs 8 LSTMs to scan a volume, while a PyraMiD-LSTM needs only 6, since it takes 8 cubes or 6
pyramids to fill a volume. Given dimension d, the number of LSTMs grows as 2d for an MD-LSTM
(exponentially) and 2 ? d for a PyraMiD-LSTM (linearly).
A similar connection strategy has been previously used to speed up non-Euclidian distance computations on surfaces [14]. There are however important differences:
? We can exploit efficient GPU-based CUDA convolution operations, but in a way unlike what
is done in CNNs, as will be explained below.
? As a result of these operations, input filters that are bigger than the necessary 3 ? 3 filters
arise naturally, creating overlapping contexts. Such redundancy turns out to be beneficial
and is used in our experiments.
? We apply several layers of complex processing with multi-channelled outputs and several
state-variables for each pixel, instead of having a single value per pixel as in distance
computations.
? Our application is focused on volumetric data.
2.2
PyraMiD-LSTM
C-LSTM
C-LSTM
C-LSTM
?
C-LSTM
C-LSTM
Fully
Connected
Layer
tanh
Fully
Connected
Layer softmax
C-LSTM
PyraMiD-LSTM
Input Data
Figure 3: PyraMiD-LSTM network architecture. Randomly rotated and flipped inputs are sampled from random
locations, then fed to six C-LSTMs over three axes. The outputs from all C-LSTMs are combined and sent to the
fully-connected layer. tanh is used as a squashing function in the hidden layer. Several PyraMiD-LSTM layers
can be applied. The last layer is fully-connected and uses a softmax function to compute probabilities for each
class for each pixel.
Here we explain the PyraMiD-LSTM network architecture for 3D volumes (see Figure 3). The
working horses are six convolutional LSTMs (C-LSTM) layers, one for each direction to create the
full context of each pixel. Note that each of these C-LSTMs is a entire LSTM RNN, processing the
3
entire volume in one direction. The directions D are formally defined over the three axes (x, y, z):
D = {(?, ?, 1), (?, ?, ?1), (?, 1, ?), (?, ?1, ?), (1, ?, ?), (?1, ?, ?)}. They essentially choose which axis is
the time direction; i.e. with (?, ?, 1) the positive direction of the z-axis represents the time.
Each C-LSTM performs computations in a plane moving in the defined direction. The input is
x ? RW ?H?D?C , where W is the width, H the height, D the depth, and C the number of channels
of the input, or hidden units in the case of second- and higher layers. Similarly, we define the volumes
f d , id , od , c?d , cd , hd , h ? RW ?H?D?O , where d ? D is a direction and O is the number of hidden
units per pixel. Since each direction needs a separate volume, we denote volumes with (?)d .
The time index t selects a slice in direction d. For instance, for direction d = (?, ?, 1), vtd refers to the
plane x, y, z, c for x = 1..X, y = 1..Y, c = 1..C, and z = t. For a negative direction d = (?, ?, ?1),
the plane is the same but moves in the opposite direction: z = Z ? t. A special case is the first plane in
each direction, which does not have a previous plane, hence we omit the corresponding computation.
C-LSTM equations:
d
d
+ ?idbias ),
idt = ?(xdt ? ?xi
+ hdt-1 ? ?hi
ftd
d
d
= ?(xdt ? ?xf
+ hdt-1 ? ?hf
+ ?fdbias ),
d
d
d
d
c?dt = tanh(xdt ? ?x?
c + ht-1 ? ?h?
c + ?c?bias ),
cdt = c?dt idt + cdt-1 ftd ,
d
d
+ ?odbias ),
odt = ?(xdt ? ?xo
+ hdt-1 ? ?ho
hdt = odt tanh(cdt ),
X
d
h=
h ,
(7)
(8)
(9)
(10)
(11)
(12)
(13)
d?D
where (?) is a convolution3 , and h is the output of the layer. All biases are the same for all LSTM
units (i.e., no positional biases are used). The outputs hd for all directions are summed together.
Fully-Connected Layer: The output of our PyraMiD-LSTM layer is connected to a pixel-wise
fully-connected layer, which output is squashed by the hyperbolic tangent (tanh) function. This step
is used to increase the number of channels for the next layer. The final classification is done using a
?h(x,y,z,c)
pixel-wise softmax function: y(x, y, z, c) = Pe e?h(x,y,z,c) giving pixel-wise probabilities for each
c
class.
3
Experiments
We evaluate our approach on two 3D biomedical image segmentation datasets: electron microscopy
(EM) and MR Brain images.
EM dataset The EM dataset [12] is provided by the ISBI 2012 workshop on Segmentation of
Neuronal Structures in EM Stacks [15]. Two stacks consist of 30 slices of 512 ? 512 pixels obtained
from a 2 ? 2 ? 1.5 ?m3 microcube with a resolution of 4 ? 4 ? 50 nm3 /pixel and binary labels.
One stack is used for training, the other for testing. Target data consists of binary labels (membrane
and non-membrane).
MR Brain dataset The MR Brain images are provided by the ISBI 2015 workshop on Neonatal
and Adult MR Brain Image Segmentation (ISBI NEATBrainS15) [13]. The dataset consists of twenty
fully annotated high-field (3T) multi-sequences: 3D T1-weighted scan (T1), T1-weighted inversion
recovery scan (IR), and fluid-attenuated inversion recovery scan (FLAIR). The dataset is divided into
a training set with five volumes and a test set with fifteen volumes. All scans are bias-corrected and
aligned. Each volume includes 48 slices with 240 ? 240 pixels (3mm slice thickness). The slices
3
In 3D volumes, convolutions are performed in 2D; in general an n-D volume requires n-1-D convolutions.
All convolutions have stride 1, and their filter sizes should at least be 3 ? 3 in each dimension to create the full
context.
4
are manually segmented through nine labels: cortical gray matter, basal ganglia, white matter, white
matter lesions, cerebrospinal fluid in the extracerebral space, ventricles, cerebellum, brainstem, and
background. Following the ISBI NEATBrainS15 workshop procedure, all labels are grouped into four
classes and background: 1) cortical gray matter and basal ganglia (GM), 2) white matter and white
matter lesions (WM), 3) cerebrospinal fluid and ventricles (CSF), and 4) cerebellum and brainstem.
Class 4) is ignored for the final evaluation as required.
Sub-volumes and Augmentation The full dataset requires more than the 12 GB of memory
provided by our GPU, hence we train and test on sub-volumes. We randomly pick a position in the
full data and extract a smaller cube (see the details in Bootstrapping). This cube is possibly rotated at
a random angle over some axis and can be flipped over any axis. For EM images, we rotate over the
z-axis and flipped sub-volumes with 50% chance along x, y, and z axes. For MR brain images, rotation
is disabled; only flipping along the x direction is considered, since brains are (mostly) symmetric in
this direction.
During test-time, rotations and flipping are disabled and the results of all sub-volumes are stitched
together using a Gaussian kernel, providing the final result.
Pre-processing We normalise each input slice towards a mean of zero and variance of one, since
the imaging methods sometimes yield large variability in contrast and brightness. We do not apply
the complex pre-processing common in biomedical image segmentation [10].
We apply simple pre-processing on the three datatypes of the MR Brain dataset, since they contain
large brightness changes under the same label (even within one slice; see Figure 5). From all slices
we subtract the Gaussian smoothed images (filter size: 31 ? 31, ? = 5.0), then a Contrast-Limited
Adaptive Histogram Equalisation (CLAHE) [16] is applied to enhance the local contrast (tile size:
16 ? 16, contrast limit: 2.0). An example of the images after pre-processing is shown in Figure 5.
The original and pre-processed images are all used, except the original IR images (Figure 5b), which
have high variability.
?
? b to be an = ?an + (1 ? ?)bn ,
Training We apply RMS-prop [17] with momentum. We define a ?
where a, b ? RN . The following equations hold for every epoch:
E = (y ? ? y)2 ,
(14)
?MSE
MSE ???? ?2? E,
?? E
,
G= ?
MSE +
(15)
(16)
?M
M ??? G,
? = ? ? ?lr M,
(17)
(18)
where y ? is the target, y is the output from the networks, E is the squared loss, MSE a running
average of the variance of the gradient, ?2 is the element-wise squared gradient, G the normalised
gradient, M the smoothed gradient, and ? the weights. The squared loss was chosen as it produced
better results than using the log-likelihood as an error function. This algorithm normalises the gradient
of each weight, such that even weights with small gradients get updated. This also helps to deal with
vanishing gradients [18].
epoch
We use a decaying learning rate: ?lr = 10?6 + 10?2 ? 12 100 , which starts at ?lr ? 10?2 and halves
every 100 epochs asymptotically towards ?lr = 10?6 . Other hyper-parameters used are = 10?5 ,
?MSE = 0.9, and ?M = 0.9.
Bootstrapping To speed up training, we run three learning procedures with increasing sub-volume
sizes: first, 3000 epochs with size 64 ? 64 ? 8, then 2000 epochs with size 128 ? 128 ? 15. Finally,
for the EM-dataset, we train 1000 epochs with size 256 ? 256 ? 20, and for the MR Brain dataset
1000 epochs with size 240 ? 240 ? 25. After each epoch, the learning rate ?lr is reset.
5
Table 1: Performance comparison on EM images. Some of the competing methods reported in the ISBI 2012
website are not yet published. Comparison details can be found under http://brainiac2.mit.edu/
isbi_challenge/leaders-board.
Rand Err.
Warping Err.(?10?3 )
Pixel Err.
Human
Simple Thresholding
0.002
0.450
0.0053
17.14
0.001
0.225
IDSIA [11]
DIVE
PyraMiD-LSTM
0.050
0.048
0.047
0.420
0.374
0.462
0.061
0.058
0.062
IDSIA-SCI
DIVE-SCI
0.0189
0.0178
0.617
0.307
0.103
0.058
Group
Experimental Setup All experiments are performed on a desktop computer with an NVIDIA GTX
TITAN X 12GB GPU. Due to the pyramidal topology all major computations can be done using
convolutions with NVIDIA?s cuDNN library [19], which has reported 20? speedup over an optimised
implementation on a modern 16 core CPU. On the MR brain dataset, training took around three days,
and testing per volume took around 2 minutes.
We use exactly the same hyper-parameters and architecture for both datasets. Our networks contain
three PyraMiD-LSTM layers. The first PyraMiD-LSTM layer has 16 hidden units followed by a
fully-connected layer with 25 hidden units. In the next PyraMiD-LSTM layer, 32 hidden units are
connected to a fully-connected layer with 45 hidden units. In the last PyraMiD-LSTM layer, 64
hidden units are connected to the fully-connected output layer whose size equals the number of
classes.
The convolutional filter size for all PyraMiD-LSTM layers is set to 7 ? 7. The total number of weights
is 10,751,549, and all weights are initialised according to a uniform distribution: U(?0.1, 0.1).
3.1
Neuronal Membrane Segmentation
(a) Input
(b) PyraMiD-LSTM
Figure 4: Segmentation results on EM dataset (slice 26)
Membrane segmentation is evaluated through an online system provided by the ISBI 2012 organisers.
The measures used are the Rand error, warping error and pixel error [15]. Comparisons to other
methods are reported in Table 1. The teams IDSIA and DIVE provide membrane probability maps
for each pixel. The IDSIA team uses a state-of-the-art deep convolutional network [11], the method
of DIVE was not provided.
These maps are adapted by the post-processing technique of the teams SCI [20], which directly
optimises the rand error (DIVE-SCI (top-1) and IDSIA-SCI (top-2)); this is most important in this
particular segmentation task.
6
Table 2: The performance comparison on MR brain images.
Structure
Metric
BIGR2
KSOM GHMF
MNAB2
ISI-Neonatology
UNC-IDEA
GM
WM
CSF
DC MD AVD DC MD AVD DC MD AVD Rank
(%) (mm) (%) (%) (mm) (%) (%) (mm) (%)
84.65
84.12
84.50
85.77
84.36
1.88
1.92
1.69
1.62
1.62
6.14
5.44
7.10
6.62
7.04
88.42
87.96
88.04
88.66
88.69
2.36
2.49
2.12
2.06
2.06
6.02
6.59
7.73
6.96
6.46
78.31
82.10
82.30
81.08
82.81
3.19
2.71
2.27
2.66
2.35
22.8
12.8
8.73
9.77
10.5
6
5
4
3
2
PyraMiD-LSTM 84.82 1.69 6.77 88.33 2.07 7.05 83.72 2.14 7.10
1
Without post-processing, PyraMiD-LSTM networks outperform other methods in rand error, and
are competitive in wrapping and pixel errors. Of course, performance could be further improved by
applying post-processing techniques. Figure 4 shows an example segmentation result.
3.2
MR Brain Segmentation
The results are compared using the DICE overlap (DC), the modified Hausdorff distance (MD), and
the absolute volume difference (AVD) [13]. MR brain image segmentation results are evaluated by the
ISBI NEATBrain15 organisers [13] who provided the extensive comparison to other approaches on
http://mrbrains13.isi.uu.nl/results.php. Table 2 compares our results to those of
the top five teams. The organisers compute nine measures in total and rank all teams for each of them
separately. These ranks are then summed per team, determining the final ranking (ties are broken
using the standard deviation). PyraMiD-LSTM leads the final ranking with a new state-of-the-art
result and outperforms other methods for CSF in all metrics.
We also tried regularisation through dropout [21]. Following earlier work [22], the dropout operator
is applied only to non-recurrent connections (50% dropout on fully connected layers and/or 20% on
input layer). However, this did not improve performance.
4
Conclusion
Since 2011, GPU-trained max-pooling CNNs have dominated classification contests [23, 24, 25] and
segmentation contests [11]. MD-LSTM, however, may pose a serious challenge to such CNNs, at
least for segmentation tasks. Unlike CNNs, MD-LSTM has an elegant recursive way of taking each
pixel?s entire spatio-temporal context into account, in both images and videos. Previous MD-LSTM
implementations, however, could not exploit the parallelism of modern GPU hardware. This has
changed through our work presented here. Although our novel highly parallel PyraMiD-LSTM
has already achieved state-of-the-art segmentation results in challenging benchmarks, we feel we
have only scratched the surface of what will become possible with such PyraMiD-LSTM and other
MD-RNNs.
5
Acknowledgements
We would like to thank Klaus Greff and Alessandro Giusti for their valuable discussions, and
Jan Koutnik and Dan Ciresan for their useful feedback. We also thank the ISBI NEATBrain15
organisers [13] and the ISBI 2012 organisers, in particular Adri?enne Mendrik and Ignacio ArgandaCarreras. Lastly we thank NVIDIA for generously providing us with hardware to perform our research.
This research was funded by the NASCENCE EU project (EU/FP7-ICT-317662).
7
(a) T1
(b) IR
(c) FLAIR
(d) T1 (pre-processed)
(e) IR (pre-processed)
(f) FLAIR (pre-processed)
(g) segmentation result from PyraMiD-LSTM
Figure 5: Slice 19 of the test image 1. (a)-(c) are examples of three scan methods used in the MR brain
dataset, and (d)-(f) show the corresponding images after our pre-processing procedure (see pre-processing in
Section ). Input (b) is omitted due to strong artefacts in the data ? the other datatypes are all used as input to the
PyraMiD-LSTM. The segmentation result is shown in (g).
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5,129 | 5,643 | Learning From Small Samples: An Analysis of
Simple Decision Heuristics
? ur
? S?ims?ek and Marcus Buckmann
Ozg
Center for Adaptive Behavior and Cognition
Max Planck Institute for Human Development
Lentzeallee 94, 14195 Berlin, Germany
{ozgur, buckmann}@mpib-berlin.mpg.de
Abstract
Simple decision heuristics are models of human and animal behavior that use few
pieces of information?perhaps only a single piece of information?and integrate
the pieces in simple ways, for example, by considering them sequentially, one at
a time, or by giving them equal weight. We focus on three families of heuristics:
single-cue decision making, lexicographic decision making, and tallying. It is
unknown how quickly these heuristics can be learned from experience. We show,
analytically and empirically, that substantial progress in learning can be made with
just a few training samples. When training samples are very few, tallying performs
substantially better than the alternative methods tested. Our empirical analysis is
the most extensive to date, employing 63 natural data sets on diverse subjects.
1
Introduction
You may remember that, on January 15, 2009, in New York City, a commercial passenger plane
struck a flock of geese within two minutes of taking off from LaGuardia Airport. The plane immediately and completely lost thrust from both engines, leaving the crew facing a number of critical
decisions, one of which was whether they could safely return to LaGuardia. The answer depended
on many factors, including the weight, velocity, and altitude of the aircraft, as well as wind speed and
direction. None of these factors, however, are directly involved in how pilots make such decisions.
As copilot Jeffrey Skiles discussed in a later interview [1], pilots instead use a single piece of visual
information: whether the desired destination is staying stationary in the windshield. If the destination is rising or descending, the plane will undershoot or overshoot the destination, respectively.
Using this visual cue, the flight crew concluded that LaGuardia was out of reach, deciding instead
to land on the Hudson River. Skiles reported that subsequent simulation experiments consistently
showed that the plane would indeed have crashed before reaching the airport.
Simple decision heuristics, such as the one employed by the flight crew, can provide effective solutions to complex problems [2, 3]. Some of these heuristics use a single piece of information; others
use multiple pieces of information but combine them in simple ways, for example, by considering
them sequentially, one at a time, or by giving them equal weight.
Our work is concerned with two questions: How effective are simple decision heuristics? And
how quickly can they be learned from experience? We focus on problems of comparison, where the
objective is to decide which of a given set of objects has the highest value on an unobserved criterion.
These problems are of fundamental importance in intelligent behavior. Humans and animals spend
much of their time choosing an object to act on, with respect to some criterion whose value is
unobserved at the time. Choosing a mate, a prey to chase, an investment strategy for a retirement
fund, or a publisher for a book are just a few examples. Earlier studies on this problem have shown
1
that simple heuristics are surprisingly accurate in natural environments [4, 5, 6, 7, 8, 9], especially
when learning from small samples [10, 11].
We present analytical and empirical results on three families of heuristics: lexicographic decision
making, tallying, and single-cue decision making. Our empirical analysis is the most extensive
to date, employing 63 natural environments on diverse subjects. Our main contributions are as
follows: (1) We present analytical results on the rate of learning heuristics from experience. (2) We
show that very few learning instances can yield effective heuristics. (3) We empirically investigate
single-cue decision making and find that its performance is remarkable. (4) We find that the most
robust decision heuristic for small sample sizes is tallying. Collectively, our results have important
implications for developing more successful heuristics and for studying how well simple heuristics
capture human and animal decision making.
2
Background
The comparison problem asks which of a given set of objects has the highest value on an unobserved
criterion, given a number of attributes of the objects. We focus on pairwise comparisons, where
exactly two objects are being compared. We consider a decision to be accurate if it selects the object
with the higher criterion value (or either object if they are equal in criterion value). In the heuristics
literature, attributes are called cues; we will follow this custom when discussing heuristics.
The heuristics we consider decide by comparing the objects on one or more cues, asking which
object has the higher cue value. Importantly, they do not require the difference in cue value to be
quantified. For example, if we use height of a person as a cue, we need to be able to determine which
of two people is taller but we do not need to know the height of either person or the magnitude of the
difference. Each cue is associated with a direction of inference, also known as cue direction, which
can be positive or negative, favoring the object with the higher or lower cue value, respectively. Cue
directions (and other components of heuristics) can be learned in a number of ways, including social
learning. In our analysis, we learn them from training examples.
Single-cue decision making is perhaps the simplest decision method one can imagine. It compares
the objects on a single cue, breaking ties randomly. We learn the identity of the cue and its direction
from a training sample. Among the 2k possible models, where k is the number of cues, we choose
the hcue, directioni combination that has the highest accuracy in the training sample, breaking ties
randomly.
Lexicographic heuristics consider the cues one at a time, in a specified order, until they find a cue
that discriminates between the objects, that is, one whose value differs on the two objects. The
heuristic then decides based on that cue alone. An example is take-the-best [12], which orders cues
with respect to decreasing validity on the training sample, where validity is the accuracy of the cue
among pairwise comparisons on which the cue discriminates between the objects.
Tallying is a voting model. It determines how each cue votes on its own (selecting one or the other
object or abstaining from voting) and selects the object with the highest number of votes, breaking
ties randomly. We set cue directions to the direction with highest validity in the training set.
Paired comparison can also be formulated as a classification problem. Let yA denote the criterion
value of object A, xA the vector of attribute values of object A, and ?y AB = yA ? yB the difference
in criterion values of objects A and B. We can define the class f of a pair of objects as a function of
the difference in their criterion values:
(
1 if ?y AB > 0
?1 if ?y AB < 0
f (?y AB ) =
0 if ?y AB = 0
A class value of 1 denotes that object A has the higher criterion value, ?1 that object B has the
higher criterion value, and 0 that the objects are equal in criterion value. The comparison problem
is intrinsically symmetrical: comparing A to B should give us the same decision as comparing B to
A. That is, f (?y AB ) should equal ?f (?y BA ). Because the latter equals ?f (??y AB ), we have
the following symmetry constraint: f (z) = ?f (?z), for all z. We can expect better classification
accuracy if we impose this symmetry constraint on our classifier.
2
3
Building blocks of decision heuristics
We first examine two building blocks of learning heuristics from experience: assigning cue direction
and determining which of two cues has the higher predictive accuracy. The former is important for
all three families of heuristics whereas the latter is important for lexicographic heuristics when
determining which cue should be placed first. Both components are building blocks of heuristics in
a broader sense?their use is not limited to the three families of heuristics considered here.
Let A and B be the objects being compared, xA and xB denote their cue values, yA and yB denote
their criterion values, and sgn denote the mathematical sign function: sgn(x) is 1 if x > 0, 0 if
x = 0, and ?1 if x < 0. A single training instance is the tuple hsgn(xA ? xB ), sgn(yA ? yB )i,
corresponding to a single pairwise comparison, indicating whether the cue and the criterion change
from one object to the other, along with the direction of change. For example, if xA = 1, yA = 10,
xB = 2, yB = 5, the training instance is h?1, +1i.
Learning cue direction. We assume, without loss of generality, that cue direction in the population
is positive (we ignore the case where the cue direction in the population is neutral). Let p denote the
success rate of the cue in the population, where success is the event that the cue decides correctly.
We examine two probabilities, e1 and e2 . The former is the probability of correctly inferring the cue
direction from a set of training instances. The latter is the probability of deciding correctly on a new
(unseen) instance using the direction inferred from the training instances.
We define an informative instance to be one in which the objects differ both in their cue values and
in their criterion values, a positive instance to be one in which the cue and the criterion change in
the same direction (h1, 1i or h?1, ?1i), and a negative instance to be one in which the cue and the
criterion change in the opposite direction (h1, ?1i or h?1, 1i).
Let n be the number of training instances, n+ the number of positive training instances, and n?
the number of negative training instances. Our estimate of cue direction is positive if n+ > n? ,
negative if n+ < n? , and a random choice between positive and negative if n+ = n? .
Given a set of independent, informative training instances, n+ follows the binomial distribution with
n trials and success probability p, allowing us to write e1 as follows:
e1
1
= P (n+ > n? ) + P (n+ = n? )
2
n
X
n k
1 n
n?k
=
p (1 ? p)
+ I(n is even)
pn/2 (1 ? p)n/2 ,
k
2 n/2
k=bn/2c+1
where I is the indicator function. After one training instance, e1 equals p. After one more instance,
e1 remains the same. This is a general property: After an odd number of training instances, an
additional instance does not increase the probability of inferring the direction correctly.
On a new (test) instance, the cue decides correctly with probability p if cue direction is inferred
correctly and with probability 1 ? p otherwise. Consequently, e2 = pe1 + (1 ? p)(1 ? e1 ).
Simple algebra yields the following expected learning rates: After 2k + 1 training instances, with
two additional instances, the increase in the probability of inferring cue direction correctly is (2p ?
1)(p(1 ? p))k+1 and the increase in the probability of deciding correctly is
(2p ? 1)2 (p(1 ? p))k+1 .
Figure 1 shows e1 and e2 as a function of training-set size n and success rate p. The more predictive
the cue is, the smaller the sample needs to be for a desired level of accuracy in both e1 and e2 . This is
of course a desirable property: The more useful the cue is, the faster we learn how to use it correctly.
The figure also shows that there are highly diminishing returns, from one odd training-set size to the
next, as the size of the training set increases. In fact, just a few instances make great progress toward
the maximum possible. The third plot in the figure reveals this property more clearly. It shows e2
divided by its maximum possible value (p) showing how quickly we reach the maximum possible
accuracy for cues of various predictive ability. The minimum value depicted in this figure is 0.83,
observed at n = 1. This means that even after a single training instance, our expected accuracy is at
least 83% of the maximum accuracy we can reach. And this value rises quickly with each additional
pair of training instances.
3
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Figure 1: Learning cue direction.
Learning to order two cues. Assume we have two cues with success rates p and q in the population,
with p > q. We expand the definition of informative instance to require that the objects differ on
the second cue as well. We examine two probabilities, e3 and e4 . The former is the probability of
ordering the two cues correctly, which means placing the cue with higher success rate above the
other one. The latter is the probability of deciding correctly with the inferred order. We chose to
examine learning to order cues independently of learning cue directions. One reason is that people
do not necessarily learn the cue directions from experience. In many cases, they can guess the cue
direction correctly through causal reasoning, social learning, past experience in similar problems, or
other means. In the analysis below, we assume that the directions are assigned correctly.
Let s1 and s2 be the success rates of the two cues in the training set. If instances are informative and
independent, s1 and s2 follow the binomial distribution with parameters (n, p) and (n, q), allowing
us to write e3 as follows:
1
e3 = P (s1 > s2 ) + P (s1 = s2 ) =
2
n
X
P (s1 = i)P (s2 = j) +
0?j<i?n
1X
P (s1 = i)P (s2 = i)
2 i=0
After one training instance, e3 is 0.5+0.5(p?q), which is a linear function of the difference between
the two success rates.
If we order cues correctly, a decision on a test instance is correct with probability p, otherwise with
probability q. Thus, e4 = pe3 + q(1 ? e3 ).
1.0
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After 3 training instances:
e 4 / max(p,q)
1.00
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p
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p
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1.0
After 3 training instances:
Probability of correctly deciding
(e 4)
0.5
1.0
After 3 training instances:
Probability of correctly ordering
(e 3)
1.0
Figure 2 shows e3 and e4 as a function of p and q after three training instances. In general, larger
values of p, as well as larger differences between p and q, require smaller training sets for a desired
level of accuracy. In other words, learning progresses faster where it is more useful. The third plot
in the figure shows e4 relative to the maximum value it can take, the maximum of p and q. The
minimum value depicted in this figure is 90.9%. If we examine the same figure after only a single
training instance, we see that this minimum value is 86.6% (figure not shown).
0.5
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q
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q
Figure 2: Learning cue order.
4
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4
Empirical analysis
We next present an empirical analysis of 63 natural data sets, most from two earlier studies [4, 13].
Our primary objective is to examine the empirical learning rates of heuristics. From the analytical
results of the preceding section, we expect learning to progress rapidly. A secondary objective is to
examine the effectiveness of different ways cues can be ordered in a lexicographic heuristic.
The data sets were gathered from a wide variety of sources, including online data repositories, textbooks, packages for R statistical software, statistics and data mining competitions, research publications, and individual scientists collecting field data. The subjects were diverse, including biology,
business, computer science, ecology, economics, education, engineering, environmental science,
medicine, political science, psychology, sociology, sports, and transportation. The data sets varied
in size, ranging from 13 to 601 objects. Many of the smaller data sets contained the entirety of the
population of objects, for example, all 29 islands in the Gal?apagos archipelago. The data sets are
described in detail in the supplementary material.
We present results on lexicographic heuristics, tallying, single-cue decision making, logistic regression, and decision trees trained by CART [14]. We used the CART implementation in rpart [15]
with the default splitting criterion Gini, cp=0, minsplit=2, minbucket=1, and 10-fold cross-validated
cost-complexity pruning. There is no explicit way to implement the symmetry constraint for decision trees; we simply augmented the training set with its mirror image with respect to the direction
of comparison. For logistic regression, we used the glm function of R, setting the intercept to zero
to implement the symmetry constraint. To the glm function, we input the cues in the order of decreasing correlation with the criterion so that the weakest cues were dropped first when the number
of training instances was smaller than the number of cues.
Ordering cues in lexicographic heuristics. We first examine the different ways lexicographic
heuristics can order the cues. With k cues, there are k! possible cue orders. Combined with the possibility of using each cue with a positive or negative direction, there are 2k k! possible lexicographic
models, a number that increases very rapidly with k. How should we choose one if our top criterion
is accuracy but we also want to pay attention to computational cost and memory requirements?
We consider three methods. The first is a greedy search, where we start by deciding on the first
cue to be used (along with its direction), then the second, and so on, until we have a fully specified
lexicographic model. When deciding on the first cue, we select the one that has the highest validity
in the training examples. When deciding on the mth cue, m ? 2, we select the cue that has the
highest validity in the examples left over after using the first m ? 1 cues, that is, those examples
where the first m ? 1 cues did not discriminate between the two objects. The second method is to
order cues with respect to their validity in the training examples, as take-the-best does. Evaluating
cues independently of each other substantially reduces computational and memory requirements but
perhaps at the expense of accuracy. The third method is to use the lexicographic model?among the
2k k! possibilities?that gives the highest accuracy in the training examples. Identifying this rule is
NP-complete [16, 17], and it is unlikely to generalize well, but it will be informative to examine it.
The three methods have been compared earlier [18] on a data set consisting of German cities [12],
where the fitting accuracy of the best, greedy, validity, and random ordering was 0.758, 0.756, 0.742,
and 0.700, respectively.
Figure 3 (top panel) shows the fitting accuracy of each method in each of the 63 data sets when all
possible pairwise comparisons were conducted among all objects. Because of the long simulation
time required, we show an approximation of the best ordering in data sets with seven or more cues.
In these data sets, we started with the two lexicographic rules generated by the greedy and the
validity ordering, kept intact the cues that were placed seventh or later in the sequence, and tested
all possible permutations of their first six cues, trying out both possible cue directions. The figure
also shows the mean accuracy of random ordering, where cues were used in the direction of higher
validity. In all data sets, greedy ordering was identical or very close in accuracy to the best ordering.
In addition, validity ordering was very close to greedy ordering except in a handful of data sets.
One explanation is that a continuous cue that is placed first in a lexicographic model makes all (or
almost all) decisions and therefore the order of the remaining cues does not matter. We therefore
also examine the binary version of each data set where numerical cues were dichotomizing around
the median (Figure 3 bottom panel). There was little difference in the relative positions of greedy
and optimal ordering except in one data set. There was more of a drop in the relative accuracy of
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Figure 3: Fitting accuracy of lexicographic models, with and without dichotomizing the cues.
the validity ordering, but this method still achieved accuracy close to that of the best ordering in the
majority of the data sets.
We next examine predictive accuracy. Figure 4 shows accuracies when the models were trained on
50% of the objects and tested on the remaining 50%, conducting all possible pairwise comparisons
within each group. Mean accuracy across data sets was 0.747 for logistic regression, 0.746 for
CART, 0.743 for greedy lexicographic and take-the-best, 0.734 for single-cue, and 0.725 for tallying.
Figure 5 shows learning curves, where we grew the training set one pairwise comparison at a time.
Two individual objects provided a single instance for training or testing and were never used again,
neither in training nor in testing. Consequently, the training instances were independent of each
other but they were not always informative (as defined in Section 3). The figure shows the mean
learning curve across all data sets as well as individual learning curves on 16 data sets. We present
the graphs without error bars for legibility; the highest standard error of the data points displayed is
0.0014 in Figure 4 and 0.0026 in Figure 5.
A few observations are noteworthy: (1) Heuristics were indeed learned rapidly. (2) In the early part
of the learning curve, tallying generally had the highest accuracy. (3) The performance of single-cue
was remarkable. When trained on 50% of the objects, its mean performance was better than tallying,
0.9 percentage points behind take-the-best, and 1.3 percentage points behind logistic regression. (4)
Take-the-best performed better than or as well as greedy lexicographic in most data sets. A detailed
comparison of the two methods is provided below.
Validity versus greedy ordering in lexicographic decision making. The learning curves on individual data sets took one of four forms: (1) There was no difference in any part of the learning
curve. This is the case when a continuous cue is placed first: This cue almost always discriminates
between the objects, and cues further down in the sequence are seldom (if ever) used. Because
greedy and validity ordering always agree on the first cue, the learning curves are identical or nearly
so. Twenty-two data sets were in this first category. (2) Validity ordering was better than greedy
ordering in some parts of the learning curve and never worse. This category included 35 data sets.
(3) Learning curves crossed: Validity ordering generally started with higher accuracy than greedy
ordering; the difference diminished with increasing training-set size, and eventually greedy ordering
exceeded validity ordering in accuracy (2 data sets). (4) Greedy ordering was better than validity or6
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Take-the-best
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Tallying
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Logistic regression
CART
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Number of cues
Figure 4: Predictive accuracy when models are trained with 50% of the objects in each data set and
tested on the remaining 50%.
dering in some parts of the learning curve and never worse (4 data sets). To draw these conclusions,
we considered a difference to be present if the error bars (? 2 SE) did not overlap.
5
Discussion
We isolated two building blocks of decision heuristics and showed analytically that they require very
few training instances to learn under conditions that matter the most: when they add value to the
ultimate predictive ability of the heuristic. Our empirical analysis confirmed that heuristics typically
make substantial progress early in learning.
Among the algorithms we considered, the most robust method for very small training sets is tallying.
Earlier work [11] concluded that take-the-best (with undichotomized cues) is the most robust model
for training sets with 3 to 10 objects but tallying (with undichotomized cues) was absent from this
earlier study. In addition, we found that the performance of single-cue decision making is truly
remarkable. This heuristic has been analyzed [19] by assuming that the cues and the criterion follow
the normal distribution; we are not aware of an earlier analysis of its empirical performance on
natural data sets.
Our analysis of learning curves differs from earlier studies. Most earlier studies [20, 10, 21, 11,
22] examined performance as a function of number of objects in the training set, where training
instances are all possible pairwise comparisons among those objects. Others increased the training
set one pairwise comparison at a time but did not keep the pairwise comparisons independent of each
other [23]. In contrast, we increased the training set one pairwise comparison at a time and kept all
pairwise comparisons independent of each other. This makes it possible to examine the incremental
value of each training instance.
There is criticism of decision heuristics because of their computational requirements. For instance, it
has been argued that take-the-best can be described as a simple algorithm but its successful execution
relies on a large amount of precomputation [24] and that the computation of cue validity in the
German city task ?would require 30,627 pairwise comparisons just to establish the cue validity
hierarchy for predicting city size? [25]. Our results clearly show that the actual computational needs
of heuristics can be very low if independent pairwise comparisons are used for training. A similar
result?that just a few samples may suffice?exists within the context of Bayesian inference [26].
Acknowledgments
Thanks to Gerd Gigerenzer, Konstantinos Katsikopoulos, Malte Lichtenberg, Laura Martignon,
Perke Jacobs, and the ABC Research Group for their comments on earlier drafts of this article.
? ur S?ims?ek from the Deutsche ForschungsgeThis work was supported by Grant SI 1732/1-1 to Ozg?
meinschaft (DFG) as part of the priority program ?New Frameworks of Rationality? (SPP 1516).
7
0.80
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Logistic regression
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63
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Figure 5: Learning curves.
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80
References
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9
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5,130 | 5,644 | 3D Object Proposals for Accurate Object Class
Detection
Xiaozhi Chen1
Kaustav Kundu 2
Huimin Ma1
Yukun Zhu2
Sanja Fidler2
Andrew Berneshawi2
Raquel Urtasun2
2
1
Department of Computer Science
University of Toronto
Department of Electronic Engineering
Tsinghua University
[email protected], {kkundu, yukun}@cs.toronto.edu,
[email protected], [email protected],
{fidler, urtasun}@cs.toronto.edu
Abstract
The goal of this paper is to generate high-quality 3D object proposals in the context of autonomous driving. Our method exploits stereo imagery to place proposals in the form of 3D bounding boxes. We formulate the problem as minimizing an
energy function encoding object size priors, ground plane as well as several depth
informed features that reason about free space, point cloud densities and distance
to the ground. Our experiments show significant performance gains over existing
RGB and RGB-D object proposal methods on the challenging KITTI benchmark.
Combined with convolutional neural net (CNN) scoring, our approach outperforms all existing results on all three KITTI object classes.
1
Introduction
Due to the development of advanced warning systems, cameras are available onboard of almost
every new car produced in the last few years. Computer vision provides a very cost effective solution
not only to improve safety, but also to one of the holy grails of AI, fully autonomous self-driving
cars. In this paper we are interested in 2D and 3D object detection for autonomous driving.
With the large success of deep learning in the past years, the object detection community shifted
from simple appearance scoring on exhaustive sliding windows [1] to more powerful, multi-layer
visual representations [2, 3] extracted from a smaller set of object/region proposals [4, 5]. This
resulted in over 20% absolute performance gains [6, 7] on the PASCAL VOC benchmark [8].
The motivation behind these bottom-up grouping approaches is to provide a moderate number of
region proposals among which at least a few accurately cover the ground-truth objects. These
approaches typically over-segment an image into super pixels and group them based on several
similarity measures [4, 5]. This is the strategy behind Selective Search [4], which is used in most
state-of-the-art detectors these days. Contours in the image have also been exploited in order to
locate object proposal boxes [9]. Another successful approach is to frame the problem as energy
minimization where a parametrized family of energies represents various biases for grouping, thus
yielding multiple diverse solutions [10].
Interestingly, the state-of-the-art R-CNN approach [6] does not work well on the autonomous driving benchmark KITTI [11], falling significantly behind the current top performers [12, 13]. This
is due to the low achievable recall of the underlying box proposals on this benchmark. KITTI images contain many small objects, severe occlusion, high saturated areas and shadows. Furthermore,
KITTI?s evaluation requires a much higher overlap with ground-truth for cars in order for a detection to count as correct. Since most existing object/region proposal methods rely on grouping super
pixels based on intensity and texture, they fail in these challenging conditions.
1
Image
Stereo
depth-Feat
Prior
Figure 1: Features: From left to right: original image, stereo reconstruction, depth-based features and our
prior. In the third image, purple is free space (F in Eq. (2)) and occupancy is yellow (S in Eq. (1)). In the prior,
the ground plane is green and red to blue indicates distance to the ground.
In this paper, we propose a new object proposal approach that exploits stereo information as well
as contextual models specific to the domain of autonomous driving. Our method reasons in 3D and
places proposals in the form of 3D bounding boxes. We exploit object size priors, ground plane, as
well as several depth informed features such as free space, point densities inside the box, visibility
and distance to the ground. Our experiments show a significant improvement in achievable recall
over the state-of-the-art at all overlap thresholds and object occlusion levels, demonstrating that our
approach produces highly accurate object proposals. In particular, we achieve a 25% higher recall for
2K proposals than the state-of-the-art RGB-D method MCG-D [14]. Combined with CNN scoring,
our method outperforms all published results on object detection for Car, Cyclist and Pedestrian on
KITTI [11]. Our code and data are online: http://www.cs.toronto.edu/?3dop.
2
Related Work
With the wide success of deep networks [2, 3], which typically operate on a fixed spatial scope,
there has been increased interest in object proposal generation. Existing approaches range from
purely RGB [4, 9, 10, 5, 15, 16], RGB-D [17, 14, 18, 19], to video [20]. In RGB, most approaches
combine superpixels into larger regions based on color and texture similarity [4, 5]. These approaches produce around 2,000 proposals per image achieving nearly perfect achievable recall on
the PASCAL VOC benchmark [8]. In [10], regions are proposed by defining parametric affinities
between pixels and solving the energy using parametric min-cut. The proposed solutions are then
scored using simple Gestalt-like features, and typically only 150 top-ranked proposals are needed
to succeed in consequent recognition tasks [21, 22, 7]. [16] introduces learning into proposal generation with parametric energies. Exhaustively sampled bounding boxes are scored in [23] using
several ?objectness? features. BING [15] proposals also score windows based on an object closure
measure as a proxy for ?objectness?. Edgeboxes [9] score millions of windows based on contour
information inside and on the boundary of each window. A detailed comparison is done in [24].
Fewer approaches exist that exploit RGB-D. [17, 18] extend CPMC [10] with additional affinities
that encourage the proposals to respect occlusion boundaries. [14] extends MCG [5] to 3D by
an additional set of depth-informed features. They show significant improvements in performance
with respect to past work. In [19], RGB-D videos are used to propose boxes around very accurate
point clouds. Relevant to our work is Sliding Shapes [25], which exhaustively evaluates 3D cuboids
in RGB-D scenes. This approach, however, utilizes an object scoring function trained on a large
number of rendered views of CAD models, and uses complex class-based potentials that make the
method run slow in both training and inference. Our work advances over prior work by exploiting the
typical sizes of objects in 3D, the ground plane and very efficient depth-informed scoring functions.
Related to our work are also detection approaches for autonomous driving. In [26], objects are predetected via a poselet-like approach and a deformable wireframe model is then fit using the image
information inside the box. Pepik et al. [27] extend the Deformable Part-based Model [1] to 3D by
linking parts across different viewpoints and using a 3D-aware loss function. In [28], an ensemble
of models derived from visual and geometrical clusters of object instances is employed. In [13],
Selective Search boxes are re-localized using top-down, object level information. [29] proposes a
holistic model that re-reasons about DPM detections based on priors from cartographic maps. In
KITTI, the best performing method so far is the recently proposed 3DVP [12] which uses the ACF
detector [30] and learned occlusion patters in order to improve performance of occluded cars.
3
3D Object Proposals
The goal of our approach is to output a diverse set of object proposals in the context of autonomous
driving. Since 3D reasoning is of crucial importance in this domain, we place our proposals in 3D
and represent them as cuboids. We assume a stereo image pair as input and compute depth via the
2
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0.4
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0.2
0.1
10 3
0.7
0.9
0.8
0.6
0.5
0.4
0.3
0.2
0.9
0.8
recall at IoU threshold 0.5
Cyclist
recall at IoU threshold 0.5
BING
SS
EB
MCG
MCG-D
Ours
0.5
0.4
0.3
0.2
0.1
# candidates
10 2
0.7
10
3
10
4
10 3
10 4
0.7
10 3
10 4
10 3
10 4
BING
SS
EB
MCG
MCG-D
Ours
0.6
0.5
0.4
0.3
0.2
0
10 1
10 4
10 2
# candidates
1
BING
SS
EB
MCG
MCG-D
Ours
0.9
0.8
0.6
0.5
0.4
0.3
0.2
0
10 1
10 3
0.1
10 2
0.1
10
0.2
0
10 1
1
2
0.3
# candidates
0.6
0
10 1
0.4
# candidates
BING
SS
EB
MCG
MCG-D
Ours
0
10 1
10 4
1
0.7
10
4
0.1
10 2
0.5
1
# candidates
0.8
10
3
recall at IoU threshold 0.5
BING
SS
EB
MCG
MCG-D
Ours
0
10 1
0.6
# candidates
0.6
0.9
2
1
recall at IoU threshold 0.5
Pedestrian
recall at IoU threshold 0.5
0.7
0.7
BING
SS
EB
MCG
MCG-D
Ours
0.1
0
10 1
4
1
0.8
0.8
0.1
0
10 1
0.9
0.9
recall at IoU threshold 0.5
0.7
0.9
recall at IoU threshold 0.7
Car
recall at IoU threshold 0.7
0.8
1
BING
SS
EB
MCG
MCG-D
Ours
recall at IoU threshold 0.7
1
0.9
0.7
BING
SS
EB
MCG
MCG-D
Ours
0.6
0.5
0.4
0.3
0.2
0.1
10
2
10
3
# candidates
10
4
0
10 1
10 2
# candidates
(a) Easy
(b) Moderate
(c) Hard
Figure 2: Proposal recall: We use 0.7 overlap threshold for Car , and 0.5 for Pedestrian and Cyclist.
state-of-the-art approach by Yamaguchi et al. [31]. We use depth to compute a point-cloud x and
conduct all our reasoning in this domain. We next describe our notation and present our framework.
3.1
Proposal Generation as Energy Minimization
We represent each object proposal with a 3D bounding box, denoted by y, which is parametrized
by a tuple, (x, y, z, ?, c, t), where (x, y, z) denotes the center of the 3D box and ?, represents its
azimuth angle. Note that each box y in principle lives in a continuous space, however, for efficiency
we reason in a discretized space (details in Sec. 3.2). Here, c denotes the object class of the box and
t ? {1, . . . , Tc } indexes the set of 3D box ?templates? which represent the physical size variations
of each object class c. The templates are learned from the training data.
We formulate the proposal generation problem as inference in a Markov Random Field (MRF)
which encodes the fact that the proposal y should enclose a high density region in the point cloud.
Furthermore, since the point cloud represents only the visible portion of the 3D space, y should not
overlap with the free space that lies within the rays between the points in the point cloud and the
camera. If that was the case, the box would in fact occlude the point cloud, which is not possible.
We also encode the fact that the point cloud should not extend vertically beyond our placed 3D box,
and that the height of the point cloud in the immediate vicinity of the box should be lower than the
box. Our MRF energy thus takes the following form:
>
>
>
>
E(x, y) = wc,pcd
?pcd (x, y) + wc,f
s ?f s (x, y) + wc,ht ?ht (x, y) + wc,ht?contr ?ht?contr (x, y)
Note that our energy depends on the object class via class-specific weights wc> , which are trained
using structured SVM [32] (details in Sec. 3.4). We now explain each potential in more detail.
Point Cloud Density: This potential encodes the density of the point cloud within the box
P
p??(y) S(p)
?pcd (x, y) =
|?(y)|
3
(1)
where S(p) indicates whether the voxel p is occupied or not (contains point cloud points), and ?(y)
denotes the set of voxels inside the box defined by y. Fig. 1 visualizes the potential. This potential
simply counts the fraction of occupied voxels inside the box. It can be efficiently computed in
constant time via integral accumulators, which is a generalization of integral images to 3D.
Free Space: This potential encodes the constraint that the free space between the point cloud and
the camera cannot be occupied by the box. Let F represent a free space grid, where F (p) = 1 means
that the ray from the camera to the voxel p does not hit an occupied voxel, i.e., voxel p lies in the
free space. We define the potential as follows:
P
p??(y) (1 ? F (p))
?f s (x, y) =
(2)
|?(y)|
This potential thus tries to minimize the free space inside the box, and can also be computed efficiently using integral accumulators.
Height Prior: This potential encodes the fact that the height of the point cloud inside the box
should be close to the mean height of the object class c. This is encoded in the following way:
X
1
?ht (x, y) =
Hc (p)
(3)
|?(y)|
p??(y)
with
?
?
?
"
1
exp ?
Hc (p) =
2
?
?
0,
dp ? ?c,ht
?c,ht
2 #
, if S(p) = 1
(4)
o.w.
where, dp indicates the height of the road plane lying below the voxel p. Here, ?c,ht , ?c,ht are the
MLE estimates of the mean height and standard deviation by assuming a Gaussian distribution of
the data. Integral accumulators can be used to efficiently compute these features.
Height Contrast: This potential encodes the fact that the point cloud that surrounds the bounding
box should have a lower height than the height of the point cloud inside the box. This is encoded as:
?ht?contr (x, y) =
?ht (x, y)
?ht (x, y+ ) ? ?ht (x, y)
(5)
where y+ represents the cuboid obtained by extending y by 0.6m in the direction of each face.
3.2
Discretization and Accumulators
Our point cloud is defined with respect to a left-handed coordinate system, where the positive Z-axis
is along the viewing direction of the camera and the Y-axis is along the direction of gravity. We
discretize the continuous space such that the width of each voxel is 0.2m in each dimension. We
compute the occupancy, free space and height prior grids in this discretized space. Following the
idea of integral images, we compute our accumulators in 3D.
3.3
Inference
Inference in our model is performed by minimizing the energy defined in Eq. (??):
y? = argminy E(x, y)
Due to the efficient computation of the features using integral accumulators evaluating each configuration y takes constant time. Still, evaluating exhaustively in the entire grid would be slow. In
order to reduce the search space, we carve certain regions of the grid by skipping configurations
which do not overlap with the point cloud. We further reduce the search space along the vertical
dimension by placing all our bounding boxes on the road plane, y = yroad . We estimate the road
by partitioning the image into super pixels, and train a road classifier using a neural net with several
2D and 3D features. We then use RANSAC on the predicted road pixels to fit the ground plane. Using the ground-plane considerably reduces the search space along the vertical dimension. However
since the points are noisy at large distances from the camera, we sample additional proposal boxes
at locations farther than 20m from the camera. We sample these boxes at heights y = yroad ? ?road ,
where ?road is the MLE estimate of the standard deviation by assuming a Gaussian distribution of
4
1
1
BING 12.3
SS 26.7
EB 37.5
MCG 45.1
MCG-D 49.6
Ours 65.6
0.9
0.8
0.7
0.8
0.7
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.6
0.7
0.8
0.9
0
0.5
1
0.1
0.6
IoU overlap threshold
1
0.8
0.7
0.7
0.8
0.9
0
0.5
1
1
0.8
0.7
0.7
recall
recall
0.5
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.8
0.9
0
0.5
1
0.1
0.6
IoU overlap threshold
1
0.8
0.7
0.7
0.8
0.9
0
0.5
1
1
0.8
0.7
0.7
recall
recall
0.5
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.8
0.9
IoU overlap threshold
(a) Easy
1
0
0.5
1
0.6
0.5
0.4
0.7
0.9
BING 4.4
SS 6.1
EB 4.4
MCG 8.1
MCG-D 10.7
Ours 40.8
0.8
0.4
0.6
0.8
0.9
0.6
0.5
0.7
IoU overlap threshold
1
BING 4.1
SS 6
EB 4.3
MCG 8
MCG-D 10.2
Ours 40.8
0.9
0.6
0
0.5
0.6
IoU overlap threshold
BING 6.2
SS 7.5
EB 4.9
MCG 10.8
MCG-D 10.8
Ours 55.2
0.9
1
0.6
0.5
0.4
0.7
0.9
BING 6.1
SS 5
EB 6.9
MCG 12.2
MCG-D 14
Ours 39.7
0.8
0.4
0.6
0.8
0.9
0.6
0.5
0.7
IoU overlap threshold
1
BING 6.6
SS 5.1
EB 7.7
MCG 13.3
MCG-D 16.1
Ours 44.8
0.9
0.6
0
0.5
0.6
IoU overlap threshold
BING 7.6
SS 5.4
EB 9.2
MCG 15
MCG-D 19.6
Ours 49.9
0.9
Pedestrian
recall
0.7
0.4
0
0.5
Cyclist
recall
0.8
recall
0.5
BING 7.1
SS 16.5
EB 23
MCG 31.3
MCG-D 32.8
Ours 57.8
0.9
0.6
recall
Car
recall
0.6
1
BING 7.7
SS 18
EB 26.5
MCG 36.1
MCG-D 38.8
Ours 58.3
0.9
0.1
0.6
0.7
0.8
0.9
1
0
0.5
0.6
0.7
0.8
IoU overlap threshold
IoU overlap threshold
(b) Moderate
(c) Hard
0.9
1
Figure 3: Recall vs IoU for 500 proposals. The number next to the labels indicates the average recall (AR).
the distance between objects and the estimated ground plane. Using our sampling strategy, scoring
all possible configurations takes only a fraction of a second.
Note that by minimizing our energy we only get one, best object candidate. In order to generate
N diverse proposals, we sort the values of E(x, y) for all y, and perform greedy inference: we
pick the top scoring proposal, perform NMS, and iterate. The entire inference process and feature
computation takes on average 1.2s per image for N = 2000 proposals.
3.4
Learning
We learn the weights {wc,pcd , wc,f s , wc,ht , wc,ht?contr } of the model using structured SVM [32].
Given N ground truth input-output pairs, {x(i) , y(i) }i=1,??? ,N , the parameters are learnt by solving
the following optimization problem:
N
1
C X
?i
min ||w||2 +
N i=1
w?RD 2
s.t.:
wT (?(x(i) , y) ? ?(x(i) , y(i) )) ? ?(y(i) , y) ? ?i , ?y \ y(i)
We use the parallel cutting plane of [33] to solve this minimization problem. We use Intersectionover-Union (IoU) between the set of GT boxes, y(i) , and candidates y as the task loss ?(y(i) , y).
We compute IoU in 3D as the volume of intersection of two 3D boxes divided by the volume of their
union. This is a very strict measure that encourages accurate 3D placement of the proposals.
3.5
Object Detection and Orientation Estimation Network
We use our object proposal method for the task of object detection and orientation estimation. We
score bounding box proposals using CNN. Our network is built on Fast R-CNN [34], which share
convolutional features across all proposals and use ROI pooling layer to compute proposal-specific
5
LSVM-MDPM-sv [35, 1]
SquaresICF [36]
DPM-C8B1 [37]
MDPM-un-BB [1]
DPM-VOC+VP [27]
OC-DPM [38]
AOG [39]
SubCat [28]
DA-DPM [40]
Fusion-DPM [41]
R-CNN [42]
FilteredICF [43]
pAUCEnsT [44]
MV-RGBD-RF [45]
3DVP [12]
Regionlets [13]
Ours
Cars
Easy Moderate Hard
68.02
56.48
44.18
74.33
60.99
47.16
71.19
62.16
48.43
74.95
64.71
48.76
74.94
65.95
53.86
84.36
71.88
59.27
84.14
75.46
59.71
87.46
75.77
65.38
84.75
76.45
59.70
93.04
88.64
79.10
Pedestrians
Easy Moderate Hard
47.74
39.36
35.95
57.33
44.42
40.08
38.96
29.03
25.61
59.48
44.86
40.37
54.67
42.34
37.95
56.36
45.51
41.08
59.51
46.67
42.05
61.61
50.13
44.79
61.14
53.98
49.29
65.26
54.49
48.60
70.21
54.56
51.25
73.14
61.15
55.21
81.78
67.47
64.70
Cyclists
Easy Moderate Hard
35.04
27.50
26.21
43.49
29.04
26.20
42.43
31.08
28.23
51.62
38.03
33.38
54.02
39.72
34.82
70.41
58.72
51.83
78.39
68.94
61.37
Table 1: Average Precision (AP) (in %) on the test set of the KITTI Object Detection Benchmark.
AOG [39]
DPM-C8B1 [37]
LSVM-MDPM-sv [35, 1]
DPM-VOC+VP [27]
OC-DPM [38]
SubCat [28]
3DVP [12]
Ours
Cars
Easy Moderate Hard
43.81
38.21
31.53
59.51
50.32
39.22
67.27
55.77
43.59
72.28
61.84
46.54
73.50
64.42
52.40
83.41
74.42
58.83
86.92
74.59
64.11
91.44 86.10
76.52
Pedestrians
Easy Moderate Hard
31.08
23.37
20.72
43.58
35.49
32.42
53.55
39.83
35.73
44.32
34.18
30.76
72.94
59.80
57.03
Cyclists
Easy Moderate Hard
27.25
19.25
17.95
27.54
22.07
21.45
30.52
23.17
21.58
70.13
58.68
52.35
Table 2: AOS scores (in %) on the test set of KITTI?s Object Detection and Orientation Estimation Benchmark.
features. We extend this basic network by adding a context branch after the last convolutional layer,
and an orientation regression loss to jointly learn object location and orientation. Features output
from the original and the context branches are concatenated and fed to the prediction layers. The
context regions are obtained by enlarging the candidate boxes by a factor of 1.5. We used smooth
L1 loss [34] for orientation regression. We use OxfordNet [3] trained on ImageNet to initialize the
weights of convolutional layers and the branch for candidate boxes. The parameters of the context
branch are initialized by copying the weights from the original branch. We then fine-tune it end to
end on the KITTI training set.
4
Experimental Evaluation
We evaluate our approach on the challenging KITTI autonomous driving dataset [11], which contains three object classes: Car, Pedestrian, and Cyclist. KITTI?s object detection benchmark has
7,481 training and 7,518 test images. Evaluation is done in three regimes: easy, moderate and hard,
containing objects at different occlusion and truncation levels. The moderate regime is used to rank
the competing methods in the benchmark. Since the test ground-truth labels are not available, we
split the KITTI training set into train and validation sets (each containing half of the images). We
ensure that our training and validation set do not come from the same video sequences, and evaluate
the performance of our bounding box proposals on the validation set.
Following [4, 24], we use the oracle recall as metric. For each ground-truth (GT) object we find
the proposal that overlaps the most in IoU (i.e., ?best proposal?). We say that a GT instance has
been recalled if IoU exceeds 70% for cars, and 50% for pedestrians and cyclists. This follows the
standard KITTI?s setup. Oracle recall thus computes the percentage of recalled GT objects, and thus
the best achievable recall. We also show how different number of generated proposals affect recall.
Comparison to the State-of-the-art: We compare our approach to several baselines: MCGD [14], MCG [5], Selective Search (SS) [4], BING [15], and Edge Boxes (EB) [9]. Fig. 2 shows
recall as a function of the number of candidates. We can see that by using 1000 proposals, we
achieve around 90% recall for Cars in the moderate and hard regimes, while for easy we need
6
Best prop. Ground truth Top 100 prop. Images
Figure 4: Qualitative results for the Car class. We show the original image, 100 top scoring proposals, groundBest prop. Ground truth Top 100 prop. Images
truth 3D boxes, and our best set of proposals that cover the ground-truth.
Figure 5: Qualitative examples for the Pedestrian class.
Method
Time (seconds)
BING
0.01
Selective Search
15
Edge Boxes (EB)
1.5
MCG
100
MCG-D
160
Ours
1.2
Table 3: Running time of different proposal methods
only 200 candidates to get the same recall. Notice that other methods saturate or require orders
of magnitude more candidates to reach 90% recall. For Pedestrians and Cyclists our results show
similar improvements over the baselines. Note that while we use depth-based features, MCG-D uses
both depth and appearance based features, and all other methods use only appearance features. This
shows the importance of 3D information in the autonomous driving scenario. Furthermore, the other
methods use class agnostic proposals to generate the candidates, whereas we generate them based
on the object class. This allows us to achieve higher recall values by exploiting size priors tailored
to each class. Fig. 3 shows recall for 500 proposals as a function of the IoU overlap. Our approach
significantly outperforms the baselines, particularly for Cyclists.
Running Time: Table 3 shows running time of different proposal methods. Our approach is fairly
efficient and can compute all features and proposals in 1.2s on a single core.
Qualitative Results: Figs. 4 and 5 show qualitative results for cars and pedestrians. We show the
input RGB image, top 100 proposals, the GT boxes in 3D, as well as proposals from our method
with the best 3D IoU (chosen among 2000 proposals). Our method produces very precise proposals
even for the more difficult (far away or occluded) objects.
7
Object Detection: To evaluate our full object detection pipeline, we report results on the test set
of the KITTI benchmark. The results are presented in Table 1. Our approach outperforms all the
competitors significantly across all categories. In particular, we achieve 12.19%, 6.32% and 10.22%
improvement in AP for Cars, Pedestrians, and Cyclists, in the moderate setting.
Object Orientation Estimation: Average Orientation Similarity (AOS) [11] is used as the evaluation metric in object detection and orientation estimation task. Results on KITTI test set are shown
in Table 2. Our approach again outperforms all approaches by a large margin. Particularly, our
approach achieves ?12% higher scores than 3DVP [12] on Cars in moderate and hard data. The improvement on Pedestrians and Cyclists are even more significant as they are more than 20% higher
than the second best method.
Suppl. material: We refer the reader to supplementary material for many additional results.
5
Conclusion
We have presented a novel approach to object proposal generation in the context of autonomous
driving. In contrast to most existing work, we take advantage of stereo imagery and reason directly
in 3D. We formulate the problem as inference in a Markov random field encoding object size priors,
ground plane and a variety of depth informed features. Our approach significantly outperforms existing state-of-the-art object proposal methods on the challenging KITTI benchmark. In particular,
for 2K proposals our approach achieves a 25% higher recall than the state-of-the-art RGB-D method
MCG-D [14]. Combined with CNN scoring our method significantly outperforms all previous published object detection results for all three object classes on the KITTI [11] benchmark.
Acknowledgements: The work was partially supported by NSFC 61171113, NSERC and Toyota
Motor Corporation.
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9
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5,131 | 5,645 | The Poisson Gamma Belief Network
Mingyuan Zhou
McCombs School of Business
The University of Texas at Austin
Austin, TX 78712, USA
Yulai Cong
National Laboratory of RSP
Xidian University
Xi?an, Shaanxi, China
Bo Chen
National Laboratory of RSP
Xidian University
Xi?an, Shaanxi, China
Abstract
To infer a multilayer representation of high-dimensional count vectors, we propose the Poisson gamma belief network (PGBN) that factorizes each of its layers
into the product of a connection weight matrix and the nonnegative real hidden
units of the next layer. The PGBN?s hidden layers are jointly trained with an
upward-downward Gibbs sampler, each iteration of which upward samples Dirichlet distributed connection weight vectors starting from the first layer (bottom data
layer), and then downward samples gamma distributed hidden units starting from
the top hidden layer. The gamma-negative binomial process combined with a
layer-wise training strategy allows the PGBN to infer the width of each layer given
a fixed budget on the width of the first layer. The PGBN with a single hidden layer
reduces to Poisson factor analysis. Example results on text analysis illustrate interesting relationships between the width of the first layer and the inferred network
structure, and demonstrate that the PGBN, whose hidden units are imposed with
correlated gamma priors, can add more layers to increase its performance gains
over Poisson factor analysis, given the same limit on the width of the first layer.
1
Introduction
There has been significant recent interest in deep learning. Despite its tremendous success in supervised learning, inferring a multilayer data representation in an unsupervised manner remains a
challenging problem [1, 2, 3]. The sigmoid belief network (SBN), which connects the binary units
of adjacent layers via the sigmoid functions, infers a deep representation of multivariate binary vectors [4, 5]. The deep belief network (DBN) [6] is a SBN whose top hidden layer is replaced by the
restricted Boltzmann machine (RBM) [7] that is undirected. The deep Boltzmann machine (DBM)
is an undirected deep network that connects the binary units of adjacent layers using the RBMs [8].
All these deep networks are designed to model binary observations. Although one may modify the
bottom layer to model Gaussian and multinomial observations, the hidden units of these networks
are still typically restricted to be binary [8, 9, 10]. One may further consider the exponential family
harmoniums [11, 12] to construct more general networks with non-binary hidden units, but often at
the expense of noticeably increased complexity in training and data fitting.
Moving beyond conventional deep networks using binary hidden units, we construct a deep directed
network with gamma distributed nonnegative real hidden units to unsupervisedly infer a multilayer
representation of multivariate count vectors, with a simple but powerful mechanism to capture the
correlations among the visible/hidden features across all layers and handle highly overdispersed
counts. The proposed model is called the Poisson gamma belief network (PGBN), which factorizes
the observed count vectors under the Poisson likelihood into the product of a factor loading matrix
and the gamma distributed hidden units (factor scores) of layer one; and further factorizes the shape
parameters of the gamma hidden units of each layer into the product of a connection weight matrix
and the gamma hidden units of the next layer. Distinct from previous deep networks that often utilize
binary units for tractable inference and require tuning both the width (number of hidden units) of
each layer and the network depth (number of layers), the PGBN employs nonnegative real hidden
1
units and automatically infers the widths of subsequent layers given a fixed budget on the width of
its first layer. Note that the budget could be infinite and hence the whole network can grow without
bound as more data are being observed. When the budget is finite and hence the ultimate capacity
of the network is limited, we find that the PGBN equipped with a narrower first layer could increase
its depth to match or even outperform a shallower network with a substantially wider first layer.
The gamma distribution density function has the highly desired strong non-linearity for deep learning, but the existence of neither a conjugate prior nor a closed-form maximum likelihood estimate
for its shape parameter makes a deep network with gamma hidden units appear unattractive. Despite
seemingly difficult, we discover that, by generalizing the data augmentation and marginalization
techniques for discrete data [13], one may propagate latent counts one layer at a time from the bottom data layer to the top hidden layer, with which one may derive an efficient upward-downward
Gibbs sampler that, one layer at a time in each iteration, upward samples Dirichlet distributed connection weight vectors and then downward samples gamma distributed hidden units.
In addition to constructing a new deep network that well fits multivariate count data and developing
an efficient upward-downward Gibbs sampler, other contributions of the paper include: 1) combining the gamma-negative binomial process [13, 14] with a layer-wise training strategy to automatically infer the network structure; 2) revealing the relationship between the upper bound imposed on
the width of the first layer and the inferred widths of subsequent layers; 3) revealing the relationship
between the network depth and the model?s ability to model overdispersed counts; 4) and generating
a multivariate high-dimensional random count vector, whose distribution is governed by the PGBN,
by propagating the gamma hidden units of the top hidden layer back to the bottom data layer.
1.1
Useful count distributions and their relationships
Let the Chinese restaurant table (CRT)
Pn distribution l ? CRT(n, r) represent the distribution of
a random count generated as l =
i=1 bi , bi ? Bernoulli [r/(r + i ? 1)] . Its probability mass
l
?(r)r
function (PMF) can be expressed as P (l | n, r) = ?(n+r)
|s(n, l)|, where l ? Z, Z := {0, 1, . . . , n},
and |s(n, l)| are unsigned Stirling numbers of the first kind. Let u ? Log(p) denote the logarithmic
pu
1
distribution with PMF P (u | p) = ? ln(1?p)
u , where u ? {1, 2, . . .}. Let n ? NB(r, p) denote
n
r
the negative binomial (NB) distribution with PMF P (n | r, p) = ?(n+r)
n!?(r) p (1 ? p) , where n ? Z.
The NB distribution n ? NB(r, p) can be generated as a gamma mixed Poisson distribution as n ?
Pois(?), ? ? Gam [r, p/(1 ? p)] , where p/(1 ? p) is the gamma scale parameter. As shown in [13],
the joint distribution of n and l given r and p in l ? CRT(n, r), n ? NB(r, p), where l ? {0, . . . , n}
Pl
and n ? Z, is the same as that in n = t=1 ut , ut ? Log(p), l ? Pois[?r ln(1 ? p)], which is
called the Poisson-logarithmic bivariate distribution, with PMF P (n, l | r, p) =
2
|s(n,l)|r l n
p (1 ? p)r .
n!
The Poisson Gamma Belief Network
(1)
Assuming the observations are multivariate count vectors xj ? ZK0 , the generative model of the
Poisson gamma belief network (PGBN) with T hidden layers, from top to bottom, is expressed as
(T +1)
(T )
? j ? Gam r, 1 cj
,
?
?
?
(t)
(t+1) (t+1)
? j ? Gam ?(t+1) ? j
, 1 cj
,
?
?
?
(2)
(1)
(1)
(1)
(2) (2)
, ? j ? Gam ?(2) ? j , pj
1 ? pj
.
(1)
xj ? Pois ?(1) ? j
(1)
The PGBN factorizes the count observation xj
into the product of the factor loading ?(1) ?
(1)
K0 ?K1
1
R+
and hidden units ? j ? RK
+ of layer one under the Poisson likelihood, where R+ = {x :
x ? 0}, and for t = 1, 2, . . . , T ?1, factorizes the shape parameters of the gamma distributed hidden
K ?K
(t)
(t+1)
t
units ? j ? RK
? R+ t t+1
+ of layer t into the product of the connection weight matrix ?
(t+1)
and the hidden units ? j
K
(T )
? R+ t+1 of layer t + 1; the top layer?s hidden units ? j
2
share the same
(2)
vector r = (r1 , . . . , rKT )0 as their gamma shape parameters; and the pj are probability parameters
(2)
(2) (2)
and {1/c(t) }3,T +1 are gamma scale parameters, with cj := 1 ? pj
pj .
K
?K
For scale identifiabilty and ease of inference, each column of ?(t) ? R+ t?1 t is restricted to have
a unit L1 norm. To complete the hierarchical model, for t ? {1, . . . , T ? 1}, we let
(t)
?k ? Dir ? (t) , . . . , ? (t) , rk ? Gam ?0 /KT , 1/c0
(2)
and impose c0 ? Gam(e0 , 1/f0 ) and ?0 ? Gam(a0 , 1/b0 ); and for t ? {3, . . . , T + 1}, we let
(2)
pj
(t)
? Beta(a0 , b0 ), cj ? Gam(e0 , 1/f0 ).
(3)
(1)
(1)
(x1 , . . . , xJ )
We expect the correlations between the rows (features) of
to be captured by the
(t)
(t)
(1)
columns of ? , and the correlations between the rows (latent features) of (? 1 , . . . , ? J ) to be
(t+1)
(t)
captured by the columns of ?
. Even if all ? for t ? 2 are identity matrices, indicating no
correlations between latent features, our analysis will show that a deep structure with T ? 2 could
(1)
still benefit data fitting by better modeling the variability of the latent features ? j .
Sigmoid and deep belief networks. Under the hierarchical model in (1), given the connection
weight matrices, the joint distribution of the count observations and gamma hidden units of the
PGBN can be expressed, similar to those of the sigmoid and deep belief networks [3], as
i
hQ
(1)
(t)
(1)
(1)
(t) (t+1)
(t+1)
(T )
T ?1
P xj , {? j }t {?(t) }t = P xj ?(1) , ? j
, ?j
P ?j
.
t=1 P ? j ?
(t)
With ?v: representing the vth row ?, for the gamma hidden units ?vj we have
P
(t)
?vj
(t+1) (t+1) (t+1)
, cj+1 =
?v: , ? j
(t+1) ?(t+1)
(t+1) ?v:
j
cj+1
(t+1) (t+1)
? ?v:
?j
(t)
?vj
(t+1)
?(t+1)
?j
?1
v:
(t+1) (t)
?vj
e?cj+1
,
(4)
which are highly nonlinear functions that are strongly desired in deep learning. By contrast, with the
(t+1)
, a sigmoid/deep belief network would
sigmoid function ?(x) = 1/(1 + e?x ) and bias terms bv
(t)
connect the binary hidden units ?vj ? {0, 1} of layer t (for deep belief networks, t < T ? 1 ) to the
product of the connection weights and binary hidden units of the next layer with
(t+1) (t+1)
(t+1)
(t)
, ?j
, bv
?j
.
(5)
= ? b(t+1)
+ ?(t+1)
P ?vj = 1 ?(t+1)
v:
v:
v
Comparing (4) with (5) clearly shows the differences between the gamma nonnegative hidden units
and the sigmoid link based binary hidden units. Note that the rectified linear units have emerged as
powerful alternatives of sigmoid units to introduce nonlinearity [15]. It would be interesting to use
the gamma units to introduce nonlinearity in the positive region of the rectified linear units.
Deep Poisson factor analysis. With T = 1, the PGBN specified by (1)-(3) reduces to Poisson factor
analysis (PFA) using the (truncated) gamma-negative binomial process [13], which is also related
(1)
(1)
to latent Dirichlet allocation [16] if the Dirichlet priors are imposed on both ?k and ? j . With
T ? 2, the PGBN is related to the gamma Markov chain hinted by Corollary 2 of [13] and realized
in [17], the deep exponential family of [18], and the deep PFA of [19]. Different from the PGBN,
in [18], it is the gamma scale but not shape parameters that are chained and factorized; in [19], it is
the correlations between binary topic usage indicators but not the full connection weights that are
captured; and neither [18] nor [19] provide a principled way to learn the network structure. Below
we break the PGBN of T layers into T related submodels that are solved with the same subroutine.
2.1
The propagation of latent counts and model properties
(1)
Lemma 1 (Augment-and-conquer the PGBN). With pj := 1 ? e?1 and
.h
i
(t+1)
(t)
(t+1)
(t)
pj
? ln(1 ? pj )
:= ? ln(1 ? pj ) cj
(6)
(t)
for t = 1, . . . , T , one may connect the observed (if t = 1) or some latent (if t ? 2) counts xj ?
(t)
ZKt?1 to the product ?(t) ? j at layer t under the Poisson likelihood as
h
i
(t)
(t)
(t)
xj ? Pois ??(t) ? j ln 1 ? pj
.
3
(7)
Proof. By definition (7) is true for layer t = 1. Suppose that (7) is true for layer t ? 2, then we can
(t)
augment each count xvj into the summation of Kt latent counts that are smaller or equal as
h
i
PKt (t)
(t)
(t) (t)
(t)
(t)
xvjk , xvjk ? Pois ??vk ?kj ln 1 ? pj
,
(8)
xvj = k=1
(t)(t+1)
where v ? {1, . . . , Kt?1 }. With mkj
(t)
:= x?jk :=
PKt?1
v=1
(t)
xvjk representing the num(t)(t+1)
ber of times that factor k ? {1, . . . , Kt } of layer t appears in observation j and mj
PKt?1 (t)
(t)
(t) 0
?vk = 1, we can marginalize out ?(t) as in [20], leading to
x?j1 , . . . , x?jKt , since v=1
h
i
(t)(t+1)
(t)
(t)
? Pois ?? j ln 1 ? pj
.
mj
:=
(t)
Further marginalizing out the gamma distributed ? j from the above Poisson likelihood leads to
(t)(t+1)
(t+1) (t+1)
mj
? NB ?(t+1) ? j
, pj
.
(9)
(t)(t+1)
The kth element of mj
(t)(t+1)
mkj
can be augmented under its compound Poisson representation as
h
i
Px(t+1)
(t+1)
(t+1)
(t+1) (t+1)
(t+1)
kj
= `=1
u` , u` ? Log(pj
), xkj ? Pois ??k: ? j
ln 1 ? pj
.
Thus if (7) is true for layer t, then it is also true for layer t + 1.
Corollary 2 (Propagate the latent counts upward). Using Lemma 4.1 of [20] on (8) and Theorem 1
(t)
of [13] on (9), we can propagate the latent counts xvj of layer t upward to layer t + 1 as
o
n
(t)
(t)
(t) (t)
?vK ?K j
?v1 ?1j
(t)
(t)
(t)
(t)
(t)
t
t
,
?
?
Mult
x
,
xvj1 , . . . , xvjKt xvj , ?(t)
,
.
.
.
,
, (10)
P
P
K
K
(t) (t)
(t) (t)
v:
j
vj
t
t
k=1 ?vk ?kj
k=1 ?vk ?kj
(t+1)
(t)(t+1)
(t+1)
(t+1)
(t)(t+1)
(t+1) (t+1)
xkj mkj
, ?k: , ? j
? CRT mkj
, ?k: ? j
.
(11)
(t)
(t)(t+1)
(t+1)
(t)(t+1)
As x?j = m?j
and xkj is in the same order as ln mkj
, the total count of layer t + 1,
P (t+1)
P (t)
expressed as j x?j , would often be much smaller than that of layer t, expressed as j x?j .
P (T )
Thus the PGBN may use j x?j as a simple criterion to decide whether to add more layers.
2.2
Modeling overdispersed counts
In comparison to a single-layer shallow model with T = 1 that assumes the hidden units of layer
one to be independent in the prior, the multilayer deep model with T ? 2 captures the correlations
between them. Note that for the extreme case that ?(t) = IKt for t ? 2 are all identity matrices,
(t?1)
which indicates that there are no correlations between the features of ? j
left to be captured, the
(1)(2)
deep structure could still provide benefits as it helps model latent counts mj
that may be highly
overdispersed. For example, supposing ?(t) = IK2 for all t ? 2, then from (1) and (9) we have
(1)(2)
mkj
(2)
(2)
(t)
(t+1)
? NB(?kj , pj ), . . . , ?kj ? Gam(?kj
(t+1)
, 1/cj
(T )
(T +1)
), . . . , ?kj ? Gam(rk , 1/cj
).
(t)
For simplicity, let us further assume cj = 1 for all t ? 3. Using the laws of total expectation and
(2)
(2)
total variance, we have E ?kj | rk = rk and Var ?kj | rk = (T ? 1)rk , and hence
h
i
(1)(2)
(1)(2)
(2)
(2)
(2)
(2) ?2
(2)
E mkj | rk = rk pj /(1 ? pj ), Var mkj | rk = rk pj 1 ? pj
1 + (T ? 1)pj .
(1)(2)
In comparison to PFA with mkj
(2)
| rk ? NB(rk , pj ), with a variance-to-mean ratio of 1/(1 ?
(2)
(1)(2)
pj ), the PGBN with T hidden layers, which mixes the shape of mkj
(2)
(2)
? NB(?kj , pj ) with a
(1)(2)
chain of gamma random variables, increases the variance-to-mean ratio of the latent count mkj
(2)
given rk by a factor of 1 + (T ? 1)pj , and hence could better model highly overdispersed counts.
4
2.3
Upward-downward Gibbs sampling
With Lemma 1 and Corollary 2 and the width of the first layer being bounded by K1 max , we develop
an upward-downward Gibbs sampler for the PGBN, each iteration of which proceeds as follows:
(t)
(t)
Sample xvjk . We can sample xvjk for all layers using (10). But for the first hidden layer, we may
(1)
treat each observed count xvj as a sequence of word tokens at the vth term (in a vocabulary of size
(1)
V := K0 ) in the jth document, and assign the x?j words {vji }i=1,x(1) one after another to the
?j
(1)
latent factors (topics), with both the topics ?(1) and topic weights ? j marginalized out, as
P (zji = k | ?) ?
(1)?ji
ji ?k
(1)?ji
? (1) +xv
V ? (1) +x??k
(1)?ji
x?jk
(2) (2)
+ ?k: ? j
, k ? {1, . . . , K1 max },
(12)
P
(1)
where zji is the topic index for vji and xvjk := i ?(vji = v, zji = k) counts the number of times
that term v appears in document j; we use the ? symbol to represent summing over the correspondP (t)
(t)
ing index, e.g., x?jk := v xvjk , and use x?ji to denote the count x calculated without considering
word i in document j. The collapsed Gibbs sampling update equation shown above is related to the
one developed in [21] for latent Dirichlet allocation, and the one developed in [22] for PFA using the
(2) (2)
beta-negative binomial process. When T = 1, we would replace the terms ?k: ? j with rk for PFA
built on the gamma-negative binomial process [13] (or with ??k for the hierarchical Dirichlet process latent Dirichlet allocation, see [23] and [22] for details), and add an additional term to account
for the possibility of creating an additional topic [22]. For simplicity, in this paper, we truncate the
nonparametric Bayesian model with K1 max factors and let rk ? Gam(?0 /K1 max , 1/c0 ) if T = 1.
(t)
(t)
Sample ?k . Given these latent counts, we sample the factors/topics ?k as
(t)
(t)
(t)
(?k | ?) ? Dir ? (t) + x1?k , . . . , ? (t) + xKt?1 ?k .
(13)
(t+1)
Sample xvj
(t+1)
. We sample xj
(T +1)
using (11), replacing ?(T +1) ? j
with r := (r1 , . . . , rKT )0 .
(t)
Sample ? j . Using (7) and the gamma-Poisson conjugacy, we sample ? j as
h
i?1
(t)
(t+1)
(t)(t+1)
(t+1)
(t)
(? j | ?) ? Gamma ?(t+1) ? j
+ mj
, cj
? ln 1 ? pj
.
(14)
Sample r. Both ?0 and c0 are sampled using related equations in [13]. We sample r as
h
i?1
P
(T +1)
(T +1)
(rv | ?) ? Gam ?0 /KT + xv?
, c0 ? j ln 1 ? pj
.
(15)
PKt (t)
(t)
(t)
(T +1)
(2)
(t)
Sample cj . With ??j := k=1
?kj for t ? T and ??j
:= r? , we sample pj and {cj }t?3 as
h
i?1
(2)
(1)(2)
(2)
(t)
(t)
(t?1)
(pj | ?) ? Beta a0 +m?j
, b0 +??j , (cj | ?) ? Gamma e0 +??j , f0 +??j
, (16)
(2)
(t)
and calculate cj and {pj }t?3 with (6).
2.4
Learning the network structure with layer-wise training
As jointly training all layers together is often difficult, existing deep networks are typically trained
using a greedy layer-wise unsupervised training algorithm, such as the one proposed in [6] to train
the deep belief networks. The effectiveness of this training strategy is further analyzed in [24]. By
contrast, the PGBN has a simple Gibbs sampler to jointly train all its hidden layers, as described in
Section 2.3, and hence does not require greedy layer-wise training. Yet the same as commonly used
deep learning algorithms, it still needs to specify the number of layers and the width of each layer.
In this paper, we adopt the idea of layer-wise training for the PGBN, not because of the lack of
an effective joint-training algorithm, but for the purpose of learning the width of each hidden layer
in a greedy layer-wise manner, given a fixed budget on the width of the first layer. The proposed
layer-wise training strategy is summarized in Algorithm 1. With a PGBN of T ? 1 layers that has
already been trained, the key idea is to use a truncated gamma-negative binomial process [13] to
(T )(T +1)
(T +1)
model the latent count matrix for the newly added top layer as mkj
? NB(rk , pj
), rk ?
5
Algorithm 1 The PGBN upward-downward Gibbs sampler that uses a layer-wise training strategy to train a set
of networks, each of which adds an additional hidden layer on top of the previously inferred network, retrains
all its layers jointly, and prunes inactive factors from the last layer. Inputs: observed counts {xvj }v,j , upper
bound of the width of the first layer K1 max , upper bound of the number of layers Tmax , and hyper-parameters.
Outputs: A total of Tmax jointly trained PGBNs with depths T = 1, T = 2, . . ., and T = Tmax .
1: for T = 1, 2, . . . , Tmax do Jointly train all the T layers of the network
2:
Set KT ?1 , the inferred width of layer T ? 1, as KT max , the upper bound of layer T ?s width.
3:
for iter = 1 : BT + CT do Upward-downward Gibbs sampling
(1)
(2)
4:
Sample {zji }j,i using collapsed inference; Calculate {xvjk }v,k,j ; Sample {xvj }v,j ;
5:
for t = 2, 3, . . . , T do
(t)
(t)
(t+1)
6:
Sample {xvjk }v,j,k ; Sample {?k }k ; Sample {xvj }v,j ;
7:
end for
(2)
(2)
(t)
(t)
8:
Sample pj and Calculate cj ; Sample {cj }j,t and Calculate {pj }j,t for t = 3, . . . , T + 1
9:
for t = T, T ? 1, . . . , 2 do
(t)
10:
Sample r if t = T ; Sample {? j }j ;
11:
end for
12:
if iter = BT then
P
(T )
(T )
13:
Prune layer T ?s inactive factors {?k }k:x(T ) =0 , let KT = k ?(x??k > 0), and update r;
??k
end if
end for
(t)
Output the posterior means (according to the last MCMC sample) of all remaining factors {?k }k,t as
KT
the inferred network of T layers, and {rk }k=1 as the gamma shape parameters of layer T ?s hidden units.
17: end for
14:
15:
16:
Gam(?0 /KT max , 1/c0 ), and rely on that stochastic process?s shrinkage mechanism to prune inactive
factors (connection weight vectors) of layer T , and hence the inferred KT would be smaller than
KT max if KT max is sufficiently large. The newly added layer and the layers below it would be
jointly trained, but with the structure below the newly added layer kept unchanged. Note that when
T = 1, the PGBN would infer the number of active factors if K1 max is set large enough, otherwise,
it would still assign the factors with different weights rk , but may not be able to prune any of them.
3
Experimental Results
We apply the PGBNs for topic modeling of text corpora, each document of which is represented
as a term-frequency count vector. Note that the PGBN with a single hidden layer is identical to
the (truncated) gamma-negative binomial process PFA of [13], which is a nonparametric Bayesian
algorithm that performs similarly to the hierarchical Dirichlet process latent Dirichlet allocation
[23] for text analysis, and is considered as a strong baseline that outperforms a large number of
topic modeling algorithms. Thus we will focus on making comparison to the PGBN with a single
layer, with its layer width set to be large to approximate the performance of the gamma-negative
binomial process PFA. We evaluate the PGBNs? performance by examining both how well they
unsupervisedly extract low-dimensional features for document classification, and how well they
predict heldout word tokens. Matlab code will be available in http://mingyuanzhou.github.io/.
We use Algorithm 1 to learn, in a layer-wise manner, from the training data the weight matrices
?(1) , . . . , ?(Tmax ) and the top-layer hidden units? gamma shape parameters r: to add layer T to
a previously trained network with T ? 1 layers, we use BT iterations to jointly train ?(T ) and r
together with {?(t) }1,T ?1 , prune the inactive factors of layer T , and continue the joint training with
another CT iterations. We set the hyper-parameters as a0 = b0 = 0.01 and e0 = f0 = 1. Given
the trained network, we apply the upward-downward Gibbs sampler to collect 500 MCMC samples
(1) (1)
after 500 burnins to estimate the posterior mean of the feature usage proportion vector ? j /??j at
the first hidden layer, for every document in both the training and testing sets.
Feature learning for binary classification.
We consider the 20 newsgroups dataset
(http://qwone.com/?jason/20Newsgroups/) that consists of 18,774 documents from 20 different
news groups, with a vocabulary of size K0 = 61,188. It is partitioned into a training set of 11,269
documents and a testing set of 7,505 ones. We first consider two binary classification tasks that distinguish between the comp.sys.ibm.pc.hardware and comp.sys.mac.hardware, and between the
sci.electronics and sci.med news groups. For each binary classification task, we remove a standard
list of stop words and only consider the terms that appear at least five times, and report the classification accuracies based on 12 independent random trials. With the upper bound of the first layer?s
6
(b) sci.electronics vs sci.med
85
84.5
84
83.5
83
82.5
94
93.5
93
92.5
92
91.5
1
2
3
4
5
6
Number of layers T
7
8
1
2
3
4
5
6
Number of layers T
7
84
83
82
81
80
79
77
8
(d) sci.electronics vs sci.med
94.5
94
93.5
K 1max
K 1max
K 1max
K 1max
K 1max
K 1max
K 1max
93
92.5
92
78
91
82
95
85
Classification accuracy
86
85.5
(c) ibm.pc.hardware vs mac.hardware
86
Classification accuracy
95
94.5
Classification accuracy
Classification accuracy
(a) ibm.pc.hardware vs mac.hardware
87
86.5
2
4
6
8
Number of layers T
91.5
2
4
6
Number of layers T
= 25
= 50
= 100
= 200
= 400
= 600
= 800
8
Figure 1: Classification accuracy (%) as a function of the network depth T for two 20newsgroups binary
classification tasks, with ? (t) = 0.01 for all layers. (a)-(b): the boxplots of the accuracies of 12 independent
runs with K1 max = 800. (c)-(d): the average accuracies of these 12 runs for various K1 max and T . Note that
K1 max = 800 is large enough to cover all active first-layer topics (inferred to be around 500 for both binary
classification tasks), whereas all the first-layer topics would be used if K1 max = 25, 50, 100, or 200.
(a)
79
78
K 1max
K 1max
K 1max
K 1max
K 1max
K 1max
76
75
74
Classification accuracy
Classification accuracy
77
= 50
= 100
= 200
= 400
= 600
= 800
73
77
76
75
T=1
T=2
T=3
T=4
T=5
74
73
72
72
71
(b)
79
78
71
1
2
3
4
Number of layers T
5
6
100
7
200
300
400
500
600
700
800
K 1max
Figure 2:
Classification accuracy (%) of the PGBNs for 20newsgroups multi-class classification (a) as a
function of the depth T with various K1 max and (b) as a function of K1 max with various depths, with ? (t) =
0.05 for all layers. The widths of hidden layers are automatically inferred, with K1 max = 50, 100, 200, 400,
600, or 800. Note that K1 max = 800 is large enough to cover all active first-layer topics, whereas all the
first-layer topics would be used if K1 max = 50, 100, or 200.
width set as K1 max ? {25, 50, 100, 200, 400, 600, 800}, and Bt = Ct = 1000 and ? (t) = 0.01 for
? j as the estiall t, we use Algorithm 1 to train a network with T ? {1, 2, . . . , 8} layers. Denote ?
mated K1 dimensional feature vector for document j, where K1 ? K1 max is the inferred number of
active factors of the first layer that is bounded by the pre-specified truncation level K1 max . We use
the L2 regularized logistic regression provided by the LIBLINEAR package [25] to train a linear
? j in the training set and use it to classify ?
? j in the test set, where the regularization
classifier on ?
parameter is five-folder cross-validated on the training set from (2?10 , 2?9 , . . . , 215 ).
As shown in Fig. 1, modifying the PGBN from a single-layer shallow network to a multilayer deep one clearly improves the qualities of the unsupervisedly extracted feature vectors.
In a random trial, with K1 max = 800, we infer a network structure of (K1 , . . . , K8 ) =
(512, 154, 75, 54, 47, 37, 34, 29) for the first binary classification task, and (K1 , . . . , K8 ) =
(491, 143, 74, 49, 36, 32, 28, 26) for the second one. Figs. 1(c)-(d) also show that increasing the
network depth in general improves the performance, but the first-layer width clearly plays an important role in controlling the ultimate network capacity. This insight is further illustrated below.
Feature learning for multi-class classification. We test the PGBNs for multi-class classification
on 20newsgroups. After removing a standard list of stopwords and the terms that appear less than
five times, we obtain a vocabulary with K0 = 33, 420. We set Ct = 500 and ? (t) = 0.05 for all
t. If K1 max ? 400, we set Bt = 1000 for all t, otherwise we set B1 = 1000 and Bt = 500 for
t ? 2. We use all 11,269 training documents to infer a set of networks with Tmax ? {1, . . . , 5} and
K1 max ? {50, 100, 200, 400, 600, 800}, and mimic the same testing procedure used for binary classification to extract low-dimensional feature vectors, with which each testing document is classified
to one of the 20 news groups using the L2 regularized logistic regression. Fig. 2 shows a clear trend
of improvement in classification accuracy by increasing the network depth with a limited first-layer
width, or by increasing the upper bound of the width of the first layer with the depth fixed. For example, a single-layer PGBN with K1 max = 100 could add one or more layers to slightly outperform
a single-layer PGBN with K1 max = 200, and a single-layer PGBN with K1 max = 200 could add
layers to clearly outperform a single-layer PGBN with K1 max as large as 800. We also note that
each iteration of jointly training multiple layers costs moderately more than that of training a single
layer, e.g., with K1 max = 400, a training iteration on a single core of an Intel Xeon 2.7 GHz CPU
on average takes about 5.6, 6.7, 7.1 seconds for the PGBN with 1, 3, and 5 layers, respectively.
Examining the inferred network structure also reveals interesting details.
For example, in a random trial with Algorithm 1, the inferred network widths (K1 , . . . , K5 ) are
7
(a)
750
12
650
10
Perplexity
700
Perplexity
(b)
14
T=1
T=2
T=3
T=4
T=5
600
8
T=1
T=2
T=3
T=4
T=5
6
4
2
550
0
500
-2
25
100
200
400
600
800
25
K 1max
100
200
400
600
800
K 1max
Figure 3: (a) per-heldout-word perplexity (the lower the better) for the NIPS12 corpus (using the 2000 most
frequent terms) as a function of the upper bound of the first layer width K1 max and network depth T , with
30% of the word tokens in each document used for training and ? (t) = 0.05 for all t. (b) for visualization, each
curve in (a) is reproduced by subtracting its values from the average perplexity of the single-layer network.
(50, 50, 50, 50, 50), (200, 161, 130, 94, 63), (528, 129, 109, 98, 91), and (608, 100, 99, 96, 89), for
K1 max = 50, 200, 600, and 800, respectively. This indicates that for a network with an insufficient
budget on its first-layer width, as the network depth increases, its inferred layer widths decay more
slowly than a network with a sufficient or surplus budget on its first-layer width; and a network with
a surplus budget on its first-layer width may only need relatively small widths for its higher hidden
layers. In the Appendix, we provide comparisons of accuracies between the PGBN and other related
algorithms, including these of [9] and [26], on similar multi-class document classification tasks.
Perplexities for holdout words. In addition to examining the performance of the PGBN for unsupervised feature learning, we also consider a more direct approach that we randomly choose 30% of
the word tokens in each document as training, and use the remaining ones to calculate per-heldoutword perplexity. We consider the NIPS12 (http://www.cs.nyu.edu/?roweis/data.html) corpus, limiting the vocabulary to the 2000 most frequent terms. We set ? (t) = 0.05 and Ct = 500 for all t, set
B1 = 1000 and Bt = 500 for t ? 2, and consider five random trials. Among the Bt + Ct Gibbs
sampling iterations used to train layer t, we collect one sample per five iterations during the last 500
(1)
(1)
iterations, for each of which we draw the topics {?k }k and topics weights ? j , to compute the
per-heldout-word perplexity using Equation (34) of [13]. As shown in Fig. 3, we observe a clear
trend of improvement by increasing both K1 max and T .
Qualitative analysis and document simulation. In addition to these quantitative experiments, we
Qt?1 (`) (t)
have also examined the topics learned at each layer. We use
?k to project topic k of
`=1 ?
layer t as a V -dimensional word probability vector. Generally speaking, the topics at lower layers
are more specific, whereas those at higher layers are more general. E.g., examining the results used
to produce Fig. 3, with K1 max = 200 and T = 5, the PGBN infers a network with (K1 , . . . , K5 ) =
(200, 164, 106, 60, 42). The ranks (by popularity) and top five words of three example topics for
layer T = 5 are ?6 network units input learning training,? ?15 data model learning set image,? and
?34 network learning model input neural;? while these of five example topics of layer T = 1 are ?19
likelihood em mixture parameters data,? ?37 bayesian posterior prior log evidence,? ?62 variables
belief networks conditional inference,? ?126 boltzmann binary machine energy hinton,? and ?127
(T +1)
speech speaker acoustic vowel phonetic.? We have also tried drawing ? (T ) ? Gam r, 1/cj
and downward passing it through the T -layer network to generate synthetic documents, which are
found to be quite interpretable and reflect various general aspects of the corpus used to train the network. We provide in the Appendix a number of synthetic documents generated from a PGBN trained
on the 20newsgroups corpus, whose inferred structure is (K1 , . . . , K5 ) = (608, 100, 99, 96, 89).
4
Conclusions
The Poisson gamma belief network is proposed to extract a multilayer deep representation for highdimensional count vectors, with an efficient upward-downward Gibbs sampler to jointly train all
its layers and a layer-wise training strategy to automatically infer the network structure. Example
results clearly demonstrate the advantages of deep topic models. For big data problems, in practice
one may rarely has a sufficient budget to allow the first-layer width to grow without bound, thus
it is natural to consider a belief network that can use a deep representation to not only enhance its
representation power, but also better allocate its computational resource. Our algorithm achieves a
good compromise between the widths of hidden layers and the depth of the network.
Acknowledgements. M. Zhou thanks TACC for computational support. B. Chen thanks the support
of the Thousand Young Talent Program of China, NSC-China (61372132), and NCET-13-0945.
8
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5,132 | 5,646 | Semi-Supervised Factored Logistic Regression for
High-Dimensional Neuroimaging Data
Danilo Bzdok, Michael Eickenberg, Olivier Grisel, Bertrand Thirion, Ga?el Varoquaux
INRIA, Parietal team, Saclay, France
CEA, Neurospin, Gif-sur-Yvette, France
[email protected]
Abstract
Imaging neuroscience links human behavior to aspects of brain biology in everincreasing datasets. Existing neuroimaging methods typically perform either discovery of unknown neural structure or testing of neural structure associated with
mental tasks. However, testing hypotheses on the neural correlates underlying
larger sets of mental tasks necessitates adequate representations for the observations. We therefore propose to blend representation modelling and task classification into a unified statistical learning problem. A multinomial logistic regression
is introduced that is constrained by factored coefficients and coupled with an autoencoder. We show that this approach yields more accurate and interpretable
neural models of psychological tasks in a reference dataset, as well as better generalization to other datasets.
keywords: Brain Imaging, Cognitive Science, Semi-Supervised Learning, Systems Biology
1
Introduction
Methods for neuroimaging research can be grouped by discovering neurobiological structure or assessing the neural correlates associated with mental tasks. To discover, on the one hand, spatial
distributions of neural activity structure across time, independent component analysis (ICA) is often
used [6]. It decomposes the BOLD (blood-oxygen level-dependent) signals into the primary modes
of variation. The ensuing spatial activity patterns are believed to represent brain networks of functionally interacting regions [26]. Similarly, sparse principal component analysis (SPCA) has been
used to separate BOLD signals into parsimonious network components [28]. The extracted brain
networks are probably manifestations of electrophysiological oscillation frequencies [17]. Their
fundamental organizational role is further attested by continued covariation during sleep and anesthesia [10]. Network discovery by applying ICA or SPCA is typically performed on task-unrelated
(i.e., unlabeled) ?resting-state? data. These capture brain dynamics during ongoing random thought
without controlled environmental stimulation. In fact, a large portion of the BOLD signal variation
is known not to correlate with a particular behavior, stimulus, or experimental task [10].
To test, on the other hand, the neural correlates underlying mental tasks, the general linear model
(GLM) is the dominant approach [13]. The contribution of individual brain voxels is estimated according to a design matrix of experimental tasks. Alternatively, psychophysiological interactions
(PPI) elucidate the influence of one brain region on another conditioned by experimental tasks [12].
As a last example, an increasing number of neuroimaging studies model experimental tasks by training classification algorithms on brain signals [23]. All these methods are applied to task-associated
(i.e., labeled) data that capture brain dynamics during stimulus-guided behavior. Two important
conclusions can be drawn. First, the mentioned supervised neuroimaging analyses typically yield
results in a voxel space. This ignores the fact that the BOLD signal exhibits spatially distributed
1
patterns of coherent neural activity. Second, existing supervised neuroimaging analyses cannot exploit the abundance of easily acquired resting-state data [8]. These may allow better discovery of the
manifold of brain states due to the high task-rest similarities of neural activity patterns, as observed
using ICA [26] and linear correlation [9].
Both these neurobiological properties can be conjointly exploited in an approach that is mixed
(i.e., using rest and task data), factored (i.e., performing network decomposition), and multitask (i.e., capitalize on neural representations shared across mental operations). The integration of brain-network discovery into supervised classification can yield a semi-supervised learning framework. The most relevant neurobiological structure should hence be identified for
the prediction problem at hand. Autoencoders suggest themselves because they can emulate
variants of most unsupervised learning algorithms, including PCA, SPCA, and ICA [15, 16].
Autoencoders (AE) are layered learning models that condense the input data to local and global representations
via reconstruction under compression prior. They behave
like a (truncated) PCA in case of one linear hidden layer
and a squared error loss [3]. Autoencoders behave like a
SPCA if shrinkage terms are added to the model weights
in the optimization objective. Moreover, they have the
characteristics of an ICA in case of tied weights and
adding a nonlinear convex function at the first layer [18].
These authors further demonstrated that ICA, sparse autoencoders, and sparse coding are mathematically equivalent under mild conditions. Thus, autoencoders may flexibly project the neuroimaging data onto the main directions of variation.
In the present investigation, a linear autoencoder will
be fit to (unlabeled) rest data and integrated as a rankreducing bottleneck into a multinomial logistic regression
fit to (labeled) task data. We can then solve the compound statistical problem of unsupervised data representation and supervised classification, previously studied in
isolation. From the perspective of dictionary learning, the Figure 1: Model architecture Linear
first layer represents projectors to the discovered set of ba- autoencoders find an optimized comsis functions which are linearly combined by the second pression of 79,941 brain voxels into n
layer to perform predictions [20]. Neurobiologically, this unknown activity patterns by improving
allows delineating a low-dimensional manifold of brain reconstruction from them. The decomnetwork patterns and then distinguishing mental tasks by position matrix equates with the bottletheir most discriminative linear combinations. Theoreti- neck of a factored logistic regression.
cally, a reduction in model variance should be achieved Supervised multi-class learning on task
by resting-state autoencoders that privilege the most neu- data (Xtask ) can thus be guided by unrobiologically valid models in the hypothesis set. Practi- supervised decomposition of rest data
cally, neuroimaging research frequently suffers from data (Xrest ).
scarcity. This limits the set of representations that can be
extracted from GLM analyses based on few participants. We therefore contribute a computational
framework that 1) analyzes many problems simultaneously (thus finds shared representations by
?multi-task learning?) and 2) exploits unlabeled data (since they span a space of meaningful configurations).
2
Methods
Data. As the currently biggest openly-accessible reference dataset, we chose resources from the
Human Connectome Project (HCP) [4]. Neuroimaging task data with labels of ongoing cognitive
processes were drawn from 500 healthy HCP participants (cf. Appendix for details on datasets). 18
HCP tasks were selected that are known to elicit reliable neural activity across participants (Table
1). In sum, the HCP task data incorporated 8650 first-level activity maps from 18 diverse paradigms
administered to 498 participants (2 removed due to incomplete data). All maps were resampled to a
common 60 ? 72 ? 60 space of 3mm isotropic voxels and gray-matter masked (at least 10% tissue
2
probability). The supervised analyses were thus based on labeled HCP task maps with 79,941 voxels
of interest representing z-values in gray matter.
Cognitive Task
1 Reward
2 Punish
3 Shapes
4 Faces
5 Random
6 Theory of mind
7 Mathematics
8 Language
9 Tongue movement
10 Food movement
11 Hand movement
12 Matching
13 Relations
14 View Bodies
15 View Faces
16 View Places
17 View Tools
18 Two-Back
Stimuli
Instruction for participants
Card game
Guess the number of a mystery card for gain/loss of money
Shape pictures
Face pictures
Decide which of two shapes matches another shape geometrically
Decide which of two faces matches another face emotionally
Videos with objects
Decide whether the objects act randomly or intentionally
Spoken numbers
Auditory stories
Complete addition and subtraction problems
Choose answer about the topic of the story
Move tongue
Squeezing of the left or right toe
Tapping of the left or right finger
Decide whether two objects match in shape or texture
Decide whether object pairs differ both along either shape or texture
Passive watching
Passive watching
Passive watching
Passive watching
Indicate whether current stimulus is the same as two items earlier
Visual cues
Shapes with textures
Pictures
Pictures
Pictures
Pictures
Various pictures
Table 1: Description of psychological tasks to predict.
These labeled data were complemented by unlabeled activity maps from HCP acquisitions of unconstrained resting-state activity [25]. These reflect brain activity in the absence of controlled thought.
In sum, the HCP rest data concatenated 8000 unlabeled, noise-cleaned rest maps with 40 brain maps
from each of 200 randomly selected participants.
We were further interested in the utility of the optimized low-rank projection in one task dataset
for dimensionality reduction in another task dataset. To this end, the HCP-derived network decompositions were used as preliminary step in the classification problem of another large sample. The
ARCHI dataset [21] provides activity maps from diverse experimental tasks, including auditory and
visual perception, motor action, reading, language comprehension and mental calculation. Analogous to HCP data, the second task dataset thus incorporated 1404 labeled, grey-matter masked, and
z-scored activity maps from 18 diverse tasks acquired in 78 participants.
Linear autoencoder. The labeled and unlabeled data were fed into a linear statistical model composed of an autoencoder and dimensionality-reducing logistic regression. The affine autoencoder
takes the input x, projects it into a coordinate system of latent representations z and reconstructs it
back to x0 by
x0 = W1 z + b1 ,
z = W0 x + b0
(1)
where x 2 Rd denotes the vector of d = 79,941 voxel values from each rest map, z 2 Rn is the ndimensional hidden state (i.e., distributed neural activity patterns), and x0 2 Rd is the reconstruction
vector of the original activity map from the hidden variables. Further, W0 denotes the weight matrix
that transforms from input space into the hidden space (encoder), W1 is the weight matrix for backprojection from the hidden variables to the output space (decoder). b0 and b1 are corresponding
bias vectors. The model parameters W0 , b0 , b1 are found by minimizing the expected squared
reconstruction error
?
?
E [LAE (x)] = E kx (W1 (W0 x + b0 ) + b1 )k2 .
(2)
Here we choose W0 and W1 to be tied, i.e. W0 = W1T . Consequently, the learned weights are
forced to take a two-fold function: That of signal analysis and that of signal synthesis. The first
layer analyzes the data to obtain the cleanest latent representation, while the second layer represents
building blocks from which to synthesize the data using the latent activations. Tying these processes
together makes the analysis layer interpretable and pulls all non-zero singular values towards 1.
Nonlinearities were not applied to the activations in the first layer.
Factored logistic regression. Our factored logistic regression model is best described as a variant
of a multinomial logistic regression. Specifically, the weight matrix is replaced by the product
3
of two weight matrices with a common latent dimension. The later is typically much lower than
the dimension of the data. Alternatively, this model can be viewed as a single-hidden-layer feedforward neural network with a linear activation function for the hidden layer and a softmax function
on the output layer. As the dimension of the hidden layer is much lower than the input layer, this
architecture is sometimes referred to as a ?linear bottleneck? in the literature. The probability of an
input x to belong to a class i 2 {1, . . . , l} is given by
P (Y = i|x; V0 , V1 , c0 , c1 ) = softmaxi (fLR (x)),
(3)
where fLR
P(x) = V1 (V0 x + c0 ) + c1 computes multinomial logits and softmaxi (x) =
exp(xi )/ j exp(xj ). The matrix V0 2 Rdxn transforms the input x 2 Rd into n latent components and the matrix V1 2 Rnxl projects the latent components onto hyperplanes that reflect l
label probabilities. c0 and c1 are bias vectors. The loss function is given by
E [LLR (x, y)] ?
1
NXtask
NXtask
X
k=0
log(P (Y = y (k) |x(k) ; V0 , V1 , c0 , c1 )).
(4)
Layer combination. The optimization problem of the linear autoencoder and the factored logistic
regression are linked in two ways. First, their transformation matrices mapping from input to the
latent space are tied
(5)
V0 = W 0 .
We hence search for a compression of the 79,941 voxel values into n unknown components that
represent a latent code optimized for both rest and task activity data. Second, the objectives of the
autoencoder and the factored logistic regression are interpolated in the common loss function
L(?, ) = LLR + (1
)
1
NXrest
LAE + ?.
(6)
In so doing, we search for the combined model parameters ? = {V0 , V1 , c0 , c1 , b0 , b1 } with
respect to the (unsupervised) reconstruction error and the (supervised) task detection. LAE is devided by NXrest to equilibrate both loss terms to the same order of magnitude. ? represents an
ElasticNet-type regularization that combines `1 and `2 penalty terms.
Optimization. The common objective was optimized by gradient descent in the SSFLogReg parameters. The required gradients were obtained by using the chain rule to backpropagate error
derivatives. We chose the rmsprop solver [27], a refinement of stochastic gradient descent. Rmsprop
dictates an adaptive learning rate for each model parameter by scaled gradients from a running average. The batch size was set to 100 (given much expected redundancy in Xrest and Xtask ), matrix
parameters were initalized by Gaussian random values multiplied by 0.004 (i.e., gain), and bias
parameters were initalized to 0.
The normalization factor and the update rule for ? are given by
?
?2
v(t+1) = ?v(t) + (1 ?) r? f (x(t) , y (t) , ?(t) )
?(t+1) = ?(t) + ?
r? f (x(t) , y (t) , ?(t) )
p
,
v(t+1) + ?
(7)
where f is the loss function computed on a minibatch sample at timestep t, ? is the learning rate
(0.00001), ? a global damping factor (10 6 ), and ? the decay rate (0.9 to deemphasize the magnitude of the gradient). Note that we have also experimented with other solvers (stochastic gradient
descent, adadelta, and adagrad) but found that rmsprop converged faster and with similar or higher
generalization performance.
Implementation. The analyses were performed in Python. We used nilearn to handle the large
quantities of neuroimaging data [1] and Theano for automatic, numerically stable differentiation
of symbolic computation graphs [5, 7]. All Python scripts that generated the results are accessible
online for reproducibility and reuse (http://github.com/banilo/nips2015).
4
3
Experimental Results
Serial versus parallel structure discovery and classification. We first tested whether there is a
substantial advantage in combining unsupervised decomposition and supervised classification learning. We benchmarked our approach against performing data reduction on the (unlabeled) first half
of the HCP task data by PCA, SPCA, ICA, and AE (n = 5, 20, 50, 100 components) and learning classification models in the (labeled) second half by ordinary logistic regression. PCA reduced
the dimensionality of the task data by finding orthogonal network components (whitening of the
data). SPCA separated the task-related BOLD signals into network components with few regions
by a regression-type optimization problem constrained by `1 penalty (no orthogonality assumptions,
1000 maximum iterations, per-iteration tolerance of 10-8 , ? = 1). ICA performed iterative blind
source separation by a parallel FASTICA implementation (200 maximum iterations, per-iteration
tolerance of 0.0001, initialized by random mixing matrix, whitening of the data). AE found a code
of latent representations by optimizing projection into a bottleneck (500 iterations, same implementation as below for rest data). The second half of the task data was projected onto the latent
components discovered in its first half. Only the ensuing component loadings were submitted to
ordinary logistic regression (no hidden layer, `1 = 0.1, `2 = 0.1, 500 iterations). These serial twostep approaches were compared against parallel decomposition and classification by SSFLogReg
(one hidden layers, = 1, `1 = 0.1, `2 = 0.1, 500 iterations). Importantly, all trained classification models were tested on a large, unseen test set (20% of data) in the present analyses. Across
choices for n, SSFLogReg achieved more than 95% out-of-sample accuracy, whereas supervised
learning based on PCA, SPCA, ICA, and AE loadings ranged from 32% to 87% (Table 2). This
experiment establishes the advantage of directly searching for classification-relevant structure in the
fMRI data, rather than solving the supervised and unsupervised problems independently. This effect
was particularly pronounced when assuming few hidden dimensions.
n
5
20
50
100
PCA + LogReg
45.1 %
78.1 %
81.7 %
81.3 %
SPCA + LogReg
32.2 %
78.2 %
84.0 %
82.2 %
ICA + LogReg
37.5 %
81.0 %
84.2 %
87.3 %
AE + LogReg
44.2 %
63.2 %
77.0 %
76.6 %
SSFLogReg
95.7%
97.3%
97.6%
97.4%
Table 2: Serial versus parallel dimensionality reduction and classification. Chance is at 5,6%.
Model performance. SSFLogReg was subsequently trained (500 epochs) across parameter
choices for the hidden components (n = 5, 20, 100) and the balance between autoencoder and
logistic regression ( = 0, 0.25, 0.5, 0.75, 1). Assuming 5 latent directions of variation should yield
models with higher bias and smaller variance than SSFLogReg with 100 latent directions. Given the
18-class problem of HCP, setting to 0 consistently yields generalization performance at chancelevel (5,6%) because only the unsupervised layer of the estimator is optimized. At each epoch (i.e.,
iteration over the data), the out-of-sample performance of the trained classifier was assessed on 20%
of unseen HCP data. Additionally, the ?out-of-study? performance of the learned decomposition
(W0 ) was assessed by using it as dimensionality reduction of an independent labeled dataset (i.e.,
ARCHI) and conducting ordinary logistic regression on the ensuing component loadings.
=0
Out-of-sample
accuracy
6.0%
Precision (mean)
5.9%
Recall (mean)
5.6%
F1 score (mean)
4.1%
Reconstr. error (norm.) 0.76
Out-of-study
accuracy
39.4%
= 0.25
n=5
= 0.5
= 0.75
=1
=0
95.7%
95.4%
95.7%
95.4%
1.79
5.5%
5.1%
4.6%
3.8%
0.64
= 0.25
88.9%
87.0%
88.3%
86.6%
0.85
95.1%
94.9%
95.2%
94.9%
0.87
96.5%
96.3%
96.6%
96.4%
1.01
60.8%
54.3%
60.7% 62.9% 77.0% 79.7%
n = 20
= 0.5
97.4% 97.8%
97.4% 97.1%
97.5% 97.5%
97.4% 97.2%
0.67
0.69
81.9%
= 0.75
=1
=0
97.3%
97.0%
97.4%
97.1%
1.22
6.1%
5.9%
7.2%
5.3%
0.60
= 0.25
n = 100
= 0.5
= 0.75
=1
97.2%
96.9%
97.2%
97.0%
0.65
97.0%
96.5%
97.2%
96.7%
0.68
97.8%
97.5%
97.9%
97.7%
0.73
79.7% 79.4% 79.2% 82.2%
81.7%
81.3% 75.8%
97.3%
97.0%
97.4%
97.1%
0.77
97.4%
96.9%
97.4%
97.2%
1.08
Table 3: Performance of SSFLogReg across model parameter choices. Chance is at 5.6%.
We made three noteworthy observations (Table 3). First, the most supervised estimator ( = 1)
achieved in no instance the best accuracy, precision, recall, or f1 scores on HCP data. Classification
by SSFLogReg is therefore facilitated by imposing structure from the unlabeled rest data. Confirmed
by the normalized reconstruction error (E = kx x
?k/kxk), little weight on the supervised term is
sufficient for good model performance while keeping E low and task-map decomposition rest-like.
5
Figure 2: Effect of bottleneck in a 38-task classificaton problem Depicts the f1 prediction scores
for each of 38 psychological tasks. Multinomial logistic regression operating in voxel space (blue
bars) was compared to SSFLogReg operating in 20 (left plot) and 100 (right plot) latent modes
(grey bars). Autoencoder or rest data were not used for these analyses ( = 1). Ordinary logistic
regression yielded 77.7% accuracy out of sample, while SSFLogReg scored at 94.4% (n = 20) and
94.2% (n = 100). Hence, compressing the voxel data into a component space for classification
achieves higher task separability. Chance is at 2, 6%.
Second, the higher the number of latent components n, the higher the out-of-study performance with
small values of . This suggests that the presence of more rest-data-inspired hidden components
results in more effective feature representations in unrelated task data. Third, for n = 20 and 100
(but not 5) the purely rest-data-trained decomposition matrix ( = 0) resulted in noninferior out-ofstudy performance of 77.0% and 79.2%, respectively (Table 3). This confirms that guiding model
learning by task-unrelated structure extracts features of general relevance beyond the supervised
problem at hand.
Individual effects of dimensionality reduction and rest data. We first quantified the impact of
introducing a bottleneck layer disregarding the autoencoder. To this end, ordinary logistic regression
was juxtaposed with SSFLogReg at = 1. For this experiment, we increased the difficulty of the
classification problem by including data from all 38 HCP tasks. Indeed, increased class separability
in component space, as compared to voxel space, entails differences in generalization performance
of ? 17% (Figure 2). Notably, the cognitive tasks on reward and punishment processing are among
the least predicted with ordinary but well predicted with low-rank logistic regression (tasks 1 and
2 in Figure 2). These experimental conditions have been reported to exhibit highly similar neural
activity patterns in GLM analyses of that dataset [4]. Consequently, also local activity differences (in
the striatum and visual cortex in this case) can be successfully captured by brain-network modelling.
We then contemplated the impact of rest structure (Figure 3) by modulating its influence ( =
0.25, 0.5, 0.75) in data-scarce and data-rich settings (n = 20, `1 = 0.1, `2 = 0.1). At the beginning
of every epoch, 2000 task and 2000 rest maps were drawn with replacement from same amounts of
task and rest maps. In data-scarce scenarios (frequently encountered by neuroimaging practitioners),
the out-of-sample scores improve as we depart from the most supervised model ( = 1). In data-rich
scenarios, we observed the same trend to be apparent.
Feature identification. We finally examined whether the models were fit for purpose (Figure 4).
To this end, we computed Pearson?s correlation between the classifier weights and the averaged
neural activity map for each of the 18 tasks. Ordinary logistic regression thus yielded a mean correlation of ? = 0.28 across tasks. For SSFLogReg ( = 0.25, 0.5, 0.75, 1), a per-class-weight map
was computed by matrix multiplication of the two inner layers. Feature identification performance
thus ranged between ? = 0.35 and ? = 0.55 for n = 5, between ? = 0.59 and ? = 0.69 for n = 20,
and between ? = 0.58 and ? = 0.69 for n = 100. Consequently, SSFLogReg puts higher absolute
weights on relevant structure. This reflects an increased signal-to-noise ratio, in part explained by
6
Figure 3: Effect of rest structure Model performance of SSFLogReg (n = 20, `1 = 0.1, `2 = 0.1)
for different choices of in data-scarce (100 task and 100 rest maps, hot color) and data-rich (1000
task and 1000 rest maps, cold color) scenarios. Gradient descent was performed on 2000 task and
2000 rest maps. At the begining of each epoch, these were drawn with replacement from a pool of
100 or 1000 different task and rest maps, respectively. Chance is at 5.6%.
Figure 4: Classification weight maps The voxel predictors corresponding to 5 exemplary (of 18
total) psychological tasks (rows) from the HCP dataset [4]. Left column: multinomial logistic regression (same implementation but without bottleneck or autoencoder), middle column: SSFLogReg
(n = 20 latent components, = 0.5, `1 = 0.1, `2 = 0.1), right column: voxel-wise average across
all samples of whole-brain activity maps from each task. SSFLogReg a) puts higher absolute weights
on relevant structure, b) lower ones on irrelevant structure, and c) yields BOLD-typical local contiguity (without enforcing an explicit spatial prior). All values are z-scored and thresholded at the
75th percentile.
the more BOLD-typical local contiguity. Conversely, SSFLogReg puts lower probability mass on
irrelevant structure. Despite lower interpretability of the results from ordinary logistic regression,
the salt-and-pepper-like weight maps were sufficient for good classification performance. Hence,
SSFLogReg yielded class weights that were much more similar to features of the respective training
samples for all choices of n and . SSFLogReg therefore captures genuine properties of task activity
patterns, rather than participant- or study-specific artefacts.
7
Miscellaneous observations. For the sake of completeness, we informally report modifications of
the statistical model that did not improve generalization performance. a) Introducing stochasticity
into model learning by input corruption of Xtask deteriorated model performance in all scenarios.
Adding b) rectified linear units (ReLU) to W0 or other commonly used nonlinearities (c) sigmoid,
d) softplus, e) hyperbolic tangent) all led to decreased classification accuracies, probably due to
sample size limits. Further, f ) ?pretraining? of the bottleneck W0 (i.e., non-random initialization)
by either corresponding PCA, SPCA or ICA loadings did not exhibit improved accuracies, neither
did g) autoencoder pretraining. Moreover, introducing an additional h) overcomplete layer (100
units) after the bottleneck was not advantageous. Finally, imposing either i) only `1 or j) only `2
penalty terms was disadvantageous in all tested cases. This favored ElasticNet regularization chosen
in the above analyses.
4
Discussion and Conclusion
Using the flexibility of factored models, we learn the low-dimensional representation from highdimensional voxel brain space that is most important for prediction of cognitive task sets. From
a machine-learning perspective, factorization of the logistic regression weights can be viewed as
transforming a ?multi-class classification problem? into a ?multi-task learning problem?. The higher
generalization accuracy and support recovery, comparing to ordinary logistic regression, hold potential for adoption in various neuroimaging analyses. Besides increased performance, these models
are more interpretable by automatically learning a mapping to and from a brain-network space. This
domain-specific learning algorithm encourages departure from the artificial and statistically less attractive voxel space. Neurobiologically, brain activity underlying defined mental operations can be
explained by linear combinations of the main activity patterns. That is, fMRI data probably concentrate near a low-dimensional manifold of characteristic brain network combinations. Extracting
fundamental building blocks of brain organization might facilitate the quest for the cognitive primitives of human thought. We hope that these first steps stimulate development towards powerful
semi-supervised representation extraction in systems neuroscience.
In the future, automatic reduction of brain maps to their neurobiological essence may leverage dataintense neuroimaging investigations. Initiatives for data collection are rapidly increasing in neuroscience [22]. These promise structured integration of neuroscientific knowledge accumulating
in databases. Tractability by condensed feature representations can avoid the ill-posed problem
of learning the full distribution of activity patterns. This is not only relevant to the multi-class
challenges spanning the human cognitive space [24] but also the multi-modal combination with
high-resolution 3D models of brain anatomy [2] and high-throughput genomics [19]. The biggest
socioeconomic potential may lie in across-hospital clinical studies that predict disease trajectories
and drug responses in psychiatric and neurological populations [11].
Acknowledgment The research leading to these results has received funding from the European Union
Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 604102 (Human Brain Project).
Data were provided by the Human Connectome Project. Further support was received from the German National Academic Foundation (D.B.) and the MetaMRI associated team (B.T., G.V.).
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5,133 | 5,647 | BinaryConnect: Training Deep Neural Networks with
binary weights during propagations
Matthieu Courbariaux
?
Ecole
Polytechnique de Montr?eal
[email protected]
Yoshua Bengio
Universit?e de Montr?eal, CIFAR Senior Fellow
[email protected]
Jean-Pierre David
?
Ecole
Polytechnique de Montr?eal
[email protected]
Abstract
Deep Neural Networks (DNN) have achieved state-of-the-art results in a wide
range of tasks, with the best results obtained with large training sets and large
models. In the past, GPUs enabled these breakthroughs because of their greater
computational speed. In the future, faster computation at both training and test
time is likely to be crucial for further progress and for consumer applications on
low-power devices. As a result, there is much interest in research and development of dedicated hardware for Deep Learning (DL). Binary weights, i.e., weights
which are constrained to only two possible values (e.g. -1 or 1), would bring great
benefits to specialized DL hardware by replacing many multiply-accumulate operations by simple accumulations, as multipliers are the most space and powerhungry components of the digital implementation of neural networks. We introduce BinaryConnect, a method which consists in training a DNN with binary
weights during the forward and backward propagations, while retaining precision
of the stored weights in which gradients are accumulated. Like other dropout
schemes, we show that BinaryConnect acts as regularizer and we obtain near
state-of-the-art results with BinaryConnect on the permutation-invariant MNIST,
CIFAR-10 and SVHN.
1
Introduction
Deep Neural Networks (DNN) have substantially pushed the state-of-the-art in a wide range of tasks,
especially in speech recognition [1, 2] and computer vision, notably object recognition from images [3, 4]. More recently, deep learning is making important strides in natural language processing,
especially statistical machine translation [5, 6, 7]. Interestingly, one of the key factors that enabled
this major progress has been the advent of Graphics Processing Units (GPUs), with speed-ups on the
order of 10 to 30-fold, starting with [8], and similar improvements with distributed training [9, 10].
Indeed, the ability to train larger models on more data has enabled the kind of breakthroughs observed in the last few years. Today, researchers and developers designing new deep learning algorithms and applications often find themselves limited by computational capability. This along, with
the drive to put deep learning systems on low-power devices (unlike GPUs) is greatly increasing the
interest in research and development of specialized hardware for deep networks [11, 12, 13].
Most of the computation performed during training and application of deep networks regards the
multiplication of a real-valued weight by a real-valued activation (in the recognition or forward
propagation phase of the back-propagation algorithm) or gradient (in the backward propagation
phase of the back-propagation algorithm). This paper proposes an approach called BinaryConnect
1
to eliminate the need for these multiplications by forcing the weights used in these forward and
backward propagations to be binary, i.e. constrained to only two values (not necessarily 0 and 1). We
show that state-of-the-art results can be achieved with BinaryConnect on the permutation-invariant
MNIST, CIFAR-10 and SVHN.
What makes this workable are two ingredients:
1. Sufficient precision is necessary to accumulate and average a large number of stochastic
gradients, but noisy weights (and we can view discretization into a small number of values
as a form of noise, especially if we make this discretization stochastic) are quite compatible
with Stochastic Gradient Descent (SGD), the main type of optimization algorithm for deep
learning. SGD explores the space of parameters by making small and noisy steps and
that noise is averaged out by the stochastic gradient contributions accumulated in each
weight. Therefore, it is important to keep sufficient resolution for these accumulators,
which at first sight suggests that high precision is absolutely required. [14] and [15] show
that randomized or stochastic rounding can be used to provide unbiased discretization.
[14] have shown that SGD requires weights with a precision of at least 6 to 8 bits and
[16] successfully train DNNs with 12 bits dynamic fixed-point computation. Besides, the
estimated precision of the brain synapses varies between 6 and 12 bits [17].
2. Noisy weights actually provide a form of regularization which can help to generalize better,
as previously shown with variational weight noise [18], Dropout [19, 20] and DropConnect [21], which add noise to the activations or to the weights. For instance, DropConnect
[21], which is closest to BinaryConnect, is a very efficient regularizer that randomly substitutes half of the weights with zeros during propagations. What these previous works show
is that only the expected value of the weight needs to have high precision, and that noise
can actually be beneficial.
The main contributions of this article are the following.
? We introduce BinaryConnect, a method which consists in training a DNN with binary
weights during the forward and backward propagations (Section 2).
? We show that BinaryConnect is a regularizer and we obtain near state-of-the-art results on
the permutation-invariant MNIST, CIFAR-10 and SVHN (Section 3).
? We make the code for BinaryConnect available 1 .
2
BinaryConnect
In this section we give a more detailed view of BinaryConnect, considering which two values to
choose, how to discretize, how to train and how to perform inference.
+1 or ?1
2.1
Applying a DNN mainly consists in convolutions and matrix multiplications. The key arithmetic
operation of DL is thus the multiply-accumulate operation. Artificial neurons are basically multiplyaccumulators computing weighted sums of their inputs.
BinaryConnect constraints the weights to either +1 or ?1 during propagations. As a result, many
multiply-accumulate operations are replaced by simple additions (and subtractions). This is a huge
gain, as fixed-point adders are much less expensive both in terms of area and energy than fixed-point
multiply-accumulators [22].
2.2
Deterministic vs stochastic binarization
The binarization operation transforms the real-valued weights into the two possible values. A very
straightforward binarization operation would be based on the sign function:
+1 if w ? 0,
wb =
(1)
?1 otherwise.
1
https://github.com/MatthieuCourbariaux/BinaryConnect
2
Where wb is the binarized weight and w the real-valued weight. Although this is a deterministic operation, averaging this discretization over the many input weights of a hidden unit could compensate
for the loss of information. An alternative that allows a finer and more correct averaging process to
take place is to binarize stochastically:
+1 with probability p = ?(w),
(2)
wb =
?1 with probability 1 ? p.
where ? is the ?hard sigmoid? function:
?(x) = clip(
x+1
x+1
, 0, 1) = max(0, min(1,
))
2
2
(3)
We use such a hard sigmoid rather than the soft version because it is far less computationally expensive (both in software and specialized hardware implementations) and yielded excellent results in our
experiments. It is similar to the ?hard tanh? non-linearity introduced by [23]. It is also piece-wise
linear and corresponds to a bounded form of the rectifier [24].
2.3
Propagations vs updates
Let us consider the different steps of back-propagation with SGD udpates and whether it makes
sense, or not, to discretize the weights, at each of these steps.
1. Given the DNN input, compute the unit activations layer by layer, leading to the top layer
which is the output of the DNN, given its input. This step is referred as the forward propagation.
2. Given the DNN target, compute the training objective?s gradient w.r.t. each layer?s activations, starting from the top layer and going down layer by layer until the first hidden
layer. This step is referred to as the backward propagation or backward phase of backpropagation.
3. Compute the gradient w.r.t. each layer?s parameters and then update the parameters using
their computed gradients and their previous values. This step is referred to as the parameter
update.
Algorithm 1 SGD training with BinaryConnect. C is the cost function for minibatch and the functions binarize(w) and clip(w) specify how to binarize and clip weights. L is the number of layers.
Require: a minibatch of (inputs, targets), previous parameters wt?1 (weights) and bt?1 (biases),
and learning rate ?.
Ensure: updated parameters wt and bt .
1. Forward propagation:
wb ? binarize(wt?1 )
For k = 1 to L, compute ak knowing ak?1 , wb and bt?1
2. Backward propagation:
?C
Initialize output layer?s activations gradient ?a
L
?C
?C
For k = L to 2, compute ?ak?1 knowing ?ak and wb
3. Parameter update:
?C
?C
Compute ?w
and db?C
knowing ?a
and ak?1
t?1
b
k
?C
wt ? clip(wt?1 ? ? ?w
)
b
?C
bt ? bt?1 ? ? ?bt?1
A key point to understand with BinaryConnect is that we only binarize the weights during the forward and backward propagations (steps 1 and 2) but not during the parameter update (step 3), as
illustrated in Algorithm 1. Keeping good precision weights during the updates is necessary for SGD
to work at all. These parameter changes are tiny by virtue of being obtained by gradient descent, i.e.,
SGD performs a large number of almost infinitesimal changes in the direction that most improves
the training objective (plus noise). One way to picture all this is to hypothesize that what matters
3
most at the end of training is the sign of the weights, w? , but that in order to figure it out, we perform
a lot of small changes to a continuous-valued quantity w, and only at the end consider its sign:
X
w? = sign(
gt )
(4)
t
t?1 ,bt?1 ),yt )
where gt is a noisy estimator of ?C(f (xt ,w
, where C(f (xt , wt?1 , bt?1 ), yt ) is the value
?wt?1
of the objective function on (input,target) example (xt , yt ), when wt?1 are the previous weights and
w? is its final discretized value of the weights.
Another way to conceive of this discretization is as a form of corruption, and hence as a regularizer,
and our empirical results confirm this hypothesis. In addition, we can make the discretization errors
on different weights approximately cancel each other while keeping a lot of precision by randomizing
the discretization appropriately. We propose a form of randomized discretization that preserves the
expected value of the discretized weight.
Hence, at training time, BinaryConnect randomly picks one of two values for each weight, for each
minibatch, for both the forward and backward propagation phases of backprop. However, the SGD
update is accumulated in a real-valued variable storing the parameter.
An interesting analogy to understand BinaryConnect is the DropConnect algorithm [21]. Just like
BinaryConnect, DropConnect only injects noise to the weights during the propagations. Whereas
DropConnect?s noise is added Gaussian noise, BinaryConnect?s noise is a binary sampling process.
In both cases the corrupted value has as expected value the clean original value.
2.4
Clipping
Since the binarization operation is not influenced by variations of the real-valued weights w when its
magnitude is beyond the binary values ?1, and since it is a common practice to bound weights (usually the weight vector) in order to regularize them, we have chosen to clip the real-valued weights
within the [?1, 1] interval right after the weight updates, as per Algorithm 1. The real-valued weights
would otherwise grow very large without any impact on the binary weights.
2.5
A few more tricks
Optimization
No learning rate scaling
Learning rate scaling
SGD
Nesterov momentum
ADAM
15.65%
12.81%
11.45%
11.30%
10.47%
Table 1: Test error rates of a (small) CNN trained on CIFAR-10 depending on optimization method
and on whether the learning rate is scaled with the weights initialization coefficients from [25].
We use Batch Normalization (BN) [26] in all of our experiments, not only because it accelerates
the training by reducing internal covariate shift, but also because it reduces the overall impact of
the weights scale. Moreover, we use the ADAM learning rule [27] in all of our CNN experiments.
Last but not least, we scale the weights learning rates respectively with the weights initialization
coefficients from [25] when optimizing with ADAM, and with the squares of those coefficients
when optimizing with SGD or Nesterov momentum [28]. Table 1 illustrates the effectiveness of
those tricks.
2.6
Test-Time Inference
Up to now we have introduced different ways of training a DNN with on-the-fly weight binarization.
What are reasonable ways of using such a trained network, i.e., performing test-time inference on
new examples? We have considered three reasonable alternatives:
1. Use the resulting binary weights wb (this makes most sense with the deterministic form of
BinaryConnect).
4
2. Use the real-valued weights w, i.e., the binarization only helps to achieve faster training but
not faster test-time performance.
3. In the stochastic case, many different networks can be sampled by sampling a wb for each
weight according to Eq. 2. The ensemble output of these networks can then be obtained by
averaging the outputs from individual networks.
We use the first method with the deterministic form of BinaryConnect. As for the stochastic form
of BinaryConnect, we focused on the training advantage and used the second method in the experiments, i.e., test-time inference using the real-valued weights. This follows the practice of Dropout
methods, where at test-time the ?noise? is removed.
Method
MNIST
CIFAR-10
SVHN
No regularizer
BinaryConnect (det.)
BinaryConnect (stoch.)
50% Dropout
1.30 ? 0.04%
1.29 ? 0.08%
1.18 ? 0.04%
1.01 ? 0.04%
10.64%
9.90%
8.27%
2.44%
2.30%
2.15%
Maxout Networks [29]
Deep L2-SVM [30]
Network in Network [31]
DropConnect [21]
Deeply-Supervised Nets [32]
0.94%
0.87%
11.68%
2.47%
10.41%
2.35%
1.94%
1.92%
9.78%
Table 2: Test error rates of DNNs trained on the MNIST (no convolution and no unsupervised
pretraining), CIFAR-10 (no data augmentation) and SVHN, depending on the method. We see
that in spite of using only a single bit per weight during propagation, performance is not worse
than ordinary (no regularizer) DNNs, it is actually better, especially with the stochastic version,
suggesting that BinaryConnect acts as a regularizer.
Figure 1: Features of the first layer of an MLP trained on MNIST depending on the regularizer. From left to right: no regularizer, deterministic BinaryConnect, stochastic BinaryConnect
and Dropout.
3
Benchmark results
In this section, we show that BinaryConnect acts as regularizer and we obtain near state-of-the-art
results with BinaryConnect on the permutation-invariant MNIST, CIFAR-10 and SVHN.
3.1
Permutation-invariant MNIST
MNIST is a benchmark image classification dataset [33]. It consists in a training set of 60000 and
a test set of 10000 28 ? 28 gray-scale images representing digits ranging from 0 to 9. Permutationinvariance means that the model must be unaware of the image (2-D) structure of the data (in other
words, CNNs are forbidden). Besides, we do not use any data-augmentation, preprocessing or unsupervised pretraining. The MLP we train on MNIST consists in 3 hidden layers of 1024 Rectifier
Linear Units (ReLU) [34, 24, 3] and a L2-SVM output layer (L2-SVM has been shown to perform
better than Softmax on several classification benchmarks [30, 32]). The square hinge loss is minimized with SGD without momentum. We use an exponentially decaying learning rate. We use Batch
5
Figure 2: Histogram of the weights of the first layer of an MLP trained on MNIST depending on
the regularizer. In both cases, it seems that the weights are trying to become deterministic to reduce
the training error. It also seems that some of the weights of deterministic BinaryConnect are stuck
around 0, hesitating between ?1 and 1.
Figure 3: Training curves of a CNN on CIFAR-10 depending on the regularizer. The dotted lines
represent the training costs (square hinge losses) and the continuous lines the corresponding validation error rates. Both versions of BinaryConnect significantly augment the training cost, slow down
the training and lower the validation error rate, which is what we would expect from a Dropout
scheme.
Normalization with a minibatch of size 200 to speed up the training. As typically done, we use the
last 10000 samples of the training set as a validation set for early stopping and model selection. We
report the test error rate associated with the best validation error rate after 1000 epochs (we do not
retrain on the validation set). We repeat each experiment 6 times with different initializations. The
results are in Table 2. They suggest that the stochastic version of BinaryConnect can be considered
a regularizer, although a slightly less powerful one than Dropout, in this context.
3.2
CIFAR-10
CIFAR-10 is a benchmark image classification dataset. It consists in a training set of 50000 and
a test set of 10000 32 ? 32 color images representing airplanes, automobiles, birds, cats, deers,
dogs, frogs, horses, ships and trucks. We preprocess the data using global contrast normalization
and ZCA whitening. We do not use any data-augmentation (which can really be a game changer for
this dataset [35]). The architecture of our CNN is:
(2?128C3)?M P 2?(2?256C3)?M P 2?(2?512C3)?M P 2?(2?1024F C)?10SV M (5)
Where C3 is a 3 ? 3 ReLU convolution layer, M P 2 is a 2 ? 2 max-pooling layer, F C a fully
connected layer, and SVM a L2-SVM output layer. This architecture is greatly inspired from VGG
[36]. The square hinge loss is minimized with ADAM. We use an exponentially decaying learning
6
rate. We use Batch Normalization with a minibatch of size 50 to speed up the training. We use the
last 5000 samples of the training set as a validation set. We report the test error rate associated with
the best validation error rate after 500 training epochs (we do not retrain on the validation set). The
results are in Table 2 and Figure 3.
3.3
SVHN
SVHN is a benchmark image classification dataset. It consists in a training set of 604K and a test set
of 26K 32 ? 32 color images representing digits ranging from 0 to 9. We follow the same procedure
that we used for CIFAR-10, with a few notable exceptions: we use half the number of hidden units
and we train for 200 epochs instead of 500 (because SVHN is quite a big dataset). The results are in
Table 2.
4
Related works
Training DNNs with binary weights has been the subject of very recent works [37, 38, 39, 40]. Even
though we share the same objective, our approaches are quite different. [37, 38] do not train their
DNN with Backpropagation (BP) but with a variant called Expectation Backpropagation (EBP).
EBP is based on Expectation Propagation (EP) [41], which is a variational Bayes method used to do
inference in probabilistic graphical models. Let us compare their method to ours:
? It optimizes the weights posterior distribution (which is not binary). In this regard, our
method is quite similar as we keep a real-valued version of the weights.
? It binarizes both the neurons outputs and weights, which is more hardware friendly than
just binarizing the weights.
? It yields a good classification accuracy for fully connected networks (on MNIST) but not
(yet) for ConvNets.
[39, 40] retrain neural networks with ternary weights during forward and backward propagations,
i.e.:
? They train a neural network with high-precision,
? After training, they ternarize the weights to three possible values ?H, 0 and +H and adjust
H to minimize the output error,
? And eventually, they retrain with ternary weights during propagations and high-precision
weights during updates.
By comparison, we train all the way with binary weights during propagations, i.e., our training procedure could be implemented with efficient specialized hardware avoiding the forward and backward
propagations multiplications, which amounts to about 2/3 of the multiplications (cf. Algorithm 1).
5
Conclusion and future works
We have introduced a novel binarization scheme for weights during forward and backward propagations called BinaryConnect. We have shown that it is possible to train DNNs with BinaryConnect on
the permutation invariant MNIST, CIFAR-10 and SVHN datasets and achieve nearly state-of-the-art
results. The impact of such a method on specialized hardware implementations of deep networks
could be major, by removing the need for about 2/3 of the multiplications, and thus potentially allowing to speed-up by a factor of 3 at training time. With the deterministic version of BinaryConnect
the impact at test time could be even more important, getting rid of the multiplications altogether
and reducing by a factor of at least 16 (from 16 bits single-float precision to single bit precision)
the memory requirement of deep networks, which has an impact on the memory to computation
bandwidth and on the size of the models that can be run. Future works should extend those results to
other models and datasets, and explore getting rid of the multiplications altogether during training,
by removing their need from the weight update computation.
7
6
Acknowledgments
We thank the reviewers for their many constructive comments. We also thank Roland Memisevic for
helpful discussions. We thank the developers of Theano [42, 43], a Python library which allowed
us to easily develop a fast and optimized code for GPU. We also thank the developers of Pylearn2
[44] and Lasagne, two Deep Learning libraries built on the top of Theano. We are also grateful for
funding from NSERC, the Canada Research Chairs, Compute Canada, and CIFAR.
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5,134 | 5,648 | Learning to Transduce with Unbounded Memory
Edward Grefenstette
Google DeepMind
[email protected]
Karl Moritz Hermann
Google DeepMind
[email protected]
Mustafa Suleyman
Google DeepMind
[email protected]
Phil Blunsom
Google DeepMind and Oxford University
[email protected]
Abstract
Recently, strong results have been demonstrated by Deep Recurrent Neural Networks on natural language transduction problems. In this paper we explore the
representational power of these models using synthetic grammars designed to exhibit phenomena similar to those found in real transduction problems such as machine translation. These experiments lead us to propose new memory-based recurrent networks that implement continuously differentiable analogues of traditional
data structures such as Stacks, Queues, and DeQues. We show that these architectures exhibit superior generalisation performance to Deep RNNs and are often able
to learn the underlying generating algorithms in our transduction experiments.
1 Introduction
Recurrent neural networks (RNNs) offer a compelling tool for processing natural language input in
a straightforward sequential manner. Many natural language processing (NLP) tasks can be viewed
as transduction problems, that is learning to convert one string into another. Machine translation is
a prototypical example of transduction and recent results indicate that Deep RNNs have the ability
to encode long source strings and produce coherent translations [1, 2]. While elegant, the application of RNNs to transduction tasks requires hidden layers large enough to store representations
of the longest strings likely to be encountered, implying wastage on shorter strings and a strong
dependency between the number of parameters in the model and its memory.
In this paper we use a number of linguistically-inspired synthetic transduction tasks to explore the
ability of RNNs to learn long-range reorderings and substitutions. Further, inspired by prior work on
neural network implementations of stack data structures [3], we propose and evaluate transduction
models based on Neural Stacks, Queues, and DeQues (double ended queues). Stack algorithms are
well-suited to processing the hierarchical structures observed in natural language and we hypothesise
that their neural analogues will provide an effective and learnable transduction tool. Our models
provide a middle ground between simple RNNs and the recently proposed Neural Turing Machine
(NTM) [4] which implements a powerful random access memory with read and write operations.
Neural Stacks, Queues, and DeQues also provide a logically unbounded memory while permitting
efficient constant time push and pop operations.
Our results indicate that the models proposed in this work, and in particular the Neural DeQue, are
able to consistently learn a range of challenging transductions. While Deep RNNs based on long
short-term memory (LSTM) cells [1, 5] can learn some transductions when tested on inputs of the
same length as seen in training, they fail to consistently generalise to longer strings. In contrast,
our sequential memory-based algorithms are able to learn to reproduce the generating transduction
algorithms, often generalising perfectly to inputs well beyond those encountered in training.
1
2 Related Work
String transduction is central to many applications in NLP, from name transliteration and spelling
correction, to inflectional morphology and machine translation. The most common approach leverages symbolic finite state transducers [6, 7], with approaches based on context free representations
also being popular [8]. RNNs offer an attractive alternative to symbolic transducers due to their simple algorithms and expressive representations [9]. However, as we show in this work, such models
are limited in their ability to generalise beyond their training data and have a memory capacity that
scales with the number of their trainable parameters.
Previous work has touched on the topic of rendering discrete data structures such as stacks continuous, especially within the context of modelling pushdown automata with neural networks [10, 11, 3].
We were inspired by the continuous pop and push operations of these architectures and the idea of
an RNN controlling the data structure when developing our own models. The key difference is that
our work adapts these operations to work within a recurrent continuous Stack/Queue/DeQue-like
structure, the dynamics of which are fully decoupled from those of the RNN controlling it. In our
models, the backwards dynamics are easily analysable in order to obtain the exact partial derivatives
for use in error propagation, rather than having to approximate them as done in previous work.
In a parallel effort to ours, researchers are exploring the addition of memory to recurrent networks.
The NTM and Memory Networks [4, 12, 13] provide powerful random access memory operations,
whereas we focus on a more efficient and restricted class of models which we believe are sufficient
for natural language transduction tasks. More closely related to our work, [14] have sought to
develop a continuous stack controlled by an RNN. Note that this model?unlike the work proposed
here?renders discrete push and pop operations continuous by ?mixing? information across levels of
the stack at each time step according to scalar push/pop action values. This means the model ends up
compressing information in the stack, thereby limiting its use, as it effectively loses the unbounded
memory nature of traditional symbolic models.
3 Models
In this section, we present an extensible memory enhancement to recurrent layers which can be set
up to act as a continuous version of a classical Stack, Queue, or DeQue (double-ended queue). We
begin by describing the operations and dynamics of a neural Stack, before showing how to modify
it to act as a Queue, and extend it to act as a DeQue.
3.1
Neural Stack
Let a Neural Stack be a differentiable structure onto and from which continuous vectors are pushed
and popped. Inspired by the neural pushdown automaton of [3], we render these traditionally discrete operations continuous by letting push and pop operations be real values in the interval (0, 1).
Intuitively, we can interpret these values as the degree of certainty with which some controller wishes
to push a vector v onto the stack, or pop the top of the stack.
Vt [i] =
st [i] =
rt =
?
8
<
:
t
X
i=1
Vt
vt
1 [i]
if 1 ? i < t
if i = t
max(0, st
1 [i]
(Note that Vt [i] = vi for all i ? t)
max(0, ut
tP1
st
1 [j]))
j=i+1
dt
(min(st [i], max(0, 1
t
X
j=i+1
st [j]))) ? Vt [i]
if 1 ? i < t
(1)
(2)
if i = t
(3)
Formally, a Neural Stack, fully parametrised by an embedding size m, is described at some timestep
t by a t ? m value matrix Vt and a strength vector st 2 Rt . These form the core of a recurrent layer
which is acted upon by a controller by receiving, from the controller, a value vt 2 Rm , a pop signal
ut 2 (0, 1), and a push signal dt 2 (0, 1). It outputs a read vector rt 2 Rm . The recurrence of this
2
layer comes from the fact that it will receive as previous state of the stack the pair (Vt 1 , st 1 ), and
produce as next state the pair (Vt , st ) following the dynamics described below. Here, Vt [i] represents
the ith row (an m-dimensional vector) of Vt and st [i] represents the ith value of st .
Equation 1 shows the update of the value component of the recurrent layer state represented as a
matrix, the number of rows of which grows with time, maintaining a record of the values pushed to
the stack at each timestep (whether or not they are still logically on the stack). Values are appended
to the bottom of the matrix (top of the stack) and never changed.
Equation 2 shows the effect of the push and pop signal in updating the strength vector st 1 to
produce st . First, the pop operation removes objects from the stack. We can think of the pop value
ut as the initial deletion quantity for the operation. We traverse the strength vector st 1 from the
highest index to the lowest. If the next strength scalar is less than the remaining deletion quantity, it
is subtracted from the remaining quantity and its value is set to 0. If the remaining deletion quantity
is less than the next strength scalar, the remaining deletion quantity is subtracted from that scalar and
deletion stops. Next, the push value is set as the strength for the value added in the current timestep.
Equation 3 shows the dynamics of the read operation, which are similar to the pop operation. A
fixed initial read quantity of 1 is set at the top of a temporary copy of the strength vector st which
is traversed from the highest index to the lowest. If the next strength scalar is smaller than the
remaining read quantity, its value is preserved for this operation and subtracted from the remaining
read quantity. If not, it is temporarily set to the remaining read quantity, and the strength scalars of
all lower indices are temporarily set to 0. The output rt of the read operation is the weighted sum
of the rows of Vt , scaled by the temporary scalar values created during the traversal. An example
of the stack read calculations across three timesteps, after pushes and pops as described above, is
illustrated in Figure 1a. The third step shows how setting the strength s3 [2] to 0 for V3 [2] logically
removes v2 from the stack, and how it is ignored during the read.
This completes the description of the forward dynamics of a neural Stack, cast as a recurrent layer,
as illustrated in Figure 1b. All operations described in this section are differentiable1 . The equations
describing the backwards dynamics are provided in Appendix A of the supplementary materials.
stack grows upwards
t = 1 u1 = 0 d1 = 0.8
t = 3 u3 = 0.9 d3 = 0.9
t = 2 u2 = 0.1 d2 = 0.5
row 3
row 2
v1
row 1
0.8
r1 = 0.8 ? v1
v3
0.9
v2
0.5
v2
0
v1
0.7
v1
0.3
r2 = 0.5 ? v2 + 0.5 ? v1
v2 removed
from stack
r3 = 0.9 ? v3 + 0 ? v2 + 0.1 ? v1
(a) Example Operation of a Continuous Neural Stack
(Vt-1, st-1)
prev. values (Vt-1)
next values (Vt)
next state
previous state
previous
state
prev. strengths (st-1)
push (dt)
input
pop (ut)
Neural
Stack
value (vt)
output (rt)
R
N
N
rt-1
next strengths (st)
input
it
Split
Vt-1
ht-1
Ht-1
(it, rt-1)
Vt
ht
next
st-1
(ot, ?)
ot
?
dt
ut
Neural
Stack
(Vt, st)
st
Ht
rt
vt
Join
(b) Neural Stack as a Recurrent Layer
state
output
ot
(c) RNN Controlling a Stack
Figure 1: Illustrating a Neural Stack?s Operations, Recurrent Structure, and Control
3.2
Neural Queue
A neural Queue operates the same way as a neural Stack, with the exception that the pop operation
reads the lowest index of the strength vector st , rather than the highest. This represents popping and
1
The max(x, y) and min(x, y) functions are technically not differentiable for x = y. Following the work
on rectified linear units [15], we arbitrarily take the partial differentiation of the left argument in these cases.
3
reading from the front of the Queue rather than the top of the stack. These operations are described
in Equations 4?5.
st [i] =
8
<
:
max(0, st
1 [i]
iP1
max(0, ut
dt
rt =
t
X
1 [j]))
if 1 ? i < t
(4)
if i = t
i 1
X
(min(st [i], max(0, 1
i=1
3.3
st
j=1
j=1
(5)
st [j]))) ? Vt [i]
Neural DeQue
A neural DeQue operates likes a neural Stack, except it takes a push, pop, and value as input for
both ?ends? of the structure (which we call top and bot), and outputs a read for both ends. We write
utop
and ubot
instead of ut , vttop and vtbot instead of vt , and so on. The state, Vt and st are now
t
t
a 2t ? m-dimensional matrix and a 2t-dimensional vector, respectively. At each timestep, a pop
from the top is followed by a pop from the bottom of the DeQue, followed by the pushes and reads.
The dynamics of a DeQue, which unlike a neural Stack or Queue ?grows? in two directions, are
described in Equations 6?11, below. Equations 7?9 decompose the strength vector update into three
steps purely for notational clarity.
8
< vtbot
Vt [i] =
vtop
: t
Vt 1 [i
stop
t [i] = max(0, st
1]
if i = 1
if i = 2t
if 1 < i < 2t
(6)
2(t 1) 1
1 [i]
X
max(0, utop
t
st
1 [j]))
j=i+1
sboth
[i] = max(0, stop
t [i]
t
8 both
< st [i
dbot
st [i] =
: ttop
dt
rtop
=
t
2t
X
2t
X
stop
t [j]))
j=1
1]
if 1 < i < 2t
if i = 1
if i = 2t
(min(st [i], max(0, 1
i=1
rbot
=
t
i 1
X
max(0, ubot
t
2t
X
i=1
i 1
X
j=1
if 1 ? i < 2(t
1)
1)
(7)
(8)
(9)
j=i+1
(min(st [i], max(0, 1
if 1 ? i < 2(t
st [j]))) ? Vt [i]
st [j]))) ? Vt [i]
(10)
(11)
To summarise, a neural DeQue acts like two neural Stacks operated on in tandem, except that the
pushes and pops from one end may eventually affect pops and reads on the other, and vice versa.
3.4
Interaction with a Controller
While the three memory modules described can be seen as recurrent layers, with the operations being
used to produce the next state and output from the input and previous state being fully differentiable,
they contain no tunable parameters to optimise during training. As such, they need to be attached
to a controller in order to be used for any practical purposes. In exchange, they offer an extensible
memory, the logical size of which is unbounded and decoupled from both the nature and parameters
of the controller, and from the size of the problem they are applied to. Here, we describe how any
RNN controller may be enhanced by a neural Stack, Queue or DeQue.
We begin by giving the case where the memory is a neural Stack, as illustrated in Figure 1c. Here
we wish to replicate the overall ?interface? of a recurrent layer?as seen from outside the dotted
4
lines?which takes the previous recurrent state Ht 1 and an input vector it , and transforms them
to return the next recurrent state Ht and an output vector ot . In our setup, the previous state Ht 1
of the recurrent layer will be the tuple (ht 1 , rt 1 , (Vt 1 , st 1 )), where ht 1 is the previous state
of the RNN, rt 1 is the previous stack read, and (Vt 1 , st 1 ) is the previous state of the stack
as described above. With the exception of h0 , which is initialised randomly and optimised during
training, all other initial states, r0 and (V0 , s0 ), are set to 0-valued vectors/matrices and not updated
during training.
The overall input it is concatenated with previous read rt 1 and passed to the RNN controller as
input along with the previous controller state ht 1 . The controller outputs its next state ht and a
controller output o0t , from which we obtain the push and pop scalars dt and ut and the value vector
vt , which are passed to the stack, as well as the network output ot :
dt = sigmoid(Wd o0t + bd )
ut = sigmoid(Wu o0t + bu )
vt = tanh(Wv o0t + bv )
ot = tanh(Wo o0t + bo )
where Wd and Wu are vector-to-scalar projection matrices, and bd and bu are their scalar biases;
Wv and Wo are vector-to-vector projections, and bd and bu are their vector biases, all randomly
intialised and then tuned during training. Along with the previous stack state (Vt 1 , st 1 ), the stack
operations dt and ut and the value vt are passed to the neural stack to obtain the next read rt and
next stack state (Vt , st ), which are packed into a tuple with the controller state ht to form the next
state Ht of the overall recurrent layer. The output vector ot serves as the overall output of the
recurrent layer. The structure described here can be adapted to control a neural Queue instead of a
stack by substituting one memory module for the other.
The only additional trainable parameters in either configuration, relative to a non-enhanced RNN,
are the projections for the input concatenated with the previous read into the RNN controller, and the
projections from the controller output into the various Stack/Queue inputs, described above. In the
case of a DeQue, both the top read rtop and bottom read rbot must be preserved in the overall state.
They are both concatenated with the input to form the input to the RNN controller. The output of the
controller must have additional projections to output push/pop operations and values for the bottom
of the DeQue. This roughly doubles the number of additional tunable parameters ?wrapping? the
RNN controller, compared to the Stack/Queue case.
4 Experiments
In every experiment, integer-encoded source and target sequence pairs are presented to the candidate
model as a batch of single joint sequences. The joint sequence starts with a start-of-sequence (SOS)
symbol, and ends with an end-of-sequence (EOS) symbol, with a separator symbol separating the
source and target sequences. Integer-encoded symbols are converted to 64-dimensional embeddings
via an embedding matrix, which is randomly initialised and tuned during training. Separate wordto-index mappings are used for source and target vocabularies. Separate embedding matrices are
used to encode input and output (predicted) embeddings.
4.1
Synthetic Transduction Tasks
The aim of each of the following tasks is to read an input sequence, and generate as target sequence a
transformed version of the source sequence, followed by an EOS symbol. Source sequences are randomly generated from a vocabulary of 128 meaningless symbols. The length of each training source
sequence is uniformly sampled from unif {8, 64}, and each symbol in the sequence is drawn with
replacement from a uniform distribution over the source vocabulary (ignoring SOS, and separator).
A deterministic task-specific transformation, described for each task below, is applied to the source
sequence to yield the target sequence. As the training sequences are entirely determined by the
source sequence, there are close to 10135 training sequences for each task, and training examples
are sampled from this space due to the random generation of source sequences. The following steps
are followed before each training and test sequence are presented to the models, the SOS symbol
(hsi) is prepended to the source sequence, which is concatenated with a separator symbol (|||) and
the target sequences, to which the EOS symbol (h/si) is appended.
5
Sequence Copying The source sequence is copied to form the target sequence. Sequences have
the form:
hsia1 . . . ak |||a1 . . . ak h/si
Sequence Reversal The source sequence is deterministically reversed to produce the target sequence. Sequences have the form:
hsia1 a2 . . . ak |||ak . . . a2 a1 h/si
Bigram flipping The source side is restricted to even-length sequences. The target is produced
by swapping, for all odd source sequence indices i 2 [1, |seq|] ^ odd(i), the ith symbol with the
(i + 1)th symbol. Sequences have the form:
hsia1 a2 a3 a4 . . . ak
4.2
1 ak |||a2 a1 a4 a3
. . . ak ak
1 h/si
ITG Transduction Tasks
The following tasks examine how well models can approach sequence transduction problems where
the source and target sequence are jointly generated by Inversion Transduction Grammars (ITG) [8],
a subclass of Synchronous Context-Free Grammars [16] often used in machine translation [17]. We
present two simple ITG-based datasets with interesting linguistic properties and their underlying
grammars. We show these grammars in Table 1, in Appendix C of the supplementary materials. For
each synchronised non-terminal, an expansion is chosen according to the probability distribution
specified by the rule probability p at the beginning of each rule. For each grammar, ?A? is always the
root of the ITG tree.
We tuned the generative probabilities for recursive rules by hand so that the grammars generate left
and right sequences of lengths 8 to 128 with relatively uniform distribution. We generate training
data by rejecting samples that are outside of the range [8, 64], and testing data by rejecting samples
outside of the range [65, 128]. For terminal symbol-generating rules, we balance the classes so
that for k terminal-generating symbols in the grammar, each terminal-generating non-terminal ?X?
generates a vocabulary of approximately 128/k, and each each vocabulary word under that class is
equiprobable. These design choices were made to maximise the similarity between the experimental
settings of the ITG tasks described here and the synthetic tasks described above.
Subj?Verb?Obj to Subj?Obj?Verb A persistent challenge in machine translation is to learn to
faithfully reproduce high-level syntactic divergences between languages. For instance, when translating an English sentence with a non-finite verb into German, a transducer must locate and move
the verb over the object to the final position. We simulate this phenomena with a synchronous
grammar which generates strings exhibiting verb movements. To add an extra challenge, we also
simulate simple relative clause embeddings to test the models? ability to transduce in the presence
of unbounded recursive structures.
A sample output of the grammar is presented here, with spaces between words being included for
stylistic purposes, and where s, o, and v indicate subject, object, and verb terminals respectively, i
and o mark input and output, and rp indicates a relative pronoun:
si1 vi28 oi5 oi7 si15 rpi si19 vi16 oi10 oi24 ||| so1 oo5 oo7 so15 rpo so19 vo16 oo10 oo24 vo28
Genderless to gendered grammar We design a small grammar to simulate translations from a
language with gender-free articles to one with gender-specific definite and indefinite articles. A
real world example of such a translation would be from English (the, a) to German (der/die/das,
ein/eine/ein).
The grammar simulates sentences in (N P/(V /N P )) or (N P/V ) form, where every noun phrase
can become an infinite sequence of nouns joined by a conjunction. Each noun in the source language
has a neutral definite or indefinite article. The matching word in the target language then needs to be
preceeded by its appropriate article. A sample output of the grammar is presented here, with spaces
between words being included for stylistic purposes:
we11 the en19 and the em17 ||| wg11 das gn19 und der gm17
6
4.3
Evaluation
For each task, test data is generated through the same procedure as training data, with the key difference that the length of the source sequence is sampled from unif {65, 128}. As a result of this
change, we not only are assured that the models cannot observe any test sequences during training,
but are also measuring how well the sequence transduction capabilities of the evaluated models generalise beyond the sequence lengths observed during training. To control for generalisation ability,
we also report accuracy scores on sequences separately sampled from the training set, which given
the size of the sample space are unlikely to have ever been observed during actual model training.
For each round of testing, we sample 1000 sequences from the appropriate test set. For each sequence, the model reads in the source sequence and separator symbol, and begins generating the
next symbol by taking the maximally likely symbol from the softmax distribution over target symbols produced by the model at each step. Based on this process, we give each model a coarse
accuracy score, corresponding to the proportion of test sequences correctly predicted from beginning until end (EOS symbol) without error, as well as a fine accuracy score, corresponding to the
average proportion of each sequence correctly generated before the first error. Formally, we have:
coarse =
#correct
#seqs
f ine =
#seqs
X #correcti
1
#seqs i=1 |targeti |
where #correct and #seqs are the number of correctly predicted sequences (end-to-end) and the
total number of sequences in the test batch (1000 in this experiment), respectively; #correcti is the
number of correctly predicted symbols before the first error in the ith sequence of the test batch, and
|targeti | is the length of the target segment that sequence (including EOS symbol).
4.4
Models Compared and Experimental Setup
For each task, we use as benchmarks the Deep LSTMs described in [1], with 1, 2, 4, and 8 layers.
Against these benchmarks, we evaluate neural Stack-, Queue-, and DeQue-enhanced LSTMs. When
running experiments, we trained and tested a version of each model where all LSTMs in each model
have a hidden layer size of 256, and one for a hidden layer size of 512. The Stack/Queue/DeQue
embedding size was arbitrarily set to 256, half the maximum hidden size. The number of parameters
for each model are reported for each architecture in Table 2 of the appendix. Concretely, the neural
Stack-, Queue-, and DeQue-enhanced LSTMs have the same number of trainable parameters as a
two-layer Deep LSTM. These all come from the extra connections to and from the memory module,
which itself has no trainable parameters, regardless of its logical size.
Models are trained with minibatch RMSProp [18], with a batch size of 10. We grid-searched learning
rates across the set {5 ? 10 3 , 1 ? 10 3 , 5 ? 10 4 , 1 ? 10 4 , 5 ? 10 5 }. We used gradient clipping
[19], clipping all gradients above 1. Average training perplexity was calculated every 100 batches.
Training and test set accuracies were recorded every 1000 batches.
5 Results and Discussion
Because of the impossibility of overfitting the datasets, we let the models train an unbounded number
of steps, and report results at convergence. We present in Figure 2a the coarse- and fine-grained
accuracies, for each task, of the best model of each architecture described in this paper alongside
the best performing Deep LSTM benchmark. The best models were automatically selected based on
average training perplexity. The LSTM benchmarks performed similarly across the range of random
initialisations, so the effect of this procedure is primarily to try and select the better performing
Stack/Queue/DeQue-enhanced LSTM. In most cases, this procedure does not yield the actual bestperforming model, and in practice a more sophisticated procedure such as ensembling [20] should
produce better results.
For all experiments, the Neural Stack or Queue outperforms the Deep LSTM benchmarks, often by
a significant margin. For most experiments, if a Neural Stack- or Queue-enhanced LSTM learns
to partially or consistently solve the problem, then so does the Neural DeQue. For experiments
where the enhanced LSTMs solve the problem completely (consistent accuracy of 1) in training,
the accuracy persists in longer sequences in the test set, whereas benchmark accuracies drop for
7
Training
Experiment
Testing
Model
4-layer LSTM
Stack-LSTM
Queue-LSTM
DeQue-LSTM
Coarse
0.98
0.89
1.00
1.00
Fine
0.98
0.94
1.00
1.00
Coarse
0.01
0.00
1.00
1.00
Fine
0.50
0.22
1.00
1.00
Sequence
Reversal
8-layer LSTM
Stack-LSTM
Queue-LSTM
DeQue-LSTM
0.95
1.00
0.44
1.00
0.98
1.00
0.61
1.00
0.04
1.00
0.00
1.00
0.13
1.00
0.01
1.00
Bigram
Flipping
2-layer LSTM
Stack-LSTM
Queue-LSTM
DeQue-LSTM
0.54
0.44
0.55
0.55
0.93
0.90
0.94
0.94
0.02
0.00
0.55
0.53
0.52
0.48
0.98
0.98
SVO to SOV
8-layer LSTM
Stack-LSTM
Queue-LSTM
DeQue-LSTM
0.98
1.00
1.00
1.00
0.99
1.00
1.00
1.00
0.98
1.00
1.00
1.00
0.99
1.00
1.00
1.00
Gender
Conjugation
8-layer LSTM
Stack-LSTM
Queue-LSTM
DeQue-LSTM
0.98
0.93
1.00
1.00
0.99
0.97
1.00
1.00
0.99
0.93
1.00
1.00
0.99
0.97
1.00
1.00
Sequence
Copying
(a) Comparing Enhanced LSTMs to Best Benchmarks
(b) Comparison of Model Convergence during Training
Figure 2: Results on the transduction tasks and convergence properties
all experiments except the SVO to SOV and Gender Conjugation ITG transduction tasks. Across
all tasks which the enhanced LSTMs solve, the convergence on the top accuracy happens orders of
magnitude earlier for enhanced LSTMs than for benchmark LSTMs, as exemplified in Figure 2b.
The results for the sequence inversion and copying tasks serve as unit tests for our models, as the
controller mainly needs to learn to push the appropriate number of times and then pop continuously.
Nonetheless, the failure of Deep LSTMs to learn such a regular pattern and generalise is itself
indicative of the limitations of the benchmarks presented here, and of the relative expressive power
of our models. Their ability to generalise perfectly to sequences up to twice as long as those attested
during training is also notable, and also attested in the other experiments. Finally, this pair of
experiments illustrates how while the neural Queue solves copying and the Stack solves reversal, a
simple LSTM controller can learn to operate a DeQue as either structure, and solve both tasks.
The results of the Bigram Flipping task for all models are consistent with the failure to consistently
correctly generate the last two symbols of the sequence. We hypothesise that both Deep LSTMs and
our models economically learn to pairwise flip the sequence tokens, and attempt to do so half the
time when reaching the EOS token. For the two ITG tasks, the success of Deep LSTM benchmarks
relative to their performance in other tasks can be explained by their ability to exploit short local
dependencies dominating the longer dependencies in these particular grammars.
Overall, the rapid convergence, where possible, on a general solution to a transduction problem
in a manner which propagates to longer sequences without loss of accuracy is indicative that an
unbounded memory-enhanced controller can learn to solve these problems procedurally, rather than
memorising the underlying distribution of the data.
6 Conclusions
The experiments performed in this paper demonstrate that single-layer LSTMs enhanced by an unbounded differentiable memory capable of acting, in the limit, like a classical Stack, Queue, or
DeQue, are capable of solving sequence-to-sequence transduction tasks for which Deep LSTMs
falter. Even in tasks for which benchmarks obtain high accuracies, the memory-enhanced LSTMs
converge earlier, and to higher accuracies, while requiring considerably fewer parameters than all
but the simplest of Deep LSTMs. We therefore believe these constitute a crucial addition to our neural network toolbox, and that more complex linguistic transduction tasks such as machine translation
or parsing will be rendered more tractable by their inclusion.
8
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9
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5,135 | 5,649 | Spectral Representations for
Convolutional Neural Networks
Oren Rippel
Department of Mathematics
Massachusetts Institute of Technology
Jasper Snoek
Twitter and Harvard SEAS
[email protected]
[email protected]
Ryan P. Adams
Twitter and Harvard SEAS
[email protected]
Abstract
Discrete Fourier transforms provide a significant speedup in the computation of
convolutions in deep learning. In this work, we demonstrate that, beyond its advantages for efficient computation, the spectral domain also provides a powerful
representation in which to model and train convolutional neural networks (CNNs).
We employ spectral representations to introduce a number of innovations to CNN
design. First, we propose spectral pooling, which performs dimensionality reduction by truncating the representation in the frequency domain. This approach
preserves considerably more information per parameter than other pooling strategies and enables flexibility in the choice of pooling output dimensionality. This
representation also enables a new form of stochastic regularization by randomized modification of resolution. We show that these methods achieve competitive
results on classification and approximation tasks, without using any dropout or
max-pooling.
Finally, we demonstrate the effectiveness of complex-coefficient spectral parameterization of convolutional filters. While this leaves the underlying model unchanged, it results in a representation that greatly facilitates optimization. We
observe on a variety of popular CNN configurations that this leads to significantly
faster convergence during training.
1
Introduction
Convolutional neural networks (CNNs) (LeCun et al., 1989) have been used to achieve unparalleled results across a variety of benchmark machine learning problems, and have been applied
successfully throughout science and industry for tasks such as large scale image and video classification (Krizhevsky et al., 2012; Karpathy et al., 2014). One of the primary challenges of CNNs,
however, is the computational expense necessary to train them. In particular, the efficient implementation of convolutional kernels has been a key ingredient of any successful use of CNNs at scale.
Due to its efficiency and the potential for amortization of cost, the discrete Fourier transform has long
been considered by the deep learning community to be a natural approach to fast convolution (Bengio & LeCun, 2007). More recently, Mathieu et al. (2013); Vasilache et al. (2014) have demonstrated
that convolution can be computed significantly faster using discrete Fourier transforms than directly
in the spatial domain, even for tiny filters. This computational gain arises from the convenient property of operator duality between convolution in the spatial domain and element-wise multiplication
in the frequency domain.
In this work, we argue that the frequency domain offers more than a computational trick for convolution: it also provides a powerful representation for modeling and training CNNs. Frequency
decomposition allows studying an input across its various length-scales of variation, and as such
provides a natural framework for the analysis of data with spatial coherence. We introduce two
1
applications of spectral representations. These contributions can be applied independently of each
other.
Spectral parametrization We propose the idea of learning the filters of CNNs directly in the
frequency domain. Namely, we parametrize them as maps of complex numbers, whose discrete
Fourier transforms correspond to the usual filter representations in the spatial domain.
Because this mapping corresponds to unitary transformations of the filters, this reparametrization
does not alter the underlying model. However, we argue that the spectral representation provides an
appropriate domain for parameter optimization, as the frequency basis captures typical filter structure well. More specifically, we show that filters tend to be considerably sparser in their spectral representations, thereby reducing the redundancy that appears in spatial domain representations. This
provides the optimizer with more meaningful axis-aligned directions that can be taken advantage of
with standard element-wise preconditioning.
We demonstrate the effectiveness of this reparametrization on a number of CNN optimization tasks,
converging 2-5 times faster than the standard spatial representation.
Spectral pooling Pooling refers to dimensionality reduction used in CNNs to impose a capacity
bottleneck and facilitate computation. We introduce a new approach to pooling we refer to as spectral pooling. It performs dimensionality reduction by projecting onto the frequency basis set and
then truncating the representation.
This approach alleviates a number of issues present in existing pooling strategies. For example,
while max pooling is featured in almost every CNN and has had great empirical success, one major
criticism has been its poor preservation of information (Hinton, 2014b,a). This weakness is exhibited in two ways. First, along with other stride-based pooling approaches, it implies a very sharp
dimensionality reduction by at least a factor of 4 every time it is applied on two-dimensional inputs.
Moreover, while it encourages translational invariance, it does not utilize its capacity well to reduce
approximation loss: the maximum value in each window only reflects very local information, and
often does not represent well the contents of the window.
In contrast, we show that spectral pooling preserves considerably more information for the same
number of parameters. It achieves this by exploiting the non-uniformity of typical inputs in their
signal-to-noise ratio as a function of frequency. For example, natural images are known to have an
expected power spectrum that follows an inverse power law: power is heavily concentrated in the
lower frequencies ? while higher frequencies tend to encode noise (Torralba & Oliva, 2003). As
such, the elimination of higher frequencies in spectral pooling not only does minimal damage to the
information in the input, but can even be viewed as a type of denoising.
In addition, spectral pooling allows us to specify any arbitrary output map dimensionality. This permits reduction of the map dimensionality in a slow and controlled manner as a function of network
depth. Also, since truncation of the frequency representation exactly corresponds to reduction in
resolution, we can supplement spectral pooling with stochastic regularization in the form of randomized resolution.
Spectral pooling can be implemented at a negligible additional computational cost in convolutional
neural networks that employ FFT for convolution kernels, as it only requires matrix truncation. We
also note that these two ideas are both compatible with the recently-introduced method of batch
normalization (Ioffe & Szegedy, 2015), permitting even better training efficiency.
2
The Discrete Fourier Transform
The discrete Fourier transform (DFT) is a powerful way to decompose a spatiotemporal signal. In
this section, we provide an introduction to a number of components of the DFT drawn upon in
this work. We confine ourselves to the two-dimensional DFT, although all properties and results
presented can be easily extended to other input dimensions.
Given an input x ? CM ?N (we address the constraint of real inputs in Subsection 2.1), its 2D
DFT F (x) ? CM ?N is given by
F (x)hw = ?
M
?1 N
?1
X
X
mh
nw
1
xmn e?2?i( M + N )
M N m=0 n=0
?h ? {0, . . . , M ? 1}, ?w ? {0, . . . , N ? 1} .
The DFT is linear and unitary, and so its inverse transform is given by F ?1 (?) = F (?)? , namely the
conjugate of the transform itself.
2
(a) DFT basis functions.
(b) Examples of input-transform pairs.
(c)
Conjugate Symm.
Figure 1: Properties of discrete Fourier transforms. (a) All discrete Fourier basis functions of map
size 8 ? 8. Note the equivalence of some of these due to conjugate symmetry. (b) Examples of
input images and their frequency representations, presented as log-amplitudes. The frequency maps
have been shifted to center the DC component. Rays in the frequency domain correspond to spatial
domain edges aligned perpendicular to these. (c) Conjugate symmetry patterns for inputs with odd
(top) and even (bottom) dimensionalities. Orange: real-valuedness constraint. Blue: no constraint.
Gray: value fixed by conjugate symmetry.
Intuitively, the DFT coefficients resulting from projections onto the different frequencies can be
thought of as measures of correlation of the input with basis functions of various length-scales. See
Figure 1(a) for a visualization of the DFT basis functions, and Figure 1(b) for examples of inputfrequency map pairs.
The widespread deployment of the DFT can be partially attributed to the development of the
Fast Fourier Transform (FFT), a mainstay of signal processing and a standard component of
most math libraries. The FFT is an efficient implementation of the DFT with time complexity O (M N log (M N )).
Convolution using DFT One powerful property of frequency analysis is the operator duality between convolution in the spatial domain and element-wise multiplication in the spectral domain.
Namely, given two inputs x, f ? RM ?N , we may write
F (x ? f ) = F (x) F (f )
(1)
where by ? we denote a convolution and by an element-wise product.
Approximation error The unitarity of the Fourier basis makes it convenient for the analysis of
approximation loss. More specifically, Parseval?s Theorem links the `2 loss between any input x and
? to the corresponding loss in the frequency domain:
its approximation x
? k22 = kF (x) ? F (?
kx ? x
x)k22 .
(2)
An equivalent statement also holds for the inverse DFT operator. This allows us to quickly assess
how an input is affected by any distortion we might make to its frequency representation.
2.1
Conjugate symmetry constraints
In the following sections of the paper, we will propagate signals and their gradients through DFT
and inverse DFT layers. In these layers, we will represent the frequency domain in the complex
field. However, for all layers apart from these, we would like to ensure that both the signal and its
gradient are constrained to the reals. A necessary and sufficient condition to achieve this is conjugate
symmetry in the frequency domain. Namely, for any transform y = F (x) of some input x, it must
hold that
?
ymn = y(M
?m ? {0, . . . , M ? 1}, ?n ? {0, . . . , N ? 1} . (3)
?m) modM,(N ?n) modN
Thus, intuitively, given the left half of our frequency map, the diminished number of degrees of
freedom allows us to reconstruct the right. In effect, this allows us to store approximately half the
parameters that would otherwise be necessary. Note, however, that this does not reduce the effective dimensionality, since each element consists of real and imaginary components. The conjugate
symmetry constraints are visualized in Figure 1(c). Given a real input, its DFT will necessarily meet
these. This symmetry can be observed in the frequency representations of the examples in Figure
1(b). However, since we seek to optimize over parameters embedded directly in the frequency domain, we need to pay close attention to ensure the conjugate symmetry constraints are enforced upon
inversion back to the spatial domain (see Subsection 2.2).
3
2.2
Differentiation
Here we discuss how to propagate the gradient through a Fourier transform layer. This analysis can
be similarly applied to the inverse DFT layer. Define x ? RM ?N and y = F (x) to be the input and
output of a DFT layer respectively, and R : RM ?N ? R a real-valued loss function applied to y
which can be considered as the remainder of the forward pass. Since the DFT is a linear operator,
its gradient is simply the transformation matrix itself. During back-propagation, then, this gradient
is conjugated, and this, by DFT unitarity, corresponds to the application of the inverse transform:
?R
= F ?1
?x
?R
?y
.
(4)
There is an intricacy that makes matters a bit more complicated. Namely, the conjugate symmetry
condition discussed in Subsection 2.1 introduces redundancy. Inspecting the conjugate symmetry constraints in Equation (3), we note their enforcement of the special case y00 ? R for N odd,
and y00 , y N ,0 , y0, N , y N , N ? R for N even. For all other indices they enforce conjugate equality
2
2
2 2
of pairs of distinct elements. These conditions imply that the number of unconstrained parameters
is about half the map in its entirety.
3
Spectral Pooling
The choice of a pooling technique boils down to the selection of an appropriate set of basis functions to project onto, and some truncation of this representation to establish a lower-dimensionality
approximation to the original input. The idea behind spectral pooling stems from the observation
that the frequency domain provides an ideal basis for inputs with spatial structure. We first discuss
the technical details of this approach, and then its advantages.
Spectral pooling is straightforward to understand and to implement. We assume we are given an
input x ? RM ?N , and some desired output map dimensionality H ? W . First, we compute the discrete Fourier transform of the input into the frequency domain as y = F (x) ? CM ?N , and assume
that the DC component has been shifted to the center of the domain as is standard practice. We
then crop the frequency representation by maintaining only the central H ? W submatrix of fre? ? CH?W . Finally, we map this approximation back into the spatial
quencies, which we denote as y
? = F ?1 (?
domain by taking its inverse DFT as x
y) ? RH?W . These steps are listed in Algorithm
1. Note that some of the conjugate symmetry special cases described in Subsection 2.2 might be
? is real-valued, we must treat these individually
broken by this truncation. As such, to ensure that x
with T REAT C ORNER C ASES, which can be found in the supplementary material.
Figure 2 demonstrates the effect of this pooling for various choices of H ? W . The backpropagation procedure is quite intuitive, and can be found in Algorithm 2 (R EMOVE R EDUNDANCY
and R ECOVER M AP can be found in the supplementary material). In Subsection 2.2, we addressed
the nuances of differentiating through DFT and inverse DFT layers. Apart from these, the last component left undiscussed is differentiation through the truncation of the frequency matrix, but this
corresponds to a simple zero-padding of the gradient maps to the appropriate dimensions.
In practice, the DFTs are the computational bottlenecks of spectral pooling. However, we note that in
convolutional neural networks that employ FFTs for convolution computation, spectral pooling can
be implemented at a negligible additional computational cost, since the DFT is performed regardless.
We proceed to discuss a number of properties of spectral pooling, which we then test comprehensively in Section 5.
Algorithm 1: Spectral pooling
M ?N
Input: Map x ? R
, output size H ?W
? ? RH?W
Output: Pooled map x
1: y ? F (x)
? ? C ROP S PECTRUM(y, H ? W )
2: y
? ? T REAT C ORNER C ASES(?
3: y
y)
? ? F ?1 (?
4: x
y)
Algorithm 2: Spectral pooling back-propagation
Input: Gradient w.r.t output ?R
?
?x
Output: Gradient w.r.t input ?R
?x
? ? F ?R
1: z
?
?x
? ? R EMOVE R EDUNDANCY(?
2: z
z)
3: z ? PAD S PECTRUM(?
z, M ? N )
4: z ? R ECOVER M AP(z)
5: ?R
? F ?1 (z)
?x
4
Figure 2: Approximations for different pooling schemes, for different factors of dimensionality
reduction. Spectral pooling projects onto the Fourier basis and truncates it as desired. This retains
significantly more information and permits the selection of any arbitrary output map dimensionality.
3.1
Information preservation
Spectral pooling can significantly increase the amount of retained information relative to maxpooling in two distinct ways. First, its representation maintains more information for the same
number of degrees of freedom. Spectral pooling reduces the information capacity by tuning the
resolution of the input precisely to match the desired output dimensionality. This operation can also
be viewed as linear low-pass filtering and it exploits the non-uniformity of the spectral density of
the data with respect to frequency. That is, that the power spectra of inputs with spatial structure,
such as natural images, carry most of their mass on lower frequencies. As such, since the amplitudes
of the higher frequencies tend to be small, Parseval?s theorem from Section 2 informs us that their
elimination will result in a representation that minimizes the `2 distortion after reconstruction.
Second, spectral pooling does not suffer from the sharp reduction in output dimensionality exhibited by other pooling techniques. More specifically, for stride-based pooling strategies such as max
pooling, the number of degrees of freedom of two-dimensional inputs is reduced by at least 75% as
a function of stride. In contrast, spectral pooling allows us to specify any arbitrary output dimensionality, and thus allows us to reduce the map size gradually as a function of layer.
3.2
Regularization via resolution corruption
We note that the low-pass filtering radii, say RH and RW , can be chosen to be smaller than the output
map dimensionalities H, W . Namely, while we truncate our input frequency map to size H ? W ,
we can further zero-out all frequencies outside the central RH ? RW square. While this maintains
the output dimensionality H ? W of the input domain after applying the inverse DFT, it effectively
reduces the resolution of the output. This can be seen in Figure 2.
This allows us to introduce regularization in the form of random resolution reduction. We apply this
stochastically by assigning a distribution pR (?) on the frequency truncation radius (for simplicity
we apply the same truncation on both axes), sampling from this a random radius at each iteration,
and wiping out all frequencies outside the square of that size. Note that this can be regarded as an
application of nested dropout (Rippel et al., 2014) on both dimensions of the frequency decomposition of our input. In practice, we have had success choosing pR (?) = U[Hmin ,H] (?), i.e., a uniform
distribution stretching from some minimum value all the way up to the highest possible resolution.
4
Spectral Parametrization of CNNs
Here we demonstrate how to learn the filters of CNNs directly in their frequency domain representations. This offers significant advantages over the traditional spatial representation, which we show
empirically in Section 5.
Let us assume that for some layer of our convolutional neural network we seek to learn filters
of size H ? W . To do this, we parametrize each filter f ? CH?W in our network directly in
the frequency domain. To attain its spatial representation, we simply compute its inverse DFT
5
Normalized count
1.0
0.6
0.4
0.2
0.0 ?4
10
(a) Filters over time.
(b) Sparsity patterns.
Spatial
Spectral
0.8
10?3
10?2
10?1
Element momentum
(c) Momenta distributions.
Figure 3: Learning dynamics of CNNs with spectral parametrization. The histograms have been
produced after 10 epochs of training on CIFAR-10 by each method, but are similar throughout.
(a) Progression over several epochs of filters parametrized in the frequency domain. Each pair of
columns corresponds to the spectral parametrization of a filter and its inverse transform to the spatial
domain. Filter representations tend to be more local in the Fourier basis. (b) Sparsity patterns for the
different parametrizations. Spectral representations tend to be considerably sparser. (c) Distributions
of momenta across parameters for CNNs trained with and without spectral parametrization. In the
spectral parametrization considerably fewer parameters are updated.
as F ?1 (f ) ? RH?W . From this point on, we proceed as we would for any standard CNN by computing the convolution of the filter with inputs in our mini-batch, and so on.
The back-propagation through the inverse DFT is virtually identical to the one of spectral pooling
described in Section 3. We compute the gradient as outlined in Subsection 2.2, being careful to obey
the conjugate symmetry constraints discussed in Subsection 2.1.
We emphasize that this approach does not change the underlying CNN model in any way ? only
the way in which it is parametrized. Hence, this only affects the way the solution space is explored
by the optimization procedure.
4.1
Leveraging filter structure
This idea exploits the observation that CNN filters have a very characteristic structure that reappears
across data sets and problem domains. That is, CNN weights can typically be captured with a small
number of degrees of freedom. Represented in the spatial domain, however, this results in significant
redundancy.
The frequency domain, on the other hand, provides an appealing basis for filter representation: characteristic filters (e.g., Gabor filters) are often very localized in their spectral representations. This
follows from the observation that filters tend to feature very specific length-scales and orientations.
Hence, they tend to have nonzero support in a narrow set of frequency components. This hypothesis
can be observed qualitatively in Figure 3(a) and quantitatively in Figure 3(b).
Empirically, in Section 5 we observe that spectral representations of filters leads to a convergence
speedup by 2-5 times. We remark that, had we trained our network with standard stochastic gradient
descent, the linearity of differentiation and parameter update would have resulted in exactly the same
filters regardless of whether they were represented in the spatial or frequency domain during training
(this is true for any invertible linear transformation of the parameter space).
However, as discussed, this parametrization corresponds to a rotation to a more meaningful axis
alignment, where the number of relevant elements has been significantly reduced. Since modern
optimizers implement update rules that consist of adaptive element-wise rescaling, they are able to
leverage this axis alignment by making large updates to a small number of elements. This can be
seen quantitatively in Figure 3(c), where the optimizer ? Adam (Kingma & Ba, 2015), in this case
? only touches a small number of elements in its updates.
There exist a number of extensions of the above approach we believe would be quite promising in
future work; we elaborate on these in the discussion.
5
Experiments
We demonstrate the effectiveness of spectral representations in a number of different experiments.
We ran all experiments on code optimized for the Xeon Phi coprocessor. We used Spearmint (Snoek
et al., 2015) for Bayesian optimization of hyperparameters with 5-20 concurrent evaluations.
6
20
2?1
2?2
Max pooling
Spectral pooling
Method
Stochastic pooling
Maxout
Network-in-network
Deeply supervised
kf ?f?k
kf k
2?3
2
?4
2?5
2?6
2?7
0.0
Spectral pooling
0.2
0.4
0.6
0.8
Fraction of parameters kept
(a) Approximation loss for the ImageNet validation set.
CIFAR-10
CIFAR-100
15.13%
11.68%
10.41%
9.78%
41.51%
38.57%
35.68%
34.57%
8.6%
31.6%
(b) Classification rates.
Figure 4: (a) Average information dissipation for the ImageNet validation set as a function of fraction of parameters kept. This is measured in `2 error normalized by the input norm. The red horizontal line indicates the best error rate achievable by max pooling. (b) Test errors on CIFAR-10/100
without data augmentation of the optimal spectral pooling architecture, as compared to current stateof-the-art approaches: stochastic pooling (Zeiler & Fergus, 2013), Maxout (Goodfellow et al., 2013),
network-in-network (Lin et al., 2013), and deeply-supervised nets (Lee et al., 2014).
5.1 Spectral pooling
Information preservation We test the information retainment properties of spectral pooling on
the validation set of ImageNet (Russakovsky et al., 2015). For the different pooling strategies we plot
the average approximation loss resulting from pooling to different dimensionalities. This can be seen
in Figure 4. We observe the two aspects discussed in Subsection 3.1: first, spectral pooling permits
significantly better reconstruction for the same number of parameters. Second, for max pooling,
the only knob controlling the coarseness of approximation is the stride, which results in severe
quantization and a constraining lower bound on preserved information (marked in the figure as a
horizontal red line). In contrast, spectral pooling permits the selection of any output dimensionality,
thereby producing a smooth curve over all frequency truncation choices.
Classification with convolutional neural networks We test spectral pooling on different classification tasks. We hyperparametrize and optimize the following CNN architecture:
M
10/100
96+32m
? C1?1
? GA ? Softmax
C3?3
? SP?b?Hm c?b?Hm c m=1 ? C96+32M
1?1
(5)
Here, by CF
S we denote a convolutional layer with F filters each of size S, by SP?S a spectral
pooling layer with output dimensionality S, and GA the global averaging layer described in Lin
et al. (2013). We upper-bound the number of filters per layer as 288. Every convolution and pooling
layer is followed by a ReLU nonlinearity. We let Hm be the height of the map of layer m. Hence,
each spectral pooling layer reduces each output map dimension by factor ? ? (0, 1). We assign
frequency dropout distribution pR (?; m, ?, ?) = U[bcm Hm c,Hm ] (?) for layer m, total layers M and
m
with cm (?, ?) = ? + M
(? ? ?) for some constants ?, ? ? R. This parametrization can be thought
of as some linear parametrization of the dropout rate as a function of the layer.
We perform hyperparameter optimization on the dimensionality decay rate ? ? [0.25, 0.85], number
of layers M ? {1, . . . , 15}, resolution randomization hyperparameters ?, ? ? [0, 0.8], weight decay
rate in [10?5 , 10?2 ], momentum in [1 ? 0.10.5 , 1 ? 0.12 ] and initial learning rate in [0.14 , 0.1]. We
train each model for 150 epochs and anneal the learning rate by a factor of 10 at epochs 100 and 140.
We intentionally use no dropout nor data augmentation, as these introduce a number of additional
hyperparameters which we want to disambiguate as alternative factors for success.
Perhaps unsurprisingly, the optimal hyperparameter configuration assigns the slowest possible
layer map decay rate ? = 0.85. It selects randomized resolution reduction constants of about
? ? 0.30, ? ? 0.15, momentum of about 0.95 and initial learning rate 0.0088. These settings allow us to attain classification rates of 8.6% on CIFAR-10 and 31.6% on CIFAR-100. These are
competitive results among approaches that do not employ data augmentation: a comparison to stateof-the-art approaches from the literature can be found in Table 4(b).
5.2
Spectral parametrization of CNNs
We demonstrate the effectiveness of spectral parametrization on a number of CNN optimization
tasks, for different architectures and for different filter sizes. We use the notation MPTS to denote a
max pooling layer with size S and stride T , and FCF is a fully-connected layer with F filters.
7
Size 5
e1
e1
e0
e?1
e?1
e?2
e?1
e1
?2
e1
e
e1
e0
0
e
0
e
e?1
e?2
Spatial
Spectral
e0
e0
e
Size 3
e1
?1
e?2
0
40 80 120 160 200
Deep
e?1
0
40 80 120 160 200
Generic
0
30 60 90 120 150
Sp. Pooling
Architecture
Filter
size
Speedup
factor
Deep (7)
Deep (7)
Generic (6)
Generic (6)
Sp. Pooling (5)
Sp. Pooling (5)
3?3
5?5
3?3
5?5
3?3
5?5
2.2
4.8
2.2
5.1
2.4
4.8
(b) Speedup factors.
(a) Training curves.
Figure 5: Optimization of CNNs via spectral parametrization. All experiments include data augmentation. (a) Training curves for the various experiments. The remainder of the optimization
past the matching point is marked in light blue. The red diamonds indicate the relative epochs in
which the asymptotic error rate of the spatial approach is achieved. (b) Speedup factors for different
architectures and filter sizes. A non-negligible speedup is observed even for tiny 3 ? 3 filters.
The first architecture is the generic one used in a variety of deep learning papers, such as Krizhevsky
et al. (2012); Snoek et al. (2012); Krizhevsky (2009); Kingma & Ba (2015):
1024
192
2
2
? FC512 ? Softmax
C96
3?3 ? MP3?3 ? C3?3 ? MP3?3 ? FC
(6)
The second architecture we consider is the one employed in Snoek et al. (2015), which was shown
to attain competitive classification rates. It is deeper and more complex:
96
C3?3
10/100
2
192
192
192
2
192
? C96
3?3 ? MP3?3 ? C3?3 ? C3?3 ? C3?3 ? MP3?3 ? C1?1 ? C1?1
? GA ? Softmax (7)
The third architecture considered is the spectral pooling network from Equation 5. To increase
the difficulty of optimization and reflect real training conditions, we supplemented all networks with
data augmentation in the form of translations, horizontal reflections, HSV perturbations and dropout.
We initialized both spatial and spectral filters in the spatial domain as the same values; for the
spectral parametrization experiments we then computed the Fourier transform of these to attain their
frequency representations. We optimized all networks using the Adam (Kingma & Ba, 2015) update
rule, a variant of RMSprop that we find to be a fast and robust optimizer.
The training curves can be found in Figure 5(a) and the respective factors of convergence speedup
in Table 5. Surprisingly, we observe non-negligible speedup even for tiny filters of size 3 ? 3, where
we did not expect the frequency representation to have much room to exploit spatial structure.
6
Discussion and remaining open problems
In this work, we demonstrated that spectral representations provide a rich spectrum of applications.
We introduced spectral pooling, which allows pooling to any desired output dimensionality while
retaining significantly more information than other pooling approaches. In addition, we showed that
the Fourier functions provide a suitable basis for filter parametrization, as demonstrated by faster
convergence of the optimization procedure.
One possible future line of work is to embed the network in its entirety in the frequency domain.
In models that employ Fourier transforms to compute convolutions, at every convolutional layer the
input is FFT-ed and the elementwise multiplication output is then inverse FFT-ed. These back-andforth transformations are very computationally intensive, and as such it would be desirable to strictly
remain in the frequency domain. However, the reason for these repeated transformations is the
application of nonlinearities in the forward domain: if one were to propose a sensible nonlinearity
in the frequency domain, this would spare us from the incessant domain switching.
Acknowledgements We would like to thank Prabhat, Michael Gelbart and Matthew Johnson for
useful discussions and assistance throughout this project. Jasper Snoek was a fellow in the Harvard
Center for Research on Computation and Society. This work is supported by the Applied Mathematics Program within the Office of Science Advanced Scientific Computing Research of the U.S.
Department of Energy under contract No. DE-AC02-05CH11231. This work used resources of the
National Energy Research Scientific Computing Center (NERSC). We thank Helen He and Doug
Jacobsen for providing us with access to the Babbage Xeon-Phi testbed at NERSC.
8
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corr1301.html#abs-1301-3557.
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5,136 | 565 | Retinogeniculate Development:
The Role of Competition and Correlated Retinal
Activity
Ron Keesing*
David G. Stork
*Ricoh
California
Research Center
Dept. of Physiology
2882 Sand Hill Rd., Suite 115
U.C. San Francisco
San Francisco, CA 94143
Menlo Park, CA 94025
[email protected]
[email protected]
Carla J. Shatz
Dept. of Neurobiology
Stanford University
Stanford, CA
94305
Abstract
During visual development, projections from retinal ganglion cells
(RGCs) to the lateral geniculate nucleus (LGN) in cat are refined to
produce ocular dominance layering and precise topographic mapping.
Normal development depends upon activity in RGCs, suggesting a key
role for activity-dependent synaptic plasticity. Recent experiments on
prenatal retina show that during early development, "waves" of activity
pass across RGCs (Meister, et aI., 1991). We provide the first
simulations to demonstrate that such retinal waves, in conjunction with
Hebbian synaptic competition and early arrival of contralateral axons,
can account for observed patterns of retinogeniculate projections in
normal and experimentally-treated animals.
1 INTRODUCTION
During the development of the mammalian visual system, initially diffuse axonal inputs
are refined to produce the precise and orderly projections seen in the adult. In the lateral
geniculate nucleus (LGN) of the cat, projections arriving from retinal ganglion cells
(RGCs) of both eyes are initially intermixed, and they gradually segregate before birth to
form alternating layers containing axons from only one eye. At the same time, the
branching patterns of individual axons are refined, with increased growth in
topographically correct locations. Axonal segregation and refinement depends upon
91
92
Keesing, Stork, and Schatz
presynaptic activity - blocking such activity disrupts nonnal development (Sretavan, et
al., 1988; Shatz & Stryker, 1988). These and fmdings in other vertebrates (Cline, et a1.,
1987) suggest that synaptic plasticity may be an essential factor in segregation and
modification of RGC axons (Shatz, 1990).
Previous models of visual development based on synaptic plasticity (Miller. et aI .? 1989;
Whitelaw & Cowan. 1981) required an assumption of spatial correlations in RGC activity
for nonnal development This assumption may have been justified for geniculocortical
development. since much of this occurs postnatally: visual stimulation provides the
correlations. Th~ assumption was more difficult to justify for retinogeniculate
development. since this occurs prenatally - before any optical stimulation.
The first strong evidence for correlated activity before birth has recently emerged in the
retinogenculate system: wave-like patterns of synchronized activity pass across the
prenatal retina. generating correlations between neighboring cells' activity (Meister, et aI .?
1991). We believe our model is the first to incorporate these important results.
We propose that during visual development. projections from both eyes compete to
innervate LGN neurons. Contralateral projections. which reach the LGN earlier. may
have a slight advantage in competing to innervate cells of the LGN located farthest from
the optic tract. Retinal waves of activity could reinforce this segregation and improve the
precision of topographic mapping by causing weight changes within the same eye - and
particularly within the same region of the same eye - to be highly correlated. Unlike
similar models of cortical development. our model does not require lateral interactions
between post-synaptic cells - available evidence suggests that lateral inhibition is not
present during early development (Shotwell. et at, 1986). Our model also incorporates
axon growth - an essential aspect of retinogeniculate development. since the growth and
branching ofaxons toward their ultimate targets occurs simultaneously with synaptic
competition. Moreover. synaptic competition may provide cues for growth (Shatz &
Stryker, 1988). We consider the possibility that diffusing intracellular signals indicating
local synaptic strength guide axon growth.
Below we present simulations which show that this model can account for development
in nonnal and experimentally-treated animals. We also predict the outcomes of novel
experiments currently underway.
2
SIMULATIONS
Although the LGN is. of course, three-dimensional, in our model we represent just a
single two-dimensional LGN slice, ten cells wide and eight cells high. The retina is then
one-dimensional: 50 cells long in our simulations. (This ratio of widths, 50/10, is
roughly that found in the developing cat.) In order to eliminate edge effects, we "wrap"
the retina into a ring; likewise we wrap the LGN into a cylinder.
Development of projections to the LGN is modelled in the following way: projections
from all fifty RGCs of the contralateral eye arrive at the base of the LGN before those of
the ipsilateral eye. A very rough topographic map is imposed, corresponding to coarse
topography which might be supplied by chemical gradients (Wolpert, 1978).
Development is then modelled as a series of growth steps, each separated by a period of
Hebb-style synaptic competition (Wig strom & Gustafson. 1985). During competition.
synapses are strengthened when pre- and post-synaptic activity are sufficiently correlated,
Retinogeniculate Development: The Role of Competition and Correlated Retinal Activity
and they are weakened otherwise. More specifically. for a given ROC cell i with activity
~. the strength of synapse Wij to LON cell j is changed according to:
~w,,=e(a, -a)(a.-~)
1J
1
J
[1]
where (l and ~ are threshholds and e a learning rate. If a "wave" of retinal activity is
present. the activity of ROC cells is detennined as a probability of firing based on a
Oaussian function of the distance from the center of the wavefront. LON cell activity is
equal to the sum of weighted inputs from ROC cells.
Mter each wave. the total synaptic weight supported by each ROC cell i is renormalized
linearly:
w .. (t)
1J
w .,(t+l)= ~
1J
k W, (t)
k
[2]
1k
The weights supported by each LON cell are also renonnalized. gradually driving them
toward some target value T:
w .. (t+l)=w .. (t)+[T-Lw k ,(t)]
[3]
1J
1J
k
J
Renonnalization reflects the notion that there is a limited amount of synaptic weight
which can be supported by any neuron.
During growth steps, connections are modified based on the strength of neighboring
synapses from the same ROC cell. After normalization. connections grow or retract
according to:
w .. (t+I)=w .. (t)+'Y
L w?k(t)
1J
1J
neighbors
1
[4]
where 'Y is a constant term. Equation 4 shows that weights in areas of high synaptic
strength will increase more than those elsewhere.
3 RESULTS
Synaptic competition. in conjunction with waves of pre-synaptic activity and early arrival
of contralateral axons. can account for pattens of growth seen in normal and
experimentally-treated animals. In the presence of synaptic competition. modelled axons
from each eye segregate to occupy discrete layers of the LGN - precisely what is seen in
nonnal development. In the absence of competition, as in treatment with the drug TTX.
axons arborize throughout the LON (Figure I).
The segregation and refinement of retinal inputs to the LON is best illustrated by the
fonnation of ocular dominance patterns and topographic ordering. In simulations of
normal development, where retinal waves are combined with early arrival of contalateral
inputs, strong ocular dominance layers are formed: LON neurons farthest from the optic
tract receive synaptic inputs solely from the contralateral eye and those closer receive only
ipsilateral inputs (Figure2, Competition). The development of these ocular dominance
patterns is gradual: early in development, a majority of LON neurons receive inputs from
both eyes. When synaptic competition is eliminated, there is no segregation into eyespecific layers - LON neurons receive significant synaptic inputs from both eyes. These
results are consistent with labelling studies of cat development (Shatz & Stryker. 1988).
93
94
Keesing, Stork, and Schatz
contralateral
ipsilateral
.?.?
--e
c::::
Q
8Q
CJ
.?=
Q=
=8Q
e
8
Figure 1: Retinogeniculate projections in vivo (adapted from Sretavan, et at, 1988.)
(left), and simulation results (right). In the presence of competition (top), arbors are
narrow and spatially localized, confined to the appropriate ocular dominance layer. In the
absence of such competition (bottom), contralateral and ipsilateral projections are diffuse;
there is no discernible ocular dominance pattern. During simulations, projections are
represented by synapses throughout the LGN slice, shown as squares; the particular
arborization patterns shown above are inferred from related simulations.
Competition
Simultaneous
No Competition
8
8
6
6
4
4
4
2IH-+-+-+-+-+-+-+-+-II
2
2
. .
0
o
2
4
6
8 10
0
2
4
0
6
8 10
0
2
4
6
8 10
Figure 2: Ocular dominance at the end of development. Dark color indicates strongest
synapses from the contralateral eye, light indicates strongest synapses from ipsilateral,
and gray indicates significant synapses from both eyes. In the presence of competition,
LGN cells segregate into eye-specific layers, with the contralateral eye dominating cells
which are farthest from the optic tract (base). When competition is eliminated (No
Competition), as in the addition of the drug TTX, there is no segregation into layers and
LGN cells receive significant inputs from both eyes. These simulations reproduced
results from cat development. When inputs from both retinae arrive simultaneously
(Simultaneous), ocular dominance "patches" are established, similar to those observed in
normal cortical development.
Retinogeniculate Development: The Role of Competition and Correlated Retinal Activity
Retinal waves cause the activity of neighboring ROCs to be highly correlated. When
combined with synaptic competition, these waves lead to a refinement of topographic
ordering of retinogeniculate projections. During development, the coarse topography
imposed as ROC axons enter the LON is refined to produce an accurate, precise mapping
of retinal inputs (Figure 3, Competition). Without competition, there is no refinement
of topography, and the coarse initial mapping remains.
Competition
? . . . . ? . . . ? . . . . ? a 0
. . . . . . . . . . ,. . ? ? a [lO' I ?
I
cCCIl:J co?
la ? ? . . . ,. .
. . . . . . . . . . . . .? aORb'?COI ? .
? ? ? ? ? ? ? ? ? ? - ? D Ccl __ _ [J a - . . .
? ? ? ? ? ? ? a cOO I 0 0 . ? . . . . . . . .
? ? ? ? ? ? ? ? ? ? ? ??
? ? ? 0
? . . . . . . . . . . . ? ? a C! I I
?
?
?
?
?
? I
a
cc[[]Jc [J
CCCDJco ? ? . . . . . . . . . . . .
??
[j a
a - ? ? ? . .
? . ? a cCO I po a? ? . . . . . . . . . .
I 10 ?? . ? ? . . . . ? . . . . . . ?
?
to
? ?
ooC
No Competition
aC- a? a a a? ? aC? . Ca. ?
a? . .
a?
??.c? ..
. - .? a? . - . . aC? - a
a?
? a? . ? .
a ? . ? a? . c? c?
0
0
0
??
.?.
CaCa aaCC- aC
a? aC. c. a a a? a
0
. a???????
?C.??. ?C. a ? ? . ? a? aD. a ? ? ? ? aaa?? ?C??
. ? . . ? a"
.? aa?? a' aaC" a?. a?CaCa? .? C.
. a'
. . . . a" ?C???? D' ?? a.Ca ?C? .CD? 'C' 'Co aCaC???? a"
.?.
?.. 'C'"
a - ? ? -C?aaC?aC? aa? a - ? ? a . ? a? ? a ? a ? ? ? ? ? ? . ? ? ? ? ?
? . - . a? . . - a? . a a -[JCa. - a . - C? - . . . - - . - .
. - ........... .
? . ? ? ? ? ? . . C? . C? ? ? ??
,......
.?..?.
,. . ? . . . . ? ,. . . ? ,.
? .Ca. a a? . D D? a ?C ? ? . . . . a?? . . . a ?. C?
. . . . ? a ? . . . a ??. a?
? a a C . CC? . ? Ca. . a . . . . . . C . .
Ca? D. . ? . . . . a Ca.
C DD . C a
0
?
?
0
?
Figure 3: Topographic mapping with and without competition. The vertical axis
represents ten LON cells within one section, and the horizontal axis 50 ROC cells. The
size of each box indicates strength of the synapse connecting corresponding ROC and
LON cells. If the system is topographically ordered, this connection matrix should
contain only connections forming a diagonal from lower left to upper right, as is found in
normal development in our model (Competition). When competition is eliminated, the
topographic map is coarse and non-contiguous.
4 PREDICTIONS
In addition to replicating current experimental findings, our model makes several
interesting predictions about the outcome of novel experiments. If inputs from each eye
arrive simultaneously, so that contralateral projections have no advantage in competing to
innervate specific regions of the LON, synaptic competition and retinal waves lead to a
pattern of ocular dominance "patches" similar to that observed in visual cortex (Figure 2,
Simultaneous). Topography is refined, but in this case a continuous map is formed
between the two eyes (Figure 4) - again similar to patterns observed in visual cortex.
95
96
Keesing, Stork, and Schatz
..
-. .-
.........
I I I DC II
? ? ? ? ? _._......"r"T""T""'1.:..,.:,11 CU
. ..?
.
1.1 , ,
..
.
. ..., ....
II
?
, , , DC D
?
?
?
?
?
?
?
?
?
??????
~....,_II.'
?
10 [JCD .? ..
.........
? ? . ? .?' ? - DQ.J
LiiP -Ii ~ ~ :
. . .. .
? ? ? ? ? ? ? ? ? D CD
? ? ? ? ? ? ? ?
. . .. ,
-CD
. . . .. ..?
. .....
Figure 4: Topographic mapping with synchronous arrival of projections from both eyes.
Light boxes represent contralateral inputs, dark boxes represent ipsilateral. Synaptic
competition and retinal waves cause ocular segregation and topo:sraphic refinement, but in
this case the continuous map is formed using both eyes rather than a single eye.
Our model predicts that the width of retinal waves - the distribution of activity around
the moving wavefront - is an essential factor in determining both the rate of ocular
segregation and topographic refinement. Wide waves, which cause many RGes within
the same eye to be active, will lead to most rapid ocular segregation as a result of
competition. However, wide waves can lead to poor topography: RGes in distant
regions of the retina are just as likely to be simultaneously active as neighboring RGes
(Figure 5).
100
-;
~
-
(IIJ
5
~QC
Segregation
80
4
BQ
0 ....
(IIJ
-
60
3
til
-QI"0
QI
u ....(IIJ
....o
QI
~~
0
~
u QC
-~
.=
(IIJ
=-Q
40
2
20
1 0=-
.......
(IIJ
~(IIJ
~
...
...
......
0
E-
rJ:J
0
O
0.0 0.2 0.4 0.6 0.8 1.0
Average Activity in
Neighboring RGCs
Figure 5: The width of retinal "waves" determines ocular dominance and topography in
normal development in our model. Width of retinal waves is represented by the average
activity in RGC cells adjacent to the Gaussian wavefront: high activity indicates wide
waves. Topographic error (scale at right) represents the average distance from an RGCs
target position multiplied by the strength of the synaptic connection. LGN cells are
considered ocularly segregated when they receive .9 or more of their total synaptic input
from one eye. Wide waves lead to rapid ocular segregation - many RGes within the
same retina are simulaneously active. An intermediate width, however, leads to lower
topographic error - wide waves cause spurious correlations, while narrow waves don't
provide enough information about neighboring RGCs to significantly refine topography.
Retinogeniculate Development: The Role of Competition and Correlated Retinal Activity
5 SUMMARY
Our biological model differs from more developed models of cortical development in its
inclusion of 1) differences in the time of arrival of RGC axons from the two eyes, 2) lack
of intra-target (LGN) inhibitory connections, 3) absence of visual stimulation, and 4)
inclusion of a growth rule. The model can account for the development of topography
and ocular dominance layering in studies of normal and experimental-treated cats, and
makes predictions concerning the role of retinal waves in both segregation and
topography. These neurobiological experiments are currently underway.
Acknowledgements
Thanks to Michael Stryker for helpful suggestions and to Steven Lisberger for his
generous support of this work.
References
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97
PART II
NEURO-DYNAMI CS
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5,137 | 5,650 | A Theory of Decision Making Under Dynamic
Context
Michael Shvartsman
Princeton Neuroscience Institute
Princeton University
Princeton, NJ, 08544
[email protected]
Vaibhav Srivastava
Department of Mechanical and Aerospace Engineering
Princeton University
Princeton, NJ, 08544
[email protected]
Jonathan D. Cohen
Princeton Neuroscience Institute
Princeton University
Princeton, NJ, 08544
[email protected]
Abstract
The dynamics of simple decisions are well understood and modeled as a class
of random walk models [e.g. 1?4]. However, most real-life decisions include a
dynamically-changing influence of additional information we call context. In this
work, we describe a computational theory of decision making under dynamically
shifting context. We show how the model generalizes the dominant existing model
of fixed-context decision making [2] and can be built up from a weighted combination of fixed-context decisions evolving simultaneously. We also show how the
model generalizes recent work on the control of attention in the Flanker task [5].
Finally, we show how the model recovers qualitative data patterns in another task
of longstanding psychological interest, the AX Continuous Performance Test [6],
using the same model parameters.
1
Introduction
In the late 1940s, Wald and colleagues developed a sequential test called the sequential probability
ratio test (SPRT; [7]). This test accumulates evidence in favor of one of two simple hypotheses until
a log likelihood threshold is crossed and one hypothesis is selected, forming a random walk to a
decision bound. This test was quickly applied as a model of human decision making behavior both
in its discrete form [e.g. 1] and in a continuous realization as biased Wiener process (the Diffusion
Decision Model or DDM; [2]). This work has seen a recent revival due to evidence of neurons that
appear to reflect ramping behavior consistent with evidence accumulation [e.g. 8], cortical circuits
implementing a decision process similar to the SPRT in the basal ganglia in rats [9], and the finding
correlations between DDM parameters and activity in EEG [10] and fMRI [11].
Bolstered by this revival, a number of groups investigated extension models. Some of these models
tackle complex hypothesis spaces [e.g. 12], or greater biological realism [e.g. 13]. Others focus
on relaxing stationarity assumptions about the task setting, whether by investigating multi-stimulus
integration [5], deadlines [14], or different evidence distribution by trial [15].
We engage with the latter literature by providing a theory of multi-alternative decision making under
dynamically changing context. We define context simply as additional information that may bear
upon a decision, whether from perception or memory. Such a theory is important because even
simple tasks that use apparently-fixed contexts such as prior biases may require inference on the
1
context itself before it can bear on the decision. The focus on dynamics is what distinguishes our
work from efforts on context-dependent changes in preferences [e.g. 16] and internal context updating [e.g. 17]. The admission of evidence from memory distinguishes it from work on multisensory
integration [e.g. 18].
We illustrate such decisions with an example: consider seeing someone that looks like a friend (a
target stimulus), and a decision: to greet or not greet this person. A context can be external (e.g.
a concert hall) or internal (e.g. the memory that the friend went on vacation, and therefore this
person is likely a lookalike). The context can strictly constrain the decision (e.g. greeting a friend
in the street vs. the middle of a film), or only bias it (guessing whether this is a friend or lookalike
after retrieving the memory of them on vacation). Regardless, context affects the decision, and we
assume it needs to be inferred, either before or alongside the greeting decision itself. We aim to
build a normative theory of this context processing component of decision making. We show that
our theory generalizes the discrete-time context-free SPRT (and therefore a Wiener process DDM in
continuous time) and how context-dependent decisions can be optimally built up from a dynamically
weighted combination of context-free decisions.
Our theory is general enough to consider a range of existing empirical paradigms in the literature,
including the Stroop, Flanker, Simon, and the AX-CPT [6, 19?21]. We choose to mention these in
particular because they reside on the bounds of the task space our theory considers on two different
dimensions, and describe a discretization of task space on those dimensions that accommodates
those existing paradigms. We show that in spite of the framework?s generality, it can provide wellbehaved zero-parameter predictions across qualitatively different tasks. We do this by using our
framework to derive a notational variant of an existing Flanker model [5], and using parameter values
from this previous model to simultaneously generate qualitatively accurate predictions in both the
flanker and AX-CPT paradigms. That is, our theory generates plausible behavior in qualitatively
different tasks, using the same parameters.
2
The theoretical framework
We assume that dynamic context decision making, like fixed context decision making, can be understood as a sequential Bayesian inference process. Our theory therefore uses sequentially drawn
samples from external input and/or internal memory to compute the joint posterior probability over
the identity of the true context and decision target over time. It maps from this joint probability to
a response probability using a fixed response mapping, and uses a fixed threshold rule defined over
the response probability to stop sampling and respond. We make a distinction between our theory
of decision making and individual task models that can be derived from the theory by picking points
in task space that the theory accommodates.
Formally, we assume the decider conditions a decision based on its best estimate of two pieces of
information: some unknown true context taking on one of the values {ci }ni=0 , and some unknown
true target taking on one of the values {gj }m
j=0 . This intentionally abstracts from richer views of
context (e.g. ones which assume that the context is qualitatively different from the target, or that the
relevant context to sample from is unknown). We denote by C, G random variables representing the
possible draws of context and target, and r(?) a deterministic function from the distribution P (C, G)
to a distribution over responses. We define an abstract context sensor and target sensor selectively
tuned to context or target information, such that eC is a discrete piece of evidence drawn from the
context sensor, and eG one drawn from the target sensor. The goal of the decider is to average over
the noise in the sensors to estimate the pair (C, G) sufficiently to determine the correct response,
and we assume that this inference is done optimally using Bayes? rule.
of f
We denote by ton
? ton
c the time at which the context appears and tc
c the time at which it disapon
of f
pears, and likewise tg ? tg the time at which the target appears and disappears. We also restrict
on
these times such that ton
c ? tg ; this is the primary distinction between context and target, which
can otherwise be two arbitrary stimuli. The onsets and offsets define one dimension in a continuous
space of tasks over which our theory can make predictions.
The form of r(?) defines a second dimension in the space of possible tasks where our theory makes
predictions. We use a suboptimal but simple threshold heuristic for the decision rule: when the a
2
posteriori probability of any response crosses some adaptively set threshold, sampling ends and the
response is made in favor of that response.
For the purposes of this paper, we restrict ourselves to two extremes on both of these dimensions.
For stimulus onset and offset times, we consider one setting where the context and target appear
on
of f
f
and disappear together (perfect overlap, i.e. ton
= tof
c = tg and tc
g ), and one where the target
of f
appears some time after the context disappears (no overlap, i.e. tc ? ton
g ). We label the former the
external context model, because the contextual information is immediately available, and the latter
the internal context model, because the information must be previously encoded and maintained.
The external context model is like the ongoing film context from the introduction, and the internal
context is like knowing that the friend is on vacation.
For the response mapping function r(?) we consider one setting where the response is solely conditioned on the perceptual target (context-independent response) and one where the response is is
conditioned jointly on the context-target pair (context-dependent response). The context-dependent
response is like choosing to greet or not greet the friend at the movie theater, and the contextindependent one is like choosing to greet or not greet the friend on the street.
In the lab, classic tasks like the Stroop, Flanker, and Simon [19?21] fall into the taxonomy as
external-context tasks with a context-independent response, because the response is solely conditioned on the perceptual target. On the other side of both dimensions are tasks like the N-back task
and the AX Continuous Performance Test [6]. In our consideration of these tasks, we restrict our
attention to the case where there are only two possible context and target hypotheses. The sequential
inference procedure we use can be performed for other numbers of potentially-dependent hypotheses
and responses, though the analysis we show later in the paper relies on the n = m = 2 assumption
and on indepednence between the two sensors.
3
External context update
First we describe the inference procedure in the case of perfect overlap of context and target. At the
current timestep ? , the decider has available evidence samples from both the context and the target
(eC and eG ) and uses Bayes? rule to compute the posterior probability P (C, G | eC , eG ):
C
G
C
G
P? (C = c, G = g | e , e ) ? P (e , e
| C = c, G = g)P? ?1 (C = c, G = g)
(1)
The first term is the likelihood of the evidence given the joint context-target hypothesis, and the
second term is the prior, which we take to be the posterior from the previous time step. We use
the flanker task as a concrete example. In this task, participants are shown a central target (e.g. an
S or an H) surrounded on both sides by distractors (?flankers?, more S or H stimuli) that are either
congruent or incongruent with it. Participants are told to respond to the target only but show a
number of indications of influence of the distractor, most notably an early period of below-chance
performance and a slowdown or reduced accuracy with incongruent relative to congruent flankers
[20]. We label the two possible target identities {g0 = S, g1 = H} and the possible flanker identities
{c0 = S_S, c1 = H_H} with the underscore representing the position of the target. This gives us
the two congruent possibilities {[C = c0 , G = g0 ], [C = c1 , G = g1 ]} or [SSS,HHH] and the
two incongruent possibilities {[C = c0 , G = g1 ], [C = c1 , G = g0 ]} or [SHS,HSH]. The response
mapping function marginalize over context identities at each timestep:
(
r(P (C, G)) =
r0
r1
P
with probability
P (C = c, G = g0 )
Pc
with probability
c P (C = c, G = g1 )
(2)
The higher of the two response probabilities is compared to a threshold ? and when this threshold
is crossed, the model responds. What remains is to define the prior P0 (C, G) and the likelihood
function P (eC , eG |C, G) or its inverse, the sample generation function. For sample generation, we
assume that the context and target are represented as two Gaussian distributions:
2
2
2
2
C
? N (?c + ?? ?g , ?c + ?? ?g )
G
? N (?g + ?? ?c , ?g + ?? ?c )
e
e
(3)
(4)
?c2
?g2
Here ?c and ?g are baseline means for the distributions of context and target,
and
are their
variances, and the ? scaling factors mix them, potentially reflecting perceptual overlap in the sensors.
This formulation is a notational variant of an earlier flanker model [5], but we are able to derive it by
describing the task in our formalism (we describe the exact mapping in the supplementary material).
Moreover, we later show how this notational equivalence lets us reproduce both Yu and colleagues?
results and data patterns in another task, using the same parameter settings.
3
4
Comparison to a constant-drift model
We now write the model in terms of a likelihood ratio test to facilitate comparison to Wald?s SPRT
and Wiener diffusion models. This is complementary to an earlier approach performing dynamical
analysis on the problem in probability space [22]. First we write the likelihood ratio Z of the full
response posteriors for the two responses. Since the likelihood ratio and the max a posteriori probability are monotonically related, thresholding on Z maps onto the threshold over the probability of
the most probable response we described above.
Z =
p(r(P (C, G)) = r0 |eC , eG )
p(r(P (C, G)) = r1 |eC , eG )
P (eC , eG | C = c0 , G = g0 )P? ?1 (C = c0 , G = g0 ) + P (eC , eG | C = c1 , G = g0 )P? ?1 (C = c1 , G = g0 )
=
P (eC , eG | C = c0 , G = g1 )P? ?1 (C = c0 , G = g1 ) + P (eC , eG | C = c1 , G = g1 )P? ?1 (C = c1 , G = g1 )
(5)
(6)
For this analysis we assume that context and target samples are drawn independently from each
other, i.e. that ?? = ?? = 0 and therefore that P (eC , eG | C, G) = P (eC | C)P (eG | T ). We also
index the evidence samples by time to remove the prior terms P? ?1 (?), and introduce the notation
C
lt (tx ) = P (eG
t | G = gx ) and lt (cx ) = P (et | C = cx ) for the likelihoods, with x ? {0, 1}
indexing stimuli and t ? {tcon = tgon . . . ? } indexing evidence samples over time. Now we can
rewrite:
Q
Q
Z
?
=
=
P0 (C = c0 , G = g0 )
P0 (C = c0 , G = g1 )
t lt (c0 )lt (g0 )
Q
t lt (c0 )lt (g1 )
+ P0 (C = c1 , G = g0 )
+ P0 (C = c1 , G = g1 )
?
(7)
t lt (c1 )lt (g1 )
P0 (C = c0 )P (G = g0 | C = c0 )
Q
+ P0 (C = c1 )P (G = g0 | C = c1 )
Q
P0 (C = c0 )P (G = g1 | C = c0 )
Q
+ P0 (C = c1 )P (G = g1 | C = c1 )
Q
t lt (c0 )lt (g0 )
t lt (c0 )lt (g1 )
Divide both the numerator and the denominator by
Z
t lt (c1 )lt (g0 )
Q
P0 (C = c0 )P (G = g0 | C = c0 )
=
P0 (C = c0 )P (G = g1 | C =
t lt (c1 )lt (g0 )
t lt (c1 )lt (g1 )
(8)
Q
t lt (c1 ):
Q lt (c0 )
t lt (c1 ) lt (g0 )
Q l (c )
c0 ) t lt (c0 ) lt (g1 )
t 1
+ P0 (C = c1 )P (G = g0 | C = c1 )
Q
+ P0 (C = c1 )P (G = g1 | C = c1 )
Q
t lt (g0 )
(9)
t lt (g1 )
Separate out the target likelihood product and take logs:
log Z
?
=
?
X
log
t=1
lt (g0 )
lt (g1 )
+ log
P (G = g0 | C = c0 ) P0 (C=c0 )
P (C=c )
Q lt (c0 )
+ P (G = g0 | C = c1 )
P (G = g1 | C =
0
1
P (C=c )
c0 ) P0 (C=c0 )
0
1
Q lt (c0 )
+ P (G = g1 | C = c1 )
t lt (c1 )
t lt (c1 )
(10)
Now, the first term is the Wald?s sequential probability ratio test [7] with zg? =
P
t
log
lt (g0 )
lt (g1 ) .
In the
l(g0 )
]
continuum limit, it is equal to a Wiener diffusion process dzg = ag dt+bg dW with ag = E[log l(g
1)
P
l(g0 )
lt (g0 )
2
?
and bg = Var[log l(g1 ) ] [1, 4]. We can relabel the SPRT for the target zg = t log lt (g1 ) and do
the same for the context drift that appears on both numerator and denominator of the final term:
P
P0 (C=c0 )
0)
0
z?c = t log lltt (c
(c1 ) and zc = log P0 (C=c1 ) . Then the expression is as follows:
0
log Z
?
?
= zg + log
?
P (G = g0 | C = c0 )ezc +zc + P (G = g0 | C = c1 )
P (G = g1 | C = c0 )e
0 +z ?
zc
c
(11)
+ P (G = g1 | C = c1 )
log Z ? in equation (11) comprises two terms. The first is the unbiased SPRT statistic, while the
second is a nonlinear function of the SPRT statistic for the decision on the context. The nonlinear
term plays the role of bias in the SPRT for decision on target. This rational dynamic prior bias is an
advance over previous heuristic approaches to dynamic biases [e.g. 23].
Several limits of (11)
if the context and the target are independent, then the second
are of interest:
(G=g0 )
,
and
(11)
reduces to the biased SPRT for the target. If each target
term reduces to log P
P (G=g1 )
is equally likely given a context, then the nonlinear term in (11) reduces to zero and (11) reduces
to the SPRT for the target. If each context deterministically determines a different target, then any
piece of evidence on the context is equally informative about the target. Accordingly, (11) reduces
to the sum of statistic for context and target, i.e., zg? ? (zc? + zc0 ). If the magnitude of drift rate for
the context is much higher than the magnitude of drift rate for the target, or the magnitude of the
bias z0c is high, then the nonlinear term saturates at a faster timescale comparedto the decision time.
(G=g0 |C=c0 )
In this limit, the approximate contribution of the nonlinear term is either log P
P (G=g1 |C=c0 ) , or
(G=g0 |C=c1 )
log P
P (G=g1 |C=c1 ) . Finally, in the limit of large thresholds, or equivalently, large decision times
?
0
|z?c + z0c | will be a large, e?|zc +zc | will be small, and the nonlinear term in (11) can be approximated
by a linear function of zc? + z0c obtained using the first order Taylor series expansion. In all these
cases, (11) can be approximated by a sum of two SPRTs. However, this approximation may not hold
4
in general and we suspect many interesting cases will require us to consider the nonlinear model
in (11). In those cases, the signal and noise characteristics of context and target will have different ?
and we think distinguishable ? effects on the RT distributions we measure.
5
The internal-context update and application to a new task
Recall our promise to explore two extremes on the dimension of context and onset timing, and
two extremes on the dimension of context-response dependence. The flanker task is an external
context task with a context-independent response, so we now turn to an internal context task with
context-dependent response. This task is the AX Continuous Performance Test (AX-CPT), a task
with origins in the psychiatry literature now applied to cognitive control [6]. In this task, subjects
are asked to make a response to a probe (target) stimulus, by convention labeled ?X? or ?Y?, where
the response mapping is determined by a previously seen cue (context) stimulus, ?A? or ?B?. In our
notation: g0 = X, g1 = Y, c0 = A, c1 = B. Unlike the flanker, where all stimuli pairs are equally
likely, in the AX-CPT AX trials are usually the most common (appearing 50% of the time or more),
and BY trials least common. AY and BX trials appear with equal frequency, but have dramatically
different conditional probabilities due to the preponderance of AX trials.
Two response mappings are used in the literature: an asymmetric one where one response is made
on AX trials and the other response otherwise; and a symmetric variant where one response is made
to AX and BY trials, and the other to AY and BX trials. We focus on the symmetric variant, since
in this case the response is always context-dependent (in the asymmetric variant the response is is
context-independent on Y trials). We can use the definition of the task to write a new form for r(?):
(
r(P (C, G)) =
r0 = ?lef t0
r1 = ?right0
with probability P (G = g0 , C = c0 ) + P (G = g1 , C = c1 )
with probability P (G = g0 , C = c1 ) + P (G = g1 , C = c0 )
(12)
We assume for simplicity that the inference process on the context models the maintenance of context information and retrieval of the response rule (though the model could be extended to perceptual
f
encoding of the context as well). That is, we start the inference machine at tof
c , using the following
f
on
:
update when tof
?
?
?
t
c
g
C
P? (C, G | e ) ? P (e
C
| C, G)P? ?1 (C, G)
(13)
Then, once the target appears the update becomes:
C
G
C
G
P? (C, G | e , e ) ? P (e , e
| C, G)P? ?1 (C, G)
(14)
For samples after the context disappears, we introduce a simple decay mechanism wherein the probability with which the context sensor provides a sample from the true context decays exponentially.
of f
A sample is drawn from the true context with probability e?d(? ?tc ) , and drawn uniformly othP (eC |C=c0 )
erwise. The update takes this into account, such that as ? grows the ratio P
approaches
(eC |C=c1 )
1 and the context sensor stops being informative (notation omitted for space). This means that the
unconditional posterior of context can saturate at values other than 1. The remainder of the model
is exactly as described above. This provides an opportunity to generate predictions of both tasks in
a shared model, something we take up in the final portion of the paper. But first, as in the flanker
model, we reduce this model to a combination of multiple instances of the well-understood DDM.
6
Relating the internal context model to the fixed-context drift model
We sketch an intuition for how our internal context model can be built up from a combination of
fixed-context drift models (again assuming sensor independence). The derivation uses the same trick
of dividing numerator and denominator by the likelihood as the flanker expressions, and is included
in the supplementary material, as is the asymmetric variant. We state the final expression for the
symmetric case here:
?
log Z = log
?
z
P0 (C = c0 , G = g0 )ezc e g + P0 (C = c1 , G = g1 )
P0 (C = c0 , G =
?
g1 )ezc
?
z
+ P0 (C = c1 , G = g0 )e g
(15)
Equation (15) combines the SPRT statistic associated with the context and the target in a nonlinear
fashion which is more complicated than in (11), further complicated by the fact that the memory
decay turns the context random walk into an Ornstein-Uhlenbeck process in expectation (notation
omitted for space, but follows from the relationship between continuous O-U and discrete AR(1)
processes). The reduction of these equations to a SPRT or the sum of two SPRTs is subtle, and is
valid only in rather contrived settings. For example, if the drift rate for the target is much higher
5
than the drift rate for the context, then in the limit of large thresholds (15) can be approximated by
P0 (C=c1 ,G=g1 )
0 (C=c0 ,G=g0 )
?
?
either log P
P0 (C=c1 ,G=g0 ) + zc , or log P0 (C=c0 ,G=g1 ) ? zc . As with (11), we think it will be highly
instructive to further invesgate the cases where the reductions cannot apply.
7
Simulation results for both tasks using the same model and parameters
With the relationship between both tasks established via our theory, we can now simulate behavior in
both tasks under nearly the same model parameters. The one difference is in the memory component,
governed by the memory decay parameter d and the target onset time ?ton . Longer intervals between
context disappearance and target appearance have the same effect as higher values of d: they make
context retrieved more poorly. We use d = 0.0001 for the decay and a 2000-timestep interval, which
results in approximately 82% probability of drawing a correct sample by the time the target comes
on. The effect of both parameters is equivalent in the results we show, since we do not explore
variable context-target delays, but could be explored by varying this duration.
For simplicity we assume the sampling distribution for eC and eG is identical for both tasks, though
this need not hold except for identical stimuli sampled from perception. For flanker simulations we
use the model no spatial uncertainty, i.e. ?? = ?? = 0, to best match the AX-CPT model and
our analytical connections to the SPRT. We assume the model has a high congruence prior for the
flanker model, and the correct prior for the AX-CPT, as detailed in Table 1.
Context
Target
Prior
Flanker
AX-CPT
Flanker
AX-CPT
Flanker
AX-CPT
S_S
S_S
H_H
H_H
A
A
B
B
S
H
S
H
X
Y
X
Y
0.45
0.05
0.05
0.45
0.5
0.2
0.2
0.1
Table 1: Priors for the inference process for the Flanker and AX-CPT instantiation of our theory.
The remainder of parameters are identical across both task simulations: ?c = ?g = 9, ? = 0.9,
?c = ?g = 0 for c0 and g0 , and ?c = ?g = 1 for c1 and g1 . To replicate the flanker results,
we followed [5] by introducing a non-decision error parameter ? = 0.03: this is the probability of
making a random response immediately at the first timestep. We simulated 100,000 trials for each
model. Figure 1 shows results from the simulation of the flanker task, recovering the characteristic
early below-chance performance in incongruent trials. This simulation supports the assertion that
our theory generalizes the flanker model of [5], though we are not sure why our scale on timesteps
appears different by about 5x in spite of using what we think are equivalent parameters. A library
for simulating tasks that fit in our framework and code for generating all simulation figures in this
paper can be found at https://github.com/mshvartsman/cddm.
For the AX-CPT behavior, we compare qualitative patterns from our model to a heterogeneous
dataset of humans performing this task (n=59) across 4 different manipulations with 200 trials per
subject [24]. The manipulations were different variants of ?proactive?-behavior inducing manipulations in the sense of [25]. This is the most apt comparison to our model: proactive strategies are
argued to involve response preparation of the sort that our model reflects in its accumulation over
the context before the target appears.
Figure 2 shows mean RTs and accuracies produced by our model for the AX-CPT, under the same
parameters that we used for the flanker model. This model recovers the qualitative pattern of behavior seen in human subjects in this task, both RT and error proportion by condition showing the same
pattern. Moreover, if we examine the conditional RT plot (Figure 3) we see that the model predicts
a region of below-chance performance early in AY trials but not other trials. This effect appears
isomorphic to the early congruence effect in the flanker task, in the sense that both are caused by a
strong prior biased away from the correct response: on incongruent trials given a high congruence
prior in the flanker, and on AY trials given a high AX prior in AX-CPT. More generally, the model
recovers conditional accuracy curves that look very similar to those in the human data.
6
1.00
0.006
?
? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ?
? ? ? ? ?
? ?
? ? ? ?
?
?
? ?
?
? ? ?
0.75
0.50
0.25
?
Congruent
?
Incongruent
Congruent
density
Accuracy
?
0.00
0.004
Incongruent
0.002
0.000
0
250
500
750
1000
0
250
Timesteps
500
750
1000
Timesteps
RT by condition (model)
550
500
450
400
350
RT by condition (humans)
?
?
RT (ms)
RT (timesteps)
Figure 1: Model recovers characteristic flanker pattern. Left: response time computed by 50timestep RT bin for congruent and incongruent trials, showing early below-chance performance.
Right: response time distributions for congruent and incongruent trials, showing the same mode but
fatter tail for incongruent relative to congruent trials. Both are signature phenomena in the flanker
task previously recovered by the model of Yu and colleagues, consistent with our theory being a
generalization of their model.
?
?
A:X
A:Y
B:X
470
460
450
440
430
420
B:Y
?
AX
Error Proportion
Error Proportion
?
?
?
0.10
0.05
?
A:X
A:Y
B:X
AY
BX
BY
Trial Type
Errors by condition (model)
0.15
?
?
Trial Type
0.20
?
B:Y
Trial Type
Errors by condition (humans)
0.20
?
?
0.15
?
0.10
0.05
?
AX
AY
BX
BY
Trial Type
Figure 2: Model recovers gross RT patterns in human behavior. Left: RT and error rates by trial
type in the model, using the same parameters as the flanker model. Right: RT and error rates by trial
type in 59 human participants. Error bars are standard errors (where not visible, they are smaller
than the dots). Both RT and error patterns are quite similar (note that the timestep-to-ms mapping
need not be one-to-one).
8
Discussion
In this paper, we have provided a theoretical framework for understanding decision making under
dynamically shifting context. We used this framework to derive models of two distinct tasks from
the cognitive control literature, one a notational equivalent of a previous model and the other a novel
model of a well-established task. We showed how we can write these models in terms of combinations of constant-drift random walks. Most importantly, we showed how two models derived
from our theoretical framing can recover RT, error, and RT-conditional accuracy patterns seen in
human data without a change of parameters between tasks and task models. Our results are quantitatively robust to small changes in the prior because equations 12 and 16 are smooth functions of
the prior. The early incongruent errors in flanker are also robust to larger changes, as long as the
congruence prior is above 0.5. The ordering of RTs and error rates for AX-CPT rely on assuming
that participants at least learn the correct ordering of trial frequencies ? we think an uncontroversial
assumption.
One natural next step should be to generate direct quantitative predictions of behavior in one task
based on a model trained on another task ? ideally on an individual subject level, and in a task
7
Conditional Accuracy (humans)
1.00
0.75
0.75
Accuracy
Accuracy
Conditional Accuracy (model)
1.00
0.50
A:X
A:Y
B:X
B:Y
0.25
0.00
0
250
500
750
0.50
AX
AY
BX
BY
0.25
0.00
1000
Timesteps
0
250
500
750
1000
RT (ms)
Figure 3: Model recovers conditional accuracy pattern in human behavior. Left: response
time computed by 50-timestep bin for the four trial types, using same parameters as the flanker
model. Right: same plot from 59 human participants (see text for details). Bins with fewer than
50 observations omitted. Error bars are standard errors (where not visible, they are smaller than
the dots). Both plots show qualitatively similar patterns. Two discrepancies are of note: first, the
model predicts very early AY responses to be more accurate than slightly later responses, and early
B responses to be close to chance. We think at least part of this is due to the non-decision error ?,
but we retained it for consistency with the flanker model. Second, the humans show slightly better
BY than BX performance early on, something the model does not recover. We think this may have
to do with a global left-response bias that the model is somehow not capturing. Note: the abscissae
are in different units (though they correspond surprisingly well).
that fits in our framework that has not been extensively explored (for example, an internal-context
Flanker variant, or a context-dependent response congruence judgment task). The main challenge in
pursuing this kind of analysis is our ability to efficiently estimate and explore these models which,
unlike the fixed-context models, have no closed-form analytic expressions or fast approximations.
We believe that approximations such as those provided for the flanker model [22] can and should
be applied within our framework, both as a way to generate more efficient data fits, and as a way
to apply the tools of dynamical systems analysis to the overall behavior of a system. Of particular interest is whether some points in the task space defined in our framework map onto existing
descriptive decision models [e.g. 3].
Another natural next step is to seek evidence of our proposed form of integrator in neural data,
or investigate plausible neural implementations or approximations to it. One way of doing so is
computing time-varying tuning curves of neural populations in different regions to the individual
components of the accumulators we propose in equations (11) and (15). Another is to find connectivity patterns that perform the log-sum computation we hypothesize happens in the integrator. A
third is to look for components correlated with them in EEG data. All of these methods have some
promise, as they have been successfully applied to the fixed context model [9, 10, 26]. Such neural
data would not only test a prediction of our theory, but also ? via the brain locations found to be
correlated ? address questions we presently do not address, such as whether the dynamic weighting
happens at the sampler or further upstream (i.e. whether unreliable evidence is gated at the sampler
or discounted at the integrator).
A second key challenge given our focus on optimal inference is the fact that the fixed threshold
decision rule we use is suboptimal for the case of non identically distributed observations. While
the likelihoods of context and target are independent in our simulations, the likelihoods of the two
responses are not identically distributed. The optimal threshold is generally time-varying for this
case [27], though the specific form is not known.
Finally, while our model recovers RT-conditional accuracies and stimulus-conditional RT and accuracy patterns, it fails to recover the correct pattern of accuracy-conditional RTs. That is, it predicts
much faster errors than corrects on average. Future work will need to investigate whether this is
caused by qualitative or quantitative aspects of the theoretical framework and model.
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5,138 | 5,651 | Bidirectional Recurrent Neural Networks as
Generative Models
Mathias Berglund
Aalto University, Finland
Leo K?arkk?ainen
Nokia Labs, Finland
Tapani Raiko
Aalto University, Finland
Akos Vetek
Nokia Labs, Finland
Mikko Honkala
Nokia Labs, Finland
Juha Karhunen
Aalto University, Finland
Abstract
Bidirectional recurrent neural networks (RNN) are trained to predict both in the
positive and negative time directions simultaneously. They have not been used
commonly in unsupervised tasks, because a probabilistic interpretation of the
model has been difficult. Recently, two different frameworks, GSN and NADE,
provide a connection between reconstruction and probabilistic modeling, which
makes the interpretation possible. As far as we know, neither GSN or NADE
have been studied in the context of time series before. As an example of an unsupervised task, we study the problem of filling in gaps in high-dimensional time
series with complex dynamics. Although unidirectional RNNs have recently been
trained successfully to model such time series, inference in the negative time direction is non-trivial. We propose two probabilistic interpretations of bidirectional
RNNs that can be used to reconstruct missing gaps efficiently. Our experiments on
text data show that both proposed methods are much more accurate than unidirectional reconstructions, although a bit less accurate than a computationally complex
bidirectional Bayesian inference on the unidirectional RNN. We also provide results on music data for which the Bayesian inference is computationally infeasible,
demonstrating the scalability of the proposed methods.
1
Introduction
Recurrent neural networks (RNN) have recently been trained successfully for time series modeling, and have been used to achieve state-of-the-art results in supervised tasks including handwriting
recognition [12] and speech recognition [13]. RNNs have also been used successfully in unsupervised learning of time series [26, 8].
Recently, RNNs have also been used to generate sequential data [1] in a machine translation context,
which further emphasizes the unsupervised setting. Bahdanau et al. [1] used a bidirectional RNN
to encode a phrase into a vector, but settled for a unidirectional RNN to decode it into a translated
phrase, perhaps because bidirectional RNNs have not been studied much as generative models. Even
more recently, Maas et al. [18] used a deep bidirectional RNN in speech recognition, generating text
as output.
Missing value reconstruction is interesting in at least three different senses. Firstly, it can be used to
cope with data that really has missing values. Secondly, reconstruction performance of artificially
missing values can be used as a measure of performance in unsupervised learning [21]. Thirdly,
reconstruction of artificially missing values can be used as a training criterion [9, 11, 27]. While
traditional RNN training criterions correspond to one-step prediction, training to reconstruct longer
gaps can push the model towards concentrating on longer-term predictions. Note that the one-step
1
Figure 1: Structure of the simple RNN (left) and the bidirectional RNN (right).
prediction criterion is typically used even in approaches that otherwise concentrate on modelling
long-term dependencies [see e.g. 19, 17].
When using unidirectional RNNs as generative models, it is straightforward to draw samples from
the model in sequential order. However, inference is not trivial in smoothing tasks, where we want to
evaluate probabilities for missing values in the middle of a time series. For discrete data, inference
with gap sizes of one is feasible - however, inference with larger gap sizes becomes exponentially
more expensive. Even sampling can be exponentially expensive with respect to the gap size.
One strategy used for training models that are used for filling in gaps is to explicitly train the model
with missing data [see e.g. 9]. However, such a criterion has not to our knowledge yet been used and
thoroughly evaluated compared with other inference strategies for RNNs.
In this paper, we compare different methods of using RNNs to infer missing values for binary
time series data. We evaluate the performance of two generative models that rely on bidirectional
RNNs, and compare them to inference using a unidirectional RNN. The proposed methods are very
favourable in terms of scalability.
2
Recurrent Neural Networks
Recurrent neural networks [24, 14] can be seen as extensions of the standard feedforward multilayer
perceptron networks, where the inputs and outputs are sequences instead of individual observations.
Let us denote the input to a recurrent neural network by X = {xt } where xt 2 RN is an input
vector for each time step t. Let us further denote the output as Y = {yt } where yt 2 RM is an
output vector for each time step t. Our goal is to model the distribution P (Y|X). Although RNNs
map input sequences to output sequences, we can use them in an unsupervised manner by letting the
RNN predict the next input. We can do so by setting Y = {yt = xt+1 }.
2.1
Unidirectional Recurrent Neural Networks
The structure of a basic RNN with one hidden layer is illustrated in Figure 1, where the output yt is
determined by
P yt | {xd }td=1 =
(Wy ht + by )
(1)
+ Wx xt + bh )
(2)
where
ht = tanh (Wh ht
1
and Wy , Wh , and Wx are the weight matrices connecting the hidden to output layer, hidden to
hidden layer, and input to hidden layer, respectively. by and bh are the output and hidden layer
bias vectors, respectively. Typical options for the final nonlinearity are the softmax function
for classification or categorical prediction tasks, or independent Bernoulli variables with sigmoid
functions for other binary prediction tasks. In this form, the RNN therefore evaluates the output yt
based on information propagated through the hidden layer that directly or indirectly depends on the
observations {xd }td=1 = {x1 , . . . , xt }.
2
2.2
Bidirectional Recurrent Neural Networks
Bidirectional RNNs (BRNN) [25, 2] extend the unidirectional RNN by introducing a second hidden layer, where the hidden to hidden connections flow in opposite temporal order. The model is
therefore able to exploit information both from the past and the future.
The output yt is traditionally determined by
P (yt | {xd }d6=t ) = Wyf hft + Wyb hbt + by ,
but we propose the use of
P (yt | {xd }d6=t ) = Wyf hft 1 + Wyb hbt+1 + by
where
hft = tanh Whf hft 1 + Wxf xt + bfh
hbt
= tanh
Whb hbt+1
+
Wxb xt
+
bbh
.
(3)
(4)
(5)
The structure of the BRNN is illustrated in Figure 1 (right). Compared with the regular RNN,
the forward and backward directions have separate non-tied weights and hidden activations, and are
denoted by the superscript f and b for forward and backward, respectively. Note that the connections
are acyclic. Note also that in the proposed formulation, yt does not get information from xt . We
can therefore use the model in an unsupervised manner to predict one time step given all other time
steps in the input sequence simply by setting Y = X.
3
Probabilistic Interpretation for Unsupervised Modelling
Probabilistic unsupervised modeling for sequences using a unidirectional RNN is straightforward,
as the joint distribution for the whole sequence is simply the product of the individual predictions:
T
Y
1
Punidirectional (X) =
P (xt | {xd }td=1
).
(6)
t=1
For the BRNN, the situation is more complicated. The network gives predictions for individual
outputs given all the others, and the joint distribution cannot be written as their product. We propose
two solutions for this, denoted by GSN and NADE.
GSN Generative Stochastic Networks (GSN) [6] use a denoising auto-encoder to estimate the data
distribution as the asymptotic distribution of the Markov chain that alternates between corruption
and denoising. The resulting distribution is thus defined only implicitly, and cannot be written
analytically. We can define a corruption function that masks xt as missing, and a denoising function
that reconstructs it from the others. It turns out that one feedforward pass of the BRNN does exactly
that.
Our first probabilistic interpretation is thus that the joint distribution defined by a BRNN is the
asymptotic distribution of a process that replaces one observation vector xt at a time in X by sampling it from PBRNN (xt | {xd }d6=t ). In practice, we will start from a random initialization and use
Gibbs sampling.
NADE The Neural Autoregressive Distribution Estimator (NADE) [27] defines a probabilistic model
by reconstructing missing components of a vector one at a time in a random order, starting from a
fully unobserved vector. Each reconstruction is given by an auto-encoder network that takes as input
the observations so far and an auxiliary mask vector that indicates which values are missing.
We extend the same idea for time series. Firstly, we concatenate an auxiliary binary element to
input vectors to indicate a missing input. The joint distribution of the time series is defined by first
drawing a random permutation od of time indices 1 . . . T and then setting data points observed one
by one in that order, starting from a fully missing sequence:
T
Y
PNADE (X | od ) =
P (xod | {xoe }de=11 ).
(7)
d=1
In practice, the BRNN will be trained with some inputs marked as missing, while all the outputs are
observed. See Section 5.1 for more training details.
3
4
Filling in gaps with Recurrent Neural Networks
The task we aim to solve is to fill in gaps of multiple consecutive data points in high-dimensional
binary time series data. The inference is not trivial for two reasons: firstly, we reconstruct multiple
consecutive data points, which are likely to depend on each other, and secondly, we fill in data in the
middle of a time series and hence need to consider the data both before and after the gap.
For filling in gaps with the GSN approach, we first train a bidirectional RNN to estimate PBRNN (xt |
{xd }d6=t ). In order to achieve that, we use the structure presented in Section 2.2. At test time,
the gap is first initialized to random values, after which the missing values are sampled from the
distribution PBRNN (xt | {xd }d6=t ) one by one in a random order repeatedly to approximate the
stationary distribution. For the RNN structures used in this paper, the computational complexity of
this approach at test time is O((dc + c2 )(T + gM )) where d is the dimensionality of a data point, c
is the number of hidden units in the RNN, T is the number of time steps in the data, g is the length
of the gap and M is the number of Markov chain Monte Carlo (MCMC) steps used for inference.
For filling in gaps with the NADE approach, we first train a bidirectional RNN where some of the
inputs are set to a separate missing value token. At test time, all data points in the gap are first
initialized with this token, after which each missing data point is reconstructed once until the whole
gap is filled. Computationally, the main difference to GSN is that we do not have to sample each
reconstructed data point multiple times, but the reconstruction is done in as many steps as there
are missing data points in the gap. For the RNN structures used in this paper, the computational
complexity of this approach at test time is O((dc + c2 )(T + g)) where d is the dimensionality of a
data point, c is the number of hidden units in the RNN, g is the length of the gap and T is the number
of time steps in the data.
In addition to the two proposed methods, one can use a unidirectional RNN to solve the same task.
We call this method Bayesian MCMC. Using a unidirectional RNN for the task of filling in gaps is
not trivial, as we need to take into account the probabilities of the values after the gap, which the
model does not explicitly do. We therefore resort to a similar approach as the GSN approach, where
we replace the PBRNN (xt | {xd }d6=t ) with a unidirectional equivalent for the Gibbs sampling. As
1
the unidirectional RNN models conditional probabilities of the form PRNN (xt | {xd }td=1
), we can
use Bayes? theorem to derive:
PRNN (xt = a | {xd }d6=t )
/ PRNN xt = a |
=
T
Y
? =t
1
{xd }td=1
PRNN (x? | {xd }?d=11 )
(8)
PRNN {xe }Te=t+1
xt =a
| xt =
1
a, {xd }td=1
(9)
(10)
where PRNN (x? | {xd }?d=11 ) is directly the output of the unidirectional RNN given an input sequence X, where one time step t, i.e. the one we Gibbs sample, is replaced by a proposal a. The
problem is that we have to go through all possible proposals a separately to evaluate the probability
P (xt = a|{xd }d6=t ). We therefore have to evaluate the product of the outputs of the unidirectional
RNN for time steps t . . . T for each possible a.
In some cases this is feasible to evaluate. For categorical data, e.g. text, there are as many possible
values for a as there are dimensions1 . However, for other binary data the number of possibilities
grows exponentially, and is clearly not feasible to evaluate. For the RNN structures used in this
paper, the computational complexity of this approach at test time is O((dc + c2 )(T + aT M )) where
a is the number of different values a data point can have, d is the dimensionality of a data point,
c is the number of hidden units in the RNN, T is the number of time steps in the data, and M is
the number of MCMC steps used for inference. The critical difference in complexity to the GSN
approach is the coefficient a, that for categorical data takes the value d, for binary vectors 2d and for
continuous data is infinite.
As a simple baseline model, we also evaluate the one-gram log-likelihood of the gaps. The one-gram
model assumes a constant context-independent categorical distribution for the categorical task, or a
1
For character-based text, the number of dimensions is the number of characters in the model alphabet.
4
vector of factorial binomial probabilities for the structured prediction task:
Pone gram (yt ) = f (by ) .
This can be done in O(dg).
We also compare to one-way inference, where the data points in the gap are reconstructed in order
without taking the future context into account, using Equations (1) and (2) directly. The computational complexity is O((dc + c2 )T ).
5
Experiments
We run two sets of experiments: one for a categorical prediction task, and one for a binary structured
prediction task. In the categorical prediction task we fill in gaps of five characters in Wikipedia text,
while in the structural prediction task we fill in gaps of five time steps in different polyphonic music
data sets.
5.1
Training details for categorical prediction task
For the categorical prediction task, we test the performance of the two proposed methods, GSN and
NADE. In addition, we compare the performance to MCMC using Bayesian inference and one-way
inference with a unidirectional RNN. We therefore have to train three different RNNs, one for each
method.
Each RNN is trained as a predictor network, where the character at each step is predicted based
on all the previous characters (in the case of the RNN) or all the previous and following characters
(in the case of the BRNNs). We use the same data set as Sutskever et al. [26], which consists of
2GB of English text from Wikipedia. For training, we follow a similar strategy as Hermans and
Schrauwen [15]. The characters are encoded as one-hot binary vectors with a dimensionality of
d = 96 characters and the output is modelled with a softmax distribution. We train the unirectional
RNN with string lengths of T = 250 characters, where the error is propagated only from the last 200
outputs. In the BRNN we use string length of T = 300 characters, where the error is propagated
from the middle 200 outputs. We therefore avoid propagating the gradient from predictions that lack
long temporal context.
For the BRNN used in the NADE method, we add one dimension to the one-hot input which corresponds to a missing value token. During training, in each minibatch we mark g = 5 consecutive
characters every 25 time steps as a gap. During training, the error is propagated only from these
gaps. For each gap, we uniformly draw a value from 1 to 5, and set that many characters in the gap
to the missing value token. The model is therefore trained to predict the output in different stages of
inference, where a number of the inputs are still marked as missing. For comparison, we also train
a similar network, but without masking. In that variant, the error is therefore propagated from all
time steps. We refer to ?NADE? masked and ?NADE no mask?, respectively, for these two training
methods.
For all the models, the weight elements are drawn from the uniform distribution: wi,j ? U [ s, s]
where
s = 1 for the input to hidden layer, and following Glorot and Bengio [10], where s =
p
6/ (din + dout ) for the hidden-to-hidden and the hidden-to output layers. The biases are initialized to zero.
We use c = 1000 hidden units in the unidirectional RNN and c = 684 hidden units in the two hidden
layers in the BRNNs. The number of parameters in the two model types is therefore roughly the
same. In the recurrent layers, we set the recurrent activation connected to the first time step to zero.
The networks are trained using stochastic gradient descent and the gradient is calculated using backpropagation through time. We use a minibatch size of 40, i.e. each minibatch consists of 40 randomly sampled sequences of length 250. As the gradients tend to occasionally ?blow up? when
training RNNs [5, 20], we normalize the gradients at each update to have length one. The step size
is set to 0.25 for all layers in the beginning of training, and it is linearly decayed to zero during
training. As training the model is very time-consuming2 , we do not optimize the hyperparameters,
or repeat runs to get confidence intervals around the evaluated performances.
2
We used about 8 weeks of GPU time for the reported results.
5
5.2
Training Details for the Binary Structured Prediction Task
In the other set of experiments, we use four polyphonic music data sets [8]. The data sets consist of at
least 7 hours of polyphonic music each, where each data point is a binary d = 88-dimensional vector
that represents one time step of MIDI-encoded music, indicating which of the 88 keys of a piano are
pressed. We test the performance of the two proposed methods, but omit training the unidirectional
RNNs as the computational complexity of the Bayesian MCMC is prohibitive (a = 288 ).
We train all models for 50 000 updates in minibatches of ? 3 000 individual data points3 . As the
data sets are small, we select the initial learning rate on a grid of {0.0001, 0.0003, . . . , 0.3, 1} based
on the lowest validation set cost. We use no ?burn-in? as several of the scores are fairly short, and
therefore do not specifically mask out values in the beginning or end of the data set as we did for the
text data.
For the NADE method, we use an additional dimension as a missing value token in the data. For the
missing values, we set the missing value token to one and the other dimensions to zero.
Other training details are similar to the categorical prediction task.
5.3
Evaluation of Models
At test time, we evaluate the models by calculating the mean log-likelihood of the correct value of
gaps of five consecutive missing values in test data.
In the GSN and Bayesian MCMC approaches, we first set the five values in the gap to a random value
for the categorical prediction task, or to zero for the structured prediction task. We then sample all
five values in the gap in random order, and repeat the procedure for M = 100 MCMC steps4 . For
evaluating the log-likelihood of the correct value for the string, we force the last five steps to sample
the correct value, and store the probability of the model sampling those values. We also evaluate
the probability of reconstructing correctly the individual data points by not forcing the last five time
steps to sample the correct value, but by storing the probability of reconstructing the correct value
for each data point separately. We run the MCMC chain 100 times and use the log of the mean of
the likelihoods of predicting the correct value over these 100 runs.
When evaluating the performance of one-directional inference, we use a similar approach to MCMC.
However, when evaluating the log-likelihood of the entire gap, we only construct it once in sequential order, and record the probabilities of reconstructing the correct value. When evaluating the probability of reconstructing the correct value for each data point separately, we use the same approach
as for MCMC and sample the gap 100 times, recording for each step the probability of sampling the
correct value. The result for each data point is the log of the mean of the likelihoods over these 100
runs.
On the Wikipedia data, we evaluate the GSN and NADE methods on 50 000 gaps on the test data.
On the music data, all models are evaluated on all possible gaps of g = 5 on the test data, excluding
gaps that intersect with the first and last 10 time steps of a score. When evaluating the Bayesian
MCMC with the unidirectional RNN, we have to significantly limit the size of the data set, as the
method is highly computationally complex. We therefore run it on 1 000 gaps on the test data.
For NADE, we set the five time steps in the gap to the missing value token. We then reconstruct
them one by one to the correct value, and record the probability of the correct reconstruction. We
repeat this process for all possible permutations of the order in which to do the reconstruction, and
therefore acquire the exact probability of the correct reconstruction given the model and the data.
We also evaluate the individual character reconstruction probabilities by recording the probability
of sampling the correct value given all other values in the gap are set to missing.
5.4
Results
From Table 1 we can see that the Bayesian MCMC method seems to yield the best results, while
GSN or NADE outperform one-way inference. It is worth noting that in the most difficult data sets,
3
4
A minibatch can therefore consist of e.g. 100 musical scores, each of length T = 30.
M = 100 MCMC steps means that each value in the gap of g = 5 will be resampled M/g = 20 times
6
Table 1: Negative Log Likelihood (NLL) for gaps of five time steps using different models (lower
is better). In the experiments, GSN and NADE perform well, although they are outperformed by
Bayesian MCMC.
Inference strategy
Wikipedia
Nottingham
Piano
Muse
JSB
GSN
NADE masked
NADE
Bayesian MCMC
One-way inference
4.60
4.86
4.88
4.37
5.79
19.1
19.0
18.5
NA
19.2
38.8
40.4
39.4
NA
38.9
37.3
36.5
34.7
NA
37.6
43.8
44.3
44.6
NA
43.9
One-gram
23.3
145
138
147
118
2.8
10
GSN
NADE
Bayesian MCMC
One-way inference
2.6
9.5
2.2
Data point NLL
Data point NLL
2.4
2
1.8
1.6
1.4
9
8.5
8
GSN
NADE
One-way inference
1.2
1
1
1.5
2
2.5
3
3.5
Position in gap
4
4.5
7.5
5
1
1.5
2
2.5
3
3.5
Position in gap
4
4.5
5
Figure 2: Average NLL per data point using different methods with the Wikipedia data set (left)
and the Piano data set (right) for different positions in a gap of 5 consecutive missing values. The
middle data point is the most difficult to estimate for the most methods, while the one-way inference
cannot take future context into account making prediction of later positions difficult. For the leftmost position in the gap, the one-way inference performs the best since it does not require any
approximations such as MCMC.
piano and JSB, oneway inference performs very well. Qualitative examples of the reconstructions
obtained with the GSN and NADE on the Wikipedia data are shown in Table 3 (supplementary
material).
In order to get an indication of how the number of MCMC steps in the GSN approach affects
performance, we plotted the difference in NLL of GSN and NADE of the test set as a function
of the number of MCMC steps in Figure 3 (supplementary material). The figure indicates that the
music data sets mix fairly well, as the performance of GSN quickly saturates. However, for the
Wikipedia data, the performance could probably be even further improved by letting the MCMC
chain run for more than M = 100 steps.
In Figure 2 we have evaluated the NLL for the individual characters in the gaps of length five. As
expected, all methods except for one-way inference are better at predicting characters close to both
edges of the gap.
As a sanity check, we make sure our models have been successfully trained by evaluating the mean
test log-likelihood of the BRNNs for gap sizes of one. In Table 2 (supplementary material) we can
see that the BRNNs expectedly outperform previously published results with unidirectional RNNs,
which indicates that the models have been trained successfully.
6
Conclusion and Discussion
Although recurrent neural networks have been used as generative models for time series data, it has
not been trivial how to use them for inference in cases such as missing gaps in the sequential data.
7
In this paper, we proposed to use bidirectional RNNs as generative models for time series, with
two probabilistic interpretations called GSN and NADE. Both provide efficient inference in both
positive and negative directions in time, and both can be used in tasks where Bayesian inference of
a unidirectional RNN is computationally infeasible.
The model we trained for NADE differed from the basic BRNN in several ways: Firstly, we artificially marked gaps of 5 consecutive points as missing, which should help in specializing the model
for such reconstruction tasks. It would be interesting to study the effect of the missingness pattern
used in training, on the learned representations and predictions. Secondly, in addition to using all
outputs as the training signal, we tested using only the reconstructions of those missing values as the
training signal. This reduces the effective amount of training that the model went through. Thirdly,
the model had one more input (the missingness indicator) that makes the learning task more difficult.
We can see from Table 2 that the model we trained for NADE where we only used the reconstructions as the training signal has a worse performance than the BRNN for reconstructing single values.
This indicates that these differences in training have a significant impact on the quality of the final
trained probabilistic model.
We used the same number of parameters when training an RNN and a BRNN. The RNN can concentrate all the learning effort on forward prediction, and re-use the learned dependencies in backward
inference by the computationally heavy Bayesian inference. It remains an open question which
approach would work best given an optimal size of the hidden layers.
As future work, other model structures could be explored in this context, for instance the Long ShortTerm Memory [16]. Specifically to our NADE approach, it might make sense to replace the regular
additive connection from the missingness indicator input to the hidden activations in Eq. (4,5), by
a multiplicative connection that somehow gates the dynamics mappings Whf and Whb . Another
direction to extend is to use a deep architecture with more hidden layers.
The midi music data is an example of a structured prediction task: Components of the output vector
depend strongly on each other. However, our model assumes independent Bernoulli distributions
for them. One way to take those dependencies into account is to use stochastic hidden units hft and
hbt , which has been shown to improve performance on structured prediction tasks [22]. Bayer and
Osendorfer [4] explored that approach, and reconstructed missing values in the middle of motion
capture data. In their reconstruction method, the hidden stochastic variables are selected based on
an auxiliary inference model, after which the missing values are reconstructed conditioned on the
hidden stochastic variable values. Both steps are done with maximum a posteriori point selection
instead of sampling. Further quantitative evaluation of the method would be an interesting point of
comparison.
The proposed methods could be easily extended to continuous-valued data. As an example application, time-series reconstructions with a recurrent model has been shown to be effective in speech
recognition especially under impulsive noise [23].
Acknowledgements
We thank KyungHyun Cho and Yoshua Bengio for useful discussions. The software for the simulations for this paper was based on Theano [3, 7]. Nokia has supported Mathias Berglund and the
Academy of Finland has supported Tapani Raiko.
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Signal Processing, 45(11), 2673?2681.
[26] Sutskever, I., Martens, J., and Hinton, G. E. (2011). Generating text with recurrent neural networks. In
Proceedings of the 28th International Conference on Machine Learning (ICML 2011), pages 1017?1024.
[27] Uria, B., Murray, I., and Larochelle, H. (2014). A deep and tractable density estimator. In Proceedings of
The 31st International Conference on Machine Learning, pages 467?475.
9
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mcmc:20 tested:1 schuster:1 |
5,139 | 5,652 | Recognizing retinal ganglion cells in the dark
Emile Richard
Stanford University
[email protected]
Georges Goetz
Stanford University
[email protected]
E.J. Chichilnisky
Stanford University
[email protected]
Abstract
Many neural circuits are composed of numerous distinct cell types that perform
different operations on their inputs, and send their outputs to distinct targets.
Therefore, a key step in understanding neural systems is to reliably distinguish
cell types. An important example is the retina, for which present-day techniques
for identifying cell types are accurate, but very labor-intensive. Here, we develop
automated classifiers for functional identification of retinal ganglion cells, the output neurons of the retina, based solely on recorded voltage patterns on a large
scale array. We use per-cell classifiers based on features extracted from electrophysiological images (spatiotemporal voltage waveforms) and interspike intervals
(autocorrelations). These classifiers achieve high performance in distinguishing
between the major ganglion cell classes of the primate retina, but fail in achieving the same accuracy in predicting cell polarities (ON vs. OFF). We then show
how to use indicators of functional coupling within populations of ganglion cells
(cross-correlation) to infer cell polarities with a matrix completion algorithm. This
can result in accurate, fully automated methods for cell type classification.
1
Introduction
In the primate and human retina, roughly 20 distinct classes of retinal ganglion cells (RGCs) send
distinct visual information to diverse targets in the brain [18, 7, 6]. Two complementary methods for identification of these RGC types have been pursued extensively. Anatomical studies have
relied on indicators such as dendritic field size and shape, and stratification patterns in synaptic connections [8] to distinguish between cell classes. Functional studies have leveraged differences in
responses to stimulation with a variety of visual stimuli [9, 3] for the same purpose. Although successful, these methods are difficult, time-consuming and require significant expertise. Thus, they are
not suitable for large-scale, automated analysis of existing large-scale physiological recording data.
Furthermore, in some clinical settings, they are entirely inapplicable. At least two specific scientific
and engineering goals demand the development of efficient methods for cell type identification:
? Discovery of new cell types. While ?20 morphologically distinct RGC types exist, only 7
have been characterized functionally. Automated means of detecting unknown cell types
in electrophysiological recordings would make it possible to process massive amounts of
existing large-scale physiological data that would take too long to analyze manually, in
order to search for the poorly understood RGC types.
? Developing brain-machine interfaces of the future. In blind patients suffering from retinal
degeneration, RGCs no longer respond to light. Advanced retinal prostheses previously
demonstrated ex-vivo aim at electrically restoring the correct neural code in each RGC type
in a diseased retina [11], which requires cell type identification without information about
the light response properties of RGCs.
In the present paper, we introduce two novel and efficient computational methods for cell type
identification in a neural circuit, using spatiotemporal voltage signals produced by spiking cells
1
recorded with a high-density, large-scale electrode array [14]. We describe the data we used for
our study in Section 2, and we show how the raw descriptors used by our classifiers are extracted
from voltage recordings of a primate retina. We then introduce a classifier that leverages both handspecified and random-projection based features of the electrical signatures of unique RGCs, as well
as large unlabeled data sets, to identify cell types (Section 3). We evaluate its performance for
distinguishing between midget, parasol and small bistratified cells on manually annotated datasets.
Then, in Section 4, we show how matrix completion techniques can be used to identify populations
of unique cell types, and assess the accuracy of our algorithm by predicting the polarity (ON or OFF)
of RGCs on datasets where a ground truth is available. Section 5 is devoted to numerical experiments
that we designed to test our modeling choices. Finally, we discuss future work in Section 6.
2
Extracting descriptors from electrical recordings
In this section, we define the electrical signatures that we will use in cell classification, and the algorithms that allow us to perform the statistical inference of cell type are described in the subsequent
sections.
We exploit three electrical signatures of recorded neurons that are well measured in large-scale, highdensity recordings. First, the electrical image (EI) of each cell, which is the average spatiotemporal
pattern of voltage measured across the entire electrode array during the spiking of a cell. This
measure provides information about the geometric and electrical conduction properties of the cell
itself. Second, the inter-spike interval distribution (ISI), which summarizes the temporal separation
between spikes emitted by the cell. This measure reflects the specific ion channels in the cell and
their distribution across the cell. Third, the cross-correlation function (CCF) of firing between cells.
This measure captures the degree and polarity of interactions between cells in generation of a spike.
2.1
Electrophysiological image calculation, alignment and filtering
The raw data we used for our numerical experiments consist of extracellular voltage recordings of
the electrical activity of retinas from male and female macaque monkeys, which were sampled and
digitized at 20 kHz per channel over 512 channels laid out in a 60 ?m hexagonal lattice (See Appendix for a 100 ms sample movie of an electrical recording). The emission of an action potential
by a spiking neuron causes transient voltage fluctuations along its anatomical features (soma, dendritic tree, axon). By bringing an extracellular matrix of electrodes in contact with neural tissue, we
capture the 2D projection of these voltage changes onto the plane of the recording electrodes (see
Figure 1). With such dense multielectrode arrays, the voltage activity from a single cell is usually
picked up on multiple electrodes. While the literature refers to this footprint as the electrophysiological or electrical image (EI) of the cell [13], it is an inherently spatiotemporal characteristic of the
neuron, due to the transient nature of action potentials. In essence, it is a short movie (? 1.5 ms) of
the average electrical activity over the array during the emission of an action potential by a spiking
neuron, which can include the properties of other cells whose firing is correlated with this neuron.
We calculated the electrical images of each identified RGC in the recording as described in the
literature [13]. In a 30?60 minute recording, we typically detected 1,000?100,000 action potentials
per RGC. For each cell, we averaged the voltages recorded over the entire array in a 1.5 ms window
starting .25 ms before the peak negative voltage sample for each action potential. We cropped from
the electrode array the subset of electrodes that falls within a 125 ?m radius around the RGC soma
(see Figure 1) in order to represent each EI by a 30 ? 19 matrix (time points ? number of electrodes
in a 125 ?m radius), or equivalently a 570 dimensional vector. We augment the training data by
exploiting the symmetries of the (approximately) hexagonal grid of the electrode array. We form the
training data EIs from original EIs, rotating them by i?/3, i = 1, ? ? ? , 6 and the reflection of each
(12 spatial symmetries in total). The characteristic radius (125 ?m here) used to select the central
portion of the EI is a hyper-parameter of our method which controls the signal to noise ratio in the
input data (see Section 5, Figure 3 middle panel).
In the Appendix of this paper we describe 3 families (subdivided into 7 sub-families) of filters we
manually built to capture anatomical features of the cell. In particular, we included filters corresponding to various action potential propagation velocities at level of the the axon and hard-coded a
parameter which captures the soma size. These quantities are believed to be indicative of cell type.
2
0.5
1
Time (ms)
1.5
0
Distance to soma, ?m
600
0.75
0.25
0
0
120 ?m
-0.25
0
0
0.25
300
0.5
Time to spike (ms)
600
Distance to soma, ?m
0.75
0.5
Time to spike (ms)
-0.25
120 ?m
0
0.5
Time (ms)
1
1.5
>0
<0
t = ?0.2 ms
t = ?0.1 ms
t = 0 ms
t = 0.1 ms
t = 0.2 ms
t = 0.3 ms
Figure 1: EIs and cell morphology. (Top row) Multielectrode arrays record a 2D projection of
spatio-temporal action potentials, schematically illustrated here for a midget (left) and a parasol
(right) RGC. Midget cells have an asymmetric dendritic field, while parasol cells are more isotropic.
(Bottom row) Temporal evolution of the voltage recorded on the electrodes located within a 125 ?mradius around the electrode where the largest action potential was detected, which we use for cell
type classification. Amplitude of the circles materialize signal amplitude. Red circles ? positive
voltages, blue circles ? negative voltages.
We filtered the spatiotemporally aligned RGC electrical images with our hand defined filters to create a first feature set. In separate experiments we also filtered aligned EIs with iid Gaussian random
filters (as many as our features) in the fashion of [17], see Table 1 to compare performances.
2.2
Interspike Intervals
The statistics of the timing of action potential trains are another source of information about functional RGC types. Interspike intervals (ISIs) are an estimate of the probability of emission of two
consecutive action potentials within a given time difference by a spiking neuron. We build histograms of the times elapsed between two consecutive action potentials for each cell to form its ISI.
We estimate the interspike intervals over 100 ms, with a time granularity of 0.5 ms, resulting in
200 dimensional ISI vectors. ISIs always begin by a refractory period (i.e. a duration over which
no action potentials occur, following a first action potential). This period lasts the first 1-2 ms. ISIs
then increase before decaying back to zero at rates representative of the functional cell type (see
Figure 2, left hand side). We describe each ISI using the values of time differences ?t where the
smoothed ISI reaches 20, 40, 60, 80, 100% of its maximum value as well as the slopes of the linear
interpolations between each consecutive pair of points.
2.3
Cross-correlation functions and electrical coupling of cells
There is in the retina a high probability of joint emission of action potentials between neighboring
ganglion cells of the same type, while RGCs of antagonistic polarities (ON vs OFF cells) tend to
exhibit strongly negatively correlated firing patterns [16, 10]. In other words, the emission of an
action potential in the ON pathway leads to a reduced probability of observing an action potential
in the OFF pathway at the same time. The cross-correlation function of two RGCs characterizes the
probability of joint emission of action potentials for this pair of cells with a given latency, and as
such holds information about functional coupling between the two cells. Cross-correlations between
different functional RGC types have been studied extensively in the literature previously, for example in [10]. Construction of CCFs follows the same steps as ISI computation: we obtain the CCF
of pairs of cells by building histograms of time differences between their consecutive firing times.
A large CCF value near the origin is indicative of positive functional coupling whereas negative
coupling corresponds to a negative CCF at the origin (see Figure 2, the three panels on the right).
3
0.01
0.005
0
On?On Parasols
0.4
0.2
0
?0.2
0
50
100
?t (ms)
150
200
0.5
0.4
0.3
0.2
0.1
0
?0.1
?0.2
?0.3
?50
?0.4
?50
Off?Off Parasols
0.3
0
?t (ms)
50
On?Off Parasols
0.2
correlation
Frequency
0.02
0.015
0.6
correlation
Off Parasol
On Parasol
On Midget
Off Midget
SBC
correlation
Interspike Intervals
0.025
0.1
0
?0.1
?0.2
?0.3
0
?t (ms)
50
?0.4
?50
0
?t (ms)
50
Figure 2: (Left panel) Interspike Intervals for the 5 major RGC types of the primate retina. (Right
panels) Cross-correlation functions between parasol cells. Purple traces: single pairwise CCF. Red
line: population average. Green arrow: strength of the correlation.
3
3.1
Learning electrical signatures of retinal ganglion cells
Learning dictionaries from slices of unlabeled data
Learning descriptors from unlabeled data, or dictionary learning [15], has been successfully used for
classification tasks in high-dimensional data such as images, speech and texts [15, 4]. The methodology we used for learning discriminative features given a relatively large amount of unlabeled data
follows closely the steps described in [4, 5].
Extracting independent slices from the data. The first step in our approach consists of extracting
independent (as much as possible) slices from data points. One can think of a slice as a subset of
the descriptors that is (nearly) independent from other subsets. In image processing the analogue
object is named a patch, i.e. a small sub-image. In our case, we used 8 data slices. The ISI
descriptors form one such slice, the others are extracted from EIs. It is reasonable to assume ISI
features and EI descriptors are independent quantities. After aligning the EIs and filtering them
with a collection of 7 filter banks (see Appendix for a description of our biologically motivated
filters), we group each set of filtered EIs. Each group of filters reacts to specific patterns in EIs:
rotational motion driven by dendrites, radial propagation of the electrical signal along the axon and
the direction of propagation constitute behaviors captured by distinct filter banks. Thereby, we treat
the response of data to each one of them as a unique data slice. Each slice is then whitened [5],
and finally we perform sparse k-means on each slice separately, k denotes an integer which is a
parameter of our algorithm. That is, letting X ? Rn?p denote a slice of data (n: number of data
points and p: dimensionality of the slice) and Cn,k denote the set of cluster assignment matrices:
Cn,k = {U ? {0, 1}n?k : ?i ? [n] , kUi,? k0 = 1}, we consider the optimization problem
min kX ? UVT k2F + ?kVk1
s.th. U ? Cn,k , V ? Rp?k .
(1)
Warm-starting k-means with warm started NMF. In order to solve the optimization problem
(1), we propose a coarse-to-fine strategy that consists in relaxing the constraint U ? Cn,k in two
steps. We initially relax the constraint U ? Cn,k completely and set ? = 0. That is, we consider
problem (1) where we substitute Cn,k with the larger set Rn?k and run an alternate minimization
for a few steps. Then, we replace the clustering constraint Cn,k with a nonnegativity constraint U ?
Rn?k
while retaining ? = 0. After a few steps of nonnegative alternate minimization we activate
+
the constraint U ? Cn,k and finally raise the value of ?. This warm-start strategy systematically
resulted in lower values of the objective compared to random or k-means++ [1] initializations.
3.2
Building feature vectors for labeled data
In order to extract feature vectors from labeled data we first extract slice each data point: we extract
ISI features on the one hand and filter each data point with all filter families. Each slice is separately
whitened and compared to the cluster centers of its slice. For this, we use the matrices V(j) of
cluster centroids computed for the all slices j = 1, ? ? ? , 8. Letting s(?, ?) denote the soft thresholding
? (j) = s(V(j)T x(j) , ?) for each slice,
operator s(x, ?) = (sign(xi ) max{|xi | ? ?, 0})i , we compute x
which is the soft-thresholded inner products of the corresponding slice of the data point x(j) with
? (j) s from different slices and use
all cluster centroids for the same slice j. We concatenate the x
4
? = (?
the resulting encoded point to predict cell types: x
x(j) )j . The last step is performed either
? together with the corresponding label to a logistic regression
by feeding concatenated vectors x
classifier which handles multiple classes in a one-versus-all fashion, or to a random forest classifier.
4
Predicting cell polarities by completing the RGC coupling matrix
We additionally exploit pairwise spike train cross-correlations to infer RGC polarities (ON vs OFF)
and estimate the polarity vector y by using a measure of the pairwise functional coupling strength
between cells. The rationale behind this approach is that neighboring cells of the same polarity
will tend to exhibit positive correlations between their action potential spike trains, corresponding to
positive functional coupling. If the cells are of antagonistic polarities, functional coupling strength
will be negative. The coupling of two neighboring cells i, j can therefore be modeled as c{i,j} '
yi yj , where yi , yj ? {+1, ?1} denote cell polarities. Because far apart cells do not excite or
inhibit each other, to avoid incorporating noise in our model, we choose to only include estimates
of functional coupling strengths between neighboring cells. The neighborhood size is a hyperparameter of this approach that we study in Section 5.
If G denotes the graph of neighboring cells in a recording, we only use cross-correlations for spike
trains of cells which are connected with an edge in G. Since we can estimate the position of each
RGC in the lattice from its EI [13], we therefore can form the graph G, which is a 2-dimensional
regular geometric graph. If q is the number of edges in G, let P denote the linear map Rn?n ? Rq
returning the values P(C) = (Ci,j ){i,j}?E(G) for cells i and j located within a critical distance. We
use P ? to denote the adjoint (transpose) operator. The complete matrix of pairwise couplings can
then be written ? up to observation noise ? as yyT , where y ? {+1, ?1}n is the vector of cell
polarities (+1 for ON and ?1 for OFF cells). Therefore, the observation can be modeled as:
c = P(yyT ) + ? with ?
observation noise.
(2)
and the recovery of yyT is then formulated as a standard matrix completion problem.
4.1
Minimizing the nonconvex loss using warm-started Newton steps
In this section, we show how to estimate y given the observation of c = P(yyT ) + ? by minimizing
the non-convex loss `(z) = 12 kP(zzT )?ck22 . Even though minimizing this degree 4 polynomial loss
function is NP-hard in general, we propose a Newton method warm-started with a spectral heuristic
for approaching the solution (see Algorithm 1). In similar contexts, when the sampling of entries is
uniform, this type of spectral initialization followed by alternate minimization has been proven to
converge to the global minimum of a least-squared loss, analogous to ` [12].
While our sampling graph G is not an Erdos-Renyi graph, we empirically observed that its regular
structure enables us to come up with a reliable initial spectral guess that falls within the basin of
attraction of the global minimum of `. In the subsequent
Newton scheme, we iterate using the
shifted Hessian matrix H(z) = P ? 2 P(zzT ) ? c +?In where ? > 0 ensures positive definiteness
H(z) 0. Whenever computing ? and H(z)?1 is expensive due to a potentially large number of
cells n, then replacing H(z)?1 by a diagonal or scalar approximation ?/kzk22 reduces per iteration
cost while resulting in a slower convergence. We refer to this method as a first-order method for
minimizing the nonconvex objective, while ISTA [2] is a first order method applied to the convex
relaxation of the problem as presented in the Appendix (see Figure 4, middle panel). Using the same
convex relaxation we prove in the Appendix that the proposed estimator has a classification accuracy
of at least 1 ? bk?k2? with b ? 2.91.
Algorithm 1 Polarity matrix completion
Require: c observed couplings, P the projection operator
Let ?, v be the leading
eigenpair
of P ? (c)
p
?
Initialize z0 ? n ? v/ |#revealed entries|
for t = 0, 1, ? ? ? do
zt+1 ? zt ? H?1 (zt )P ? P(zt zT
\\ H(zt ) is the Hessian or an approximation
t ) ? c zt
end for
5
Input
Task
T
P
T+P
EI & ISI
our filters
k = 30
93.5 % (1.1 )
81.5 % (3.0)
78.0 % ( 3.3)
EI & ISI
rand. filters
k = 50
88.3 % (1.9)
80.0 % (2.3)
66.7 % (1.9)
EI & ISI
rand. filters
k = 10
93.1 % (1.3)
77.8 % (2.3)
72.0 % (1.7)
EI only
our filters
k = 30
86.0 % (2.6)
64.1 % (3.7)
60.4 % (2.9)
ISI only
CCF
80.6 % (2.6)
76.8 % (3.8)
64.7 % (2.9)
?
75.7 % (4.9)
?
Table 1: Comparing performance for input data sources and filters. T: cell type identification. P:
polarity identification. T+P: cell type and polarity identification. EIs cropped within 125 ?m from
the central electrode.
5
Numerical experiments
In this section, we benchmark the performance of the cell type classifiers introduced previously on
datasets where the ground truth was available. For the RGCs in those datasets, experts manually
hand-labeled the light response properties of the cells in the manner previously described in the
literature [9, 3]. Our unlabeled data contained 17,457 ? 12 (spatial symmetries) data points. The
labeled data consists of 436 OFF midget, 652 OFF parasol, 964 ON midget, 607 ON parasol and
169 small bistratified cells assembled from 10 distinct recordings.
RGC classification from their electrical features. Our numerical experiment consists in hiding
one out of 10 labeled recordings, learning cell classifiers on the 9 others and testing the classifier on
the hidden recording. We chose to test the performance of the classifier against individual recordings
for two reasons. Firstly, we wanted to compare the polarity prediction accuracy from electrical
features with the prediction made by matrix completion (see Section 4) and the matrix completion
algorithm takes as input pairwise data obtained from a single recording only. Secondly, experimental
parameters likely to influence the EIs and ISIs such as recording temperature vary from recording
to recording, but remain consistent within a recording. Since we want the reported scores to reflect
expected performance against new recordings, not including points from the test distribution gives
us a more realistic proxy to the true test error.
In Table 1 we report classification accuracies on 3 different classification tasks:
1. Cell type identification (T): midget vs. parasol vs. small bistratified cells;
2. Polarity identification (P): ON versus OFF cells;
3. Cell type and polarity (T+P): ON-midget vs. ON-parasol vs. OFF-midget vs. OFF-parasol
vs. small bistratified.
Each row of the table contains the data used as input. The first column represents the results for the
method where the dictionary learning step is performed with k = 30, and EIs are recorded within a
radius of 125 ?m from the central electrode (19 electrodes on our array). We compare our method
with an identical method where we replaced the hand-specified filters by the random Gaussian
filters of [17] (second column for k = 50 and third for k = 10). The performance of random
filters opens perspectives for learning deeper predictors using random filters in the first layer. The
impact of k on our filters can be seen in Figure 3, left-hand panel: larger k seems to bring further
information for polarity prediction but not for cell type classification, which leads to an optimal
choice k ' 20 in the 5-class problem. In the 4th and 5th columns, we used only part of the features
sets at our disposal, EIs only and ISIs only respectively. These results confirm that the joint use of
both EIs and ISIs for cell classification is beneficial. Globally, cell type identification turns out to
be an easier task than polarity prediction using per cell descriptors.
Figure 3 middle panel illustrates the impact of EI diameter on classification accuracy. While a
larger recording radius lets us make use of more signal, the amount of noise incorporated also
increases with the number of electrodes taken into account and we observe a trade-off in terms
of signal to noise ratio on all three tasks. An interesting observation is the second jump in the
accuracy of cell-type prediction around an EI diameter of 325?m, at which point we attain a peak
performance of 96.8% ? 1.0. We believe this jump takes place when axonal signals start being
incorporated in the EI, and we believe these signals to be a strong indicator of cell type because of
6
Accuracy (%)
100
100
100
95
95
90
90
90
85
85
80
80
75
75
70
80
70
60
50
70
5 10 15 20 25 30 35 40 45 50
100 150 200 250 300 350
Dictionary size (k)
Electrical image radius (?m)
100
150
200
250
300
Maximum cell distance (?m)
Figure 3: (Left panel) Effect of the dictionary size k and (Middle panel) EIs radius on per cell
classification. (Right panel) Effect of the neighborhood size on polarity prediction using matrix
completion.
1
Cell index
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
20
40
60
Cell index
80
100
-1
102
100
10-2
100
loss (t) - OPT
0.6
Coupling strength (a.u.)
0.8
loss (ut) ? OPT
P*(c): observed couplings
10
20
30
40
50
60
70
80
90
100
110
10-2
10-4
10-6
100
Newton
First order
Convex (ISTA)
PCA
101
Iteration (t)
102
10-4
10-6
10-8
10-10
100
SP-MNF
SP
NMF
++
RND
101
102
103
Iteration (t)
Figure 4: (Left panel) Observed coupling matrix. (Middle panel) Convergence of matrix completion
algorithms. (Right panel) k-means with our initialization (SP-NMF) versus other choices.
known differences in axonal conduction velocities [13]. Prediction variance is also relatively low
for cell-type prediction compared to polarity prediction, while predicting polarity turns out to be
significantly easier on some datasets than others. On average, the logistic regression classifier we
used performed slightly better (? +1%) than random forests on the various tasks and data sets at
our disposal.
Matrix completion based polarity prediction. Matrix completion resulted in > 90% accuracy
on three out of 10 datasets and in an average of 66.8% accuracy in the 7 other datasets. We report
the average performance in Table 1 even though it is inferior to the simpler classification approach
for two reasons: (a) the idea of using matrix completion for this task is new and (b) it has a high
potential, as demonstrated by Figure 3, right hand panel. On some datasets, matrix completion
has 100%accuracy. However, on other datasets, either because of issues a fragile spike-sorting, or
of other noise, the approach does not do as well. In Figure 3 (right hand side) we examine the
effect of the neighborhood size on prediction accuracy. Colors correspond to different datasets. For
sake of readability, we only show the results for 4 out of 10 datasets: the best, the worse and 2
intermediary. The sensitivity to maximum cell distance is clear on this plot. Bold curves correspond
to the prediction resulting after 100 steps of our Newton algorithm. Dashed curves correspond to
predictions by the first order (nonconvex) method stopped after 100 steps, and dots are prediction
accuracies of the leading singular vector, i.e. the spectral initialization of our algorithm. Overall, the
Newton algorithm seems to perform better than its rivals, and there appears to be an optimal radius
to choose for each dataset which corresponds to the characteristic distance between pairs of cells
(here only Parasols). This parameter varies from dataset to dataset and hence requires parameter
tuning before extracting CCF data in order to get the best performance out of the algorithm.
Warm-start strategy for dictionary learning. We refer to Figure 4, right hand panel for an illustration of our warm-start strategy for minimizing (1) as described in Section 3.1. There, we compare
dense (? = 0) k-means initialized with our double-warm start (25 steps of unconstrained alternate
minimization and 25 steps of nonnegative alternate minimization, referred to as SP-NMF), with a
single spectral warm start 50 steps unconstrained alternate minimization initialization (SP) and a
50 steps nonnegative alternate minimization (NMF) as well as with two standard baselines which
7
are random initialization and k-means++ initializations [1]. We postpone theoretical study of this
initialization choice to future work. Note that each step of the alternate minimization involves a few
matrix-matrix products and element-wise operations on matrices. Using a NVIDIA Tesla K40
GPU drastically accelerated these steps, allowing us to scale up our experiments.
6
Discussion
We developed accurate cell-type classifiers using a unique collection of labeled and unlabeled electrical recordings and employing recent advances in several areas of machine learning. The results
show strong empirical success of the methodology, which is highly scalable and adapted for major applications discussed below. Matrix completion for binary classification is novel, and the two
heuristics we used for minimizing our non-convex objectives show convincing superiority to existing
baselines. Future work will be dedicated to studying properties of these algorithms.
Recording Methods. Three major aspects of electrical recordings are critical for successful cell
type identification from electrical signatures. First, high spatial resolution is required to detect the
fine features of the EIs; much more widely spaced electrode arrays such as those often used in the
cortex may not perform as well. Second, high temporal resolution is required to measure the ISI
accurately; this suggests that optical measurements using Ca++ sensors would not be as useful as
electrical measurements. Third, large-scale recordings are required to detect many pairs of cells
and estimate their functional interactions; electrode arrays with fewer channels may not suffice.
Thus, large-scale, high-density electrophysiological recordings are uniquely well suited to the task
of identifying cell types.
Future directions. A probable source of variability in cell type classification is differences between
retinal preparations, including eccentricity in the retina, inter-animal variability, and experimental
variables such as temperature and signal-to-noise of the recording. In the present data, features
were defined and assembled across a dozen different recordings. This motivates transfer learning
to account for such variability, exploiting the fact that although the features may change somewhat
between preparations (target domains), the underlying cell types and the fundamental differences
in electrical signatures are expected to remain. We expect future work to result in models that
enjoy higher complexity thanks to training on larger datasets, thus achieving invariance to ambient
conditions (eccentricity and temperature) automatically. The model we used can be interpreted as
a single-layer neural network. A straightforward development would be to increase the number
of layers. The relative success of random filters on the first layer is a sign that one can hope to
get further automated improvement by building richer representations from the data itself and with
minimum incorporation of prior knowledge.
Application. Two major applications are envisioned. First, an extensive set of large-scale, highdensity recordings from primate retina can now be mined for information on infrequently-recorded
cell types. Manual identification of cell types using their light response properties is extremely
labor-intensive, however, the present approach promises to facilitate automated mining. Second,
the identification of cell types without light responses is fundamental for the development of highresolution retinal prostheses of the future [11]. In such devices, it is necessary to identify which
electrodes are capable of stimulating which cells, and drive spiking in RGCs according to their type
in order to deliver a meaningful visual signal to the brain. For this futuristic brain-machine interface
application, our results solve a fundamental problem. Finally, it is hoped that these applications in
the retina will also be relevant for other brain areas, where identification of neural cell types and
customized electrical stimulation for high-resolution neural implants may be equally important in
the future.
Acknowledgement
We are grateful to A. Montanari and D. Palanker for inspiring discussions and valuable comments,
and C. Rhoades for labeling the data. ER acknowledges support from grants AFOSR/DARPA
FA9550-12-1-0411 and FA9550-13-1-0036. We thank the Stanford Data Science Initiative for financial support and NVIDIA Corporation for the donation of the Tesla K40 GPU we used. Data
collection was supported by National Eye Institute grants EY017992 and EY018003 (EJC). Please
contact EJC ([email protected]) for access to the data.
8
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9
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5,140 | 5,653 | A Recurrent Latent Variable Model
for Sequential Data
Junyoung Chung, Kyle Kastner, Laurent Dinh, Kratarth Goel,
Aaron Courville, Yoshua Bengio?
Department of Computer Science and Operations Research
Universit?e de Montr?eal
?
CIFAR Senior Fellow
{firstname.lastname}@umontreal.ca
Abstract
In this paper, we explore the inclusion of latent random variables into the hidden state of a recurrent neural network (RNN) by combining the elements of the
variational autoencoder. We argue that through the use of high-level latent random variables, the variational RNN (VRNN)1 can model the kind of variability
observed in highly structured sequential data such as natural speech. We empirically evaluate the proposed model against other related sequential models on four
speech datasets and one handwriting dataset. Our results show the important roles
that latent random variables can play in the RNN dynamics.
1
Introduction
Learning generative models of sequences is a long-standing machine learning challenge and historically the domain of dynamic Bayesian networks (DBNs) such as hidden Markov models (HMMs)
and Kalman filters. The dominance of DBN-based approaches has been recently overturned by a
resurgence of interest in recurrent neural network (RNN) based approaches. An RNN is a special
type of neural network that is able to handle both variable-length input and output. By training an
RNN to predict the next output in a sequence, given all previous outputs, it can be used to model
joint probability distribution over sequences.
Both RNNs and DBNs consist of two parts: (1) a transition function that determines the evolution
of the internal hidden state, and (2) a mapping from the state to the output. There are, however, a
few important differences between RNNs and DBNs.
DBNs have typically been limited either to relatively simple state transition structures (e.g., linear
models in the case of the Kalman filter) or to relatively simple internal state structure (e.g., the HMM
state space consists of a single set of mutually exclusive states). RNNs, on the other hand, typically
possess both a richly distributed internal state representation and flexible non-linear transition functions. These differences give RNNs extra expressive power in comparison to DBNs. This expressive
power and the ability to train via error backpropagation are the key reasons why RNNs have gained
popularity as generative models for highly structured sequential data.
In this paper, we focus on another important difference between DBNs and RNNs. While the hidden
state in DBNs is expressed in terms of random variables, the internal transition structure of the
standard RNN is entirely deterministic. The only source of randomness or variability in the RNN
is found in the conditional output probability model. We suggest that this can be an inappropriate
way to model the kind of variability observed in highly structured data, such as natural speech,
which is characterized by strong and complex dependencies among the output variables at different
1
Code is available at http://www.github.com/jych/nips2015_vrnn
1
timesteps. We argue, as have others [4, 2], that these complex dependencies cannot be modelled
efficiently by the output probability models used in standard RNNs, which include either a simple
unimodal distribution or a mixture of unimodal distributions.
We propose the use of high-level latent random variables to model the variability observed in the
data. In the context of standard neural network models for non-sequential data, the variational autoencoder (VAE) [11, 17] offers an interesting combination of highly flexible non-linear mapping
between the latent random state and the observed output and effective approximate inference. In this
paper, we propose to extend the VAE into a recurrent framework for modelling high-dimensional
sequences. The VAE can model complex multimodal distributions, which will help when the underlying true data distribution consists of multimodal conditional distributions. We call this model
a variational RNN (VRNN).
A natural question to ask is: how do we encode observed variability via latent random variables?
The answer to this question depends on the nature of the data itself. In this work, we are mainly
interested in highly structured data that often arises in AI applications. By highly structured, we
mean that the data is characterized by two properties. Firstly, there is a relatively high signal-tonoise ratio, meaning that the vast majority of the variability observed in the data is due to the signal
itself and cannot reasonably be considered as noise. Secondly, there exists a complex relationship
between the underlying factors of variation and the observed data. For example, in speech, the vocal
qualities of the speaker have a strong but complicated influence on the audio waveform, affecting
the waveform in a consistent manner across frames.
With these considerations in mind, we suggest that our model variability should induce temporal
dependencies across timesteps. Thus, like DBN models such as HMMs and Kalman filters, we
model the dependencies between the latent random variables across timesteps. While we are not the
first to propose integrating random variables into the RNN hidden state [4, 2, 6, 8], we believe we are
the first to integrate the dependencies between the latent random variables at neighboring timesteps.
We evaluate the proposed VRNN model against other RNN-based models ? including a VRNN
model without introducing temporal dependencies between the latent random variables ? on two
challenging sequential data types: natural speech and handwriting. We demonstrate that for the
speech modelling tasks, the VRNN-based models significantly outperform the RNN-based models
and the VRNN model that does not integrate temporal dependencies between latent random variables.
2
2.1
Background
Sequence modelling with Recurrent Neural Networks
An RNN can take as input a variable-length sequence x = (x1 , x2 , . . . , xT ) by recursively processing each symbol while maintaining its internal hidden state h. At each timestep t, the RNN reads
the symbol xt ? Rd and updates its hidden state ht ? Rp by:
ht =f? (xt , ht?1 ) ,
(1)
where f is a deterministic non-linear transition function, and ? is the parameter set of f . The
transition function f can be implemented with gated activation functions such as long short-term
memory [LSTM, 9] or gated recurrent unit [GRU, 5]. RNNs model sequences by parameterizing a
factorization of the joint sequence probability distribution as a product of conditional probabilities
such that:
p(x1 , x2 , . . . , xT ) =
T
Y
p(xt | x<t ),
t=1
p(xt | x<t ) = g? (ht?1 ),
(2)
where g is a function that maps the RNN hidden state ht?1 to a probability distribution over possible
outputs, and ? is the parameter set of g.
One of the main factors that determines the representational power of an RNN is the output function
g in Eq. (2). With a deterministic transition function f , the choice of g effectively defines the family
of joint probability distributions p(x1 , . . . , xT ) that can be expressed by the RNN.
2
We can express the output function g in Eq. (2) as being composed of two parts. The first part ?? is
a function that returns the parameter set ?t given the hidden state ht?1 , i.e., ?t = ?? (ht?1 ), while
the second part of g returns the density of xt , i.e., p?t (xt | x<t ).
When modelling high-dimensional and real-valued sequences, a reasonable choice of an observation
model is a Gaussian mixture model (GMM) as used in [7]. For GMM, ?? returns a set of mixture
coefficients ?t , means ??,t and covariances ??,t of the corresponding mixture components. The
probability of xt under the mixture distribution is:
X
p?t ,??,t ,??,t (xt | x<t ) =
?j,t N xt ; ?j,t , ?j,t .
j
With the notable exception of [7], there has been little work investigating the structured output
density model for RNNs with real-valued sequences.
There is potentially a significant issue in the way the RNN models output variability. Given a
deterministic transition function, the only source of variability is in the conditional output probability
density. This can present problems when modelling sequences that are at once highly variable
and highly structured (i.e., with a high signal-to-noise ratio). To effectively model these types of
sequences, the RNN must be capable of mapping very small variations in xt (i.e., the only source
of randomness) to potentially very large variations in the hidden state ht . Limiting the capacity
of the network, as must be done to guard against overfitting, will force a compromise between
the generation of a clean signal and encoding sufficient input variability to capture the high-level
variability both within a single observed sequence and across data examples.
The need for highly structured output functions in an RNN has been previously noted. Boulangerlewandowski et al. [4] extensively tested NADE and RBM-based output densities for modelling
sequences of binary vector representations of music. Bayer and Osendorfer [2] introduced a sequence of independent latent variables corresponding to the states of the RNN. Their model, called
STORN, first generates a sequence of samples z = (z1 , . . . , zT ) from the sequence of independent
latent random variables. At each timestep, the transition function f from Eq. (1) computes the next
hidden state ht based on the previous state ht?1 , the previous output xt?1 and the sampled latent
random variables zt . They proposed to train this model based on the VAE principle (see Sec. 2.2).
Similarly, Pachitariu and Sahani [16] earlier proposed both a sequence of independent latent random
variables and a stochastic hidden state for the RNN.
These approaches are closely related to the approach proposed in this paper. However, there is a
major difference in how the prior distribution over the latent random variable is modelled. Unlike
the aforementioned approaches, our approach makes the prior distribution of the latent random variable at timestep t dependent on all the preceding inputs via the RNN hidden state ht?1 (see Eq. (5)).
The introduction of temporal structure into the prior distribution is expected to improve the representational power of the model, which we empirically observe in the experiments (See Table 1).
However, it is important to note that any approach based on having stochastic latent state is orthogonal to having a structured output function, and that these two can be used together to form a single
model.
2.2
Variational Autoencoder
For non-sequential data, VAEs [11, 17] have recently been shown to be an effective modelling
paradigm to recover complex multimodal distributions over the data space. A VAE introduces a
set of latent random variables z, designed to capture the variations in the observed variables x. As
an example of a directed graphical model, the joint distribution is defined as:
p(x, z) = p(x | z)p(z).
(3)
The prior over the latent random variables, p(z), is generally chosen to be a simple Gaussian distribution and the conditional p(x | z) is an arbitrary observation model whose parameters are computed
by a parametric function of z. Importantly, the VAE typically parameterizes p(x | z) with a highly
flexible function approximator such as a neural network. While latent random variable models of
the form given in Eq. (3) are not uncommon, endowing the conditional p(x | z) as a potentially
highly non-linear mapping from z to x is a rather unique feature of the VAE.
However, introducing a highly non-linear mapping from z to x results in intractable inference of the
posterior p(z | x). Instead, the VAE uses a variational approximation q(z | x) of the posterior that
3
enables the use of the lower bound:
log p(x) ? ?KL(q(z | x)kp(z)) + Eq(z|x) [log p(x | z)] ,
(4)
where KL(QkP ) is Kullback-Leibler divergence between two distributions Q and P .
In [11], the approximate posterior q(z | x) is a Gaussian N (?, diag(? 2 )) whose mean ? and variance ? 2 are the output of a highly non-linear function of x, once again typically a neural network.
The generative model p(x | z) and inference model q(z | x) are then trained jointly by maximizing
the variational lower bound with respect to their parameters, where the integral with respect to
q(z | x) is approximated stochastically. The gradient of this estimate can have a low variance
estimate, by reparameterizing z = ? + ? and rewriting:
Eq(z|x) [log p(x | z)] = Ep() [log p(x | z = ? + ? )] ,
where is a vector of standard Gaussian variables. The inference model can then be trained through
standard backpropagation technique for stochastic gradient descent.
3
Variational Recurrent Neural Network
In this section, we introduce a recurrent version of the VAE for the purpose of modelling sequences.
Drawing inspiration from simpler dynamic Bayesian networks (DBNs) such as HMMs and Kalman
filters, the proposed variational recurrent neural network (VRNN) explicitly models the dependencies between latent random variables across subsequent timesteps. However, unlike these simpler
DBN models, the VRNN retains the flexibility to model highly non-linear dynamics.
Generation The VRNN contains a VAE at every timestep. However, these VAEs are conditioned
on the state variable ht?1 of an RNN. This addition will help the VAE to take into account the
temporal structure of the sequential data. Unlike a standard VAE, the prior on the latent random
variable is no longer a standard Gaussian distribution, but follows the distribution:
zt ? N (?0,t , diag(? 20,t )) , where [?0,t , ? 0,t ] = ?prior
? (ht?1 ),
(5)
where ?0,t and ? 0,t denote the parameters of the conditional prior distribution. Moreover, the
generating distribution will not only be conditioned on zt but also on ht?1 such that:
z
xt | zt ? N (?x,t , diag(? 2x,t )) , where [?x,t , ? x,t ] = ?dec
? (?? (zt ), ht?1 ),
?prior
?
(6)
?dec
?
can be any
where ?x,t and ? x,t denote the parameters of the generating distribution,
and
highly flexible function such as neural networks. ?x? and ?z? can also be neural networks, which
extract features from xt and zt , respectively. We found that these feature extractors are crucial for
learning complex sequences. The RNN updates its hidden state using the recurrence equation:
ht =f? (?x? (xt ), ?z? (zt ), ht?1 ) ,
(7)
where f was originally the transition function from Eq. (1). From Eq. (7), we find that ht is a
function of x?t and z?t . Therefore, Eq. (5) and Eq. (6) define the distributions p(zt | x<t , z<t ) and
p(xt | z?t , x<t ), respectively. The parameterization of the generative model results in and ? was
motivated by ? the factorization:
p(x?T , z?T ) =
T
Y
p(xt | z?t , x<t )p(zt | x<t , z<t ).
(8)
t=1
Inference In a similar fashion, the approximate posterior will not only be a function of xt but also
of ht?1 following the equation:
x
zt | xt ? N (?z,t , diag(? 2z,t )) , where [?z,t , ? z,t ] = ?enc
? (?? (xt ), ht?1 ),
(9)
similarly ?z,t and ? z,t denote the parameters of the approximate posterior. We note that the encoding of the approximate posterior and the decoding for generation are tied through the RNN hidden
state ht?1 . We also observe that this conditioning on ht?1 results in the factorization:
q(z?T | x?T ) =
T
Y
t=1
4
q(zt | x?t , z<t ).
(10)
(a) Prior
(b) Generation
(c) Recurrence
(d) Inference
(e) Overall
Figure 1: Graphical illustrations of each operation of the VRNN: (a) computing the conditional
prior using Eq. (5); (b) generating function using Eq. (6); (c) updating the RNN hidden state using
Eq. (7); (d) inference of the approximate posterior using Eq. (9); (e) overall computational paths of
the VRNN.
Learning The objective function becomes a timestep-wise variational lower bound using Eq. (8)
and Eq. (10):
" T
#
X
Eq(z?T |x?T )
(?KL(q(zt | x?t , z<t )kp(zt | x<t , z<t )) + log p(xt | z?t , x<t )) .
(11)
t=1
As in the standard VAE, we learn the generative and inference models jointly by maximizing the
variational lower bound with respect to their parameters. The schematic view of the VRNN is
shown in Fig. 1, operations (a)?(d) correspond to Eqs. (5)?(7), (9), respectively. The VRNN applies
the operation (a) when computing the conditional prior (see Eq. (5)). If the variant of the VRNN
(VRNN-I) does not apply the operation (a), then the prior becomes independent across timesteps.
STORN [2] can be considered as an instance of the VRNN-I model family. In fact, STORN puts
further restrictions on the dependency structure of the approximate inference model. We include this
version of the model (VRNN-I) in our experimental evaluation in order to directly study the impact
of including the temporal dependency structure in the prior (i.e., conditional prior) over the latent
random variables.
4
Experiment Settings
We evaluate the proposed VRNN model on two tasks: (1) modelling natural speech directly from
the raw audio waveforms; (2) modelling handwriting generation.
Speech modelling We train the models to directly model raw audio signals, represented as a sequence of 200-dimensional frames. Each frame corresponds to the real-valued amplitudes of 200
consecutive raw acoustic samples. Note that this is unlike the conventional approach for modelling
speech, often used in speech synthesis where models are expressed over representations such as
spectral features [see, e.g., 18, 3, 13].
We evaluate the models on the following four speech datasets:
1. Blizzard: This text-to-speech dataset made available by the Blizzard Challenge 2013 contains 300 hours of English, spoken by a single female speaker [10].
2. TIMIT: This widely used dataset for benchmarking speech recognition systems contains
6, 300 English sentences, read by 630 speakers.
3. Onomatopoeia2 : This is a set of 6, 738 non-linguistic human-made sounds such as coughing, screaming, laughing and shouting, recorded from 51 voice actors.
4. Accent: This dataset contains English paragraphs read by 2, 046 different native and nonnative English speakers [19].
2
This dataset has been provided by Ubisoft.
5
Table 1: Average log-likelihood on the test (or validation) set of each task.
Speech modelling
Handwriting
Models
Blizzard
TIMIT
Onomatopoeia
Accent
IAM-OnDB
RNN-Gauss
3539
-1900
-984
-1293
1016
RNN-GMM
7413
26643
18865
3453
1358
VRNN-I-Gauss
? 8933
? 28340
? 19053
? 3843
? 1332
? 9188
? 29639
? 19638
? 4180
? 1353
VRNN-Gauss
? 9223
? 28805
? 20721
? 3952
? 1337
? 9516 ? 30235
? 21332
? 4223
? 1354
VRNN-GMM
? 9107
? 28982
? 20849
? 4140
? 1384
? 9392
? 29604
? 21219
? 4319
? 1384
For the Blizzard and Accent datasets, we process the data so that each sample duration is 0.5s (the
sampling frequency used is 16kHz). Except the TIMIT dataset, the rest of the datasets do not have
predefined train/test splits. We shuffle and divide the data into train/validation/test splits using a
ratio of 0.9/0.05/0.05.
Handwriting generation We let each model learn a sequence of (x, y) coordinates together with
binary indicators of pen-up/pen-down, using the IAM-OnDB dataset, which consists of 13, 040
handwritten lines written by 500 writers [14]. We preprocess and split the dataset as done in [7].
Preprocessing and training The only preprocessing used in our experiments is normalizing each
sequence using the global mean and standard deviation computed from the entire training set. We
train each model with stochastic gradient descent on the negative log-likelihood using the Adam
optimizer [12], with a learning rate of 0.001 for TIMIT and Accent and 0.0003 for the rest. We use
a minibatch size of 128 for Blizzard and Accent and 64 for the rest. The final model was chosen
with early-stopping based on the validation performance.
Models We compare the VRNN models with the standard RNN models using two different output
functions: a simple Gaussian distribution (Gauss) and a Gaussian mixture model (GMM). For each
dataset, we conduct an additional set of experiments for a VRNN model without the conditional
prior (VRNN-I).
We fix each model to have a single recurrent hidden layer with 2000 LSTM units (in the case of
Blizzard, 4000 and for IAM-OnDB, 1200). All of ?? shown in Eqs. (5)?(7), (9) have four hidden
layers using rectified linear units [15] (for IAM-OnDB, we use a single hidden layer). The standard
prior
enc
z
RNN models only have ?x? and ?dec
? , while the VRNN models also have ?? , ?? and ?? . For the
x
dec
standard RNN models, ?? is the feature extractor, and ?? is the generating function. For the RNNGMM and VRNN models, we match the total number of parameters of the deep neural networks
(DNNs), ?x,z,enc,dec,prior
, as close to the RNN-Gauss model having 600 hidden units for every layer
?
that belongs to either ?x? or ?dec
(we consider 800 hidden units in the case of Blizzard). Note that
?
we use 20 mixture components for models using a GMM as the output function.
For qualitative analysis of speech generation, we train larger models to generate audio sequences.
We stack three recurrent hidden layers, each layer contains 3000 LSTM units. Again for the RNNGMM and VRNN models, we match the total number of parameters of the DNNs to be equal to the
RNN-Gauss model having 3200 hidden units for each layer that belongs to either ?x? or ?dec
? .
5
Results and Analysis
We report the average log-likelihood of test examples assigned by each model in Table 1. For
RNN-Gauss and RNN-GMM, we report the exact log-likelihood, while in the case of VRNNs, we
report the variational lower bound (given with ? sign, see Eq. (4)) and approximated marginal
log-likelihood (given with ? sign) based on importance sampling using 40 samples as in [17].
In general, higher numbers are better. Our results show that the VRNN models have higher loglikelihood, which support our claim that latent random variables are helpful when modelling com6
Figure 2: The top row represents the difference ?t between ?z,t and ?z,t?1 . The middle row shows
the dominant KL divergence values in temporal order. The bottom row shows the input waveforms.
plex sequences. The VRNN models perform well even with a unimodal output function (VRNNGauss), which is not the case for the standard RNN models.
Latent space analysis In Fig. 2, we show an analysis of the latent random variables. We let
a VRNN model read some unseen examples and observe the transitions in the latent space. We
P
compute ?t = j (?jz,t ? ?jz,t?1 )2 at every timestep and plot the results on the top row of Fig. 2.
The middle row shows the KL divergence computed between the approximate posterior and the
conditional prior. When there is a transition in the waveform, the KL divergence tends to grow
(white is high), and we can clearly observe a peak in ?t that can affect the RNN dynamics to change
modality.
(a) Ground Truth
(b) RNN-GMM
(c) VRNN-Gauss
Figure 3: Examples from the training set and generated samples from RNN-GMM and VRNNGauss. Top three rows show the global waveforms while the bottom three rows show more zoomedin waveforms. Samples from (b) RNN-GMM contain high-frequency noise, and samples from (c)
VRNN-Gauss have less noise. We exclude RNN-Gauss, because the samples are almost close to
pure noise.
7
Speech generation We generate waveforms with 2.0s duration from the models that were trained
on Blizzard. From Fig. 3, we can clearly see that the waveforms from the VRNN-Gauss are much
less noisy and have less spurious peaks than those from the RNN-GMM. We suggest that the large
amount of noise apparent in the waveforms from the RNN-GMM model is a consequence of the
compromise these models must make between representing a clean signal consistent with the training data and encoding sufficient input variability to capture the variations across data examples. The
latent random variable models can avoid this compromise by adding variability in the latent space,
which can always be mapped to a point close to a relatively clean sample.
Handwriting generation Visual inspection of the generated handwriting (as shown in Fig. 4) from
the trained models reveals that the VRNN model is able to generate more diverse writing style while
maintaining consistency within samples.
(a) Ground Truth
(b) RNN-Gauss
(c) RNN-GMM
(d) VRNN-GMM
Figure 4: Handwriting samples: (a) training examples and unconditionally generated handwriting
from (b) RNN-Gauss, (c) RNN-GMM and (d) VRNN-GMM. The VRNN-GMM retains the writing
style from beginning to end while RNN-Gauss and RNN-GMM tend to change the writing style
during the generation process. This is possibly because the sequential latent random variables can
guide the model to generate each sample with a consistent writing style.
6
Conclusion
We propose a novel model that can address sequence modelling problems by incorporating latent
random variables into a recurrent neural network (RNN). Our experiments focus on unconditional
natural speech generation as well as handwriting generation. We show that the introduction of
latent random variables can provide significant improvements in modelling highly structured sequences such as natural speech sequences. We empirically show that the inclusion of randomness
into high-level latent space can enable the VRNN to model natural speech sequences with a simple
Gaussian distribution as the output function. However, the standard RNN model using the same
output function fails to generate reasonable samples. An RNN-based model using more powerful
output function such as a GMM can generate much better samples, but they contain a large amount
of high-frequency noise compared to the samples generated by the VRNN-based models.
We also show the importance of temporal conditioning of the latent random variables by reporting
higher log-likelihood numbers on modelling natural speech sequences. In handwriting generation,
the VRNN model is able to model the diversity across examples while maintaining consistent writing
style over the course of generation.
Acknowledgments
The authors would like to thank the developers of Theano [1]. Also, the authors thank Kyunghyun
Cho, Kelvin Xu and Sungjin Ahn for insightful comments and discussion. We acknowledge the
support of the following agencies for research funding and computing support: Ubisoft, NSERC,
Calcul Qu?ebec, Compute Canada, the Canada Research Chairs and CIFAR.
8
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5,141 | 5,654 | Deep Knowledge Tracing
Chris Piech? , Jonathan Bassen? , Jonathan Huang?? , Surya Ganguli? ,
Mehran Sahami? , Leonidas Guibas? , Jascha Sohl-Dickstein??
?
Stanford University, ? Khan Academy, ? Google
{piech,jbassen}@cs.stanford.edu, [email protected],
Abstract
Knowledge tracing?where a machine models the knowledge of a student as they
interact with coursework?is a well established problem in computer supported
education. Though effectively modeling student knowledge would have high educational impact, the task has many inherent challenges. In this paper we explore
the utility of using Recurrent Neural Networks (RNNs) to model student learning.
The RNN family of models have important advantages over previous methods
in that they do not require the explicit encoding of human domain knowledge,
and can capture more complex representations of student knowledge. Using neural networks results in substantial improvements in prediction performance on a
range of knowledge tracing datasets. Moreover the learned model can be used for
intelligent curriculum design and allows straightforward interpretation and discovery of structure in student tasks. These results suggest a promising new line of
research for knowledge tracing and an exemplary application task for RNNs.
1
Introduction
Computer-assisted education promises open access to world class instruction and a reduction in the
growing cost of learning. We can develop on this promise by building models of large scale student
trace data on popular educational platforms such as Khan Academy, Coursera, and EdX.
Knowledge tracing is the task of modelling student knowledge over time so that we can accurately
predict how students will perform on future interactions. Improvement on this task means that resources can be suggested to students based on their individual needs, and content which is predicted
to be too easy or too hard can be skipped or delayed. Already, hand-tuned intelligent tutoring systems that attempt to tailor content show promising results [28]. One-on-one human tutoring can
produce learning gains for the average student on the order of two standard deviations [5] and machine learning solutions could provide these benefits of high quality personalized teaching to anyone
in the world for free. The knowledge tracing problem is inherently difficult as human learning is
grounded in the complexity of both the human brain and human knowledge. Thus, the use of rich
models seems appropriate. However most previous work in education relies on first order Markov
models with restricted functional forms.
In this paper we present a formulation that we call Deep Knowledge Tracing (DKT) in which we
apply flexible recurrent neural networks that are ?deep? in time to the task of knowledge tracing. This
family of models represents latent knowledge state, along with its temporal dynamics, using large
vectors of artificial ?neurons?, and allows the latent variable representation of student knowledge to
be learned from data rather than hard-coded. The main contributions of this work are:
1.
2.
3.
4.
A novel way to encode student interactions as input to a recurrent neural network.
A 25% gain in AUC over the best previous result on a knowledge tracing benchmark.
Demonstration that our knowledge tracing model does not need expert annotations.
Discovery of exercise influence and generation of improved exercise curricula.
1
correct,
incorrect
1.0
Line graph intuition
Slope of a line
Solving for y-intercept
Solving for x-intercept
Graphing linear equations
Square roots
0.5
0.0
10
20
30
Exercise index
40
Predicted
Probability
Exercise attempted:
50 E[p]
Figure 1: A single student and her predicted responses as she solves 50 Khan Academy exercises. She seems to
master finding x and y intercepts and then has trouble transferring knowledge to graphing linear equations.
The task of knowledge tracing can be formalized as: given observations of interactions x0 . . . xt
taken by a student on a particular learning task, predict aspects of their next interaction xt+1 [6].
In the most ubiquitous instantiation of knowledge tracing, interactions take the form of a tuple of
xt = {qt , at } that combines a tag for the exercise being answered qt with whether or not the exercise
was answered correctly at . When making a prediction, the model is provided the tag of the exercise
being answered, qt and must predict whether the student will get the exercise correct, at . Figure 1
shows a visualization of tracing knowledge for a single student learning 8th grade math. The student
first answers two square root problems correctly and then gets a single x-intercept exercise incorrect.
In the subsequent 47 interactions the student solves a series of x-intercept, y-intercept and graphing
exercises. Each time the student answers an exercise we can make a prediction as to whether or not
she would answer an exercise of each type correctly on her next interaction. In the visualization
we only show predictions over time for a relevant subset of exercise types. In most previous work,
exercise tags denote the single ?concept? that human experts assign to an exercise. Our model
can leverage, but does not require, such expert annotation. We demonstrate that in the absence of
annotations the model can autonomously learn content substructure.
2
Related Work
The task of modelling and predicting how human beings learn is informed by fields as diverse
as education, psychology, neuroscience and cognitive science. From a social science perspective
learning has been understood to be influenced by complex macro level interactions including affect
[21], motivation [10] and even identity [4]. The challenges present are further exposed on the micro
level. Learning is fundamentally a reflection of human cognition which is a highly complex process.
Two themes in the field of cognitive science that are particularly relevant are theories that the human
mind, and its learning process, are recursive [12] and driven by analogy [13].
The problem of knowledge tracing was first posed, and has been heavily studied within the intelligent
tutoring community. In the face of aforementioned challenges it has been a primary goal to build
models which may not capture all cognitive processes, but are nevertheless useful.
2.1
Bayesian Knowledge Tracing
Bayesian Knowledge Tracing (BKT) is the most popular approach for building temporal models
of student learning. BKT models a learner?s latent knowledge state as a set of binary variables,
each of which represents understanding or non-understanding of a single concept [6]. A Hidden
Markov Model (HMM) is used to update the probabilities across each of these binary variables, as a
learner answers exercises of a given concept correctly or incorrectly. The original model formulation
assumed that once a skill is learned it is never forgotten. Recent extensions to this model include
contextualization of guessing and slipping estimates [7], estimating prior knowledge for individual
learners [33], and estimating problem difficulty [23].
With or without such extensions, Knowledge Tracing suffers from several difficulties. First, the
binary representation of student understanding may be unrealistic. Second, the meaning of the
hidden variables and their mappings onto exercises can be ambiguous, rarely meeting the model?s
expectation of a single concept per exercise. Several techniques have been developed to create and
refine concept categories and concept-exercise mappings. The current gold standard, Cognitive Task
Analysis [31] is an arduous and iterative process where domain experts ask learners to talk through
2
their thought processes while solving problems. Finally, the binary response data used to model
transitions imposes a limit on the kinds of exercises that can be modeled.
2.2
Other Dynamic Probabilistic Models
Partially Observable Markov Decision Processes (POMDPs) have been used to model learner behavior over time, in cases where the learner follows an open-ended path to arrive at a solution [29].
Although POMDPs present an extremely flexible framework, they require exploration of an exponentially large state space. Current implementations are also restricted to a discrete state space,
with hard-coded meanings for latent variables. This makes them intractable or inflexible in practice,
though they have the potential to overcome both of those limitations.
Simpler models from the Performance Factors Analysis (PFA) framework [24] and Learning Factors
Analysis (LFA) framework [3] have shown predictive power comparable to BKT [14]. To obtain
better predictive results than with any one model alone, various ensemble methods have been used
to combine BKT and PFA [8]. Model combinations supported by AdaBoost, Random Forest, linear
regression, logistic regression and a feed-forward neural network were all shown to deliver superior
results to BKT and PFA on their own. But because of the learner models they rely on, these ensemble
techniques grapple with the same limitations, including a requirement for accurate concept labeling.
Recent work has explored combining Item Response Theory (IRT) models with switched nonlinear
Kalman filters [20], as well as with Knowledge Tracing [19, 18]. Though these approaches are
promising, at present they are both more restricted in functional form and more expensive (due to
inference of latent variables) than the method we present here.
2.3
Recurrent Neural Networks
Recurrent neural networks are a family of flexible dynamic models which connect artificial neurons
over time. The propagation of information is recursive in that hidden neurons evolve based on both
the input to the system and on their previous activation [32]. In contrast to hidden Markov models
as they appear in education, which are also dynamic, RNNs have a high dimensional, continuous,
representation of latent state. A notable advantage of the richer representation of RNNs is their ability to use information from an input in a prediction at a much later point in time. This is especially
true for Long Short Term Memory (LSTM) networks?a popular type of RNN [16].
Recurrent neural networks are competitive or state-of-the-art for several time series tasks?for instance, speech to text [15], translation [22], and image captioning [17]?where large amounts of
training data are available. These results suggest that we could be much more successful at tracing
student knowledge if we formulated the task as a new application of temporal neural networks.
3
Deep Knowledge Tracing
We believe that human learning is governed by many diverse properties ? of the material, the context,
the timecourse of presentation, and the individual involved ? many of which are difficult to quantify
relying only on first principles to assign attributes to exercises or structure a graphical model. Here
we will apply two different types of RNNs ? a vanilla RNN model with sigmoid units and a Long
Short Term Memory (LSTM) model ? to the problem of predicting student responses to exercises
based upon their past activity.
3.1
Model
Traditional Recurrent Neural Networks (RNNs) map an input sequence of vectors x1 , . . . , xT , to
an output sequence of vectors y1 , . . . , yT . This is achieved by computing a sequence of ?hidden?
states h1 , . . . , hT which can be viewed as successive encodings of relevant information from past
observations that will be useful for future predictions. See Figure 2 for a cartoon illustration. The
variables are related using a simple network defined by the equations:
ht = tanh (Whx xt + Whh ht?1 + bh ) ,
yt = ? (Wyh ht + by ) ,
3
(1)
(2)
y1
h0
y2
h1
x1
y3
yT
h3
h2
x2
x3
?
hT
xT
Figure 2: The connection between variables in a simple recurrent neural network. The inputs (xt ) to the
dynamic network are either one-hot encodings or compressed representations of a student action, and the
prediction (yt ) is a vector representing the probability of getting each of the dataset exercises correct.
where both tanh and the sigmoid function, ? (?), are applied elementwise. The model is parameterized by an input weight matrix Whx , recurrent weight matrix Whh , initial state h0 , and readout
weight matrix Wyh . Biases for latent and readout units are given by bh and by .
Long Short Term Memory (LSTM) networks [16] are a more complex variant of RNNs that often
prove more powerful. In LSTMs latent units retain their values until explicitly cleared by the action
of a ?forget gate?. They thus more naturally retain information for many time steps, which is believed
to make them easier to train. Additionally, hidden units are updated using multiplicative interactions,
and they can thus perform more complicated transformations for the same number of latent units.
The update equations for an LSTM are significantly more complicated than for an RNN, and can be
found in Appendix A.
3.2
Input and Output Time Series
In order to train an RNN or LSTM on student interactions, it is necessary to convert those interactions into a sequence of fixed length input vectors xt . We do this using two methods depending on
the nature of those interactions:
For datasets with a small number M of unique exercises, we set xt to be a one-hot encoding of
the student interaction tuple ht = {qt , at } that represents the combination of which exercise was
answered and if the exercise was answered correctly, so xt ? {0, 1}2M . We found that having
separate representations for qt and at degraded performance.
For large feature spaces, a one-hot encoding can quickly become impractically large. For datasets
with a large number of unique exercises, we therefore instead assign a random vector nq,a ?
N (0, I) to each input tuple, where nq,a ? RN , and N M . We then set each input vector
xt to the corresponding random vector, xt = nqt ,at . This random low-dimensional representation
of a one-hot high-dimensional vector is motivated by compressed sensing. Compressed sensing
states that a k-sparse signal in d dimensions can be recovered exactly from k log d random linear
projections (up to scaling and additive constants) [2]. Since a one-hot encoding is a 1-sparse signal,
the student interaction tuple can be exactly encoded by assigning it to a fixed random Gaussian input
vector of length ? log 2M . Although the current paper deals only with 1-hot vectors, this technique
can be extended easily to capture aspects of more complex student interactions in a fixed length
vector.
The output yt is a vector of length equal to the number of problems, where each entry represents
the predicted probability that the student would answer that particular problem correctly. Thus the
prediction of at+1 can then be read from the entry in yt corresponding to qt+1 .
3.3
Optimization
The training objective is the negative log likelihood of the observed sequence of student responses
under the model. Let ?(qt+1 ) be the one-hot encoding of which exercise is answered at time t + 1,
and let ` be binary cross entropy. The loss for a given prediction is `(yT ? (qt+1 ) , at+1 ), and the
4
loss for a single student is:
L=
X
`(yT ? (qt+1 ) , at+1 )
(3)
t
This objective was minimized using stochastic gradient descent on minibatches. To prevent overfitting during training, dropout was applied to ht when computing the readout yt , but not when
computing the next hidden state ht+1 . We prevent gradients from ?exploding? as we backpropagate
through time by truncating the length of gradients whose norm is above a threshold. For all models
in this paper we consistently used hidden dimensionality of 200 and a mini-batch size of 100. To
facilitate research in DKTs we have published our code and relevant preprocessed data1 .
4
Educational Applications
The training objective for knowledge tracing is to predict a student?s future performance based on
their past activity. This is directly useful ? for instance formal testing is no longer necessary if a
student?s ability undergoes continuous assessment. As explored experimentally in Section 6, the
DKT model can also power a number of other advancements.
4.1
Improving Curricula
One of the biggest potential impacts of our model is in choosing the best sequence of learning items
to present to a student. Given a student with an estimated hidden knowledge state, we can query
our RNN to calculate what their expected knowledge state would be if we were to assign them a
particular exercise. For instance, in Figure 1 after the student has answered 50 exercises we can test
every possible next exercise we could show her and compute her expected knowledge state given that
choice. The predicted optimal next problem for this student is to revisit solving for the y-intercept.
We use a trained DKT to test two classic curricula rules from education literature: mixing where
exercises from different topics are intermixed, and blocking where students answer series of exercises of the same type [30]. Since choosing the entire sequence of next exercises so as to maximize
predicted accuracy can be phrased as a Markov decision problem we can also evaluate the benefits
of using the expectimax algorithm (see Appendix) to chose an optimal sequence of problems.
4.2
Discovering Exercise Relationships
The DKT model can further be applied to the task of discovering latent structure or concepts in the
data, a task that is typically performed by human experts. We approached this problem by assigning
an influence Jij to every directed pair of exercises i and j,
y (j|i)
Jij = P
,
k y (j|k)
(4)
where y (j|i) is the correctness probability assigned by the RNN to exercise j on the second timestep,
given that a student answered exercise i correctly on the first. We show that this characterization of
the dependencies captured by the RNN recovers the pre-requisites associated with exercises.
5
Datasets
We test the ability to predict student performance on three datasets: simulated data, Khan Academy
Data, and the Assistments benchmark dataset. On each dataset we measure area under the curve
(AUC). For the non-simulated data we evaluate our results using 5-fold cross validation and in all
cases hyper-parameters are learned on training data. We compare the results of Deep Knowledge
Tracing to standard BKT and, when possible to optimal variations of BKT. Additionally we compare
our results to predictions made by simply calculating the marginal probability of a student getting a
particular exercise correct.
1
https://github.com/chrispiech/DeepKnowledgeTracing
5
Overview
AU C
Dataset
Students
Exercise Tags
Answers
Marginal
BKT
BKT*
DKT
Simulated-5
Khan Math
Assistments
4,000
47,495
15,931
50
69
124
200 K
1,435 K
526 K
?
0.63
0.62
0.54
0.68
0.67
0.69
0.75
0.85
0.86
Table 1: AUC results for all datasets tested. BKT is the standard BKT. BKT* is the best reported result from
the literature for Assistments. DKT is the result of using LSTM Deep Knowledge Tracing.
Simulated Data: We simulate virtual students learning virtual concepts and test how well we can
predict responses in this controlled setting. For each run of this experiment we generate two thousand students who answer 50 exercises drawn from k ? 1 . . . 5 concepts. For this dataset only, all
students answer the same sequence of 50 exercises. Each student has a latent knowledge state ?skill?
for each concept, and each exercise has both a single concept and a difficulty. The probability of
a student getting a exercise with difficulty ? correct if the student had concept skill ? is modelled
1?c
using classic Item Response Theory [9] as: p(correct|?, ?) = c + 1+e
??? where c is the probability of a random guess (set to be 0.25). Students ?learn? over time via an increase to the concept
skill which corresponded to the exercise they answered. To understand how the different models
can incorporate unlabelled data, we do not provide models with the hidden concept labels (instead
the input is simply the exercise index and whether or not the exercise was answered correctly). We
evaluate prediction performance on an additional two thousand simulated test students. For each
number of concepts we repeat the experiment 20 times with different randomly generated data to
evaluate accuracy mean and standard error.
Khan Academy Data: We used a sample of anonymized student usage interactions from the eighth
grade Common Core curriculum on Khan Academy. The dataset included 1.4 million exercises
completed by 47,495 students across 69 different exercise types. It did not contain any personal
information. Only the researchers working on this paper had access to this anonymized dataset, and
its use was governed by an agreement designed to protect student privacy in accordance with Khan
Academy?s privacy notice [1]. Khan Academy provides a particularly relevant source of learning
data, since students often interact with the site for an extended period of time and for a variety of
content, and because students are often self-directed in the topics they work on and in the trajectory
they take through material.
Benchmark Dataset: In order to understand how our model compared to other models we evaluated
models on the Assistments 2009-2010 ?skill builder? public benchmark dataset2 . Assistments is an
online tutor that simultaneously teaches and assesses students in grade school mathematics. It is, to
the best of our knowledge, the largest publicly available knowledge tracing dataset [11].
6
Results
On all three datasets Deep Knowledge Tracing substantially outperformed previous methods. On
the Khan dataset using an LSTM neural network model led to an AUC of 0.85 which was a notable
improvement over the performance of a standard BKT (AUC = 0.68), especially when compared to
the small improvement BKT provided over the marginal baseline (AUC = 0.63). See Table 1 and
Figure 3(b). On the Assistments dataset DKT produced a 25% gain over the previous best reported
result (AUC = 0.86 and 0.69 respectively) [23]. The gain we report in AUC compared to the marginal
baseline (0.24) is more than triple the largest gain achieved on the dataset to date (0.07).
The prediction results from the synthetic dataset provide an interesting demonstration of the capacities of deep knowledge tracing. Both the LSTM and RNN models did as well at predicting student
responses as an oracle which had perfect knowledge of all model parameters (and only had to fit the
latent student knowledge variables). See Figure 3(a). In order to get accuracy on par with an oracle
the models would have to mimic a function that incorporates: latent concepts, the difficulty of each
exercise, the prior distributions of student knowledge and the increase in concept skill that happened
2
https://sites.google.com/site/assistmentsdata/home/assistment-2009-2010-data
6
0.85
1.0
0.85
0.85
Average Predicted Probability
0.7
True Positive Rate
0.75
0.75
0.65
0.65
Test Accuracy
Test
TestAccuracy
Accuracy
0.8
0.55
0.55
Oracle
Oracle
RNN
RNN
LSTM
LSTM
BKT
BKT
0.75
11
0.6
0.4
LSTM
RNN
BKT
Marginal
0.2
22
33
44
Number
of Concepts
Number
of Hidden
Concepts
(a)
55
0.6
MDP-8
MDP-1
Blocking
Mixing
0.5
0.0
0.0
0.2
0.4
0.6
False Positive Rate
0.8
1.0
0
10
20
Exercise Index
(b)
30
(c)
Figure 3: Left: PredictionOracle
results for (a) simulated data and (b) Khan Academy data. Right: (c) Predicted
0.65
knowledge on Assistments data
for different exercise curricula. Error bars are standard error of the mean.
RNN
LSTM
after each exercise. In contrast,
BKT the BKT prediction degraded substantially as the number of hidden
concepts increased as it doesn?t have a mechanism to learn unlabelled concepts.
We tested our ability to intelligently chose exercises on a subset of five concepts from the Assistment
dataset. 0.55
For each curricula method, we used our DKT model to simulate how a student would
answer questions and evaluate how much a student knew after 30 exercises. We repeated student
1 and measured
2 the average
3 predicted
4 probability5 of a student getting future
simulations 500 times
questions correct. In the Assistment
contextofthe
blocking strategy had a notable advantage over
Number
Concepts
mixing. See Figure 3(c). While blocking performs on par with solving expectimax one exercise
deep (MDP-1), if we look further into the future when choosing the next problem we come up with
curricula where students have higher predicted knowledge after solving fewer problems (MDP-8).
The prediction accuracy on the synthetic dataset suggest that it may be possible to use DKT models
to extract the latent structure between the assessments in the dataset. The graph of our model?s
conditional influences for the synthetic dataset reveals a perfect clustering of the five latent concepts
(see Figure 4), with directed edges set using the influence function in Equation 4. An interesting
observation is that some of the exercises from the same concept occurred far apart in time. For
example, in the synthetic dataset, where node numbers depict sequence, the 5th exercise in the
synthetic dataset was from hidden concept 1 and even though it wasn?t until the 22nd problem
that another problem from the same concept was asked, we were able to learn a strong conditional
dependency between the two. We analyzed the Khan dataset using the same technique. The resulting
graph is a compelling articulation of how the concepts in the 8th grade Common Core are related
to each other (see Figure 4. Node numbers depict exercise tags). We restricted the analysis to
ordered pairs of exercises {A, B} such that after A appeared, B appeared more than 1% of the
time in the remainder of the sequence). To determine if the resulting conditional relationships are a
product of obvious underlying trends in the data we compared our results to two baseline measures
(1) the transition probabilities of students answering B given they had just answered A and (2) the
probability in the dataset (without using a DKT model) of answering B correctly given a student
had earlier answered A correctly. Both baseline methods generated discordant graphs, which are
shown in the Appendix. While many of the relationships we uncovered may be unsurprising to an
education expert their discovery is affirmation that the DKT network learned a coherent model.
7
Discussion
In this paper we apply RNNs to the problem of knowledge tracing in education, showing improvement over prior state-of-the-art performance on the Assistments benchmark and Khan dataset. Two
particularly interesting novel properties of our new model are that (1) it does not need expert annotations (it can learn concept patterns on its own) and (2) it can operate on any student input that can
be vectorized. One disadvantage of RNNs over simple hidden Markov methods is that they require
large amounts of training data, and so are well suited to an online education environment, but not a
small classroom environment.
7
6
3
5 22
30 31
7 11
6 3 18
20
5 22 36
7 11 25
33
4
27
30 31
17
18
20
13
45
41
43
2933
48 25
36
4 26 32
42
27
49
44
6 3
17
37
13
45
41
43
5 22
29
48
7 11
26 32 39
42
35
6 3
30 31
18
20
37 44 49
34 38
5 22
3
6
725 11 33
39
36
4
Hidden concept 1
47
28
5
27
11
22
2
35
9 14 21
30 31
187 20
46
17
34 38
Hidden concept 2
45
41 30 31 13
1
18
43
2033
36
29
25
40 24
8 10 16
48
432
26
23
47
Hidden concept 3
42
28
27
2 9 14 21
33 46
36
25
17444 49
15
13
37
45
41
12
27 43
Hidden concept 4
1
29
19
40 24
393217 48
8 10 16
2613
45
4241
23
Hidden concept 5
43
29
35
Simulated Data
32 4948
26 44
15
37
42
34 38
12
39 44 49
37
19
Khan Data
47
28
35
2 9 3914 21
46 Scatter plots
1 Linear function intercepts
24 Interpreting function graphs
47 Constructing inconsistent system
34 38
11 1
35
39
2 Recognizing irrational numbers
25 Systems of equations w. Elim. 0
48 Pythagorean theorem proofs
444047249
34Pythagorean
38
8 10 16
28
23
152 9 14 24
3 Linear equations 3
26 Solutions to systems of equations
49 Scientific notation intuition
theorem
46
21
57
15 47
28
12 41
2 9 14 21
4 Multiplication in scientific notation
27 Views of a function
50 Line graph intuition
1
46
27
40
8
10 16 23
34 Lines
19 24
55
5
Parallel
lines
2
28
Recog
func
2
51 Multistep equations w. distribution
Angles
1
40 24
8 10 16
15
23
28
6 Systems of equations
29 Graphing proportional relationships 52 Fractions as repeating decimals
12
62
69
59
13 1946
15
48
7 Equations word problems
30 Exponent rules
53 Cube roots
12
Functions
68
56
19
65
8 Slope of a line
31 Angles 2
54 Scientific notation
19
32
5
31
7
6
17
43
10
42
22
58
Line graphs
1
8
50
36
38
29
Exponents
54
53
12
25
35
49
60
26
Systems of
Equations
14
66
51
3
67
37
16
30
4
23
64
33
63
2
Fractions
52
20
9 Linear models of bivariate data
32 Understand equations word problems 55 Pythagorean theorem 2
10 Systems of equations with elimination
33 Exponents 2
56 Functions 1
11 Plotting the line of best fit
34 Segment addition
57 Vertical angles 2
12 Integer sums
35 Systems of equations w. substitution 58 Solving for the x intercept
13 Congruent angles
36 Comparing proportional relationships 59 Recognizing functions
14 Exponents 1
37 Solutions to linear equations
15 Interpreting scatter plots
38 Finding intercepts of linear functions 61 Slope and triangle similarity
16 Repeating decimals to fractions 2
39 Midpoint of a segment
62 Distance formula
17 Graphical solutions to systems
40 Volume word problems
63 Converting decimals to fractions 2
18 Linear non linear functions
41 Constructing scatter plots
64 Age word problems
60 Square roots
19 Interpreting features of linear functions 42 Solving for the y intercept
65 Pythagorean theorem 1
20 Repeating decimals to fractions 1
43 Graphing systems of equations
66 Comparing features of functions 0
21 Constructing linear functions
44 Frequencies of bivariate data
67 Orders of magnitude
22 Graphing linear equations
45 Comparing features of functions 1
68 Angle addition postulate
23 Computing in scientific notation
46 Angles 1
69 Parallel lines 1
Figure 4: Graphs of conditional influence between exercises in DKT models. Above: We observe a perfect
clustering of latent concepts in the synthetic data. Below: A convincing depiction of how 8th grade math
Common Core exercises influence one another. Arrow size indicates connection strength. Note that nodes may
be connected in both directions. Edges with a magnitude smaller than 0.1 have been thresholded. Cluster
labels are added by hand, but are fully consistent with the exercises in each cluster.
The application of RNNs to knowledge tracing provides many directions for future research. Further investigations could incorporate other features as inputs (such as time taken), explore other
educational impacts (such as hint generation, dropout prediction), and validate hypotheses posed in
education literature (such as spaced repetition, modeling how students forget). Because DKTs take
vector input it should be possible to track knowledge over more complex learning activities. An especially interesting extension is to trace student knowledge as they solve open-ended programming
tasks [26, 27]. Using a recently developed method for vectorization of programs [25] we hope to be
able to intelligently model student knowledge over time as they learn to program.
In an ongoing collaboration with Khan Academy, we plan to test the efficacy of DKT for curriculum
planning in a controlled experiment, by using it to propose exercises on the site.
Acknowledgments
Many thanks to John Mitchell for his guidance and Khan Academy for its support. Chris Piech is
supported by NSF-GRFP grant number DGE-114747.
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9
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5,142 | 5,655 | Deep Temporal Sigmoid Belief Networks
for Sequence Modeling
Zhe Gan, Chunyuan Li, Ricardo Henao, David Carlson and Lawrence Carin
Department of Electrical and Computer Engineering
Duke University, Durham, NC 27708
{zhe.gan, chunyuan.li, r.henao, david.carlson, lcarin}@duke.edu
Abstract
Deep dynamic generative models are developed to learn sequential dependencies
in time-series data. The multi-layered model is designed by constructing a hierarchy of temporal sigmoid belief networks (TSBNs), defined as a sequential stack
of sigmoid belief networks (SBNs). Each SBN has a contextual hidden state,
inherited from the previous SBNs in the sequence, and is used to regulate its hidden bias. Scalable learning and inference algorithms are derived by introducing
a recognition model that yields fast sampling from the variational posterior. This
recognition model is trained jointly with the generative model, by maximizing its
variational lower bound on the log-likelihood. Experimental results on bouncing
balls, polyphonic music, motion capture, and text streams show that the proposed
approach achieves state-of-the-art predictive performance, and has the capacity to
synthesize various sequences.
1
Introduction
Considerable research has been devoted to developing probabilistic models for high-dimensional
time-series data, such as video and music sequences, motion capture data, and text streams. Among
them, Hidden Markov Models (HMMs) [1] and Linear Dynamical Systems (LDS) [2] have been
widely studied, but they may be limited in the type of dynamical structures they can model. An
HMM is a mixture model, which relies on a single multinomial variable to represent the history of a
time-series. To represent N bits of information about the history, an HMM could require 2N distinct
states. On the other hand, real-world sequential data often contain complex non-linear temporal
dependencies, while a LDS can only model simple linear dynamics.
Another class of time-series models, which are potentially better suited to model complex probability distributions over high-dimensional sequences, relies on the use of Recurrent Neural Networks
(RNNs) [3, 4, 5, 6], and variants of a well-known undirected graphical model called the Restricted
Boltzmann Machine (RBM) [7, 8, 9, 10, 11]. One such variant is the Temporal Restricted Boltzmann Machine (TRBM) [8], which consists of a sequence of RBMs, where the state of one or more
previous RBMs determine the biases of the RBM in the current time step. Learning and inference in
the TRBM is non-trivial. The approximate procedure used in [8] is heuristic and not derived from a
principled statistical formalism.
Recently, deep directed generative models [12, 13, 14, 15] are becoming popular. A directed graphical model that is closely related to the RBM is the Sigmoid Belief Network (SBN) [16]. In the work
presented here, we introduce the Temporal Sigmoid Belief Network (TSBN), which can be viewed
as a temporal stack of SBNs, where each SBN has a contextual hidden state that is inherited from
the previous SBNs and is used to adjust its hidden-units bias. Based on this, we further develop
a deep dynamic generative model by constructing a hierarchy of TSBNs. This can be considered
1
W1
W2
Time
Time
(c) Generative model
(d) Recognition model
W3
W4
(a) Generative model
U1
U2
U3
(b) Recognition model
Figure 1: Graphical model for the Deep Temporal Sigmoid Belief Network. (a,b) Generative and recognition
model of the TSBN. (c,d) Generative and recognition model of a two-layer Deep TSBN.
as a deep SBN [15] with temporal feedback loops on each layer. Both stochastic and deterministic
hidden layers are considered.
Compared with previous work, our model: (i) can be viewed as a generalization of an HMM with
distributed hidden state representations, and with a deep architecture; (ii) can be seen as a generalization of a LDS with complex non-linear dynamics; (iii) can be considered as a probabilistic
construction of the traditionally deterministic RNN; (iv) is closely related to the TRBM, but it has a
fully generative process, where data are readily generated from the model using ancestral sampling;
(v) can be utilized to model different kinds of data, e.g., binary, real-valued and counts.
The ?explaining away? effect described in [17] makes inference slow, if one uses traditional inference methods. Another important contribution we present here is to develop fast and scalable
learning and inference algorithms, by introducing a recognition model [12, 13, 14], that learns an
inverse mapping from observations to hidden variables, based on a loss function derived from a variational principle. By utilizing the recognition model and variance-reduction techniques from [13],
we achieve fast inference both at training and testing time.
2
2.1
Model Formulation
Sigmoid Belief Networks
Deep dynamic generative models are considered, based on the Sigmoid Belief Network (SBN) [16].
An SBN is a Bayesian network that models a binary visible vector v ? {0, 1}M , in terms of binary
hidden variables h ? {0, 1}J and weights W ? RM ?J with
>
p(vm = 1|h) = ?(wm
h + cm ),
p(hj = 1) = ?(bj ),
(1)
where v = [v1 , . . . , vM ]> , h = [h1 , . . . , hJ ]> , W = [w1 , . . . , wM ]> , c = [c1 , . . . , cM ]> ,
b = [b1 , . . . , bJ ]> , and the logistic function, ?(x) , 1/(1 + e?x ). The parameters W, b and c
characterize all data, and the hidden variables, h, are specific to particular visible data, v.
The SBN is closely related to the RBM [18], which is a Markov random field with the same bipartite structure as the SBN. The RBM defines a distribution over a binary vector that is proportional
to the exponential of its energy, defined as ?E(v, h) = v > c + v > Wh + h> b. The conditional
distributions, p(v|h) and p(h|v), in the RBM are factorial, which makes inference fast, while parameter estimation usually relies on an approximation technique known as Contrastive Divergence
(CD) [18].
P
The energy function of an SBN may be written as ?E(v, h) = v > c+v > Wh+h> b? m log(1+
>
exp(wm
h + cm )). SBNs explicitly manifest the generative process to obtain data, in which the
hidden layer provides a directed ?explanation? for patterns generated in the visible layer. However,
the ?explaining away? effect described in [17] makes inference inefficient, the latter can be alleviated
by exploiting recent advances in variational inference methods [13].
2
2.2
Temporal Sigmoid Belief Networks
The proposed Temporal Sigmoid Belief Network (TSBN) model is a sequence of SBNs arranged
in such way that at any given time step, the SBN?s biases depend on the state of the SBNs in the
previous time steps. Specifically, assume we have a length-T binary visible sequence, the tth time
step of which is denoted vt ? {0, 1}M . The TSBN describes the joint probability as
p? (V, H) = p(h1 )p(v1 |h1 ) ?
T
Y
p(ht |ht?1 , vt?1 ) ? p(vt |ht , vt?1 ),
(2)
t=2
where V = [v1 , . . . , vT ], H = [h1 , . . . , hT ], and each ht ? {0, 1}J represents the hidden state
corresponding to time step t. For t = 1, . . . , T , each conditional distribution in (2) is expressed as
>
>
p(hjt = 1|ht?1 , vt?1 ) = ?(w1j
ht?1 + w3j
vt?1 + bj ),
(3)
>
>
p(vmt = 1|ht , vt?1 ) = ?(w2m
ht + w4m
vt?1 + cm ),
(4)
where h0 and v0 , needed for the prior model p(h1 ) and p(v1 |h1 ), are defined as zero vectors,
respectively, for conciseness. The model parameters, ?, are specified as W1 ? RJ?J , W2 ?
RM ?J , W3 ? RJ?M , W4 ? RM ?M . For i = 1, 2, 3, 4, wij is the transpose of the jth row of Wi ,
and c = [c1 , . . . , cM ]> and b = [b1 , . . . , bJ ]> are bias terms. The graphical model for the TSBN is
shown in Figure 1(a).
By setting W3 and W4 to be zero matrices, the TSBN can be viewed as a Hidden Markov Model
[1] with an exponentially large state space, that has a compact parameterization of the transition and
the emission probabilities. Specifically, each hidden state in the HMM is represented as a one-hot
length-J vector, while in the TSBN, the hidden states can be any length-J binary vector. We note
that the transition matrix is highly structured, since the number of parameters is only quadratic w.r.t.
J. Compared with the TRBM [8], our TSBN is fully directed, which allows for fast sampling of
?fantasy? data from the inferred model.
2.3
TSBN Variants
Modeling real-valued data The model above can be readily extended to model real-valued sequence data, by substituting (4) with p(vt |ht , vt?1 ) = N (?t , diag(?t2 )), where
>
>
2
0
0
?mt = w2m
ht + w4m
vt?1 + cm , log ?mt
= (w2m
)> ht + (w4m
)> vt?1 + c0m ,
(5)
2
are elements of ?t and ?t2 , respectively. W20 and W40 are of the same size of
and ?mt and ?mt
W2 and W4 , respectively. Compared with the Gaussian TRBM [9], in which ?mt is fixed to 1, our
formalism uses a diagonal matrix to parameterize the variance structure of vt .
Modeling count data We also introduce an approach for modeling time-series data with count
QM
vmt
observations, by replacing (4) with p(vt |ht , vt?1 ) = m=1 ymt
, where
ymt = PM
>
>
ht + w4m
vt?1 + cm )
exp(w2m
m0 =1
> h + w> v
exp(w2m
0
t
4m0 t?1 + cm0 )
.
(6)
This formulation is related to the Replicated Softmax Model (RSM) described in [19], however, our
approach uses a directed connection from the binary hidden variables to the visible counts, while
also learning the dynamics in the count sequences.
Furthermore, rather than assuming that ht and vt only depend on ht?1 and vt?1 , in the experiments,
we also allow for connections from the past n time steps of the hidden and visible states, to the
current states, ht and vt . A sliding window is then used to go through the sequence to obtain n
frames at each time. We refer to n as the order of the model.
2.4
Deep Architecture for Sequence Modeling with TSBNs
Learning the sequential dependencies with the shallow model in (2)-(4) may be restrictive. Therefore, we propose two deep architectures to improve its representational power: (i) adding stochastic
hidden layers; (ii) adding deterministic hidden layers. The graphical model for the deep TSBN
3
(`)
is shown in Figure 1(c). Specifically, we consider a deep TSBN with hidden layers ht for
t = 1, . . . , T and ` = 1, . . . , L. Assume layer ` contains J (`) hidden units, and denote the visi(0)
(L+1)
ble layer vt = ht and let ht
= 0, for convenience. In order to obtain a proper generative
(L)
model, the top hidden layer h contains stochastic binary hidden variables.
For the middle layers, ` = 1, . . . , L?1, if stochastic hidden layers are utilized, the generative process
QJ (`)
(`)
(`) (`+1)
(`)
(`?1)
is expressed as p(ht ) = j=1 p(hjt |ht
, ht?1 , ht?1 ), where each conditional distribution
is parameterized via a logistic function, as in (4). If deterministic hidden layers are employed,
(`)
(`+1)
(`)
(`?1)
we obtain ht = f (ht
, ht?1 , ht?1 ), where f (?) is chosen to be a rectified linear function.
Although the differences between these two approaches are minor, learning and inference algorithms
can be quite different, as shown in Section 3.3.
3
Scalable Learning and Inference
Computation of the exact posterior over the hidden variables in (2) is intractable. Approximate
Bayesian inference, such as Gibbs sampling or mean-field variational Bayes (VB) inference, can
be implemented [15, 16]. However, Gibbs sampling is very inefficient, due to the fact that the
conditional posterior distribution of the hidden variables does not factorize. The mean-field VB
indeed provides a fully factored variational posterior, but this technique increases the gap between
the bound being optimized and the true log-likelihood, potentially resulting in a poor fit to the data.
To allow for tractable and scalable inference and parameter learning, without loss of the flexibility of
the variational posterior, we apply the Neural Variational Inference and Learning (NVIL) algorithm
described in [13].
3.1
Variational Lower Bound Objective
We are interested in training the TSBN model, p? (V, H), described in (2), with parameters ?.
Given an observation V, we introduce a fixed-form distribution, q? (H|V), with parameters ?, that
approximates the true posterior distribution, p(H|V). We then follow the variational principle to
derive a lower bound on the marginal log-likelihood, expressed as1
L(V, ?, ?) = Eq? (H|V) [log p? (V, H) ? log q? (H|V)] .
(7)
We construct the approximate posterior q? (H|V) as a recognition model. By using this, we avoid
the need to compute variational parameters per data point; instead we compute a set of parameters
? used for all V. In order to achieve fast inference, the recognition model is expressed as
T
Y
q? (H|V) = q(h1 |v1 ) ?
q(ht |ht?1 , vt , vt?1 ) ,
(8)
t=2
and each conditional distribution is specified as
>
>
q(hjt = 1|ht?1 , vt , vt?1 ) = ?(u>
(9)
1j ht?1 + u2j vt + u3j vt?1 + dj ) ,
where h0 and v0 , for q(h1 |v1 ), are defined as zero vectors. The recognition parameters ? are
specified as U1 ? RJ?J , U2 ? RJ?M , U3 ? RJ?M . For i = 1, 2, 3, uij is the transpose of the jth
row of Ui , and d = [d1 , . . . , dJ ]> is the bias term. The graphical model is shown in Figure 1(b).
The recognition model defined in (9) has the same form as in the approximate inference used for the
TRBM [8]. Exact inference for our model consists of a forward and backward pass through the entire
sequence, that requires the traversing of each possible hidden state. Our feedforward approximation
allows the inference procedure to be fast and implemented in an online fashion.
3.2
Parameter Learning
To optimize (7), we utilize Monte Carlo methods to approximate expectations and stochastic gradient
descent (SGD) for parameter optimization. The gradients can be expressed as
?? L(V) = Eq? (H|V) [?? log p? (V, H)],
(10)
?? L(V) = Eq? (H|V) [(log p? (V, H) ? log q? (H|V)) ? ?? log q? (H|V)].
1
This lower bound is equivalent to the marginal log-likelihood if q? (H|V) = p(H|V).
4
(11)
>
>
? jt =
Specifically, in the TSBN model, if we define v?mt = ?(w2m
ht + w4m
vt?1 + cm ) and h
>
>
>
?(u1j ht?1 + u2j vt + u3j vt?1 + dj ), the gradients for w2m and u2j can be calculated as
T
T
? log p? (V, H) X
=
(vmt ? v?mt ) ? hjt ,
?w2mj
t=1
? log q? (H|V) X
? jt ) ? vmt .
=
(hjt ? h
?u2jm
t=1
(12)
Other update equations, along with the learning details for the TSBN variants in Section 2.3, are
provided in the Supplementary Section B. We observe that the gradients in (10) and (11) share many
similarities with the wake-sleep algorithm [20]. Wake-sleep alternates between updating ? in the
wake phase and updating ? in the sleep phase. The update of ? is based on the samples generated
from q? (H|V), and is identical to (10). However, in contrast to (11), the recognition parameters ?
are estimated from samples generated by the model, i.e., ?? L(V) = Ep? (V,H) [?? log q? (H|V)].
This update does not optimize the same objective as in (10), hence the wake-sleep algorithm is not
guaranteed to converge [13].
Inspecting (11), we see that we are using l? (V, H) = log p? (V, H) ? log q? (H|V) as the learning
signal for the recognition parameters ?. The expectation of this learning signal is exactly the lower
bound (7), which is easy to evaluate. However, this tractability makes the estimated gradients of the
recognition parameters very noisy. In order to make the algorithm practical, we employ the variance
reduction techniques proposed in [13], namely: (i) centering the learning signal, by subtracting the
data-independent baseline and the data-dependent baseline; (ii) variance normalization, by dividing
the centered learning signal by a running estimate of its standard deviation. The data-dependent
baseline is implemented using a neural network. Additionally, RMSprop [21], a form of SGD where
the gradients are adaptively rescaled by a running average of their recent magnitude, were found
in practice to be important for fast convergence; thus utilized throughout all the experiments. The
outline of the NVIL algorithm is provided in the Supplementary Section A.
3.3
Extension to deep models
The recognition model corresponding to the deep TSBN is shown in Figure 1(d). Two kinds of deep
architectures are discussed in Section 2.4. We illustrate the difference of their learning algorithms
in two respects: (i) the calculation of the lower bound; and (ii) the calculation of the gradients.
The top hidden layer is stochastic. If the middle hidden layers are also stochastic, the calculation
of the lower bound is more involved, compared with the shallow model; however, the gradient
evaluation remain simple as in (12). On the other hand, if deterministic middle hidden layers (i.e.,
recurrent neural networks) are employed, the lower bound objective will stay the same as a shallow
model, since the only stochasticity in the generative process lies in the top layer; however, the
gradients have to be calculated recursively through the back-propagation through time algorithm
[22]. All details are provided in the Supplementary Section C.
4
Related Work
The RBM has been widely used as building block to learn the sequential dependencies in time-series
data, e.g., the conditional-RBM-related models [7, 23], and the temporal RBM [8]. To make exact
inference possible, the recurrent temporal RBM was also proposed [9], and further extended to learn
the dependency structure within observations [11].
In the work reported here, we focus on modeling sequences based on the SBN [16], which recently
has been shown to have the potential to build deep generative models [13, 15, 24]. Our work serves
as another extension of the SBN that can be utilized to model time-series data. Similar ideas have
also been considered in [25] and [26]. However, in [25], the authors focus on grammar learning, and
use a feed-forward approximation of the mean-field VB to carry out the inference; while in [26], the
wake-sleep algorithm was developed. We apply the model in a different scenario, and develop a fast
and scalable inference algorithm, based on the idea of training a recognition model by leveraging
the stochastic gradient of the variational bound.
There exist two main methods for the training of recognition models. The first one, termed Stochastic Gradient Variational Bayes (SGVB), is based on a reparameterization trick [12, 14], which can
be only employed in models with continuous latent variables, e.g., the variational auto-encoder [12]
5
Top: Generated from Piano midi
1
20
Topic 29
Nicaragua v. U.S.
0.5
40
0
60
1800
1850
1
80
50
100
150
200
250
300
1900
1950
2000
Topic 30
War of 1812
0.5
Iraq War
World War II
Bottom: Generated from Nottingham
0
1800
20
1850
1
40
0.5
1900
1950
2000
1950
2000
Topic 130
The age of American revolution
60
80
0
20
40
60
80
100
120
140
160
180
1800
1850
1900
Figure 2: (Left) Dictionaries learned using the HMSBN for the videos of bouncing balls. (Middle)
Samples generated from the HMSBN trained on the polyphonic music. Each column is a sample
vector of notes. (Right) Time evolving from 1790 to 2014 for three selected topics learned from the
STU dataset. Plotted values represent normalized probabilities that the topic appears in a given year.
Best viewed electronically.
and all the recent recurrent extensions of it [27, 28, 29]. The second one, called Neural Variational
Inference and Learning (NVIL), is based on the log-derivative trick [13], which is more general and
can also be applicable to models with discrete random variables. The NVIL algorithm has been
previously applied to the training of SBN in [13]. Our approach serves as a new application of this
algorithm for a SBN-based time-series model.
5
Experiments
We present experimental results on four publicly available datasets: the bouncing balls [9], polyphonic music [10], motion capture [7] and state-of-the-Union [30]. To assess the performance of the
TSBN model, we show sequences generated from the model, and report the average log-probability
that the model assigns to a test sequence, and the average squared one-step-ahead prediction error per
frame. Code is available at https://github.com/zhegan27/TSBN_code_NIPS2015.
The TSBN model with W3 = 0 and W4 = 0 is denoted Hidden Markov SBN (HMSBN), the deep
TSBN with stochastic hidden layer is denoted DTSBN-S, and the deep TSBN with deterministic
hidden layer is denoted DTSBN-D.
Model parameters were initialized by sampling randomly from N (0, 0.0012 I), except for the bias
parameters, that were initialized as 0. The TSBN model is trained using a variant of RMSprop
[6], with momentum of 0.9, and a constant learning rate of 10?4 . The decay over the root mean
squared gradients is set to 0.95. The maximum number of iterations we use is 105 . The gradient
estimates were computed using a single sample from the recognition model. The only regularization
we used was a weight decay of 10?4 . The data-dependent baseline was implemented by using a
neural network with a single hidden layer with 100 tanh units.
For the prediction of vt given v1:t?1 , we (i) first obtain a sample from q? (h1:t?1 |v1:t?1 ); (ii)
calculate the conditional posterior p? (ht |h1:t?1 , v1:t?1 ) of the current hidden state ; (iii) make a
prediction for vt using p? (vt |h1:t , v1:t?1 ). On the other hand, synthesizing samples is conceptually
simper. Sequences can be readily generated from the model using ancestral sampling.
5.1
Bouncing balls dataset
We conducted the first experiment on synthetic videos of 3 bouncing balls, where pixels are binary
valued. We followed the procedure in [9], and generated 4000 videos for training, and another 200
videos for testing. Each video is of length 100 and of resolution 30 ? 30.
The dictionaries learned using the HMSBN are shown in Figure 2 (Left). Compared with previous
work [9, 10], our learned bases are more spatially localized. In Table 1, we compare the average
squared prediction error per frame over the 200 test videos, with recurrent temporal RBM (RTRBM)
and structured RTRBM (SRTRBM). As can be seen, our approach achieves better performance
compared with the baselines in the literature. Furthermore, we observe that a high-order TSBN
reduces the prediction error significantly, compared with an order-one TSBN. This is due to the fact
6
Table 1: Average prediction error for the bounc-
Table 2: Average prediction error obtained for
ing balls dataset. () taken from [11].
the motion capture dataset. () taken from [11].
M ODEL
DTSBN- S
DTSBN- D
TSBN
TSBN
RTRBM
SRTRBM
D IM
100-100
100-100
100
100
3750
3750
O RDER
2
2
4
1
1
1
P RED . E RR .
2.79 ? 0.39
2.99 ? 0.42
3.07 ? 0.40
9.48 ? 0.38
3.88 ? 0.33
3.31 ? 0.33
M ODEL
DTSBN- S
DTSBN- D
TSBN
HMSBN
SS -SRTRBM
G -RTRBM
WALKING
4.40 ? 0.28
4.62 ? 0.01
5.12 ? 0.50
10.77 ? 1.15
8.13 ? 0.06
14.41 ? 0.38
RUNNING
2.56 ? 0.40
2.84 ? 0.01
4.85 ? 1.26
7.39 ? 0.47
5.88 ? 0.05
10.91 ? 0.27
that by using a high-order TSBN, more information about the past is conveyed. We also examine
the advantage of employing deep models. Using stochastic, or deterministic hidden layer improves
performances. More results, including log-likelihoods, are provided in Supplementary Section D.
5.2
Motion capture dataset
In this experiment, we used the CMU motion capture dataset, that consists of measured joint angles
for different motion types. We used the 33 running and walking sequences of subject 35 (23 walking
sequences and 10 running sequences). We followed the preprocessing procedure of [11], after which
we were left with 58 joint angles. We partitioned the 33 sequences into training and testing set: the
first of which had 31 sequences, and the second had 2 sequences (one walking and another running).
We averaged the prediction error over 100 trials, as reported in Table 2. The TSBN we implemented
is of size 100 in each hidden layer and order 1. It can be seen that the TSBN-based models improves
over the Gaussian (G-)RTRBM and the spike-slab (SS-)SRTRBM significantly.
Figure 3: Motion trajectories generated from the HMSBN trained on the motion capture dataset.
(Left) Walking. (Middle) Running-running-walking. (Right) Running-walking.
Another popular motion capture dataset is the MIT dataset2 . To further demonstrate the directed,
generative nature of our model, we give our trained HMSBN model different initializations, and
show generated, synthetic data and the transitions between different motion styles in Figure 3. These
generated data are readily produced from the model and demonstrate realistic behavior. The smooth
trajectories are walking movements, while the vibrating ones are running. Corresponding video files
(AVI) are provided as mocap 1, 2 and 3 in the Supplementary Material.
5.3
Polyphonic music dataset
The third experiment is based on four different polyphonic music sequences of piano [10], i.e.,
Piano-midi.de (Piano), Nottingham (Nott), MuseData (Muse) and JSB chorales (JSB). Each of these
datasets are represented as a collection of 88-dimensional binary sequences, that span the whole
range of piano from A0 to C8.
The samples generated from the trained HMSBN model are shown in Figure 2 (Middle). As can
be seen, different styles of polyphonic music are synthesized. The corresponding MIDI files are
provided as music 1 and 2 in the Supplementary Material. Our model has the ability to learn basic
harmony rules and local temporal coherence. However, long-term structure and musical melody
remain elusive. The variational lower bound, along with the estimated log-likelihood in [10], are
presented in Table 3. The TSBN we implemented is of size 100 and order 1. Empirically, adding
layers did not improve performance on this dataset, hence no such results are reported. The results
of RNN-NADE and RTRBM [10] were obtained by only 100 runs of the annealed importance sampling, which has the potential to overestimate the true log-likelihood. Our variational lower bound
provides a more conservative estimate. Though, our performance is still better than that of RNN.
2
Quantitative results on the MIT dataset are provided in Supplementary Section D.
7
Table 3: Test log-likelihood for the polyphonic
Table 4: Average prediction precision for STU.
music dataset. () taken from [10].
() taken from [31].
M ODEL
TSBN
RNN-NADE
RTRBM
RNN
5.4
P IANO .
-7.98
-7.05
-7.36
-8.37
N OTT.
-3.67
-2.31
-2.62
-4.46
M USE .
-6.81
-5.60
-6.35
-8.13
JSB.
-7.48
-5.56
-6.35
-8.71
M ODEL
HMSBN
DHMSBN- S
GP-DPFA
DRFM
D IM
25
25-25
100
25
MP
0.327 ? 0.002
0.299 ? 0.001
0.223 ? 0.001
0.217 ? 0.003
PP
0.353 ? 0.070
0.378 ? 0.006
0.189 ? 0.003
0.177 ? 0.010
State of the Union dataset
The State of the Union (STU) dataset contains the transcripts of T = 225 US State of the Union addresses, from 1790 to 2014. Two tasks are considered, i.e., prediction and dynamic topic modeling.
Prediction The prediction task is concerned with estimating the held-out words. We employ the
setup in [31]. After removing stop words and terms that occur fewer than 7 times in one document or
less than 20 times overall, there are 2375 unique words. The entire data of the last year is held-out.
For the documents in the previous years, we randomly partition the words of each document into
80%/20% split. The model is trained on the 80% portion, and the remaining 20% held-out words
are used to test the prediction at each year. The words in both held-out sets are ranked according to
the probability estimated from (6).
To evaluate the prediction performance, we calculate the precision @top-M as in [31], which is given
by the fraction of the top-M words, predicted by the model, that matches the true ranking of the word
counts. M = 50 is used. Two recent works are compared, GP-DPFA [31] and DRFM [30]. The
results are summarized in Table 4. Our model is of order 1. The column MP denotes the mean
precision over all the years that appear in the training set. The column PP denotes the predictive
precision for the final year. Our model achieves significant improvements in both scenarios.
Dynamic Topic Modeling The setup described in [30] is employed, and the number of topics is
200. To understand the temporal dynamic per topic, three topics are selected and the normalized
probability that a topic appears at each year are shown in Figure 2 (Right). Their associated top 6
words per topic are shown in Table 5. The learned trajectory exhibits different temporal patterns
across the topics. Clearly, we can identify jumps associated with some key historical events. For
instance, for Topic 29, we observe a positive jump in 1986 related to military and paramilitary
activities in and against Nicaragua brought by the U.S. Topic 30 is related with war, where the War
of 1812, World War II and Iraq War all spike up in their corresponding years. In Topic 130, we
observe consistent positive jumps from 1890 to 1920, when the American revolution was taking
place. Three other interesting topics are also shown in Table 5. Topic 64 appears to be related to
education, Topic 70 is about Iraq, and Topic 74 is Axis and World War II. We note that the words
for these topics are explicitly related to these matters.
Table 5: Top 6 most probable words associated with the STU topics.
Topic #29
family
budget
Nicaragua
free
future
freedom
6
Topic #30
officer
civilized
warfare
enemy
whilst
gained
Topic #130
government
country
public
law
present
citizens
Topic #64
generations
generation
recognize
brave
crime
race
Topic #70
Iraqi
Qaida
Iraq
Iraqis
AI
Saddam
Topic #74
Philippines
islands
axis
Nazis
Japanese
Germans
Conclusion
We have presented the Deep Temporal Sigmoid Belief Networks, an extension of SBN, that models the temporal dependencies in high-dimensional sequences. To allow for scalable inference and
learning, an efficient variational optimization algorithm is developed. Experimental results on several datasets show that the proposed approach obtains superior predictive performance, and synthesizes interesting sequences.
In this work, we have investigated the modeling of different types of data individually. One interesting future work is to combine them into a unified framework for dynamic multi-modality learning.
Furthermore, we can use high-order optimization methods to speed up inference [32].
Acknowledgements This research was supported in part by ARO, DARPA, DOE, NGA and ONR.
8
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9
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5,143 | 5,656 | Hidden Technical Debt in Machine Learning Systems
D. Sculley, Gary Holt, Daniel Golovin, Eugene Davydov, Todd Phillips
{dsculley,gholt,dgg,edavydov,toddphillips}@google.com
Google, Inc.
Dietmar Ebner, Vinay Chaudhary, Michael Young, Jean-Franc?ois Crespo, Dan Dennison
{ebner,vchaudhary,mwyoung,jfcrespo,dennison}@google.com
Google, Inc.
Abstract
Machine learning offers a fantastically powerful toolkit for building useful complex prediction systems quickly. This paper argues it is dangerous to think of
these quick wins as coming for free. Using the software engineering framework
of technical debt, we find it is common to incur massive ongoing maintenance
costs in real-world ML systems. We explore several ML-specific risk factors to
account for in system design. These include boundary erosion, entanglement,
hidden feedback loops, undeclared consumers, data dependencies, configuration
issues, changes in the external world, and a variety of system-level anti-patterns.
1
Introduction
As the machine learning (ML) community continues to accumulate years of experience with live
systems, a wide-spread and uncomfortable trend has emerged: developing and deploying ML systems is relatively fast and cheap, but maintaining them over time is difficult and expensive.
This dichotomy can be understood through the lens of technical debt, a metaphor introduced by
Ward Cunningham in 1992 to help reason about the long term costs incurred by moving quickly in
software engineering. As with fiscal debt, there are often sound strategic reasons to take on technical
debt. Not all debt is bad, but all debt needs to be serviced. Technical debt may be paid down
by refactoring code, improving unit tests, deleting dead code, reducing dependencies, tightening
APIs, and improving documentation [8]. The goal is not to add new functionality, but to enable
future improvements, reduce errors, and improve maintainability. Deferring such payments results
in compounding costs. Hidden debt is dangerous because it compounds silently.
In this paper, we argue that ML systems have a special capacity for incurring technical debt, because
they have all of the maintenance problems of traditional code plus an additional set of ML-specific
issues. This debt may be difficult to detect because it exists at the system level rather than the code
level. Traditional abstractions and boundaries may be subtly corrupted or invalidated by the fact that
data influences ML system behavior. Typical methods for paying down code level technical debt are
not sufficient to address ML-specific technical debt at the system level.
This paper does not offer novel ML algorithms, but instead seeks to increase the community?s awareness of the difficult tradeoffs that must be considered in practice over the long term. We focus on
system-level interactions and interfaces as an area where ML technical debt may rapidly accumulate.
At a system-level, an ML model may silently erode abstraction boundaries. The tempting re-use or
chaining of input signals may unintentionally couple otherwise disjoint systems. ML packages may
be treated as black boxes, resulting in large masses of ?glue code? or calibration layers that can lock
in assumptions. Changes in the external world may influence system behavior in unintended ways.
Even monitoring ML system behavior may prove difficult without careful design.
1
2
Complex Models Erode Boundaries
Traditional software engineering practice has shown that strong abstraction boundaries using encapsulation and modular design help create maintainable code in which it is easy to make isolated
changes and improvements. Strict abstraction boundaries help express the invariants and logical
consistency of the information inputs and outputs from an given component [8].
Unfortunately, it is difficult to enforce strict abstraction boundaries for machine learning systems
by prescribing specific intended behavior. Indeed, ML is required in exactly those cases when the
desired behavior cannot be effectively expressed in software logic without dependency on external
data. The real world does not fit into tidy encapsulation. Here we examine several ways that the
resulting erosion of boundaries may significantly increase technical debt in ML systems.
Entanglement. Machine learning systems mix signals together, entangling them and making isolation of improvements impossible. For instance, consider a system that uses features x1 , ...xn in
a model. If we change the input distribution of values in x1 , the importance, weights, or use of
the remaining n ? 1 features may all change. This is true whether the model is retrained fully in a
batch style or allowed to adapt in an online fashion. Adding a new feature xn+1 can cause similar
changes, as can removing any feature xj . No inputs are ever really independent. We refer to this here
as the CACE principle: Changing Anything Changes Everything. CACE applies not only to input
signals, but also to hyper-parameters, learning settings, sampling methods, convergence thresholds,
data selection, and essentially every other possible tweak.
One possible mitigation strategy is to isolate models and serve ensembles. This approach is useful
in situations in which sub-problems decompose naturally such as in disjoint multi-class settings like
[14]. However, in many cases ensembles work well because the errors in the component models are
uncorrelated. Relying on the combination creates a strong entanglement: improving an individual
component model may actually make the system accuracy worse if the remaining errors are more
strongly correlated with the other components.
A second possible strategy is to focus on detecting changes in prediction behavior as they occur.
One such method was proposed in [12], in which a high-dimensional visualization tool was used to
allow researchers to quickly see effects across many dimensions and slicings. Metrics that operate
on a slice-by-slice basis may also be extremely useful.
Correction Cascades. There are often situations in which model ma for problem A exists, but a
solution for a slightly different problem A? is required. In this case, it can be tempting to learn a
model m?a that takes ma as input and learns a small correction as a fast way to solve the problem.
However, this correction model has created a new system dependency on ma , making it significantly
more expensive to analyze improvements to that model in the future. The cost increases when
correction models are cascaded, with a model for problem A?? learned on top of m?a , and so on,
for several slightly different test distributions. Once in place, a correction cascade can create an
improvement deadlock, as improving the accuracy of any individual component actually leads to
system-level detriments. Mitigation strategies are to augment ma to learn the corrections directly
within the same model by adding features to distinguish among the cases, or to accept the cost of
creating a separate model for A? .
Undeclared Consumers. Oftentimes, a prediction from a machine learning model ma is made
widely accessible, either at runtime or by writing to files or logs that may later be consumed by
other systems. Without access controls, some of these consumers may be undeclared, silently using
the output of a given model as an input to another system. In more classical software engineering,
these issues are referred to as visibility debt [13].
Undeclared consumers are expensive at best and dangerous at worst, because they create a hidden
tight coupling of model ma to other parts of the stack. Changes to ma will very likely impact these
other parts, potentially in ways that are unintended, poorly understood, and detrimental. In practice,
this tight coupling can radically increase the cost and difficulty of making any changes to ma at all,
even if they are improvements. Furthermore, undeclared consumers may create hidden feedback
loops, which are described more in detail in section 4.
2
Undeclared consumers may be difficult to detect unless the system is specifically designed to guard
against this case, for example with access restrictions or strict service-level agreements (SLAs). In
the absence of barriers, engineers will naturally use the most convenient signal at hand, especially
when working against deadline pressures.
3
Data Dependencies Cost More than Code Dependencies
In [13], dependency debt is noted as a key contributor to code complexity and technical debt in
classical software engineering settings. We have found that data dependencies in ML systems carry
a similar capacity for building debt, but may be more difficult to detect. Code dependencies can be
identified via static analysis by compilers and linkers. Without similar tooling for data dependencies,
it can be inappropriately easy to build large data dependency chains that can be difficult to untangle.
Unstable Data Dependencies. To move quickly, it is often convenient to consume signals as input
features that are produced by other systems. However, some input signals are unstable, meaning
that they qualitatively or quantitatively change behavior over time. This can happen implicitly,
when the input signal comes from another machine learning model itself that updates over time,
or a data-dependent lookup table, such as for computing TF/IDF scores or semantic mappings. It
can also happen explicitly, when the engineering ownership of the input signal is separate from the
engineering ownership of the model that consumes it. In such cases, updates to the input signal
may be made at any time. This is dangerous because even ?improvements? to input signals may
have arbitrary detrimental effects in the consuming system that are costly to diagnose and address.
For example, consider the case in which an input signal was previously mis-calibrated. The model
consuming it likely fit to these mis-calibrations, and a silent update that corrects the signal will have
sudden ramifications for the model.
One common mitigation strategy for unstable data dependencies is to create a versioned copy of a
given signal. For example, rather than allowing a semantic mapping of words to topic clusters to
change over time, it might be reasonable to create a frozen version of this mapping and use it until
such a time as an updated version has been fully vetted. Versioning carries its own costs, however,
such as potential staleness and the cost to maintain multiple versions of the same signal over time.
Underutilized Data Dependencies. In code, underutilized dependencies are packages that are
mostly unneeded [13]. Similarly, underutilized data dependencies are input signals that provide
little incremental modeling benefit. These can make an ML system unnecessarily vulnerable to
change, sometimes catastrophically so, even though they could be removed with no detriment.
As an example, suppose that to ease the transition from an old product numbering scheme to new
product numbers, both schemes are left in the system as features. New products get only a new
number, but old products may have both and the model continues to rely on the old numbers for
some products. A year later, the code that stops populating the database with the old numbers is
deleted. This will not be a good day for the maintainers of the ML system.
Underutilized data dependencies can creep into a model in several ways.
? Legacy Features. The most common case is that a feature F is included in a model early in
its development. Over time, F is made redundant by new features but this goes undetected.
? Bundled Features. Sometimes, a group of features is evaluated and found to be beneficial.
Because of deadline pressures or similar effects, all the features in the bundle are added to
the model together, possibly including features that add little or no value.
? ?-Features. As machine learning researchers, it is tempting to improve model accuracy
even when the accuracy gain is very small or when the complexity overhead might be high.
? Correlated Features. Often two features are strongly correlated, but one is more directly
causal. Many ML methods have difficulty detecting this and credit the two features equally,
or may even pick the non-causal one. This results in brittleness if world behavior later
changes the correlations.
Underutilized dependencies can be detected via exhaustive leave-one-feature-out evaluations. These
should be run regularly to identify and remove unnecessary features.
3
Figure 1: Only a small fraction of real-world ML systems is composed of the ML code, as shown
by the small black box in the middle. The required surrounding infrastructure is vast and complex.
Static Analysis of Data Dependencies. In traditional code, compilers and build systems perform
static analysis of dependency graphs. Tools for static analysis of data dependencies are far less
common, but are essential for error checking, tracking down consumers, and enforcing migration
and updates. One such tool is the automated feature management system described in [12], which
enables data sources and features to be annotated. Automated checks can then be run to ensure that
all dependencies have the appropriate annotations, and dependency trees can be fully resolved. This
kind of tooling can make migration and deletion much safer in practice.
4
Feedback Loops
One of the key features of live ML systems is that they often end up influencing their own behavior
if they update over time. This leads to a form of analysis debt, in which it is difficult to predict the
behavior of a given model before it is released. These feedback loops can take different forms, but
they are all more difficult to detect and address if they occur gradually over time, as may be the case
when models are updated infrequently.
Direct Feedback Loops. A model may directly influence the selection of its own future training
data. It is common practice to use standard supervised algorithms, although the theoretically correct
solution would be to use bandit algorithms. The problem here is that bandit algorithms (such as
contextual bandits [9]) do not necessarily scale well to the size of action spaces typically required for
real-world problems. It is possible to mitigate these effects by using some amount of randomization
[3], or by isolating certain parts of data from being influenced by a given model.
Hidden Feedback Loops. Direct feedback loops are costly to analyze, but at least they pose a
statistical challenge that ML researchers may find natural to investigate [3]. A more difficult case is
hidden feedback loops, in which two systems influence each other indirectly through the world.
One example of this may be if two systems independently determine facets of a web page, such as
one selecting products to show and another selecting related reviews. Improving one system may
lead to changes in behavior in the other, as users begin clicking more or less on the other components
in reaction to the changes. Note that these hidden loops may exist between completely disjoint
systems. Consider the case of two stock-market prediction models from two different investment
companies. Improvements (or, more scarily, bugs) in one may influence the bidding and buying
behavior of the other.
5
ML-System Anti-Patterns
It may be surprising to the academic community to know that only a tiny fraction of the code in
many ML systems is actually devoted to learning or prediction ? see Figure 1. In the language of
Lin and Ryaboy, much of the remainder may be described as ?plumbing? [11].
It is unfortunately common for systems that incorporate machine learning methods to end up with
high-debt design patterns. In this section, we examine several system-design anti-patterns [4] that
can surface in machine learning systems and which should be avoided or refactored where possible.
4
Glue Code. ML researchers tend to develop general purpose solutions as self-contained packages.
A wide variety of these are available as open-source packages at places like mloss.org, or from
in-house code, proprietary packages, and cloud-based platforms.
Using generic packages often results in a glue code system design pattern, in which a massive
amount of supporting code is written to get data into and out of general-purpose packages. Glue
code is costly in the long term because it tends to freeze a system to the peculiarities of a specific
package; testing alternatives may become prohibitively expensive. In this way, using a generic
package can inhibit improvements, because it makes it harder to take advantage of domain-specific
properties or to tweak the objective function to achieve a domain-specific goal. Because a mature
system might end up being (at most) 5% machine learning code and (at least) 95% glue code, it may
be less costly to create a clean native solution rather than re-use a generic package.
An important strategy for combating glue-code is to wrap black-box packages into common API?s.
This allows supporting infrastructure to be more reusable and reduces the cost of changing packages.
Pipeline Jungles. As a special case of glue code, pipeline jungles often appear in data preparation. These can evolve organically, as new signals are identified and new information sources added
incrementally. Without care, the resulting system for preparing data in an ML-friendly format may
become a jungle of scrapes, joins, and sampling steps, often with intermediate files output. Managing these pipelines, detecting errors and recovering from failures are all difficult and costly [1].
Testing such pipelines often requires expensive end-to-end integration tests. All of this adds to
technical debt of a system and makes further innovation more costly.
Pipeline jungles can only be avoided by thinking holistically about data collection and feature extraction. The clean-slate approach of scrapping a pipeline jungle and redesigning from the ground
up is indeed a major investment of engineering effort, but one that can dramatically reduce ongoing
costs and speed further innovation.
Glue code and pipeline jungles are symptomatic of integration issues that may have a root cause in
overly separated ?research? and ?engineering? roles. When ML packages are developed in an ivorytower setting, the result may appear like black boxes to the teams that employ them in practice. A
hybrid research approach where engineers and researchers are embedded together on the same teams
(and indeed, are often the same people) can help reduce this source of friction significantly [16].
Dead Experimental Codepaths. A common consequence of glue code or pipeline jungles is that
it becomes increasingly attractive in the short term to perform experiments with alternative methods
by implementing experimental codepaths as conditional branches within the main production code.
For any individual change, the cost of experimenting in this manner is relatively low?none of the
surrounding infrastructure needs to be reworked. However, over time, these accumulated codepaths
can create a growing debt due to the increasing difficulties of maintaining backward compatibility
and an exponential increase in cyclomatic complexity. Testing all possible interactions between
codepaths becomes difficult or impossible. A famous example of the dangers here was Knight
Capital?s system losing $465 million in 45 minutes, apparently because of unexpected behavior
from obsolete experimental codepaths [15].
As with the case of dead flags in traditional software [13], it is often beneficial to periodically reexamine each experimental branch to see what can be ripped out. Often only a small subset of the
possible branches is actually used; many others may have been tested once and abandoned.
Abstraction Debt. The above issues highlight the fact that there is a distinct lack of strong abstractions to support ML systems. Zheng recently made a compelling comparison of the state ML
abstractions to the state of database technology [17], making the point that nothing in the machine
learning literature comes close to the success of the relational database as a basic abstraction. What
is the right interface to describe a stream of data, or a model, or a prediction?
For distributed learning in particular, there remains a lack of widely accepted abstractions. It could
be argued that the widespread use of Map-Reduce in machine learning was driven by the void of
strong distributed learning abstractions. Indeed, one of the few areas of broad agreement in recent
years appears to be that Map-Reduce is a poor abstraction for iterative ML algorithms.
5
The parameter-server abstraction seems much more robust, but there are multiple competing specifications of this basic idea [5, 10]. The lack of standard abstractions makes it all too easy to blur the
lines between components.
Common Smells. In software engineering, a design smell may indicate an underlying problem in
a component or system [7]. We identify a few ML system smells, not hard-and-fast rules, but as
subjective indicators.
? Plain-Old-Data Type Smell. The rich information used and produced by ML systems is
all to often encoded with plain data types like raw floats and integers. In a robust system,
a model parameter should know if it is a log-odds multiplier or a decision threshold, and a
prediction should know various pieces of information about the model that produced it and
how it should be consumed.
? Multiple-Language Smell. It is often tempting to write a particular piece of a system in
a given language, especially when that language has a convenient library or syntax for the
task at hand. However, using multiple languages often increases the cost of effective testing
and can increase the difficulty of transferring ownership to other individuals.
? Prototype Smell. It is convenient to test new ideas in small scale via prototypes. However, regularly relying on a prototyping environment may be an indicator that the full-scale
system is brittle, difficult to change, or could benefit from improved abstractions and interfaces. Maintaining a prototyping environment carries its own cost, and there is a significant
danger that time pressures may encourage a prototyping system to be used as a production
solution. Additionally, results found at small scale rarely reflect the reality at full scale.
6
Configuration Debt
Another potentially surprising area where debt can accumulate is in the configuration of machine
learning systems. Any large system has a wide range of configurable options, including which
features are used, how data is selected, a wide variety of algorithm-specific learning settings, potential pre- or post-processing, verification methods, etc. We have observed that both researchers and
engineers may treat configuration (and extension of configuration) as an afterthought. Indeed, verification or testing of configurations may not even be seen as important. In a mature system which is
being actively developed, the number of lines of configuration can far exceed the number of lines of
the traditional code. Each configuration line has a potential for mistakes.
Consider the following examples. Feature A was incorrectly logged from 9/14 to 9/17. Feature B is
not available on data before 10/7. The code used to compute feature C has to change for data before
and after 11/1 because of changes to the logging format. Feature D is not available in production, so
a substitute features D? and D?? must be used when querying the model in a live setting. If feature
Z is used, then jobs for training must be given extra memory due to lookup tables or they will train
inefficiently. Feature Q precludes the use of feature R because of latency constraints.
All this messiness makes configuration hard to modify correctly, and hard to reason about. However, mistakes in configuration can be costly, leading to serious loss of time, waste of computing
resources, or production issues. This leads us to articulate the following principles of good configuration systems:
? It should be easy to specify a configuration as a small change from a previous configuration.
? It should be hard to make manual errors, omissions, or oversights.
? It should be easy to see, visually, the difference in configuration between two models.
? It should be easy to automatically assert and verify basic facts about the configuration:
number of features used, transitive closure of data dependencies, etc.
? It should be possible to detect unused or redundant settings.
? Configurations should undergo a full code review and be checked into a repository.
6
7
Dealing with Changes in the External World
One of the things that makes ML systems so fascinating is that they often interact directly with the
external world. Experience has shown that the external world is rarely stable. This background rate
of change creates ongoing maintenance cost.
Fixed Thresholds in Dynamic Systems. It is often necessary to pick a decision threshold for a
given model to perform some action: to predict true or false, to mark an email as spam or not spam,
to show or not show a given ad. One classic approach in machine learning is to choose a threshold
from a set of possible thresholds, in order to get good tradeoffs on certain metrics, such as precision
and recall. However, such thresholds are often manually set. Thus if a model updates on new data,
the old manually set threshold may be invalid. Manually updating many thresholds across many
models is time-consuming and brittle. One mitigation strategy for this kind of problem appears in
[14], in which thresholds are learned via simple evaluation on heldout validation data.
Monitoring and Testing. Unit testing of individual components and end-to-end tests of running
systems are valuable, but in the face of a changing world such tests are not sufficient to provide
evidence that a system is working as intended. Comprehensive live monitoring of system behavior
in real time combined with automated response is critical for long-term system reliability.
The key question is: what to monitor? Testable invariants are not always obvious given that many
ML systems are intended to adapt over time. We offer the following starting points.
? Prediction Bias. In a system that is working as intended, it should usually be the case that
the distribution of predicted labels is equal to the distribution of observed labels. This is
by no means a comprehensive test, as it can be met by a null model that simply predicts
average values of label occurrences without regard to the input features. However, it is a
surprisingly useful diagnostic, and changes in metrics such as this are often indicative of
an issue that requires attention. For example, this method can help to detect cases in which
the world behavior suddenly changes, making training distributions drawn from historical
data no longer reflective of current reality. Slicing prediction bias by various dimensions
isolate issues quickly, and can also be used for automated alerting.
? Action Limits. In systems that are used to take actions in the real world, such as bidding
on items or marking messages as spam, it can be useful to set and enforce action limits as a
sanity check. These limits should be broad enough not to trigger spuriously. If the system
hits a limit for a given action, automated alerts should fire and trigger manual intervention
or investigation.
? Up-Stream Producers. Data is often fed through to a learning system from various upstream producers. These up-stream processes should be thoroughly monitored, tested, and
routinely meet a service level objective that takes the downstream ML system needs into
account. Further any up-stream alerts must be propagated to the control plane of an ML
system to ensure its accuracy. Similarly, any failure of the ML system to meet established
service level objectives be also propagated down-stream to all consumers, and directly to
their control planes if at all possible.
Because external changes occur in real-time, response must also occur in real-time as well. Relying
on human intervention in response to alert pages is one strategy, but can be brittle for time-sensitive
issues. Creating systems to that allow automated response without direct human intervention is often
well worth the investment.
8
Other Areas of ML-related Debt
We now briefly highlight some additional areas where ML-related technical debt may accrue.
Data Testing Debt. If data replaces code in ML systems, and code should be tested, then it seems
clear that some amount of testing of input data is critical to a well-functioning system. Basic sanity
checks are useful, as more sophisticated tests that monitor changes in input distributions.
7
Reproducibility Debt. As scientists, it is important that we can re-run experiments and get similar
results, but designing real-world systems to allow for strict reproducibility is a task made difficult by
randomized algorithms, non-determinism inherent in parallel learning, reliance on initial conditions,
and interactions with the external world.
Process Management Debt. Most of the use cases described in this paper have talked about the
cost of maintaining a single model, but mature systems may have dozens or hundreds of models
running simultaneously [14, 6]. This raises a wide range of important problems, including the
problem of updating many configurations for many similar models safely and automatically, how to
manage and assign resources among models with different business priorities, and how to visualize
and detect blockages in the flow of data in a production pipeline. Developing tooling to aid recovery
from production incidents is also critical. An important system-level smell to avoid are common
processes with many manual steps.
Cultural Debt. There is sometimes a hard line between ML research and engineering, but this
can be counter-productive for long-term system health. It is important to create team cultures that
reward deletion of features, reduction of complexity, improvements in reproducibility, stability, and
monitoring to the same degree that improvements in accuracy are valued. In our experience, this is
most likely to occur within heterogeneous teams with strengths in both ML research and engineering.
9
Conclusions: Measuring Debt and Paying it Off
Technical debt is a useful metaphor, but it unfortunately does not provide a strict metric that can be
tracked over time. How are we to measure technical debt in a system, or to assess the full cost of this
debt? Simply noting that a team is still able to move quickly is not in itself evidence of low debt or
good practices, since the full cost of debt becomes apparent only over time. Indeed, moving quickly
often introduces technical debt. A few useful questions to consider are:
? How easily can an entirely new algorithmic approach be tested at full scale?
? What is the transitive closure of all data dependencies?
? How precisely can the impact of a new change to the system be measured?
? Does improving one model or signal degrade others?
? How quickly can new members of the team be brought up to speed?
We hope that this paper may serve to encourage additional development in the areas of maintainable
ML, including better abstractions, testing methodologies, and design patterns. Perhaps the most
important insight to be gained is that technical debt is an issue that engineers and researchers both
need to be aware of. Research solutions that provide a tiny accuracy benefit at the cost of massive
increases in system complexity are rarely wise practice. Even the addition of one or two seemingly
innocuous data dependencies can slow further progress.
Paying down ML-related technical debt requires a specific commitment, which can often only be
achieved by a shift in team culture. Recognizing, prioritizing, and rewarding this effort is important
for the long term health of successful ML teams.
Acknowledgments
This paper owes much to the important lessons learned day to day in a culture that values both
innovative ML research and strong engineering practice. Many colleagues have helped shape our
thoughts here, and the benefit of accumulated folk wisdom cannot be overstated. We would like
to specifically recognize the following: Roberto Bayardo, Luis Cobo, Sharat Chikkerur, Jeff Dean,
Philip Henderson, Arnar Mar Hrafnkelsson, Ankur Jain, Joe Kovac, Jeremy Kubica, H. Brendan
McMahan, Satyaki Mahalanabis, Lan Nie, Michael Pohl, Abdul Salem, Sajid Siddiqi, Ricky Shan,
Alan Skelly, Cory Williams, and Andrew Young.
A short version of this paper was presented at the SE4ML workshop in 2014 in Montreal, Canada.
8
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5,144 | 5,657 | Statistical Model Criticism
using Kernel Two Sample Tests
James Robert Lloyd
Department of Engineering
University of Cambridge
Zoubin Ghahramani
Department of Engineering
University of Cambridge
Abstract
We propose an exploratory approach to statistical model criticism using maximum
mean discrepancy (MMD) two sample tests. Typical approaches to model criticism require a practitioner to select a statistic by which to measure discrepancies
between data and a statistical model. MMD two sample tests are instead constructed as an analytic maximisation over a large space of possible statistics and
therefore automatically select the statistic which most shows any discrepancy. We
demonstrate on synthetic data that the selected statistic, called the witness function, can be used to identify where a statistical model most misrepresents the data
it was trained on. We then apply the procedure to real data where the models being
assessed are restricted Boltzmann machines, deep belief networks and Gaussian
process regression and demonstrate the ways in which these models fail to capture
the properties of the data they are trained on.
1
Introduction
Statistical model criticism or checking1 is an important part of a complete statistical analysis. When
one fits a linear model to a data set a complete analysis includes computing e.g. Cook?s distances
[3] to identify influential points or plotting residuals against fitted values to identify non-linearity or
heteroscedasticity. Similarly, modern approaches to Bayesian statistics view model criticism as in
important component of a cycle of model construction, inference and criticism [4].
As statistical models become more complex and diverse in response to the challenges of modern
data sets there will be an increasing need for a greater range of model criticism procedures that are
either automatic or widely applicable. This will be especially true as automatic modelling methods
[e.g. 5, 6, 7] and probabilistic programming [e.g. 8, 9, 10, 11] mature.
Model criticism typically proceeds by choosing a statistic of interest, computing it on data and
comparing this to a suitable null distribution. Ideally these statistics are chosen to assess the utility
of the statistical model under consideration (see applied examples [e.g. 4]) but this can require
considerable expertise on the part of the modeller. We propose an alternative to this manual approach
by using a statistic defined as a supremum over a broad class of measures of discrepancy between two
distributions, the maximum mean discrepancy (MMD) [e.g. 12]). The advantage of this approach
is that the discrepancy measure attaining the supremum automatically identifies regions of the data
which are most poorly represented by the statistical model fit to the data.
We demonstrate MMD model criticism on toy examples, restricted Boltzmann machines and deep
belief networks trained on MNIST digits and Gaussian process regression models trained on several
time series. Our proposed method identifies discrepancies between the data and fitted models that
would not be apparent from predictive performance focused metrics. It is our belief that more effort
should be expended on attempting to falsify models fitted to data, using model criticism techniques
or otherwise. Not only would this aid research in targeting areas for improvement but it would give
greater confidence in any conclusions drawn from a model.
1
We follow Box [1] using the term ?model criticism? for similar reasons to O?Hagan [2].
1
2
Model criticism
Suppose we observe data Y obs = (yiobs )i=1...n and we attempt to fit a model M with parameters
? or an (approximate)
?. After performing a statistical analysis we will have either an estimate, ?,
posterior, p(? | Y obs , M ), for the parameters. How can we check whether any aspects of the data
were poorly modelled?
Criticising prior assumptions The classical approach to model criticism is to attempt to falsify
the null hypothesis that the data could have been generated by the model M for some value of the
parameters ? i.e. Y obs ? p(Y | ?, M ). This is typically achieved by constructing a statistic T of the
data whose distribution does not depend on the parameters ? i.e. a pivotal quantity. The extent to
which the observed data Y obs differs from expectations under the model M can then be quantified
with a tail-area based p-value
pfreq (Y obs ) = P(T (Y ) ? T (Y obs )) where
Y ? p(Y | ?, M ) for any ?.
(2.1)
Analogous quantities in a Bayesian analysis are the prior predictive p-values of Box [1]. The null
hypothesis is replaced with
R the claim that the data could have been generated from the prior predictive distribution Y obs ? p(Y | ?, M )p(? | M )d?. A tail-area p-value can then be constructed for
any statistic T of the data
Z
pprior (Y obs ) = P(T (Y ) ? T (Y obs )) where Y ? p(Y | ?, M )p(? | M )d?.
(2.2)
Both of these procedures construct a function of the data p(Y obs ) whose distribution under a suitable null hypothesis is uniform i.e. a p-value. The p-value quantifies how surprising it would be
for the data Y obs to have been generated by the model. The different null hypotheses reflect the
different uses of the word ?model? in frequentist and Bayesian analyses. A frequentist model is a
class of probability distributions over data indexed by parameters whereas a Bayesian model is a
joint probability distribution over data and parameters.
Criticising estimated models or posterior distributions A constrasting method of Bayesian
model criticism is the calculation of posterior predictive p-values ppost [e.g. 13, 14] where
the prior
predictive distribution in (2.2) is replaced with the posterior predictive distribution
R
Y ? p(Y | ?, M )p(? | Y obs , M )d?. The corresponding test for an analysis resulting in a point
? M ) to form
estimate of the parameters ?? would use the plug-in predictive distribution Y ? p(Y | ?,
the plug-in p-value pplug .
These p-values quantify how surprising the data Y obs is even after having observed it. A simple
variant of this method of model criticism is to use held out data Y ? , generated from the same
distribution as Y obs , to compute a p-value i.e. p(Y ? ) = P(T (Y ) ? T (Y ? )). This quantifies how
surprising the held out data is after having observed Y obs .
Which type of model criticism should be used? Different forms of model criticism are appropriate in different contexts, but we believe that posterior predictive and plug-in p-values will be most
often useful for highly flexible models. For example, suppose one is fitting a deep belief network
to data. Classical p-values would assume a null hypothesis that the data could have been generated
from some deep belief network. Since the space of all possible deep belief networks is very large
it will be difficult to ever falsify this hypothesis. A more interesting null hypothesis to test in this
example is whether or not our particular deep belief network can faithfully mimick the distribution
of the sample it was trained on. This is the null hypothesis of posterior or plug-in p-values.
3
Model criticism using maximum mean discrepancy two sample tests
We assume that our data Y obs are i.i.d. samples from some distribution (yiobs )i=1...n ?iid p(y | ?, M ).
? the null hypothesis
After performing inference resulting in a point estimate of the parameters ?,
obs
?
associated with a plug-in p-value is (yi )i=1...n ?iid p(y | ?, M ).
We can test this null hypothesis using a two sample test [e.g. 15, 16]. In particular, we have
samples of data (yiobs )i=1...n and we can generate samples from the plug-in predictive distribution
? M ) and then test whether or not these samples could have been generated
(yirep )i=1...m ?iid p(y | ?,
2
from the same distribution. For consistency with two sample testing literature we now switch notation; suppose we have samples X = (xi )i=1...m and Y = (yi )i=1...n drawn i.i.d. from distributions
p and q respectively. The two sample problem asks if p = q.
A way of answering the two sample problem is to consider maximum mean discrepancy (MMD)
[e.g. 12] statistics
MMD(F, p, q) = sup (Ex?p [f (x)] ? Ey?q [f (y)])
(3.1)
f ?F
where F is a set of functions. When F is a reproducing kernel Hilbert space (RKHS) the function
attaining the supremum can be derived analytically and is called the witness function
f (x) = Ex0 ?p [k(x, x0 )] ? Ex0 ?q [k(x, x0 )]
(3.2)
where k is the kernel of the RKHS. Substituting (3.2) into (3.1) and squaring yields
MMD2 (F, p, q) = Ex,x0 ?p [k(x, x0 )] + 2Ex?p,y?q [k(x, y)] + Ey,y0 ?q [k(y, y 0 )].
(3.3)
This expression only involves expectations of the kernel k which can be estimated empirically by
m,n
m
n
1 X
1 X
2 X
MMD2b (F, X, Y ) = 2
k(xi , yj ) + 2
k(xi , xj ) ?
k(yi , yj ).
(3.4)
m i,j=1
mn i,j=1
n i,j=1
One can also estimate the witness function from finite samples
m
n
1X
1 X
k(x, xi ) ?
k(x, yi )
f?(x) =
m i=1
n i=1
(3.5)
i.e. the empirical witness function is the difference of two kernel density estimates [e.g. 17, 18].
This means that we can interpret the witness function as showing where the estimated densities of
p and q are most different. While MMD two sample tests are well known in the literature the main
contribution of this work is to show that this interpretability of the witness function makes them a
useful tool as an exploratory form of statistical model criticism.
4
Examples on toy data
To illustrate the use of the MMD two sample test as a tool for model criticism we demonstrate its
properties on two simple datasets and models.
Newcomb?s speed of light data A histogram of Simon Newcomb?s 66 measurements used to
determine the speed of light [19] is shown on the left of figure 1. We fit a normal distribution to this
data by maximum likelihood and ask whether this model is a faithful representation of the data.
18
0.2
16
0.1
0.1
10
8
6
0.05
0
0
0.1
0.05
0.05
0
0
?0.05
Witness function
Count
12
Witness function
Density estimate
Density estimate
14
4
2
0
?50
?0.2
?40
?30
?20
?10
0
10
20
Deviations from 24,800 nanoseconds
30
40
?60
?40
?20
0
20
40
Deviations from 24,800 nansoeconds
60
?60
?40
?20
0
20
40
60
Deviations from 24,800 nansoeconds
Figure 1: Left: Histogram of Simon Newcomb?s speed of light measurements. Middle: Histogram
together with density estimate (red solid line) and MMD witness function (green dashed line). Right:
Histogram together with updated density estimate and witness function.
We sampled 1000 points from the fitted distribution and performed an MMD two sample test using
a radial basis function kernel2 . The estimated p-value of the test was less than 0.001 i.e. a clear
disparity between the model and data.
The data, fitted density estimate (normal distribution) and witness function are shown in the middle
of figure 1. The witness function has a trough at the centre of the data and peaks either side indicating
that the fitted model has placed too little mass in its centre and too much mass outside its centre.
2
Throughout this paper we estimate the null distribution of the MMD statistic using the bootstrap method
described in [12] using 1000 replicates. We use a radial basis function kernel and select the lengthscale by 5
fold cross validation using predictive likelihood of the kernel density estimate as the selection criterion.
3
This suggests that we should modify our model by either using a distribution with heavy tails or
explicitly modelling the possibility of outliers. However, to demonstrate some of the properties of
the MMD two sample test we make an unusual choice of fitting a Gaussian by maximum likelihood,
but ignoring the two outliers in the data. The new fitted density estimate (the normal distribution)
and witness function of an MMD test are shown on the right of figure 1. The estimated p-value
associated with the MMD two sample test is roughly 0.5 despite the fitted model being a very poor
explanation of the outliers.
The nature of an MMD test depends on the kernel defining the RKHS in equation (3.1). In this
paper we use the radial basis function kernel which encodes for smooth functions with a typical
lengthscale [e.g. 20]. Consequently the test identifies ?dense? discrepancies, only identifying outliers
if the model and inference method are not robust to them. This is not a failure; a test that can identify
too many types of discrepancy would have low statistical power (see [12] for discussion of the power
of the MMD test and alternatives).
High dimensional data The interpretability of the witness functions comes from being equal to
the difference of two kernel density estimates. In high dimensional spaces, kernel density estimation is a very high variance procedure that can result in poor density estimates which destroy the
interpretability of the method. In response, we consider using dimensionality reduction techniques
before performing two sample tests.
We generated synthetic data from a mixture of 4 Gaussians and a t-distribution in 10 dimensions3 .
We then fit a mixture of 5 Gaussians and performed an MMD two sample test. We reduced the dimensionality of the data using principal component analysis (PCA), selecting the first two principal
components. To ensure that the MMD test remains well calibrated we include the PCA dimensionality reduction within the bootstrap estimation of the null distribution. The data and plug-in predictive
samples are plotted on the left of figure 2. While we can see that one cluster is different from the
rest, it is difficult to assess by eye if these distributions are different ? due in part to the difficulty
of plotting two sets of samples on top of each other.
6
0.01
5
0.005
4
0
3
?0.005
2
?0.01
1
?0.015
0
?0.02
?1
?0.025
?2
?3
?4
?8
?0.03
?6
?4
?2
0
2
4
?0.035
6
Figure 2: Left: PCA projection of synthetic high dimensional cluster data (green circles) and projection of samples from fitted model (red circles). Right: Witness function of MMD model criticism.
The poorly fit cluster is clearly identified.
The MMD test returns a p-value of 0.05 and the witness function (right of figure 2) clearly identifies
the cluster that has been incorrectly modelled. Presented with this discrepancy a statistical modeller
might try a more flexible clustering model [e.g. 21, 22]. The p-value of the MMD statistic can also
be made non-significant by fitting a mixture of 10 Gaussians; this is a sufficient approximation to
the t-distribution such that no discrepancy can be detected with the amount of data available.
5
What exactly do neural networks dream about?
?To recognize shapes, first learn to generate images? quoth Hinton [23]. Restricted Boltzmann
Machine (RBM) pretraining of neural networks was shown by [24] to learn a deep belief network
(DBN) for the data i.e. a generative model. In agreement with this observation, as well as computing
estimates of marginal likelihoods and testing errors, it is standard to demonstrate the effectiveness
of a generative neural network by generating samples from the distribution it has learned.
3
For details see code at [redacted]
4
When trained on the MNIST handwritten digit data, samples from RBMs (see figure 3a for random
samples4 ) and DBNs certainly look like digits, but it is hard to detect any systematic anomalies
purely by visual inspection. We now use MMD model criticism to investigate how faithfully RBMs
and DBNs can capture the distribution over handwritten digits.
RBMs can consistently mistake the identity of digits We trained an RBM with architecture
(784) ? (500) ? (10)5 using 15 epochs of persistent contrastive divergence (PCD-15), a batch
size of 20 and a learning rate of 0.1 (i.e. we used the same settings as the code available at the deep
learning tutorial [25]). We generated 3000 independent samples from the learned generative model
by initialising the network with a random training image and performing 1000 gibbs updates with
the digit labels clamped6 to generate each image (as in e.g. [23]).
Since we generated digits from the class conditional distributions we compare each class separately.
Rather than show plots of the witness function for each digit we summarise the witness function
by examples of digits closest to the peaks and troughs of the witness function (the witness function
estimate is differentiable so we can find the peaks and troughs by gradient based optimisation).
We apply MMD model criticism to each class conditional distribution, using PCA to reduce to 2
dimensions as in section 4.
a)
b)
c)
d)
e)
f)
Figure 3: a) Random samples from an RBM. b) Peaks of the witness function for the RBM (digits
that are over-represented by the model). c) Peaks of the witness function for samples from 1500
RBMs (with differently initialised pseudo random number generators during training). d) Peaks of
the witness function for the DBN. e) Troughs (digits that are under-represented by the model) of the
witness function for samples from 1500 RBMs. f) Troughs of the witness function for the DBN.
Figure 3b shows the digits closest to the two most extreme peaks of the witness function for each
class; the peaks indicate where the fitted distribution over-represents the distribution of true digits.
The estimated p-value for all tests was less than 0.001. The most obvious problem with these digits
is that the first 2 and 3 look quite similar.
To test that this was not just an single unlucky RBM, we trained 1500 RBMs (with differently
initialised pseudo random number generators) and generated one sample from each and performed
the same tests. The estimated p-values were again all less than 0.001 and the summaries of the
peaks of the witness function are shown in figure 3c. On the first toy data example we observed
that the MMD statistic does not highlight outliers and therefore we can conclude that RBMs are
making consistent mistakes e.g. generating a 0 from the 7 distribution or a 5 when it should have
been generating an 8.
DBNs have nightmares about ghosts We now test the effectiveness of deep learning to represent
the distribution of MNIST digits. In particular, we fit a DBN with architecture (784) ? (500) ?
(500) ? (2000) ? (10) using RBM pre-training and a generative fine tuning algorithm described
in [24]. Performing the same tests with 3000 samples results in estimated p-values of less than 0.001
except for the digit 4 (0.150) and digit 7 (0.010). Summaries of the witness function peaks are shown
in figure 3d.
4
Specifically these are the activations of the visible units before sampling sampling binary values. This
procedure is an attempt to be consistent with the grayscale input distribution of the images. Analogous discrepancies would be discovered if we had instead sampled binary pixel values.
5
That is, 784 input pixels and 10 indicators of the class label are connected to 500 hidden neurons.
6
Without clamping the label neurons, the generative distribution is heavily biased towards certain digits.
5
The witness function no longer shows any class label mistakes (except perhaps for the digit 1 which
looks very peculiar) but the 2, 3, 7 and 8 appear ?ghosted? ? the digits fade in and out. For comparison, figure 3f shows digits closest to the troughs of the witness function; there is no trace of ghosting.
This discrepancy could be due to errors in the autoassociative memory of a DBN propogating down
the hidden layers resulting in spurious features in several visible neurons.
6
An extension to non i.i.d. data
We now describe how the MMD statistic can be used for model criticism of non i.i.d. predictive
distributions. In particular we construct a model criticism procedure for regression models.
obs
We assume that our data consists of pairs of inputs and outputs (xobs
i , yi )i=1...n . A typical formulation of the problem of regression is to estimate the conditional distribution of the outputs given
the inputs p(y | x, ?). Ignoring that our data are not i.i.d. we can generate data from the plug-in con?
ditional distribution yirep ? p(y | xobs
i , ?) and compute the empirical MMD estimate (3.4) between
obs obs
obs rep
(xi , yi )i=1...n and (xi , yi )i=1...n . The only difference between this test and the MMD two
sample test is that our data is generated from a conditional distribution, rather than being i.i.d. . The
null distribution of this statistic can be trivially estimated by sampling several sets of replicate data
from the plug-in predictive distribution.
To demonstrate this test we apply it to 4 regression algorithms and 13 time series analysed
in [7]. In this work the authors compare several methods for constructing Gaussian process
[e.g. 20] regression models. Example data sets
are shown in figures 4 and 5. While it is clear
that simple methods will fail to capture all of
the structure in this data, it is not clear a priori
how much better the more advanced methods
will fair.
To construct p-values we use held out data using the same split of training and testing data
as the interpolation experiment in [7]7 . Table 1 shows a table of p-values for 13 data sets
and 4 regression methods. The four methods
are linear regression (Lin), Gaussian process
regression using a squared exponential kernel
(SE), spectral mixture kernels [26] (SP) and the
method proposed in [7] (ABCD). Values in bold
indicate a positive discovery after a Benjamini?
Hochberg [27] procedure with a false discovery
rate of 0.05 applied to each model construction
method.
Dataset
Airline
Solar
Mauna
Wheat
Temperature
Internet
Call centre
Radio
Gas production
Sulphuric
Unemployment
Births
Wages
Lin
0.34
0.00
0.00
0.00
0.44
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
SE
0.36
0.00
0.99
0.00
0.54
0.00
0.02
0.00
0.00
0.29
0.00
0.00
0.00
SP
0.07
0.00
0.34
0.00
0.68
0.05
0.00
0.00
0.01
0.34
0.00
0.00
0.01
ABCD
0.15
0.05
0.21
0.19
0.75
0.01
0.07
0.00
0.11
0.52
0.01
0.12
0.00
Table 1: Two sample test p-values applied
to 13 time series and 4 regression algorithms.
Bold values indicate a positive discovery using a
Benjamini?Hochberg procedure with a false discovery rate of 0.05 for each method.
We now investigate the type of discrepancies found by this test by looking at the witness function
(which can still be interpreted as the difference of kernel density estimates). Figure 4 shows the
solar and gas production data sets, the posterior distribution of the SE fits to this data and the witness
functions for the SE fit. The solar witness function has a clear narrow trough, indicating that the data
is more dense than expected by the fitted model in this region. We can see that this has identified a
region of low variability in the data i.e. it has identified local heteroscedasticity not captured by the
model. Similar conclusions can be drawn about the gas production data and witness function.
Of the four methods compared here, only ABCD is able to model heteroscedasticity, explaining
why it is the only method with a substantially different set of significant p-values. However, the
procedure is still potentially failing to capture structure on four of the datasets.
7
Gaussian processes when applied to regression problems learn a joint distribution of all output values.
However this joint distribution information is rarely used; typically only the pointwise conditional distributions
?
p(y | xobs
i , ?) are used as we have done here.
6
4
Gas production
x 10
Solar
6
1361.8
0.02
20
20
0.02
0.01
1361.6
5
40
40
0
y
60
1361.2
80
?0.02
1361
100
?0.03
1360.8
120
?0.04
1360.6
140
0.01
60
?0.01
4
80
0
y
1361.4
100
3
?0.01
120
2
?0.05
140
?0.02
?0.06
1360.4
160
1360.2
180
1650
1700
1750
1800
x
1850
1900
1950
2000
200
160
1
?0.07
?0.03
180
?0.08
1960
50
100
150
200
1965
1970
1975
x
1980
1985
1990
1995
200
?0.04
50
100
150
200
Figure 4: From left to right: Solar data with SE posterior. Witness function of SE fit to solar. Gas
production data with SE posterior. Witness function of SE fit to gas production.
Figure 5 shows the unemployment and Internet data sets, the posterior distribution for the ABCD
fits to the data and the witness functions of the ABCD fits. The ABCD method has captured much
of the structure in these data sets, making it difficult to visually identify discrepancies between
model and data. The witness function for unemployment shows peaks and troughs at similar values
of the input x. Comparing to the raw data we see that at these input values there are consistent
outliers. Since ABCD is based on Gaussianity assumptions these consistent outliers have caused
the method to estimate a large variance in this region, when the true data is non-Gaussian. There
is also a similar pattern of peaks and troughs on the Internet data suggesting that non-normality has
again been detected. Indeed, the data appears to have a hard lower bound which is inconsistent with
Gaussianity.
4
x 10
Unemployment
1200
20
0.01
20
10
0.008
0.015
1000
40
40
0.006
8
60
0.004
7
80
0.002
100
0
6
100
120
?0.005
5
120
400
140
?0.01
4
140
300
160
?0.015
3
160
900
9
0.01
60
800
0.005
y
80
y
Internet
11
0.02
1100
700
600
500
200
1955
1960
1965
x
1970
1975
1980
200
100
150
?0.004
?0.008
180
?0.02
50
?0.002
?0.006
2
180
1950
0
?0.01
2004.9 2004.92 2004.94 2004.96 2004.98 2005 2005.02 2005.04 2005.06
x
200
200
50
100
150
200
Figure 5: From left to right: Unemployment data with ABCD posterior. Witness function of ABCD
fit to unemployment. Internet data with ABCD posterior. Witness function of ABCD fit to Internet.
7
Discussion of model criticism and related work
Are we criticising a particular model, or class of models? In section 2 we interpreted the differences between classical, Bayesian prior/posterior and plug-in p-values as corresponding to different
null hypotheses and interpretations of the word ?model?. In particular classical p-values test a null
hypothesis that the data could have been generated by a class of distributions (e.g. all normal distributions) whereas all other p-values test a particular probability distribution.
Robins, van der Vaart & Ventura [28] demonstrated that Bayesian and plug-in p-values are not classical p-values (frequentist p-values in their terminology) i.e. they do not have a uniform distribution
under the relevant null hypothesis. However, this was presented as a failure of these methods; in
particular they demonstrated that methods proposed by Bayarri & Berger [29] based on posterior
predictive p-values are asymptotically classical p-values.
This claimed inadequacy of posterior predictive p-values was rebutted [30] and while their usefulness is becoming more accepted (see e.g. introduction of [31]) it would appear there is still confusion
on the subject [32]. We hope that our interpretation of the differences between these methods as different null hypotheses ? appropriate in different circumstances ? sheds further light on the matter.
Should we worry about using the same data for traning and criticism? Plug-in and posterior
predictive p-values test the null hypothesis that the observed data could have been generated by the
fitted model or posterior predictive distribution. In some situations it may be more appropriate to
attempt to falsify the null hypothesis that future data will be generated by the plug-in or posterior
predictive distribution. As mentioned in section 2 this can be achieved by reserving a portion of the
data to be used for model criticism alone, rather than fitting a model or updating a posterior on the
full data. Cross validation methods have also been investigated in this context [e.g. 33, 34].
7
Other methods for evaluating statistical models Other typical methods of model evaluation include estimating the predictive performance of the model, analyses of sensitivities to modelling
parameters / priors, graphical tests, and estimates of model utility. For a recent survey of Bayesian
methods for model assessment, selection and comparison see [35] which phrases many techniques
as estimates of the utility of a model. For some discussion of sensitivity analysis and graphical
model comparison see [e.g. 4].
In this manuscript we have focused on methods that compare statistics of data with predictive distributions, ignoring parameters of the model. The discrepancy measures of [36] compute statistics
of data and parameters; examples can be found in [4]. O?Hagan [2] also proposes a method and
selectively reviews techniques for model criticism that also take model parameters into account.
In the spirit of scientific falsification [e.g. 37], ideally all methods of assessing a model should
be performed to gain confidence in any conclusions made. Of course, when performing multiple
hypothesis tests care must be taken in the intrepetation of individual p-values.
8
Conclusions and future work
In this paper we have demonstrated an exploratory form of model criticism based on two sample
tests using kernel maximum mean discrepancy. In contrast to other methods for model criticism,
the test analytically maximises over a broad class of statistics, automatically identifying the statistic
which most demonstrates the discrepancy between the model and data. We demonstrated how this
method of model criticism can be applied to neural networks and Gaussian process regression and
demonstrated the ways in which these models were misrepresenting the data they were trained on.
We have demonstrated an application of MMD two sample tests to model criticism, but they can
also be applied to any aspect of statistical modelling where two sample tests are appropriate. This
includes for example, Geweke?s tests of markov chain posterior sampler validity [38] and tests of
markov chain convergence [e.g. 39].
The two sample tests proposed in this paper naturally apply to i.i.d. data and models, but model
criticism techniques should of course apply to models with other symmetries (e.g. exchangeable
data, logitudinal data / time series, graphs, and many others). We have demonstrated an adaptation
of the MMD test to regression models but investigating extensions to a greater number of model
classes would be a profitable area for future study.
We conclude with a question. Do you know how the model you are currently working with most
misrepresents the data it is attempting to model? In proposing a new method of model criticism we
hope we have also exposed the reader unfamiliar with model criticism to its utility in diagnosing
potential inadequacies of a model.
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9
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5,145 | 5,658 | Calibrated Structured Prediction
Percy Liang
Department of Computer Science
Stanford University
Stanford, CA 94305
Volodymyr Kuleshov
Department of Computer Science
Stanford University
Stanford, CA 94305
Abstract
In user-facing applications, displaying calibrated confidence measures?
probabilities that correspond to true frequency?can be as important as obtaining
high accuracy. We are interested in calibration for structured prediction problems
such as speech recognition, optical character recognition, and medical diagnosis.
Structured prediction presents new challenges for calibration: the output space is
large, and users may issue many types of probability queries (e.g., marginals) on
the structured output. We extend the notion of calibration so as to handle various
subtleties pertaining to the structured setting, and then provide a simple recalibration method that trains a binary classifier to predict probabilities of interest. We
explore a range of features appropriate for structured recalibration, and demonstrate their efficacy on three real-world datasets.
1
Introduction
Applications such as speech recognition [1], medical diagnosis [2], optical character recognition
[3], machine translation [4], and scene labeling [5] have two properties: (i) they are instances of
structured prediction, where the predicted output is a complex structured object; and (ii) they are
user-facing applications for which it is important to provide accurate estimates of confidence. This
paper explores confidence estimation for structured prediction.
Central to this paper is the idea of probability calibration [6, 7, 8, 9], which is prominent in the
meteorology [10] and econometrics [9] literature. Calibration requires that the probability that a
system outputs for an event reflects the true frequency of that event: of the times that a system says
that it will rain with probability 0.3, then 30% of the time, it should rain. In the context of structured
prediction, we do not have a single event or a fixed set of events, but rather a multitude of events
that depend on the input, corresponding to different conditional and marginal probabilities that one
could ask of a structured prediction model. We must therefore extend the definition of calibration in
a way that deals with the complexities that arise in the structured setting.
We also consider the practical question of building a system that outputs calibrated probabilities.
We introduce a new framework for calibration in structured prediction, which involves defining
probabilities of interest, and then training binary classifiers to predict these probabilities based on a
set of features. Our framework generalizes current methods for binary and multiclass classification
[11, 12, 13], which predict class probabilities based on a single feature, the uncalibrated prediction
score. In structured prediction, the space of interesting probabilities and useful features is considerably richer. This motivates us to introduce a new concept of events as well as a range of new
features?margin, pseudomargin?which have varying computational demands. We perform a thorough study of which features yield good calibration, and find that domain-general features are quite
good for calibrating MAP and marginal estimates over three tasks?object recognition, optical character recognition, and scene understanding. Interestingly, features based on MAP inference alone
can achieve good calibration on marginal probabilities (which can be more difficult to compute).
1
Figure 1:
In the context of an
OCR system, our framework augments the structured predictor with
calibrated confidence measures for
a set of events, e.g., whether the first
letter is ?l?.
?l ?
?a?
?n?
?d ?
y1
y2
y3
y4
x
Event
Probability
[y = ?land ?]
0.8
[y1 = ?l ?]
0.8
[y2 = ?a?]
0.9
[y3 = ?n?]
0.9
[y4 = ?d ?]
0.8
(a) Structured prediction model
2
2.1
(b) Forecaster output
Background
Structured Prediction
In structured prediction, we want to assign a structured label y = (y1 , . . . , yL ) ? Y to an input
x ? X . For example, in optical character recognition (OCR), x is a sequence of images and y is the
sequence of associated characters (see Figure 1(a)); note that the number of possible outputs y for a
given x may be exponentially large.
A common approach to structured prediction is conditional random fields (CRFs), where we posit a
probabilistic model p? (y | x). We train p? by optimizing a maximum-likelihood or a max-margin
objective over a training set {(x(i) , y (i) )}ni=1 , assumed to be drawn i.i.d. from an unknown datagenerating distribution P(x, y). The promise of a probabilistic model is that in addition to computing
the most likely output y? = arg maxy p? (y | x), we can also get its probability p? (y = y? | x) ? [0, 1],
or even marginal probabilities p? (y1 = y?1 | x) ? [0, 1].
2.2
Probabilistic Forecasting
Probabilities from a CRF p? are just numbers that sum to 1. In order for these probabilities to be
useful as confidence measures, we would ideally like them to be calibrated. Calibration intuitively
means that whenever a forecaster assigns 0.7 probability to an event, it should be the case that the
event actually holds about 70% of the time. In the case of binary classification (Y = {0, 1}), we say
that a forecaster F : X ? [0, 1] is perfectly calibrated if for all possible probabilities p ? [0, 1]:
P[y = 1 | F (x) = p] = p.
(1)
Calibration by itself does not guarantee a useful confidence measure. A forecaster that always
outputs the marginal class probability F (x) = P(y = 1) is calibrated but useless for accurate
prediction. Good forecasts must also be sharp, i.e., their probabilities should be close to 0 or 1.
Calibration and sharpness. Given a forecaster F : X ? [0, 1], define T (x) = E[y | F (x)] to be
the true probability of y = 1 given a that x received a forecast F (x). We can use T to decompose
the `2 prediction loss as follows:
E[(y ? F (x))2 ] = E[(y ? T (x))2 ] + E[(T (x) ? F (x))2 ]
(2)
2
= Var[y] ? Var[T (x)] + E[(T (x) ? F (x)) ] .
| {z } | {z } |
{z
}
uncertainty
sharpness
(3)
calibration error
The first equality follows because y ? T (x) has expectation 0 conditioned on F (x), and the second
equality follows from the variance decomposition of y onto F (x).
The three terms in (3) formalize our intuitions about calibration and sharpness [7]. The calibration
term measures how close the predicted probability is to the true probability over that region and is
a natural generalization of perfect calibration (1) (which corresponds to zero calibration error). The
sharpness term measures how much variation there is in the true probability across forecasts. It does
not depend on the numerical value of the forecaster F (x), but only the induced grouping of points;
it is maximized by making F (x) closer to 0 and 1. Uncertainty does not depend on the forecaster
and can be mostly ignored; note that it is always greater than sharpness and thus ensures that the
loss stays positive.
2
input x
0
1
2
calib. sharp.
Examples. To illustrate
true P(y | x)
0
1
0.5
0
0.167
the difference between
calibrated, unsharp p? (y | x) 0.5 0.5
0.5
0
0
calibration error (lower
uncalibrated, sharp p? (y | x) 0.2 0.8
0.4
0.03
0.167
is better) and sharpness
0
0.75 0.75 0
0.125
balanced p? (y | x)
(higher is better), consider
the following binary classification example: we have a uniform distribution (P(x) = 1/3) over inputs X = {0, 1, 2}. For
x ? {0, 1}, y = x with probability 1, and for x = 2, y is either 0 or 1, each with probability 12 .
Setting p? (y | x) ? 0.5 would achieve perfect calibration (0) but not sharpness (0). We can get
excellent sharpness (0.167) but suffer in calibration (0.03) by predicting probabilities 0.2, 0.8, 0.4.
We can trade off some sharpness (0.125) for perfect calibration (0) by predicting 0 for x = 0 and
0.75 for x ? {1, 2}.
Discretized probabilities. We have assumed so far that the forecaster might return arbitrary probabilities in [0, 1]. In this case, we might need an infinite amount of data to estimate T (x) = E[y |
F (x)] accurately for each value of F (x). In order to estimate calibration and sharpness from finite
data, we use a discretized version of calibration and sharpness. Let B be a partitioning of the interval
[0, 1]; for example B = {[0, 0.1), [0.1, 0.2), . . . }. Let B : [0, 1] ? B map a probability p to the
interval B(p) containing p; e.g., B(0.15) = [0.1, 0.2). In this case, we simply redefine T (x) to be
the true probability of y = 1 given that F (x) lies in a bucket: T (x) = E[y | B(F (x))]. It is not hard
to see that discretized calibration estimates form an upper bound on the calibration error (3) [14].
3
Calibration in the Context of Structured Prediction
We have so far presented calibration in the context of binary classification. In this section, we extend
these definitions to structured prediction. Our ultimate motivation is to construct forecasters that
augment pre-trained structured models p? (y|x) with confidence estimates. Unlike in the multiclass
setting [12], we cannot learn a forecaster Fy : X ? [0, 1] that targets P(y | x) for each y ? Y
because the cardinality of Y is too large; in fact, the user will probably not be interested in every y.
Events of interest. Instead, we assume that for a given x and associated prediction y, the user is
interested in a set I(x) of events concerning x and y. An event E ? I(x) is a subset E ? Y; we
would like to determine the probability P(y ? E | x) for each E ? I(x). Here are two useful types
of events that will serve as running examples:
1. {MAP(x)}, which encodes whether MAP(x) = arg maxy p? (y | x) is correct.
2. {y : yj = MAP(x)j }, which encodes whether the label at position j in MAP(x) is correct.
In the OCR example (Figure 1), suppose we predict MAP(x) = ?land?. Define the events of
interest to be the MAP and the marginals: I(x) = {{MAP(x)}, {y : y1 = MAP(x)1 }, . . . , {y :
yL = MAP(x)L }}. Then we have I(x) = {{?land?}, {y : y1 = ?l?}, {y : y2 = ?a?}, {y : y3 =
?n?}, {y : y4 = ?d?}}. Note that the events of interest I(x) depend on x through MAP(x).
Event pooling. We now define calibration in analogy with (1). We will construct a forecaster
F (x, E) that tries to predict P(y ? E | x). As we remarked earlier, we cannot make a statement
that holds uniformly for all events E; we can only make a guarantee in expectation. Thus, let E be
drawn uniformly from I(x), so that P is extended to be a joint distribution over (x, y, E). We say
that a forecaster F : X ? 2Y 7? [0, 1] is perfectly calibrated if
P (y ? E | F (x, E) = p) = p.
(4)
In other words, averaged over all x, y and events of interest E ? I(x), whenever the forecaster
outputs probability p, then the event E actually holds with probability p. Note that this definition corresponds to perfect binary calibration (1) for the transformed pair of variables y 0 = I[y ?
E], x0 = (x, E). As an example, if I(x) = {{MAP(x)}}, then (4) says that of all the MAP predictions with confidence p, a p fraction will be correct. If I(x) = {{y : yj = MAP(x)j }}L
j=1 , then
(4) states that out of all the marginals (pooled together across all samples x and all positions j) with
confidence p, a p fraction will be correct.
3
Algorithm 1 Recalibration procedure for calibrated structured prediction.
Input: Features ?(x, E) from trained model p? , event set I(x), recalibration set S = {(xi , yi )}ni=1 .
Output: Forecaster F (x, E).
Construct the events dataset: Sbinary = {(?(x, E), I[y ? E]) : (x, y) ? S, E ? I(x)}
Train the forecaster F (e.g., k-NN or decision trees) on Sbinary .
The second example hints at an important subtlety inherent to having multiple events in structured
prediction. The confidence scores for marginals are only calibrated when averaged over all positions.
If a user only looked at the marginals for the first position, she might be sorely disappointed. As
an extreme example, suppose y = (y1 , y2 ) and y1 is 0 or 1 with probability 21 while y2 ? 1. Then
a forecaster that outputs a confidence of 0.75 for both events {y : y1 = 1} and {y : y2 = 1} will
be perfectly calibrated. However, neither event is calibrated in isolation (P(y1 = 1 | x) = 21 and
P(y2 = 1 | x) = 1). Finally, perfect calibration can be relaxed; following (3), we may define the
def
calibration error to be E[(T (x, E) ? F (x, E))2 ], where T (x, E) = P(y ? E | F (x, E)).
4
Constructing Calibrated Forecasters
Having discussed the aspects of calibration specific to structured prediction, let us now turn to the
problem of constructing calibrated (and sharp) forecasters from finite data.
Recalibration framework. We propose a framework that generalizes existing recalibration strategies to structured prediction models p? . First, the user specifies a set of events of interest I(x) as
well as features ?(x, E), which will in general depend on the trained model p? . We then train a
forecaster F to predict whether the event E holds (i.e. I[y ? E]) given features ?(x, E). We train F
by minimizing
the
P
Pempirical `2 loss over a recalibration set S (disjoint from the training examples):
minF (x,y)?S E?I(x) (F (x, E) ? I[y ? E])2 . Algorithm 1 outlines our procedure.
As an example, consider again the OCR setting in Figure 1. The margin feature ?(x, E) =
log p? (MAP(1) (x)) ? log p? (MAP(2) (x)) (where MAP(1) (x) and MAP(2) (x) are the first and second highest scoring labels for x according to p? , respectively) will typically correlate with the event
that the MAP prediction is correct. We can perform isotonic regression using this feature on the
recalibration set S to produce well-calibrated probabilities.
In the limit of infinite data, Algorithm 1 minimizes the expected loss E[(F (x, E) ? I[y ? E])2 ],
where the expectation is over (x, y, E). By (3), the calibration error E[(T (x, E) ? F (x, E))2 ] will
also be small. If there are not too many features ?, we can drive the `2 loss close to zero with
a nonparametric method such as k-NN. This is also why isotonic regression is sensible for binary
recalibration: we first project the data into a highly informative one-dimensional feature space; then
we predict labels from that space to obtain small `2 loss.
Note also that standard multiclass recalibration is a special case of this framework, where we use the
raw uncalibrated score from p? as a single feature. In the structured setting, one must invest careful
thought in the choice of classifier and features; we discuss these choices below.
Features. Calibration is possible even with a single constant feature (e.g. ?(x, E) ? 1), but
sharpness depends strongly on the features? quality. If ? collapses points of opposite labels, no
forecaster will be able to separate them and be sharp. While we want informative features, we can
only afford to have a few, since our recalibration set is typically small.
Compared to calibration for binary classification, our choice of features must also be informed
by their computational requirements: the most informative features might require performing full
inference in an intractable model. It is therefore useful to think of features as belonging to one
of three types, depending on whether they are derived from unstructured classifiers (e.g. an SVM
trained individually on each label), MAP inference, or marginal inference. In Section 5, we will
show that marginal inference produces the sharpest features, but clever MAP-based features can do
almost as well.
In Table 1, we propose several features that follow our guiding principles and that illustrate the
computational tradeoffs inherent to structured prediction.
4
Type
none
MAP
Marg.
MAP recalibration on y
Name
Definition
(yj )]
?no
minj mrgyj [sSVM
1 : SVM margin
j
mp
MAP
?1 : Label length
|y
|
?mp
I[y MAP ? G(x)]
2 : Admissibility
?mp
mrgy [p? (y | x)]
3 : Margin
?mg
1 : Margin
minj mrgyj [p? (yj | x)]
Marginal recalibration on yj
Name
Definition
(yj )]
?no
mrgyj [sSVM
2 : SVM margin
j
mp
?4 : Label freq.
% positions j 0 labeled yjMAP
?mp
:
Neighbors
% neighbors j 0 labeled yjMAP
5
mp
?6 : Label type
I[yjMAP ? L(x)]
mp
MAP
?7 : Pseudomargin mrgyj [p? (yj | y?j
, x)]
mg
?2 : Margin
mrgyj [p? (yj | x)]
?mg
I[yjMG = yjMAP ]
3 : Concordance
Table 1: Features for MAP recalibration (I(x) = {{y MAP (x)}}) and marginal recalibration (I(x) = {{y :
yj = y MAP (x)j }}L
j=1 ). We consider three types of features, requiring either unstructured, MAP, or marginal
inference. For a generic function f , define mrga f (a) , f (a(1) ) ? f (a(2) ), where a(1) and a(2) are the top
(yj ) be the score of an SVM
two inputs to f , ordered by f (a). Let yjMG , arg maxyj p? (yj | x); let sSVM
j
mp
require
domain-specific
knowledge:
defining admissible
and
?
classifier predicting label yj . Features ?mp
6
2
sets G(x), L(x). In OCR, G are all English words and L(x) are similar-looking letters. Percentages in ?mp
4 and
MAP
.
?mp
5 are relative to all the labels in y
Region-based forecasters. Recall from (4) that calibration examines the true probability of an
event (y ? E) conditioned on the forecaster?s prediction F (x, E) = p. By limiting the number of
different probabilities p that F can output, we can more accurately estimate the true probability for
each p To this end, let us partition the feature space (the range of ?) into regions R, and output a
probability FR ? [0,P
1] for each region R ? R. Formally, we consider region-based forecasters of
the form F (x, E) = R?R FR I[?(x, E) ? R], where FR is the fraction of points in region R (that
is, (x, E) for which ?(x, E) ? R) for which the event holds (y ? E). Note that the partitioning R
could itself depend on the recalibration set. Two examples of region-based forecasters are k-nearest
neighbors (k-NN) and decision trees.
Let us obtain additional insight into the performance of region-based forecasters as a function of
recalibration set size. Let S denote here a recalibration set of size n, which is used to derive a
partitioning R and probability estimates FR for each region R ? R. Let TR , P(y ? E | ?(x, E) ?
R) be the true event probability for region R, and wR , P(?(x, E) ? R) be the probability mass of
region R. We may rewrite the expected calibration error (3) of FR trained on a random S of size n
(drawn i.i.d. from P) as
"
#
X
CalibrationErrorn = ER
wR ES [(FR ? TR )2 | R] .
(5)
R?R
We see that there is a classic bias-variance tradeoff between having smaller regions (lower bias,
increased sharpness) and having more data points per region (lower variance, better calibration):
E[(FR ? TR )2 | R] = (E[FR | R] ? TR )2 + E[(FR ? E[FR | R])2 | R] .
|
{z
} |
{z
}
bias
variance
If R is a fixed partitioning independent of S, then the bias will be zero, and the variance is due to
an empirical average, falling off as 1/n. However, both k-NN and decision trees produce biased
estimates FR of TR because the regions are chosen adaptively, which is important for achieving
sharpness. In this case, we can still ensure that the calibration error vanishes to zero if we let the
P
regions grow uniformly larger: minR?R |{(x, y) ? S : ?(x, E) ? R, E ? I(x)}| ?
? ?.
5
Experiments
We test our proposed recalibrators and features on three real-world tasks.
Multiclass image classification. The task is to predict an image label given an image. This setting
is a special case of structured prediction in which we show that our framework improves over existing multiclass recalibration strategies. We perform our experiments on the CIFAR-10 dataset [15],
5
Image classification (Multi-class MAP recal.);
75% accuracy on raw uncalibrated SVM
raw (23.0)
0.8
cal (19.6)
1-vs-a (20.1)
0.6
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.0
0.0
Fraction of positives
1.0
0.2
0.4
0.6
Mean predicted value
0.8
1.0
1.0
OCR (Chain CRF MAP recalibration);
45% per-word accuracy using Viterbi decoding
Scene understanding (Graph CRF marginal recal.);
78% accuracy using mean-field marg. decoding
1.0
0.4
raw (29.5)
cal (13.6)
0.2
0.4
0.6
Mean predicted value
0.8
raw (65.9)
cal (18.6)
0.2
1.0
0.0
0.0
0.2
0.4
0.6
Mean predicted value
0.8
1.0
Figure 2: MAP recalibration in the multiclass and chain CRF settings (left and middle) and marginal
recalibration of the graph CRF (right). The legend includes the `2 loss before and after calibration.
The radius of the black balls reflects the number of points having the given forecasted and true
probabilities.
which consists of 60,000 32x32 color images of different types of animals and vehicles (ten classes
in total). We train a linear SVM on features derived from k-means clustering and that produce high
accuracies (79%) on this dataset [16]. We use 800 out of the 1600 features having the highest mutual information with the label (the drop in performance is negligible). 38,000 images were used for
training, 2,000 for calibration, and 20,000 for testing.
Optical character recognition. The task is to predict the word (sequence of characters) given a
sequence of images (Figure 1). Calibrated OCR systems can be useful for automatic sorting of mail.
This setting demonstrates calibration on a tractable linear-chain CRF. We used a dataset consisting
of ? 8-character-long words from 150 human subjects [3]. Each character is rasterized into a 16 ? 8
binary image. We chose 2000 words for training and another 2000 for testing. The remaining words
are subsampled in various ways to produce recalibration sets.
Scene understanding. Given an image divided into a set of regions, the task is to label each region
with its type (e.g. person, tree, etc.). Calibrated scene understanding is important for building
autonomous agents that try to take optimal actions in the environment, integrating over uncertainty.
This is a structured prediction setting in which inference is intractable. We conduct experiments on
a post-processed VOC Pascal dataset [5]. In brief, we train a graph CRF to predict the joint labeling
yi of superpixels yij in an image (? 100 superpixels per image; 21 possible labels). The input
xi consists of 21 node features; CRF edges connect adjacent superpixels. We use 600 examples
for training, 500 for testing and subsample the remaining ? 800 examples to produce calibration
sets. We perform MAP inference using AD3, a dual composition algorithm; we use a mean field
approximation to compute marginals.
Experimental setup. We perform both MAP and marginal calibration as described in Section 3.
We use decision trees and k-NN as our recalibration algorithms and examine the quality of our
forecasts based on calibration and sharpness (Section 2). We further discretize probabilities into
i
buckets of size 0.1: B = {[ i?1
10 , 10 ) : i = 1, . . . , 10}.
We report results using calibration curves: For each test point (xi , Ei , yi ), let fi = F (xi , Ei ) ?
[0, 1] be the forecasted probability and ti = I[yi ?PEi ] ? {0, 1} be the true outcome.
For each
P
bucket B ? B, we compute averages fB = NB?1 i:fi ?B fi and tB = NB?1 i:fi ?B ti , where
NB = |{fi ? B}| is the number of points in bucket B. A calibration curve plots the tB as a
function of fB . Perfect calibration corresponds to a straight line. See Figure 2 for an example.
5.1
?Out-of-the-Box? Recalibration
We would first like to demonstrate that our approach works well ?out of the box? with very simple
parameters: a single feature, k-NN with k = 100, and a reasonably-sized calibration set. We report
results in three settings: (i) multiclass and (ii) chain CRF MAP recalibration with the margin feature
?mg
1 (Figure 2, left, middle), as well as (iii) graph CRF marginal recalibration with the margin feature
?mg
2 (Figure 2, right). We use calibration sets of 2,000, 1,000, and 300 (respectively) and compare
to the raw CRF probabilities p? (y ? E | x).
6
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
Uncalibrated : 30.2
1.0
0.8
0.6
0.4
0.2
0.0
Unstructured SVM scores
?no
2 : 15.8
1.0
0.8
0.6
0.4
0.2
0.0
26 character indicators
mp
?6
: 16.1
1.0
0.8
0.6
0.4
0.2
0.0
Marginal probabilities
mg
?2
: 12.0
1.0
0.8
0.6
0.4
0.2
0.0
Marginal probabilities +
Marginal/MAP agreement
1.0
0.8
0.6
0.4
0.2
0.0
0.0
mg
mg
?2 , ? 3
: 10.9
All features : 10.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Per-letter OCR (Chain CRF marginal recalibration);
84% per-letter accuracy using Viterbi decoding
1.0
1.0
0.8
0.6
0.4
0.2
0.0
Uncalibrated : 21.0
1.0
0.8
0.6
0.4
0.2
0.0
Unstructured SVM scores
?no
1 : 20.5
1.0
0.8
0.6
0.4
0.2
0.0
Length + Presence in dict.
mp
mp
?1 , ? 2
: 4.2
1.0
0.8
0.6
0.4
0.2
0.0
Margin between 1st and 2nd
best
mp
?3
: 13.1
1.0
0.8
0.6
0.4
0.2
0.0
Lowest marginal probability
1.0
0.8
0.6
0.4
0.2
0.0
0.0
mg
?1
: 20.6
All features : 4.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Per-word OCR (Chain CRF MAP recalibration);
45% per-word accuracy using Viterbi decoding
1.0
Uncalibrated : 67.0
Unstructured SVM scores
?no
2 : 14.7
Pseudomargins
mp
: 17.0
mp
: 15.4
mg
: 15.9
?7
Pseudomargins, other MAP
features
mp
mp
?4 , ? 5 , ? 7
Marginals, MAP/marg.
concordance
1.0
0.8
0.6
0.4
0.2
0.0
0.0
?2
All features : 14.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Scene understanding (Graph CRF marginal recalibration);
78% accuracy using mean-field marg. decoding
Figure 3: Feature analysis for MAP and marginal recalibration of the chain CRF (left and middle,
resp.) and marginal recalibration of the graph CRF (right). Subplots show calibration curves for
various groups of features from Table 1, as well as `2 losses; dot sizes indicate relative bucket size.
Figure 2 shows that our predictions (green line) are well-calibrated in every setting. In the multiclass
setting, we outperform an existing approach which individually recalibrates one-vs-all classifiers and
normalizes their probability estimates [12]. This suggests that recalibrating for a specific event (e.g.
the highest scoring class) is better than first estimating all the multiclass probabilities.
5.2
Feature Analysis
Next, we investigate the role of features. In Figure 3, we consider three structured settings, and
in each setting evaluate performance using different sets of features from Table 1. From top to
bottom, the subplots describe progressively more computationally demanding features. Our main
takeaways are that clever inexpensive features do as well as naive expensive ones, that features may
be complementary and help each other, and that recalibration allows us to add ?global? features to a
chain CRF. We also see that features affect only sharpness.
In the intractable graph CRF setting (Figure 3, right), we observe that pseudomarginals ?mp
7 (which
require only MAP inference) fare almost as well as true marginals ?mg
,
although
they
lack
resolution.
2
mp
Augmenting with additional MAP-based features (?mp
4 , ?5 ) that capture whether a label is similar
to its neighbors and whether it occurs elsewhere in the image resolves this.
This synergistic interaction of features appears elsewhere. On marginal chain CRF recalibration
(Figure 3, left), the margin ?mg
2 between the two best classes yields calibrated forecasts that slightly
lack sharpness near zero (points with e.g. 50% and 10% confidences will have similarly small
margins). Adding the MAP-marginal concordance feature ?mg
3 improves calibration, since we can
further differentiate between low and very low confidence estimates. Similarly, individual SVM and
mp
mp
MAP-based features ?no
2 , ?6 (the ?6 are 26 binary indicators, one per character) are calibrated,
but not very sharp. They accurately identify 70%, 80% and 90% confidence sets, which may be
sufficient in practice, given that they take no additional time to compute. Adding features based on
mg
marginals ?mg
2 , ?3 improves sharpness.
mp
On MAP CRF recalibration (Figure 3, middle), we see that simple features (?mp
1 , ?2 ) can fare better
mp
mp
than more sophisticated ones like the margin ?3 (recall that ?1 is the length of a word; G in ?mp
2
encodes whether the word y MAP is in the dictionary). This demonstrates that recalibration lets us
introduce new global features beyond what?s in the original CRF, which can dramatically improve
calibration at no additional inferential cost.
7
kNN
10-1
Cal
Sha
10-2
10-3
10-4
5.3
Decision tree
10-1
10-2
10-3
0
500
1000
1500
Recalibration set size
2000
10-4
0
500
1000
1500
Recalibration set size
2000
Figure 4: Calibration error
(blue) and sharpness (green)
of k-NN (left) and decision
trees (right) as a function
of calibration set size (chain
CRF; marginal recalibration).
Effects of Recalibration Set Size and Recalibration Technique
Lastly, in Figure 4, we compare k-NN and decision trees on chain CRF marginal prediction using
feature ?mg
2 . We subsample calibration sets S of various sizes N . For each N and each algorithm
we choose a hyperparameter (minimum leaf size for decision trees, k in k-NN) by 10-fold crossvalidation on S. We tried values between 5 and 500 in increments of 5.
Figure 4 shows that for both methods, sharpness remains constant, while the calibration error decreases with N and quickly stabilizes below 10?3 ; this confirms that we can always recalibrate with
enough data. The decrease in calibration error also indicates that cross-validation successfully finds
a good model for each N . Finally, we found that k-NN fared better when using continuous features
(see also right columns of Figures 2 and 3); decision trees performed much better on categorical
features.
6
Previous Work and Discussion
Calibration and sharpness provide the conceptual basis for this work. These ideas and their connection to l2 losses have been explored extensively in the statistics literature [7, 9] in connection to
forecast evaluation; there exist generalizations to other losses as well [17, 10]. Calibration in the
online setting is a field in itself; see [8] for a starting point. Finally, calibration has been explored
extensively from a Bayesian viewpoint, starting with the seminal work of Dawid [18].
Recalibration has been mostly studied in the binary classification setting, with Platt scaling [11] and
isotonic regression [13] being two popular and effective methods. Non-binary methods typically
involve training one-vs-all predictors [12] and include extensions to ranking losses [19] and combinations of estimators [20]. Our generalization to structured prediction required us to develop the
notion of events of interest, which even in the multiclass setting works better than estimating every
class probability, and this might be useful beyond typical structured prediction problems.
Confidence estimation methods play a key role in speech recognition [21], but they require domain
specific acoustic features [1]. Our approach is more general, as it applies in any graphical model
(including ones where inference is intractable), uses domain-independent features, and guarantees
calibrated probabilities, rather than simple scores that correlate with accuracy.
The issue of calibration arises any time one needs to assess the confidence of a prediction. Its importance has been discussed and emphasized in medicine [22], natural language processing [23],
speech recognition [21], meteorology [10], econometrics [9], and psychology [24]. Unlike uncalibrated confidence measures, calibrated probabilities are formally tied to objective frequencies. They
are easy to understand by users, e.g., patients undergoing diagnosis or researchers querying a probabilistic database. Moreover, modern AI systems typically consist of a pipeline of modules [23]. In
this setting, calibrated probabilities are important to express uncertainty meaningfully across different (potentially third-party) modules. We hope our extension to the structured prediction setting can
help make calibration more accessible and easier to apply to more complex and diverse settings.
Acknowledgements. This research is supported by an NSERC Canada Graduate Scholarship to
the first author and a Sloan Research Fellowship to the second author.
Reproducibility. All code, data, and experiments for this paper are available on CodaLab at
https://www.codalab.org/worksheets/0xecc9a01cfcbc4cd6b0444a92d259a87c/.
8
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9
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5,146 | 5,659 | A Bayesian Framework for Modeling Confidence in
Perceptual Decision Making
Koosha Khalvati, Rajesh P. N. Rao
Department of Computer Science and Engineering
University of Washington
Seattle, WA 98195
{koosha, rao}@cs.washington.edu
Abstract
The degree of confidence in one?s choice or decision is a critical aspect of perceptual decision making. Attempts to quantify a decision maker?s confidence by
measuring accuracy in a task have yielded limited success because confidence and
accuracy are typically not equal. In this paper, we introduce a Bayesian framework
to model confidence in perceptual decision making. We show that this model,
based on partially observable Markov decision processes (POMDPs), is able to
predict confidence of a decision maker based only on the data available to the
experimenter. We test our model on two experiments on confidence-based decision making involving the well-known random dots motion discrimination task.
In both experiments, we show that our model?s predictions closely match experimental data. Additionally, our model is also consistent with other phenomena
such as the hard-easy effect in perceptual decision making.
1
Introduction
The brain is faced with the persistent challenge of decision making under uncertainty due to noise in
the sensory inputs and perceptual ambiguity . A mechanism for self-assessment of one?s decisions
is therefore crucial for evaluating the uncertainty in one?s decisions. This kind of decision making,
called perceptual decision making, and the associated self-assessment, called confidence, have received considerable attention in decision making experiments in recent years [9, 10, 12, 13]. One
possible way of estimating the confidence of a decision maker is to assume that it is equal to the
accuracy (or performance) on the task. However, the decision maker?s belief about the chance of
success and accuracy need not be equal because the decision maker may not have access to information that the experimenter has access to [3]. For example, in the well-known task of random dots
motion discrimination [18], on each trial, the experimenter knows the difficulty of the task (coherence or motion strength of the dots), but not the decision maker [3, 13]. In this case, when the data is
binned based on difficulty of the task, the accuracy is not equal to decision confidence. An alternate
way to estimate the subject?s confidence is to use auxiliary tasks such as post-decision wagering [15]
or asking the decision maker to estimate confidence explicitly [12]. These methods however only
provide an indirect window into the subject?s confidence and are not always applicable.
In this paper, we explain how a model of decision making based on Partially Observable Decision
Making Processes (POMDPs) [16, 5] can be used to estimate a decision maker?s confidence based on
experimental data. POMDPs provide a unifying Bayesian framework for modeling several important aspects of perceptual decision making including evidence accumulation via Bayesian updates,
the role of priors, costs and rewards of actions, etc. One of the advantages of the POMDP model
over the other models is that it can incorporate various types of uncertainty in computing the optimal
This research was supported by NSF grants EEC-1028725 and 1318733, and ONR grant N000141310817.
1
decision making strategy. Drift-diffusion and race models are able to handle uncertainty in probability updates [3] but not the costs and rewards of actions. Furthermore, these models originated as
descriptive models of observed data, where as the POMDP approach is fundamentally normative,
prescribing the optimal policy for any task requiring decision making under uncertainty. In addition,
the POMDP model can capture the temporal dynamics of a task. Time has been shown to play a
crucial role in decision making, especially in decision confidence [12, 4]. POMDPs have previously been used to model evidence accumulation and understand the role of priors [16, 5, 6]. To our
knowledge, this is the first time that it is being applied to model confidence and explain experimental
data on confidence-based decision making tasks.
In the following sections, we introduce some basic concepts in perceptual decision making and
show how a POMDP can model decision confidence. We then explore the model?s predictions
for two well-known experiments in perceptual decision making involving confidence: (1) a fixed
duration motion discrimination task with post-decision wagering [13], and (2) a reaction-time motion discrimination task with confidence report [12]. Our results show that the predictions of the
POMDP model closely match experimental data. The model?s predictions are also consistent with
the ?hard-easy? phenomena in decision making, involving over-confidence in the hard trials and
under-confidence in the easy ones [7].
2
Accuracy, Belief and Confidence in Perceptual Decision Making
Consider perceptual decision making tasks in which the subject has to guess the hidden state of the
environment correctly to get a reward. Any guess other than the correct state usually leads to no
reward. The decision maker has been trained on the task, and wants to obtain the maximum possible
reward. Since the state is hidden, the decision maker must use one or more observations to estimate
the state. For example, the state could be one of two biased coins, one biased toward heads and
the other toward tails. On each trial, the experimenter picks one of these coins randomly and flips
it. The decision maker only sees the result, heads or tails, and must guess which coin has been
picked. If she guesses correctly, she gets a reward immediately. If she fails, she gets nothing. In
this context, Accuracy is defined as the number of correct guesses divided by the total number of
trials. In a single trial, if A represents the action (or choice) of the decision maker, and S and Z
denote the state and observation respectively, then Accuracy for the choice s with observation z is
the probability P (A = as |S = s, Z = z) where as represents the action of decision maker, i.e.
choosing s, and s is the true state. This Accuracy can be measured by the experimenter. However,
from the decision maker?s perspective, her chance of success in a trial is given by the probability
of s being the correct state, given observation z: P (S = s|Z = z). We call this probability the
decision maker?s belief . After choosing an action, for example as , the conf idence for this choice
is the probability: P (S = s|A = as , Z = z). According to Bayes theorem:
P (A|S, Z)P (S|Z) = P (S|A, Z)P (A|Z).
(1)
As the goal of our decision maker is to maximize her reward, she picks the most probable
state. This means that on observing z she picks as? where s? is the most probable state, i.e.
s? = arg max(P (S = s|Z = z)). Therefore, P (A|Z = z) is equal to 1 for as? and 0 for the
rest of the actions. As a result, accuracy is 1 for the most probable state and 0 for the rest. Also
P (S|A, Z) is equal to P (S|Z) for the most probable state. This means that, given observation z,
Accuracy is equal to the confidence on the most probable state. Also, this confidence is equal to the
belief of the most probable state. As confidence cannot be defined on actions not performed, one
could consider confidence on the most probable state only, implying that accuracy, confidence, and
belief are all equal given observation z:
X
P (A = as |S = s, Z)P (S = s|Z) = P (S = s? |A = as? , Z) = P (S = s? |Z).1
(2)
s
All of the above equalities, however, depend on the ability of the decision maker to compute P (S|Z).
According to Bayes? theorem P (S|Z) = P (Z|S)P (S)/P (Z) (P (Z) 6= 0). If the decision maker
has the perfect observation model P (Z|S), she could compute P (S|Z) by estimating P (S) and
1
In the case that there are multiple states with maximum probability, Accuracy is the sum of the confidence
values on those states.
2
P (Z) beforehand by counting the total number of occurrences of each state without considering
any observations, and the total number of occurrences of observation z, respectively. Therefore,
accuracy and confidence are equal if the decision maker has the true model for observations. Sometimes, however, the decision maker does not even have access to Z. For example, in the motion
discrimination task, if the data is binned based on difficulty (i.e., motion strength), the decision
maker cannot estimate P (S|difficulty) because she does not know the difficulty of each trial. As a
result, accuracy and confidence are not equal.
In the general case, the decision maker can utilize multiple observations over time, and perform an
action on each time step. For example, in the coin toss problem, the decision maker could request a
flip multiple times to gather more information. If she requests a flip two times and then guesses the
state to be the coin biased toward heads, her actions would be Sample, Sample, Choose heads. She
also has two observations (likely to be two Heads). In the general case, the state of the environment
can also change after each action.2 In this case, the relationship between accuracy and confidence at
time t after a sequence (history Ht ) of actions and observations ht = a0 , z1 , a2 , ..., zt?1 , at?1 , is:
P (At |St , Ht )P (St |Ht ) = P (St |At , Ht )P (At |Ht ).
(3)
With the same reasoning as above, accuracy and confidence are equal if and only if the decision
maker has access to all the observations and has the true model of the task.
3
The POMDP Model
Partially Observable Markov Decision Processes (POMDPs) provide a mathematical framework
for decision making under uncertainty in autonomous agents [8]. A POMDP is formally a tuple
(S, A, Z, T, O, R, ?) with the following description: S is a finite set of states of the environment,
A is a finite set of possible actions, and Z is a finite set of possible observations. T is a transition
function defined as T : S ? S ? A ? [0, 1] which determines P (s|s0 , a), the probability of going
from a state s0 to another state s after performing a particular action a. O is an observation function
defined as O : Z ? A ? S ? [0, 1], which determines P (z|a, s), the probability of observing z
after performing an action a and ending in a particular state s. R is the reward function, defined as
R : S ? A ? R, determining the reward received by performing an action in a particular state. ?
is the discount factor, which is always between 0 and 1, and determines how much rewards in the
future are discounted compared to current rewards.
In a POMDP,
P?the goal is to find a sequence of actions to maximize the expected discounted reward, Est [ t=0 ? t R(st , at )]. The states are not fully observable and the agent must rely on its
observations to choose actions. At the time t, we have a history of actions and observations:
ht = a0 , z1 , a1 , ..., zt?1 , at?1 . The belief state [1] at time t is the posterior probability over states
given this history and the prior probability b0 over states: bt = P (st |ht , b0 ). As the system is
Markovian, the belief state captures the sufficient statistics for the history of states and actions [19]
and it is possible to obtain bt+1 using only bt , at and zt+1 :
X
bt+1 (s) ? O(s, at , zt+1 )
T (s0 , s, at )bt (s0 ).
(4)
s0
Given this definition P
of belief, the goal of the agent is to find a sequence of actions to maximize
?
the expected reward t=0 ? t R(bt , at ). The actions are picked based on the belief state, and the
resulting mapping, from belief states to actions, is called a policy (denoted by ?),Pwhich is a prob?
ability distribution over actions ?(bt ) : P (At |bt ). The policy which maximizes t=0 ? t R(bt , at )
?
is called the optimal policy, ? . It can be shown that there is always a deterministic optimal policy,
allowing the agent to always choose one action for each bt [20]. As a result, we may use a function
? ? : B ? A where B is the space of all possible beliefs. There has been considerable progress in
recent years in fast ?POMDP-solvers? which find near-optimal policies for POMDPs [14, 17, 11].
3.1
Modeling Decision Making with POMDPs
Results from experiments and theoretical models indicate that in many perceptual decision making
tasks, if the previous task state is revealed, the history beyond this state does not exert a noticeable
2
In traditional perceptual decision making tasks such as the random dots task, the state does not usually
change. However, our model is equally applicable to this situation.
3
influence on decisions [2], suggesting that the Markov assumption and the notion of belief state is
applicable to perceptual decision making. Additionally, since the POMDP model aims to maximize
the expected reward, the problem of guessing the correct state in perceptual decision making can be
converted to a reward maximization problem by simply setting the reward for the correct guess to
1 and the reward for all other actions to 0. The POMDP model also allows other costs in decision
making to be taken into account, e.g., the cost of sampling, that the brain may utilize for metabolic
or other evolutionarily-driven reasons. Finally, as there is only one correct hidden state in each
trial, the policy is deterministic (choosing the most probable state), consistent with the POMDP
model. All these facts mean that we could model the perceptual decision making with the POMDP
framework. In the cases where all observations and the true environment model are available to
the decision maker, the belief state in the POMDP is equal to both accuracy and confidence as
discussed above. When some information is hidden from the decision maker, one can use a POMDP
with that information to model accuracy and another POMDP without that information to model the
confidence. If this hidden information is independent of time, we can model the difference with the
initial belief state, b0 , i.e., we use two similar POMDPs to model accuracy and confidence but with
different initial belief states. In the well-known motion discrimination experiment, it is common to
bin the data based on the difficulty of the task. This difficulty is hidden to the decision maker and
also independent of the time. As a result, the confidence can be calculated by the same POMDP that
models accuracy but with different initial belief state. This case is discussed in the next section.
4
Experiments and Results
We investigate the applicability of the POMDP model in the context of two well-known tasks in
perceptual decision making. The first is a fixed-duration motion discrimination task with a ?sure
option,? presented in [13]. In this task, a movie of randomly moving dots is shown to a monkey for
a fixed duration. After a delay period, the monkey must correctly choose the direction of motion
(left or right) of the majority of the dots to obtain a reward. In half of the trials, a third choice
also becomes available, the ?sure option,? which always leads to a reward, though the reward is less
than the reward for guessing the direction correctly. Intuitively, if the monkey wants to maximize
reward, it should go for the sure choice only when it is very uncertain about the direction of the
dots. The second task is a reaction-time motion discrimination task in humans studied in [12]. In
this task, the subject observes the random dots motion stimuli but must determine the direction of
the motion (in this case, up or down) of the majority of the dots as fast and as accurately as possible
(rather than observing for a fixed duration). In addition to their decision regarding direction, subjects
indicated their confidence in their decision on a horizontal bar stimulus, where pointing nearer to
the left end meant less confidence and nearer to the right end meant more confidence. In both tasks,
the difficulty of the task is governed by a parameter known as ?coherence?? (or ?motion strength??),
defined as the percentage of dots moving in the same direction from frame to frame in a given trial.
In the experiments, the coherence value for a given trial was chosen to be one of the following:
0.0%, 3.2%, 6.4%, 12.8%, 25.6%, 51.2%.
4.1
Fixed Duration Task as a POMDP
The direction and the coherence of the moving dots comprise the states of the environment. In
addition, the actions which are available to the subject are dependent on the stage of the trial, namely,
random dots display, wait period, choosing the direction or the sure choice, or choosing only the
direction. As a result, the stage of the trial is also a part of the state of the POMDP. As the transition
between these stages are dependent on time, we incorporate discretized time as a part of the state.
Considering the data, we define a new state for each constant ?t, each direction, each coherence,
and each stage (when there is intersection between stages). We use dummy states to enforce the
delay period of waiting and a terminal state, which indicates termination of the trial:
S = { (direction, coherence, stage, time), waiting states, terminal }
The actions are Sample, Wait, Left, Right, and Sure. The transition function models the passage of
time and stages. The observation function models evidence accumulation only in the random dots
display stage and with the action Sample. The observations received in each ?t are governed by the
number of dots moving in the same direction. We model the observations as normally distributed
4
(a)
(b)
Figure 1: Experimental accuracy of the decision maker for each coherence is shown in (a). This plot
is from [13]. The curves with empty circles and dashed lines are the trials where the sure option was
not given to the subject. The curves with solid circles and solid lines are the trials where the sure
option was shown, but waived by the decision maker. (b) shows the accuracy curves for the POMDP
model fit to the experimental accuracy data from trials where the sure option was not given.
around a mean related to the coherence and the direction as follows:
O((d, c, display, t), Sample) = N (?d,c , ?d,c ).
The reward for choosing the correct direction is set to 1, and the other rewards were set relative to
this reward. The sure option was set to a positive reward less than 1 while the cost of sampling and
receiving a new observation was set to a negative ?reward? value. To model the unavailability of
some actions in some states, we set their resultant rewards to a large negative number to preclude
the decision making agent from picking these actions. The discount factor models how much more
immediate reward is worth relative to future rewards. In the fixed-duration task, the subject does not
have the option of terminating the trial early to get reward sooner, and therefore we used a discount
factor of 1 for this task.
4.2
Predicting the Confidence in the Fixed Duration Task
As mentioned before, confidence and accuracy are equal to each other when the same amount of
information is available to the experimenter and the decision maker. Therefore, they can be modeled by the same POMDP. However, these two are not equal when we look at a specific coherence
(difficulty), i.e. the data is binned based on coherence, because the coherence in each trial is not
revealed to the decision maker. Figure 1a shows the accuracy vs. stimulus duration, binned based
on coherence. The confidence is not equal to the accuracy in this plot. However, we could predict
the decision maker?s confidence only from accuracy data. This time, we use two POMDPs, one
for the experimenter and one for the decision maker. At time t, bt of the experimenter?s POMDP
can be related to accuracy and bt of the decision maker?s to confidence. These two POMDPs have
the same model parameters, but different initial belief state. This is because the subject knows the
environment model but does not have access to the coherence in each trial.
First, we find the set of parameters for the experimenter?s POMDP to reproduce the same accuracy
curves as in the experiment for each coherence. We only use data from the trials where the sure
option is not given i.e. dashed curves in figure 1a. As the data is binned based on the coherence, and
coherence is observable to the experimenter, the initial belief state of the experimenter?s POMDP
for coherence c is as following: .5 for each of two possible initial states (at time = 0), and 0 for
the rest. Fitting the POMDP to the accuracy data yields the mean and variance for each observation
function and the cost for sampling. Figure 1b shows the accuracy curves based on the experimenter?s
POMDP.
Now, we could apply the parameters obtained from fitting accuracy data (the experimenter?s
POMDP) to the decision maker?s POMDP to predict her confidence. The decision maker does
not know the coherence in each single trial. Therefore, the initial belief state should be a uniform
distribution over all initial states (all coherences, not only coherence of that trial). Also, neural data
from experiments and post-decision wagering experiments suggest that the decision maker does not
recognize the existence of a true zero coherence state (coherence = 0%) [13]. Therefore, the initial
5
(a)
(b)
Figure 2: The confidence predicted by the POMDP fit to the observed accuracy data in the fixedduration experiment is shown in (a). (b) shows accuracy and confidence in one plot, demonstrating
that they are not equal for this task. Curves with solid lines show the confidence (same curves as
(a)) and the ones with dashed lines show the accuracy (same as Figure 1b).
(a)
(b)
Figure 3: Experimental post-decision wagering results (plot (a)) and the wagering predicted by our
model (plot (b)). Plot (a) is from [13].
probability of 0 coherence states is set to 0. Figure 2a shows the POMDP predictions regarding the
subject?s belief. Figure 2b confirms that the predicted confidence and accuracy are not equal.
To test our prediction about the confidence of the decision maker, we use experimental data from
post-decision wagering in this experiment. If the reward for the sure option is rsure then the decision
maker chooses it if and only if b(right)rright < rsure and b(lef t)rlef t < rsure where b(direction)
is the sum of the belief states of all states in that direction. Since rsure cannot be obtained from the
fit to the accuracy data, we choose a value for rsure which makes the prediction of the confidence
consistent with the wagering data shown in Figure 3a. We found that if rsure is approximately twothirds the value of the reward for correct direction choice, the POMDP model?s prediction matches
experimental data (Figure 3b). A possible objection is that the free parameter of rsure was used to
fit the data. Although rsure is needed to fit the exact probabilities, we found that any reasonable
value for rsure generates the same trend of wagering. In general, the effect of rsure is to shift the
plots vertically. The most important phenomena here is the relatively small gap between hard trials
and easy trials in Figure 3b. Figure 4a shows what this wagering data would look like if the decision
maker knew the coherence in each trial and confidence was equal to the accuracy. The difference
between these two plots (figure 3b and figure 4a), and figure 2b which shows the confidence and the
accuracy together confirm the POMDP model?s ability to explain hard-easy effect [7], wherein the
decision maker underestimates easy trials and has overconfidence in the hard ones.
Another way of testing the predictions about the confidence is to verify if the POMDP predicts the
correct accuracy in the trials where the decision maker waives the sure option. Figure 4b shows that
the results from the POMDP closely match the experimental data both in post-decision wagering and
accuracy improvement. Our methods are presented in more detail in the supplementary document.
4.3
Reaction Time Task
The POMDP for the reaction time task is similar to the fixed duration task. The most important
components of the state are again direction and coherence. We also need some dummy states for the
6
(a)
(b)
Figure 4: (a) shows what post-decision wagering would look like if the accuracy and the confidence
were equal. (b) shows the accuracy predicted by the POMDP model in the trials where the sure
option is shown but waived (solid lines), and also in the trials where it is not shown (dashed lines).
For comparison, see experimental data in figure 1a.
waiting period between the decision command from the decision maker and reward delivery. However, the passage of stages and time are not modeled. The absence of time in the state representation
does not mean that the time is not modeled in the framework. Tracking time is a very important
component of any POMDP especially when the discount factor is less than one. The actions for this
task are Sample, Wait, Up and Down (the latter two indicating choice for the direction of motion).
The transition model and the observation model are similar to those for the Fixed duration task.
S = (direction, coherence), waiting states, terminal
O((d, c), Sample) = N (?d,c , ?d,c )
The reward for choosing the correct direction is 1 and the reward for sampling is a small negative
value adjusted to the reward of the correct choice. As the subject controls the termination of the
trial, the discount factor is less than 1. In this task, the subjects have been explicitly advised to
terminate the task as soon as they discover the direction. Therefore, there is an incentive for the
subject to terminate the trial sooner. While sampling cost is constant during the experiment, the
discount factor makes the decision making strategy dependent on time. A discount factor less than
1 means that as time passes, the effective value of the rewards decreases. Also, in a general reaction
time task, the discount factor connects the trials to each other. While models usually assume each
single trial is independent of the others, trials are actually dependent when the decision maker has
control over trial termination. Specifically, the decision maker has a motivation to terminate each
trial quickly to get the reward, and proceed to the next one. Moreover, when one is very uncertain
about the outcome of a trial, it may be prudent to terminate the trial sooner with the expectation that
the next trial may be easier.
4.4
Predicting the Confidence in the Reaction time Task
Like the fixed duration task, we want to predict the decision maker?s confidence on a specific coherence. To achieve this, we use the same technique, i.e., having two POMDPs with the same model
and different initial belief states. The control of the subject over the termination of the trial makes
estimating the confidence more difficult in the reaction time task. As the subject decides based on
her own belief, not accuracy, the relationship between the accuracy and the reaction time, binned
based on difficulty is very noisy in comparison to the fixed duration task (the plots of this relationship are illustrated in the supplementary materials of [12]). Therefore we fit the experimenter?s
POMDP to two other plots, reaction time vs. motion strength (coherence), and accuracy vs. motion
strength (coherence). The first subject (S1) of the original experiment was picked for this analysis
because the behavior of this subject was consistent with the behavior of the majority of subjects
[12]. Figures 5a and 5b show the experimental data from [12]. Figure 5c and 5d show the results
from the POMDP model fit to experimental data. As in the previous task, the initial belief state of
the POMDP for a coherence c is .5 for each direction of c, and 0 for the rest.
All the free parameters of the POMDP were extracted from this fit. Again, as in the fixed duration
task, we assume that the decision maker knows the environment model, but does not know about
the coherence of each trial and existence of 0% coherence. Figure 6a shows the reported confidence
from the experiments and figure 6b shows the prediction of our POMDP model for the belief of the
7
(a)
(b)
(c)
(d)
Figure 5: (a) and (b) show Accuracy vs. motion strength, and reaction time vs. motion strength
plots from the reaction-time random dots experiments in [12]. (c) and (d) show the results from the
POMDP model.
(a)
(b)
Figure 6: (a) illustrates the reported confidence by the human subject from [12]. (b) shows the
predicted confidence by the POMDP model.
decision maker. Although this report is not in percentile and quantitative comparison is not possible,
the general trends in these plots are similar. The two become almost identical if one maps the report
bar to the probability range.
In both tasks, we assume that the decision maker has a nearly perfect model of the environment,
apart from using 5 different coherences instead of 6 (the zero coherence state assumed not known).
This assumption is not necessarily true. Although, the decision maker understands that the difficulty
of the trials is not constant, she might not know the exact number of coherences. For example, she
may divide trials into three categories: easy, normal, and hard for each direction. However, these
differences do not significantly change the belief because the observations are generated by the true
model, not the decision maker?s model. We tested this hypothesis in our experiments. Although
using using a separate decision maker?s model makes the predictions closer to the real data, we used
the true (experimenter?s) model to avoid overfitting the data.
5
Conclusions
Our results present, to our knowledge, the first supporting evidence for the utility of a Bayesian
reward optimization framework based on POMDPs for modeling confidence judgements in subjects
engaged in perceptual decision making. We showed that the predictions of the POMDP model are
consistent with results on decision confidence in both primate and human decision making tasks, encompassing fixed-duration and reaction-time paradigms. Unlike traditional descriptive models such
as drift-diffusion or race models, the POMDP model is normative and is derived from Bayesian and
reward optimization principles. Additionally, unlike the traditional models, it allows one to model
optimal decision making across trials using the concept of a discount factor. Important directions
for future research include leveraging the ability of the POMDP framework to model intra-trial
probabilistic state transitions, and exploring predictions of the POMDP model for decision making
experiments with more sophisticated reward/cost functions.
8
References
[1] Karl J. Astrom. Optimal control of Markov decision processes with incomplete state estimation. Journal of Mathematical Analysis and Applications, pages 174?205, 1965.
[2] Jan Drugowitsch, Ruben Moreno-Bote, Anne K. Churchland, Michael N. Shadlen, and Alexandre Pouget. The cost of accumulating evidence in perceptual decision making. The Journal of
neuroscience, 32(11):3612?3628, 2012.
[3] Jan Drugowitsch, Ruben Moreno-Bote, and Alexandre Pouget. Relation between belief and
performance in perceptual decision making. PLoS ONE, 9(5):e96511, 2014.
[4] Timothy D. Hanks, Mark E. Mazurek, Roozbeh Kiani, Elisabeth Hopp, and Michael N.
Shadlen. Elapsed decision time affects the weighting of prior probability in a perceptual decision task. Journal of Neuroscience, 31(17):6339?6352, 2011.
[5] Yanping Huang, Abram L. Friesen, Timothy D. Hanks, Michael N. Shadlen, and Rajesh P. N.
Rao. How prior probability influences decision making: A unifying probabilistic model. In
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[6] Yanping Huang and Rajesh P. N. Rao. Reward optimization in the primate brain: A probabilistic model of decision making under uncertainty. PLoS ONE, 8(1):e53344, 01 2013.
[7] Peter Juslin, Henrik Olsson, and Mats Bjorkman. Brunswikian and Thurstonian origins of
bias in probability assessment: On the interpretation of stochastic components of judgment.
Journal of Behavioral Decision Making, 10(3):189?209, 1997.
[8] Leslie Pack Kaelbling, Michael L. Littman, and Anthony R. Cassandra. Planning and acting
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Sciences, 367(1594):1322?1337, 2012.
[10] Adam Kepecs, Naoshige Uchida, Hatim A. Zariwala, and Zachary F. Mainen. Neural correlates, computation and behavioural impact of decision confidence. Nature, 455(7210):227?
231, 2012.
[11] Koosha Khalvati and Alan K. Mackworth. A fast pairwise heuristic for planning under uncertainty. In Proceedings of The Twenty-Seventh AAAI Conference on Artificial Intelligence,
pages 187?193, 2013.
[12] Roozbeh Kiani, Leah Corthell, and Michael N. Shadlen. Choice certainty is informed by both
evidence and decision time. Neuron, 84(6):1329 ? 1342, 2014.
[13] Roozbeh Kiani and Michael N. Shadlen. Representation of confidence associated with a decision by neurons in the parietal cortex. Science, 324(5928):759?764, 2009.
[14] Hanna Kurniawati, David Hsu, and Wee Sun Lee. SARSOP: Efficient point-based POMDP
planning by approximating optimally reachable belief spaces. In Proceedings of The Robotics:
Science and Systems IV, 2008.
[15] Navindra Persaud, Peter McLeod, and Alan Cowey. Post-decision wagering objectively measures awareness. Nature Neuroscience, 10(2):257?261, 2007.
[16] Rajesh P. N. Rao. Decision making under uncertainty: a neural model based on partially
observable Markov decision processes. Frontiers in computational neuroscience, 4, 2010.
[17] Stphane Ross, Joelle Pineau, Sebastien Paquet, and Brahim Chaib-draa. Online planning algorithms for POMDPs. Journal of Artificial Intelligence Research, 32(1), 2008.
[18] Michael N. Shadlen and William T. Newsome. Motion perception: seeing and deciding. Proceedings of the National Academy of Sciences of the United States of America, 93(2):628?633,
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[19] Richard D. Smallwood and Edward J. Sondik. The optimal control of partially observable
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9
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5,147 | 566 | Dual Inhibitory Mechanisms for Definition of
Receptive Field Characteristics in Cat Striate
Cortex
A. B. Bonds
Dept. of Electrical Engineering
Vanderbilt University
Nashville, TN 37235
Abstract
In single cells of the cat striate cortex, lateral inhibition across orientation and/or spatial frequency is found to enhance pre-existing biases. A
contrast-dependent but spatially non-selective inhibitory component is also
found. Stimulation with ascending and descending contrasts reveals the
latter as a response hysteresis that is sensitive, powerful and rapid, suggesting that it is active in day-to-day vision. Both forms of inhibition are
not recurrent but are rather network properties. These findings suggest
two fundamental inhibitory mechanisms: a global mechanism that limits
dynamic range and creates spatial selectivity through thresholding and a
local mechanism that specifically refines spatial filter properties. Analysis
of burst patterns in spike trains demonstrates that these two mechanisms
have unique physiological origins.
1
INFORMATION PROCESSING IN STRIATE
CORTICAL CELLS
The most popular current model of single cells in the striate cortex casts them
in terms of spatial and temporal filters. The input to visual cortical cells from
lower visual areas, primarily the LGN, is fairly broadband (e.g., Soodak, Shapley &
Kaplan, 1987; Maffei & Fiorentini, 1973). Cortical cells perform significant bandwidth restrictions on this information in at least three domains: orientation, spatial
frequency and temporal frequency. The most interesting quality of these cells is
75
76
Bonds
therefore what they reject from the broadband input signal, rather than what they
pass, since the mere passage of the signal adds no information. Visual cortical cells
also show contrast-transfer, or amplitude-dependent, nonlinearities which are not
seen at lower levels in the visual pathway. The primary focus of our lab is study of
the cortical mechanisms that support both the band limitations and nonlinearities
that are imposed on the relatively unsullied signals incoming from the LGN. All of
our work is done on the cat.
2
THE ROLE OF INHIBITION IN ORIENTATION
SELECTIVITY
Orientation selectivity is one of the most dramatic demonstrations of the filtering
ability of cortical cells. Cells in the LGN are only mildly biased for stimulus orientation, but cells in cortex are completely unresponsive to orthogonal stimuli and
have tuning bandwidths that average only about 40-50? (e.g., Rose & Blakemore,
1974). How this happens remains controversial, but there is general consensus that
inhibition helps to define orientation selectivity although the schemes vary. The
concept of cross-orientation inhibition suggests that the inhibition is itself orientation selective and tuned in a complimentary way to the excitatory tuning of the cell,
being smallest at the optimal orientation and greatest at the orthogonal orientation.
More recent results, including those from our own lab, suggests that this is not the
case.
We studied the orientation dependence of inhibition by presenting two superimposed gratings, a base grating at the optimal orientation to provide a steady level
of background response activity, and a mask grating of varying orientation which
yielded either excitation or inhibition that could supplement or suppress the basegenerated response. There is some confusion when both base and mask generate
excitation. In order to separate the response components from each of these stimuli,
the two gratings were drifted at differing temporal frequencies. At least in simple
cells, the individual contributions to the response from each grating could then be
resolved by performing Fourier analysis on the response histograms.
Experiments were done on 52 cells, of which about 2/3 showed organized suppression
from the mask grating (Bonds, 1989). Fig. 1 shows that while the mask-generated
response suppression is somewhat orientation selective, it is by and large much flatter than would be required to account for the tuning of the cell. There is thus some
orientation dependence of inhibition, but not specifically at the orthogonal orientation as might be expected. Instead, the predominant component of the suppression
is constant with mask orientation, or global. This suggests that virtually any
stimulus can result in inhibition, whether or not the recorded cell actually "sees"
it. What orientation-dependent component of inhibition that might appear is expressed in suppressive side-bands near the limits of the excitatory tuning function,
which have the effect of enhancing any pre-existing orientation bias.
Thus the concept of cross-orientation inhibition is not particularly correct, since
the inhibition is found not just at the "cross" orientation but rather at all orientations. Even without orientation-selective inhibition, a scheme for establishment
of true orientation selectivity from orientation-biased LGN input can be derived
Dual Inhibitory Mechanisms
70.------------,
No INIIk (luning) ??0??
A
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------
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90
No mask (luning)
Exc:iWtion
150
210
???0??????
Sv12:A4.D11
Simple
270
330
Mask Orientation (deg)
Figure 1: Response suppression by mask gratings of varying orientation. A. Impact
of masks of 2 different contrasts on 2 Hz (base-generated) response, expressed by
decrease (negative imp/sec) from response level arising from base stimulus alone.
B. Similar example for mask orientations spanning a full 360 0 ?
by assuming that the nonselective inhibition is graded and contrast-dependent and
that it acts as a thresholding device (Bonds, 1989).
3
THE ROLE OF INHIBITION IN SPATIAL
FREQUENCYSELECTnnTY
While most retinal and LGN cells are broadly tuned and predominantly low-pass,
cortical cells generally have spatial frequency bandpasses of about 1.5-2 octaves (e.g.,
Maffei & Fiorentini, 1973). We have examined the influence of inhibition on spatial
frequency selectivity using the same strategy as the previous experiment (Bauman &
Bonds, 1991). A base grating, at the optimal orientation and spatial frequency, drove
the cell, and a superimposed mask grating, at the optimal orientation but at different
spatial and temporal frequencies, provided response facilitation or suppression.
We defined three broad categories of spatial frequency tuning functions: Low pass,
with no discernible low-frequency fall-off, band-pass, with a peak between 0.4 and
0.9 c/deg, and high pass, with a peak above 1 c/deg. About 75% of the cells
showed response suppression organized with respect to the spatial frequency of
mask gratings. For example, Fig. 2A shows a low-pass cells with high-frequency
suppression and Fig. 2B shows a band-pass cell with mixed suppression, flanking
the tuning curve at both low and high frequencies. In each case response suppression
was graded with mask contrast and some suppression was found even at the optimal
spatial frequency. Some cells showed no suppression, indicating that the suppression
was not merely a stimulus artifact. In all but 2 of 42 cases, the suppression was
appropriate to the enhancement of the tuning function (e.g., low-pass cells had highfrequency response suppression), suggesting that the design of the system is more
77
78
Bonds
than coincidental. No similar spatial-frequency-dependent suppression was found
in LGN cells.
~.-----------------------,
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0.3
0.5
1
2
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0.2
0.3
0.5
1
2
Spatial Frequency (cyc/deg)
Figure 2: Examples of spatial frequency-dependent response suppression. Upper
broken lines show excitatory tuning functions and solid lines below zero indicate
response reduction at three different contrasts. A. Low-pass cell with high frequency
inhibition. B. Band-pass cell with mixed (low and high frequency) inhibition. Note
suppression at optimal spatial frequency in both cases.
4
NON-STATIONARITY OF CONTRAST TRANSFER
PROPERTIES
The two experiments described above demonstrate the existence of intrinsic cortical
mechanisms that refine the spatial filter properties of the cells. They also reveal a
global form of inhibition that is spatially non-specific. Since it is found even with
spatially optimal stimuli, it can influence the form of the cortical contrast-response
function (usually measured with optimal stimuli). This function is essentially logarithmic, with saturation or even super-saturation at higher contrasts (e.g., Albrecht
& Hamilton, 1982), as opposed to the more linear response behavior seen in cells
earlier in the visual pathway. Cortical cells also show some degree of contrast adaptation; when exposed to high mean contrasts for long periods of time, the response
vs contrast curves move rightward (e.g., Ohzawa, Sclar & Freeman, 1985). We
addressed the question of whether contrast-response nonlinearity and adaptation
might be causally related.
In order to compensate for "intrinsic response variability" in visual cortical cells,
experimental stimulation has historically involved presentation of randomized sequences of pattern parameters, the so-called multiple histogram technique (Henry,
Bishop, Tupper & Dreher, 1973). Scrambling presentation order distributes timedependent response variability across all stimulus conditions, but this procedure can
be self-defeating by masking any stimulus-dependent response variation. We therefore presented cortical cells with ordered sequences of contrasts, first ascending then
descending in a stepwise manner (Bonds, 1991). This revealed a clear and powerful
response hysteresis. Fig. 3A shows a solid line representing the contrast-response
Dual Inhibitory Mechanisms
function measured in the usual way, with randomized parameter presentation, overlaid on an envelope outlining responses to sequentially increasing or decreasing 3-sec
contrast epochs; one sequential presentation set required 54 secs. Across 36 cells
measured in this same way, the average response hysteresis corresponded to 0.36 log
units of contrast. Some hysteresis was found in every cortical cell and in no LGN
cells, so this phenomenon is intrinsically cortical.
-
50
100
0
CD 40
80
.!e
a.
E 30
._
8
Complex
14% peak contr.
60
CD
~20
a.
40
0
en
CD
10
20
a:
0
3
5
10
2030
50
100
0
3
5
10
Contrast (%)
Figure 3: Dynamic response hysteresis. A. A response function measured in the
usual way, with randomized stimulus sequences (filled circles) is overlaid on the
function resulting from stimulation with sequential ascending (upper level) and
descending (lower level) contrasts. Each contrast was presented for 3 seconds. B.
Hysteresis resulting from peak contrast of 14%; 3 secs per datum.
Hysteresis demonstrates a clear dependence of response amplitude on the history
of stimulation: at a given contrast, the amplitude is always less if a higher contrast
was shown first. This is one manifestation of cortical contrast adaptation, which
is well-known. However, adaptation is usually measured after long periods of stimulation with high contrasts, and may not be relevant to normal behavioral vision.
Fig. 3B shows hysteresis at a modest response level and low peak contrast (14%),
suggesting that it can serve a major function in day-to-day visual processing. The
speed of hysteresis also addresses this issue, but it is not so easily measured. Some
response histogram waveforms show consistent amplitude loss over a few seconds
of stimulation (see also Albrecht, Farrar & Hamilton, 1984), but other histograms
can be flat or even show a slight rise over time despite clear contrast adaptation
(Bonds, 1991) . This suggests the possibility that, in the classical pattern of any
well-designed automatic gain control, gain reduction takes place quite rapidly, but
its effects linger for some time.
The speed of reaction of gain change is illustrated in the experiment of Fig. 4.
A "pedestal" grating of 14% contrast is introduced. After 500 msec, a contrast
increment of 14% is added to the pedestal for a variable length of time. The response
during the first and last 500 msec of the pedestal presentation is counted and the
ratio is taken. In the absence of the increment, this ratio is about 0.8, reflecting the
adaptive nature of the pedestal itself. For an increment of even 50 msec duration,
this ratio is reduced, and it is reduced monotonically-by up to half the control
79
80
Bonds
level-for increments lasting less than a second. The gain control mechanism is thus
both sensitive and rapid.
1
0
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L
2.0
Iii
~
z0
0.2
Norm. Ampl.
CV9:L11.06-7
0
0
0.2
0.4
0.6
= spikes (t2)/spikes(t1)
0.8
Blip Duration (sec)
Figure 4: Speed of gain reduction. The ratio of spikes generated during the last and
first 500 msec of a 2 sec pedestal presentation can be modified by a brief contrast
increment (see text).
5
PHYSIOLOGICAL INDEPENDENCE OF TWO
INHIBITORY MECHANISMS
The experimental observations presented above support two basic phenomena:
spatially-dependent and spatially-independent inhibition. The question remains
whether these two types of inhibition are fundamentally different, or if they stem
from the same physiological mechanisms. This question can be addressed by examining the structure of a serial spike train generated by a cortical cell. In general,
rather than being distributed continuously, cortical spikes are grouped into discrete
packets, or bursts, with some intervening isolated spikes. The burst structure can
be fundamentally characterized by two parameters: the burst frequency (bursts per
second, or BPS) and the burst duration (spikes per burst, or SPB).
We have analyzed cortical spike trains for these properties by using an adaptive
algorithm to define burst groupings; as a rule of thumb, spike intervals of 8 msec
or less were considered to belong to bursts. Both burst frequency (BPS) and structure (SPB) depend strongly on mean firing rate, but once firing rate is corrected
for, two basic patterns emerge. Consider two experiments, both yielding firing rate
variation about a similar range. In one experiment, firing rate is varied by varying stimulus contrast, while in the other, firing rate is varied by varying stimulus
orientation. Burst frequency (BPS) depends only on spike rate, regardless of the
type of experiment. In Fig. 5A, no systematic difference is seen between the experiments in which contrast (filled circles) and orientation (open squares) are varied.
To quantify the difference between the curves, polynomials were fit to each and the
quantity gamma, defined by the (shaded) area bounded by the two polynomials,
was calculated; here, it equalled about 0.03.
Dual Inhibitory Mechanisms
16,----------------------------.
0'
14
A
Gamma: 0.0290
Q)
~
CI)
12
~
10
CI)
Q)
~
? ?
1;)
III
m
2.8
?? ?
? ?
? ?
?
~
6
4
~
:J
3.0
::J
Q)
1U
~
-...
e?.8
a:
3.2
B
Gamma: 0.2485
3.4
'0..
CJ)
?
::J
,.-... 3.6
Q)
2.6
CI)
2.4
a.
_
2
--tl-
Q)
Variation of stimulus contrast
D
~
.0..
Variation of stimulus orientation
2.2
_
Variation of stimulus contrast
-t:r-
Variation of stimulus orientation
CJ)
60
2.0
0
10
60
Response (imp/sec)
Figure 5: A. Comparison of burst frequency (bursts per second) as function offiring
rate resulting from presentations of varying contrast (filled circles) and varying
orientation (open squares). B. Comparison of burst length (spikes per burst) under
similar conditions. Note that at a given firing rate, burst length is always shorter
for experiment parametric on orientation. Shaded area (gamma) is quantitative
indicator of difference between two curves.
Fig. 5B shows that at similar firing rates, burst length (SPB) is markedly shorter
when firing rate is controlled by varying orientation (open squares) rather than
contrast (filled circles). In this pair of curves, the gamma (of about 0.25) is nearly
ten times that found in the upper curve. This is a clear violation ofunivariance, since
at a given spike rate (output level) the structure of the spike train differs depending
on the type of stimulation. Analysis of cortical response merely on the basis of
overall firing rate thus does not give the signalling mechanisms the respect they are
properly due. This result also implies that the strength of signalling between nerve
cells can dynamically vary independent of firing rate. Because of post-synaptic
temporal integration, bursts of spikes with short interspike intervals will be much
more effective in generating depolarization than spikes at longer intervals. Thus,
at a given average firing rate, a cell that generates longer bursts will have more
influence on a target cell than a cell that distributes its spikes in shorter bursts, all
other factors being equal.
J
This phenomenon was consistent across a population of 59 cells. Gamma, which
reflects the degree of difference between curves measured by variation of contrast
and by variation of orientation, averaged zero for curves based on number of bursts
(BPS). For both simple and complex cells, gamma for burst duration (SPB) averaged 0.15.
At face value, these results simply mean that when lower spike rates are achieved
by use of non-optimal orientations, they result from shorter bursts than when lower
spike rates result from reduction of contrast (with the spatial configuration remaining optimal). This means that non-optimal orientations and, from some preliminary
results, non-optimal spatial frequencies, result in inhibition that acts specifically to
shorten bursts, whereas contrast manipulations for the most part act to modulate
both the number and length of bursts.
81
82
Bonds
These results suggest strongly that there are at least two distinct forms of cortical inhibition, with unique physiological bases differentiated by the burst organization in
cortical spike trains. Recent results from our laboratory (Bonds, Unpub. Obs.) confirm that burst length modulation, which seems to reflect inhibition that depends on
the spatial characteristics of the stimulus, is strongly mediated by GABA. Microiontophoretic injection of GABA shortens burst length and injection of bicuculline, a
GABA blocker, lengthens bursts. This is wholly consistent with the hypothesis that
GABA is central to definition of spatial qualities of the cortical receptive field, and
suggests that one can indirectly observe GAB A-mediated inhibition by spike train
analysis.
Acknowledgements
This work was done in collaboration with Ed DeBruyn, Lisa Bauman and Brian
DeBusk. Supported by NIH (ROI-EY03778-09).
References
D. G. Albrecht & D. B. Hamilton. (1982) Striate cortex of monkey and cat: contrast
response functions. Journal of Neurophysiology 48, 217-237.
D. G. Albrecht, S. B. Farrar & D. B. Hamilton. (1984) Spatial contrast adaptation
characteristics ofneurones recorded in the cat's visual cortex. Journal of Physiology
347,713-739.
A. B. Bonds. (1989) The role of inhibition in the specification of orientation selectivity of cells of the cat striate cortex. Visual Neuroscience 2, 41-55.
A. B. Bonds. (1991) Temporal dynamics of contrast gain control in single cells of
the cat striate cortex. Visual Neuroscience 6, 239-255.
L. A. Bauman & A. B. Bonds. (1991) Inhibitory refinement of spatial frequency
selectivity in single cells of the cat striate cortex. Vision Research 31, 933-944.
G. Henry, P. O. Bishop, R. M. Tupper & B. Dreher. (1973) Orientation specificity
of cells in cat striate cortex. Vision Research 13, 1771-1779.
L. Maffei & A. Fiorentini. (1973) The visual cortex as a spatial frequency analyzer.
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1. Ohzawa, G. Sclar & R. D. Freeman. (1985) Contrast gain control in the cat's
visual system. Journal of Neurophysiology 54, 651-667.
D. Rose & C. B. Blakemore. (1974) An analysis of orientation selectivity in the
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R. E. Soodak, R. M. Shapley & E. Kaplan. (1987) Linear mechanism of orientation
tuning in the retina and lateral geniculate of the cat. Journal of Neurophysiology
58, 267-275.
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5,148 | 5,660 | Dependent Multinomial Models Made Easy:
Stick Breaking with the P?olya-Gamma Augmentation
Scott W. Linderman?
Harvard University
Cambridge, MA 02138
[email protected]
Matthew J. Johnson?
Harvard University
Cambridge, MA 02138
[email protected]
Ryan P. Adams
Twitter & Harvard University
Cambridge, MA 02138
[email protected]
Abstract
Many practical modeling problems involve discrete data that are best represented
as draws from multinomial or categorical distributions. For example, nucleotides
in a DNA sequence, children?s names in a given state and year, and text documents
are all commonly modeled with multinomial distributions. In all of these cases,
we expect some form of dependency between the draws: the nucleotide at one
position in the DNA strand may depend on the preceding nucleotides, children?s
names are highly correlated from year to year, and topics in text may be correlated and dynamic. These dependencies are not naturally captured by the typical
Dirichlet-multinomial formulation. Here, we leverage a logistic stick-breaking
representation and recent innovations in P?olya-gamma augmentation to reformulate the multinomial distribution in terms of latent variables with jointly Gaussian
likelihoods, enabling us to take advantage of a host of Bayesian inference techniques for Gaussian models with minimal overhead.
1
Introduction
It is often desirable to model discrete data in terms of continuous latent structure. In applications involving text corpora, discrete-valued time series, or polling and purchasing decisions, we may want
to learn correlations or spatiotemporal dynamics and leverage these structures to improve inferences
and predictions. However, adding these continuous latent dependence structures often comes at the
cost of significantly complicating inference: such models may require specialized, one-off inference
algorithms, such as a non-conjugate variational optimization, or they may only admit very general
inference tools like particle MCMC [1] or elliptical slice sampling [2], which can be inefficient and
difficult to scale. Developing, extending, and applying these models has remained a challenge.
In this paper we aim to provide a class of such models that are easy and efficient. We develop models
for categorical and multinomial data in which dependencies among the multinomial parameters are
modeled via latent Gaussian distributions or Gaussian processes, and we show that this flexible class
of models admits a simple auxiliary variable method that makes inference easy, fast, and modular.
This construction not only makes these models simple to develop and apply, but also allows the
resulting inference methods to use off-the-shelf algorithms and software for Gaussian processes and
linear Gaussian dynamical systems.
The paper is organized as follows. After providing background material and defining our general
models and inference methods, we demonstrate the utility of this class of models by applying it to
three domains as case studies. First, we develop a correlated topic model for text corpora. Second,
we study an application to modeling the spatial and temporal patterns in birth names given only
sparse data. Finally, we provide a new continuous state-space model for discrete-valued sequences,
?
These authors contributed equally.
1
including text and human DNA. In each case, given our model construction and auxiliary variable
method, inference algorithms are easy to develop and very effective in experiments.
Code to use these models, write new models that leverage these inference methods, and reproduce
the figures in this paper is available at github.com/HIPS/pgmult.
2
Modeling correlations in multinomial parameters
In this section, we discuss an auxiliary variable scheme that allows multinomial observations to
appear as Gaussian likelihoods within a larger probabilistic model. The key trick discussed in the
proceeding sections is to introduce P?olya-gamma random variables into the joint distribution over
data and parameters in such a way that the resulting marginal leaves the original model intact.
The integral identity underlying the P?olya-gamma augmentation scheme [3] is
Z ?
2
(e? )a
?b ??
=
2
e??? /2 p(? | b, 0) d?,
e
?
b
(1 + e )
0
(1)
where ? = a ? b/2 and p(? | b, 0) is the density of the P?olya-gamma distribution PG(b, 0), which
does not depend on ?. Consider a likelihood function of the form
p(x | ?) = c(x)
(e? )a(x)
(1 + e? )b(x)
(2)
for some functions a, b, and c. Such likelihoods arise, e.g., in logistic regression and in binomial and
negative binomial regression [3]. Using (1) along with a prior p(?), we can write the joint density
of (?, x) as
Z ?
2
(e? )a(x)
=
p(?) c(x) 2?b(x) e?(x)? e??? /2 p(? | b(x), 0) d?. (3)
p(?, x) = p(?) c(x)
(1 + e? )b(x)
0
The integrand of (3) defines a joint density on (?, x, ?) which admits p(?, x) as a marginal density.
Conditioned on these auxiliary variables ?, we have
p(? | x, ?) ? p(?)e?(x)? e???
2
/2
(4)
which is Gaussian when p(?) is Gaussian. Furthermore, by the exponential tilting property of the
P?olya-gamma distribution, we have ? | ?, x ? PG(b(x), ?). Thus the identity (1) gives rise to a
conditionally conjugate augmentation scheme for Gaussian priors and likelihoods of the form (2).
This augmentation scheme has been used to develop Gibbs sampling and variational inference algorithms for Bernoulli, binomial [3], and negative binomial [4] regression models with logit link
functions, and to the multinomial distribution with a multi-class logistic link function [3, 5].
The multi-class logistic ?softmax? function, ? LN (?), maps a real-valued vector ? ? RK to a probaPK
bility vector ? ? [0, 1]K by setting ?k = e?k / j=1 e?j . It is commonly used in multi-class regression [6] and correlated topic modeling [7]. Correlated multinomial parameters can be modeled with
a Gaussian prior on the vector ?, though the resulting models are not conjugate. The P?olya-gamma
augmentation can be applied to such models [3, 5], but it only provides single-site Gibbs updating
of ?. This paper develops a joint augmentation in the sense that, given the auxiliary variables, the
entire vector ? is resampled as a block in a single Gibbs update.
2.1
A new P?olya-gamma augmentation for the multinomial distribution
First, rewrite the K-dimensional multinomial recursively in terms of K ? 1 binomial densities:
Mult(x | N, ?) =
K?1
Y
Bin(xk | Nk , ?
ek ),
(5)
k=1
Nk = N ?
X
j<k
xj ,
?
ek =
1?
2
?
Pk
j<k
?j
,
k = 2, 3, . . . , K,
(6)
Figure 1: Correlated 2D Gaussian priors on ? and their implied densities on ? SB (?). See text for details.
P
where N1 = N = k xk and ?
e1 = ?1 . For convenience, we define N (x) ? [N1 , . . . , NK?1 ].
This decomposition of the multinomial density is a ?stick-breaking? representation where
each ?
ek represents the fraction of the remaining probability mass assigned to the k-th component. We let ?
ek = ?(?k ), where ?(?) denotes the logistic function, and define the function, ? SB : RK?1 ? [0, 1]K , which maps a vector ? to a normalized probability vector ?.
Next, we rewrite the density into the form required by (1) by substituting ?(?k ) for ?
ek :
Mult(x | N, ?) =
K?1
Y
k=1
K?1
Y
Nk
Bin(xk | Nk , ?(?k )) =
?(?k )xk (1 ? ?(?k ))Nk ?xk
xk
k=1
K?1
Y Nk (e?k )xk
=
.
xk (1 + e?k )Nk
(7)
(8)
k=1
Choosing ak (x) = xk and bk (x) = Nk for each k = 1, 2, . . . , K ? 1, we can then introduce P?olyagamma auxiliary variables ?k corresponding to each coordinate ?k ; dropping terms that do not
depend on ? and completing the square yields
K?1
Y
2
p(x, ? | ?) ?
e(xk ?Nk /2)?k ??k ?k /2 ? N ??1 ?(x) ?, ??1 ,
(9)
k=1
where ? ? diag(?) and ?(x) ? x ? N (x)/2. That is, conditioned on ?, the likelihood of ? under
the augmented multinomial model is proportional to a diagonal Gaussian distribution.
Figure 1 shows how several Gaussian densities map to probability densities on the simplex. Correlated Gaussians (left) put most probability mass near the ?1 = ?2 axis of the simplex, and anticorrelated Gaussians (center) put mass along the sides where ?1 is large when ?2 is small and
vice-versa. Finally, a nearly isotropic Gaussian approximates a symmetric Dirichlet. Appendix A
gives a closed-form expression for the density on ? induced by a Gaussian distribution on ?, and
also an expression for a diagonal Gaussian that approximates a Dirichlet by matching moments.
3
Correlated topic models
The Latent Dirichlet Allocation (LDA) [8] is a popular model for learning topics from text corpora.
The Correlated Topic Model (CTM) [7] extends LDA by including a Gaussian correlation structure
among topics. This correlation model is powerful not only because it reveals correlations among
3
Figure 2: A comparison of correlated topic model performance. The left panel shows a subset of the inferred
topic correlations for the AP News corpus. Two examples are highlighted: a) positive correlation between topics
(house, committee, congress, law) and (Bush, Dukakis, president, campaign), and b) anticorrelation between
(percent, year, billion, rate) and (court, case, attorney, judge). The middle and right panels demonstrate the
efficacy of our SB-CTM relative to competing models on the AP News corpus and the 20 Newsgroup corpus,
respectively.
topics but also because inferring such correlations can significantly improve predictions, especially
when inferring the remaining words in a document after only a few have been revealed [7]. However, the addition of this Gaussian correlation structure breaks the Dirichlet-Multinomial conjugacy
of LDA, making estimation and particularly Bayesian inference and model-averaged predictions
more challenging. An approximate maximum likelihood approach using variational EM [7] is often
effective, but a fully Bayesian approach which integrates out parameters may be preferable, especially when making predictions based on a small number of revealed words in a document. A recent
Bayesian approach based on a P?olya-Gamma augmentation to the logistic normal CTM (LN-CTM)
[5] provides a Gibbs sampling algorithm with conjugate updates, but the Gibbs updates are limited
to single-site resampling of one scalar at a time, which can lead to slow mixing in correlated models.
In this section we show that MCMC sampling in a correlated topic model based on the stick breaking
construction (SB-CTM) can be significantly more efficient than sampling in the LN-CTM while
maintaining the same integration advantage over EM.
In the standard LDA model, each topic ? t (t = 1, 2, . . . , T ) is a distribution over a vocabulary
of V possible words, and each document d has a distribution over topics ? d (d = 1, 2, . . . , D).
The n-th word in document d is denoted wn,d for d = 1, 2, . . . , Nd . When each ? t and ? d is given
a symmetric Dirichlet prior with parameters ?? and ?? , respectively, the generative model is
? t ? Dir(?? ),
? d ? Dir(?? ),
zn,d | ? d ? Cat(? d ),
wn,d | zn,d , {? t } ? Cat(? zn,d ). (10)
The CTM replaces the Dirichlet prior on each ? d with a correlated prior induced by first
sampling a correlated Gaussian vector ? d ? N (?, ?) and then applying the logistic normal
map: ? d = ? LN (? d ) Analogously, our SB-CTM generates the correlation structure by instead applying the stick-breaking logistic map, ? d = ? SB (? d ). The goal is then to infer the posterior distribution over the topics ? t , the documents? topic allocations ? d , and their mean and correlation
structure (?, ?), where the parameters (?, ?) are given a conjugate normal-inverse Wishart (NIW)
prior. Modeling correlation structure within the topics ? can be done analogously.
For fully Bayesian inference in the SB-CTM, we develop a Gibbs sampler that exploits the block
conditional Gaussian structure provided by the stick-breaking construction. The Gibbs sampler
iteratively samples z | w, ?, ?; ? | z, w; ? | z, ?, ?, ?; and ?, ? | ? as well as the auxiliary
variables ? | ?, z. The first two are standard updates for LDA models, so we focus on the latter
three. Using the identities derived in Section 2.1, the conditional density of each ? d | z d , ?, ?, ?
can be written
e
e , ?),
p(? d | z d , ? d ) ? N (??1 ?(cd ) | ? d , ??1 ) N (? d | ?, ?) ? N (? d | ?
(11)
d
d
4
where we have defined
e ?(cd ) + ??1 ? ,
e=?
?
e = ?d + ??1 ?1 ,
?
cd,t =
X
I[zn,d = t],
?d = diag(? d ),
n
and so it is resampled as a joint Gaussian. The correlation structure parameters ? and ? are sampled
from their conditional NIW distribution. Finally, the auxiliary variables ? are sampled as P?olyaGamma random variables, with ? d | z d , ? d ? PG(N (cd ), ? d ). A feature of the stick-breaking
construction is that the the auxiliary variable update is embarrassingly parallel.
We compare the performance of this Gibbs sampling algorithm for the SB-CTM to the Gibbs sampling algorithm of the LN-CTM [5], which uses a different P?olya-gamma augmentation, as well as
the original variational EM algorithm for the CTM and collapsed Gibbs sampling in standard LDA.
Figure 2 shows results on both the AP News dataset and the 20 Newsgroups dataset, where models
were trained on a random subset of 95% of the complete documents and tested on the remaining 5%
by estimating held-out likelihoods of half the words given the other half. The collapsed Gibbs sampler for LDA is fast but because it does not model correlations its ability to predict is significantly
constrained. The variational EM algorithm for the CTM is reasonably fast but its point estimate
doesn?t quite match the performance from integrating out parameters via MCMC in this setting.
The LN-CTM Gibbs sampler continues to improve slowly but is limited by its single-site updates,
while the SB-CTM sampler seems to both mix effectively and execute efficiently due to its block
Gaussian updating.
The SB-CTM demonstrates that the stick-breaking construction and corresponding P?olya-Gamma
augmentation makes inference in correlated topic models both easy to implement and computationally efficient. The block conditional Gaussianity also makes inference algorithms modular and
compositional: the construction immediately extends to dynamic topic models (DTMs) [9], in which
the latent ? d evolve according to linear Gaussian dynamics, and inference can be implemented simply by applying off-the-shelf code for Gaussian linear dynamical systems (see Section 5). Finally,
because LDA is so commonly used as a component of other models (e.g. for images [10]), easy,
effective, modular inference for CTMs and DTMs is a promising general tool.
4
Gaussian processes with multinomial observations
Consider the United States census data, which lists the first names of children born in each state for
the years 1910-2013. Suppose we wish to predict the probability of a particular name in New York
State in the years 2012 and 2013 given observed names in earlier years. We might reasonably expect
that name probabilities vary smoothly over time as names rise and fall in popularity, and that name
probability would be similar in neighboring states. A Gaussian process naturally captures these
prior intuitions about spatiotemporal correlations, but the observed name counts are most naturally
modeled as multinomial draws from latent probability distributions over names for each combination
of state and year. We show how efficient inference can be performed in this otherwise difficult model
by leveraging the P?olya-gamma augmentation.
Let Z ? RM ?D denote the matrix of D dimensional inputs and X ? NM ?K denote the observed K
dimensional count vectors for each input. In our example, each row z m of Z corresponds to the
year, latitude, and longitude of an observation, and K is the number of names. Underlying these
observations we introduce a set of latent variables, ?m,k such that the probability vector at input z m
is ? m = ? SB (? m,: ). The auxiliary variables for the k-th name, ? :,k , are linked via a Gaussian
process with covariance matrix, C, whose entry Ci,j is the covariance between input z i and z j
under the GP prior, and mean vector ?k . The covariance matrix is shared by all names, and the
mean is empirically set to match the measured name probability. The full model is then,
? :,k ? GP(?k , C),
xm ? Mult(Nm , ? SB (? m,: )).
To perform inference, introduce auxiliary P?olya-gamma variables, ?m,k for each ?m,k . Conditioned
on these variables, the conditional distribution of ? :,k is,
?1
?1
ek
ek, ?
p(? :,k | Z, X, ?, ?, C) ? N ?k ?(X :,k ) ? :,k , ?k
N (? :,k | ?k , C) ? N ? :,k | ?
e k = C ?1 + ?k
?
?1
e k ?(X :,k ) + C ?1 ?k ,
ek = ?
?
5
Model
Static 2011
Raw GP
LNM GP
SBM GP
2012
Top 10
Bot. 10
4.2 (1.3) 0.7 (1.2)
4.9 (1.1) 0.7 (0.9)
6.7 (1.4) 4.8 (1.7)
7.3 (1.0) 4.0 (1.8)
2013
Top 10
Bot. 10
4.2 (1.4) 0.8 (1.0)
5.0 (1.0) 0.8 (0.9)
6.8 (1.4) 4.6 (1.7)
7.0 (1.0) 3.9 (1.4)
Average number of names correctly predicted
Figure 3: A spatiotemporal Gaussian process applied to the names of children born in the United States from
1960-2013. With a limited dataset of only 50 observations per state/year, the stick breaking and logistic normal
multinomial GPs (SBM GP and LNM GP) outperform na??ve approaches in predicting the top and bottom 10
names (bottom left, parentheses: std. error). Our SBM GP, which leverages the P?olya-gamma augmentation, is
considerably more efficient than the non-conjugate LNM GP (bottom right).
where ?k = diag(? :,k ). The auxiliary variables are updated according
P to their conditional distribution: ?m,k | xm , ?m,k ? PG(Nm,k , ?m,k ), where Nm,k = Nm ? j<k xm,j .
Figure 3 illustrates the power of this approach on U.S. census data. The top two plots show the
inferred probabilities under our stick-breaking multinomial GP model for the full dataset. Interesting spatiotemporal correlations in name probability are uncovered. In this large-count regime, the
posterior uncertainty is negligible since we observe thousands of names per state and year, and simply modeling the transformed empirical probabilities with a GP works well. However, in the sparse
data regime with only Nm = 50 observations per input, it greatly improves performance to model
uncertainty in the latent probabilities using a Gaussian process with multinomial observations.
The bottom panels compare four methods of predicting future names in the years 2012 and 2013
for a down-sampled dataset with Nm = 50: predicting based on the empirical probability measured
in 2011; a standard GP to the empirical probabilities transformed by ? ?1
SB (Raw GP); a GP whose
outputs are transformed by the logistic normal function, ? LN , to obtain multinomial probabilities
(LNM GP) fit using elliptical slice sampling [2]; and our stick-breaking multinomial GP (SBM
GP). In terms of ability to predict the top and bottom 10 names, the multinomial models are both
comparable and vastly superior to the naive approaches.
The SBM GP model is considerably faster than the logistic normal version, as shown in the bottom
right panel. The augmented Gibbs sampler is more efficient than the elliptical slice sampling algorithm used to handle the non-conjugacy in the LNM GP. Moreover, we are able to make collapsed
predictions in which we compute the predictive distribution test ??s given ?, integrating out the
training ?. In contrast, the LNM GP must condition on the training GP values in order to make predictions, and effectively integrate over training samples using MCMC. Appendix B goes into greater
detail on how marginal predictions are computed and why they are more efficient than predicting
conditioned on a single value of ?.
6
Figure 4: Predictive log likelihood comparison of time series models with multinomial observations.
5
Multinomial linear dynamical systems
While discrete-state hidden Markov models (HMMs) are ubiquitous for modeling time series and
sequence data, it can be preferable to use a continuous state space model. In particular, while
discrete states have no intrinsic geometry, continuous states can correspond to natural Euclidean
embeddings [11]. These considerations are particularly relevant to text, where word embeddings
[12] have proven to be a powerful tool.
Gaussian linear dynamical systems (LDS) provide very efficient learning and inference algorithms,
but they can typically only be applied when the observations are themselves linear with Gaussian
noise. While it is possible to apply a Gaussian LDS to count vectors [11], the resulting model is
misspecified in the sense that, as a continuous density, the model assigns zero probability to training
and test data. However, Belanger and Kakade [11] show that this model can still be used for several
machine learning tasks with compelling performance, and that the efficient algorithms afforded by
the misspecified Gaussian assumptions confer a significant computational advantage. Indeed, the authors have observed that such a Gaussian model is ?worth exploring, since multinomial models with
softmax link functions prevent closed-form M step updates and require expensive? computations
[13]; this paper aims to bridge precisely this gap and enable efficient Gaussian LDS computational
methods to be applied while maintaining multinomial emissions and an asymptotically unbiased
representation of the posterior. While there are other approximation schemes that effectively extend
some of the benefits of LDSs to nonlinear, non-Gaussian settings, such as the extended Kalman filter
(EKF) and unscented Kalman filter (UKF) [14, 15], these methods do not allow for asymptotically
unbiased Bayesian inference, can have complex behavior, and can make model learning a challenge.
Alternatively, particle MCMC (pMCMC) [1] is a very powerful algorithm that provides unbiased
Bayesian inference for very general state space models, but it does not enjoy the efficient block
updates or conjugacy of LDSs or HMMs.
The stick-breaking multinomial linear dynamical system (SBM-LDS) generates states via a linear
Gaussian dynamical system but generates multinomial observations via the stick-breaking map:
z 0 |?0 , ?0 ? N (?0 , ?0 ),
z t |z t?1 , A, B ? N (Az t?1 , B),
D
xt |z t , C ? Mult(Nt , ? SB (Cz t )),
K
where z t ? R is the system state at time t and xt ? N are the multinomial observations.
We suppress notation for conditioning on A, B, C, ?0 , and ?0 , which are system parameters of
appropriate sizes that are given conjugate priors. The logistic normal multinomial LDS (LNM-LDS)
is defined analogously but uses ? LN in place of ? SB .
To produce a Gibbs sampler with fully conjugate updates, we augment the observations with
P?olya-gamma random variables ?t,k . As a result, the conditional state sequence z 1:T |? 1:T , x1:T
is jointly distributed according to a Gaussian LDS in which the diagonal observation potential at
?1
time t is N (??1
t ?(xt )|Cz t , ?t ). Thus the state sequence can be jointly sampled using off7
the-shelf LDS software, and the system parameters can similarly be updated using standard algorithms. The only remaining update is to the auxiliary variables, which are sampled according to
? t |z t , C, x ? PG(N (xt ), Cz t ).
We compare the SBM-LDS and the Gibbs sampling inference algorithm to three baseline methods:
an LNM-LDS using pMCMC and ancestor resampling [16] for inference, an HMM using Gibbs
sampling, and a ?raw? LDS which treats the multinomial observation vectors as observations in
RK as in [11]. We examine each method?s performance on each of three experiments: in modeling
a sequence of 682 amino acids from human DNA with 22 dimensional observations, a set of 20
random AP news articles with an average of 77 words per article and a vocabulary size of 200
words, and an excerpt of 4000 words from Lewis Carroll?s Alice?s Adventures in Wonderland with
a vocabulary of 1000 words. We reserved the final 10 amino acids, 10 words per news article, and
100 words from Alice for computing predictive likelihoods. Each linear dynamical model had a 10dimensional state space, while the HMM had 10 discrete states (HMMs with 20, 30, and 40 states
all performed worse on these tasks).
Figure 4 (left panels) shows the predictive log likelihood for each method on each experiment, normalized by the number of counts in the test dataset and relative to the likelihood under a multinomial
model fit to the training data mean. For the DNA data, which has the smallest ?vocabulary? size, the
HMM achieves the highest predictive likelihood, but the SBM-LDS edges out the other LDS methods. On the two text datasets, the SBM-LDS outperforms the other methods, particularly in Alice
where the vocabulary is larger and the document is longer. In terms of run time, the SBM-LDS is
orders of magnitude faster than the LNM-LDS with pMCMC (right panel) because it mixes much
more efficiently over the latent trajectories.
6
Related Work
The stick-breaking transformation used herein was applied to categorical models by Khan et al. [17],
but they used local variational bound instead of the P?olya-gamma augmentation. Their promising
results corroborate our findings of improved performance using this transformation. Their generalized expectation-maximization algorithm is not fully Bayesian, and does not integrate into existing
Gaussian modeling and inference code as easily as our augmentation.
Conversely, Chen et al. [5] used the P?olya-gamma augmentation in conjunction with the logistic
normal transformation for correlated topic modeling, exploiting the conditional conjugacy of a single
entry ?k | ?k , ? ?k with a Gaussian prior. Unlike our stick-breaking transformation, which admits
block Gibbs sampling over the entire vector ? simultaneously, their approach is limited to singlesite Gibbs sampling. As shown in our correlated topic model experiments, this has dramatic effects
on inferential performance. Moreover, it precludes analytical marginalization and integration with
existing Gaussian modeling algorithms. For example, it is not immediately applicable to inference
in linear dynamical systems with multinomial observations.
7
Conclusion
These case studies demonstrate that the stick-breaking multinomial model construction paired with
the P?olya-gamma augmentation yields a flexible class of models with easy, efficient, and compositional inference. In addition to making these models easy, the methods developed here can also
enable new models for multinomial and mixed data: the latent continuous structures used here to
model correlations and state-space structure can be leveraged to explore new models for interpretable
feature embeddings, interacting time series, and dependence with other covariates.
8
Acknowledgements
S.W.L. is supported by a Siebel Scholarship and the Center for Brains, Minds and Machines
(CBMM), funded by NSF STC award CCF-1231216. M.J.J. is supported by the Harvard/MIT Joint
Research Grants Program. R.P.A. is supported by NSF IIS-1421780 as well as the Alfred P. Sloan
Foundation.
8
References
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Carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72
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[2] Iain Murray, Ryan P. Adams, and David J.C. MacKay. Elliptical slice sampling. Journal of
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[3] Nicholas G Polson, James G Scott, and Jesse Windle. Bayesian inference for logistic models
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[13] David Belanger and Sham Kakade. Embedding word tokens using a linear dynamical system.
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[14] Eric A Wan and Rudolph Van Der Merwe. The unscented Kalman filter for nonlinear estimation. In Adaptive Systems for Signal Processing, Communications, and Control Symposium
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[15] Sebastian Thrun, Wolfram Burgard, and Dieter Fox. Probabilistic robotics. MIT press, 2005.
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9
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5,149 | 5,661 | Scalable Adaptation of State Complexity for
Nonparametric Hidden Markov Models
Michael C. Hughes, William Stephenson, and Erik B. Sudderth
Department of Computer Science, Brown University, Providence, RI 02912
[email protected], [email protected], [email protected]
Abstract
Bayesian nonparametric hidden Markov models are typically learned via fixed
truncations of the infinite state space or local Monte Carlo proposals that make
small changes to the state space. We develop an inference algorithm for the sticky
hierarchical Dirichlet process hidden Markov model that scales to big datasets by
processing a few sequences at a time yet allows rapid adaptation of the state space
cardinality. Unlike previous point-estimate methods, our novel variational bound
penalizes redundant or irrelevant states and thus enables optimization of the state
space. Our birth proposals use observed data statistics to create useful new states
that escape local optima. Merge and delete proposals remove ineffective states to
yield simpler models with more affordable future computations. Experiments on
speaker diarization, motion capture, and epigenetic chromatin datasets discover
models that are more compact, more interpretable, and better aligned to ground
truth segmentations than competitors. We have released an open-source Python
implementation which can parallelize local inference steps across sequences.
1 Introduction
The hidden Markov model (HMM) [1, 2] is widely used to segment sequential data into interpretable
discrete states. Human activity streams might use walking or dancing states, while DNA transcription might be understood via promotor or repressor states [3]. The hierarchical Dirichlet process
HMM (HDP-HMM) [4, 5, 6] provides an elegant Bayesian nonparametric framework for reasoning
about possible data segmentations with different numbers of states.
Existing inference algorithms for HMMs and HDP-HMMs have numerous shortcomings: they cannot efficiently learn from large datasets, do not effectively explore segmentations with varying numbers of states, and are often trapped at local optima near their initialization. Stochastic optimization
methods [7, 8] are particularly vulnerable to these last two issues, since they cannot change the number of states instantiated during execution. The importance of removing irrelevant states has been
long recognized [9]. Samplers that add or remove states via split and merge moves have been developed for HDP topic models [10, 11] and beta process HMMs [12]. However, these Monte Carlo
proposals use the entire dataset and require all sequences to fit in memory, limiting scalability.
We propose an HDP-HMM learning algorithm that reliably transforms an uninformative, single-state
initialization into an accurate yet compact set of states. Generalizing previous work on memoized
variational inference for DP mixture models [13] and HDP topic models [14], we derive a variational
bound for the HDP-HMM that accounts for sticky state persistence and can be used for effective
Bayesian model selection. Our algorithm uses birth proposal moves to create new states and merge
and delete moves to remove states with poor predictive power. State space adaptations are validated
via a global variational bound, but by caching sufficient statistics our memoized algorithm efficiently
processes subsets of sequences at each step. Extensive experiments demonstrate the reliability and
scalability of our approach, which can be reproduced via Python code we have released online1 .
1
http://bitbucket.org/michaelchughes/x-hdphmm-nips2015/
1
(A) Initialization, K=1
!
"!!
#!!
$!!
(B) After first lap births, K=47
%!!
!
"!!
#!!
$!!
%!!
(C) After first lap merges, K=37
!
"!!
#!!
$!!
%!!
accepted merge pairs
(F) Ground truth labels, K=12
(E) After 100 laps, K=31
(D) After second lap, K=56
accepted birth
!
"!!
#!!
$!!
%!!
!
"!!
#!!
$!!
%!!
!
"!!
#!!
$!!
%!!
Figure 1: Illustration of our new birth/merge/delete variational algorithm as it learns to segment motion capture
sequences into common exercise types (Sec. 5). Each panel shows segmentations of the same 6 sequences,
with time on the horizontal axis. Starting from just one state (A), birth moves at the first sequence create useful
states. Local updates to each sequence in turn can use existing states or birth new ones (B). After all sequences
are updated once, we perform merge moves to clean up and lap is complete (C). After another complete lap of
birth updates at each sequence followed by merges and deletes, the segmentation is further refined (D). After
many laps, our final segmentation (E) aligns well to labels from a human annotator (F), with some true states
aligning to multiple learned states that capture subject-specific variability in exercises.
2 Hierarchical Dirichlet Process Hidden Markov Models
We wish to jointly model N sequences, where sequence n has data xn = [xn1 , xn2 , . . . , xnTn ] and
observation xnt is a vector representing interval or timestep t. For example, xnt ? RD could be the
spectrogram for an instant of audio, or human limb positions during a 100ms interval.
The HDP-HMM explains this data by assigning each observation xnt to a single hidden state znt .
The chosen state comes from a countably infinite set of options k ? {1, 2, . . .}, generated via
Markovian dynamics with initial state distributions ?0 and transition distributions {?k }?
k=1 :
p(zn1 = k) = ?0k ,
p(znt = ? | zn,t?1 = k) = ?k? .
(1)
We draw data xnt given assigned state znt = k from an exponential family likelihood F :
F : log p(xnt | ?k ) = sF (xdn )T ?k + cF (?k ),
H : log p(?k | ??) = ?Tk ?? + cH (?
? ).
(2)
The natural parameter ?k for each state has conjugate prior H. Cumulant functions cF , cH ensure
these distributions are normalized. The chosen exponential family is defined by its sufficient statistics sF . Our experiments consider Bernoulli, Gaussian, and auto-regressive Gaussian likelihoods.
Hierarchies of Dirichlet processes. Under the HDP-HMM prior and posterior, the number of
states is unbounded; it is possible that every observation comes from a unique state. The hierarchical
Dirichlet process (HDP) [5] encourages sharing states over time via a latent root probability vector
? over the infinite set of states (see Fig. 2). The stick-breaking representation of the prior on ? first
Qk?1
(1 ? u?).
draws independent variables uk ? Beta(1, ?) for each state k, and then sets ?k = uk ?=1
We interpret uk as the conditional probability of choosing state k among states {k, k + 1, k + 2, . . .}.
In expectation, the K most common states are first in stick-breaking
order. We represent their
P
probabilities via the vector [?1 ?2 . . . ?K ?>K ], where ?>K = ?
k=K+1 ?k . Given this (K + 1)dimensional probability vector ?, the HDP-HMM generates transition distributions ?k for each state
k from a Dirichlet with mean equal to ? and variance governed by concentration parameter ? > 0:
[?k1 . . . ?kK ?k>K ] ? Dir(??1 , ??2 , . . . , ??>K ).
(3)
We draw starting probability vector ?0 from a similar prior with much smaller variance, ?0 ?
Dir(?0 ?) with ?0 ? ?, because few starting states are observed.
Sticky self-transition bias. In many applications, we expect each segment to persist for many
timesteps. The ?sticky? parameterization of [4, 6] favors self-transition by placing extra prior mass
on the transition probability ?kk . In particular, [?k1 . . . ?k>K ] ? Dir(??1 , . . . ??k + ?, . . . ??>K )
where ? > 0 controls the degree of self-transition bias. Choosing ? ? 100 leads to long segment
lengths, while avoiding the computational cost of semi-Markov alternatives [7].
2
uk
??k
??k
?
?k
?50
?20
?k
?40
?
?
?100
?k
??k
zn1
s?n1
s?
s?
?60
T -1
zn2 n2? ? ? n znT
?
xn1
xn2 ? ? ?
?80
?150
xnT
cD sticky
lower bound
?200
0.0
0.5
1.0
1.5
alpha
N
2.0
cD sticky
lower bound
?100
?120
0
5
10
15
20
25
30
num states K
Figure 2: Left: Graphical representation of the HDP hidden Markov model. Variational parameters are shown
in red. Center: Our surrogate bound for the sticky Dirichlet cumulant function cD (Eq. 9) as a function of ?,
computed with ? = 100 and uniform ? with K = 20 active states. Right: Surrogate bound vs. K, with fixed
? = 100, ? = 0.5. This bound remains tight when our state adaptation moves insert or remove states.
3 Memoized and Stochastic Variational Inference
After observing data x, our inferential goal is posterior knowledge of top-level conditional probabilities u, HMM parameters ?, ?, and assignments z. We refer to u, ?, ? as global parameters because
they generalize to new data sequences. In contrast, the states zn are local to a specific sequence xn .
3.1 A Factorized Variational Lower Bound
We seek a distribution q over the unobserved variables that is close to the true posterior, but lies in the
simpler factorized family q(?) , q(u)q(?)q(?)q(z). Each factor has exponential family form with
free parameters denoted by hats, and our inference algorithms update these parameters to minimize
the Kullback-Leibler (KL) divergence KL(q || p). Our chosen factorization for q is similar to [7],
but includes a substantially more accurate approximation to q(u) as detailed in Sec. 3.2.
Factor q(z). For each sequence n, we use an independent factor q(zn ) with Markovian structure:
"K
# ?1 K K
Y ? (z ) TY
Y Y s?ntk? ?k (znt )?? (zn,t+1 )
n1
k
q(zn ) ,
r?n1k
(4)
r?ntk
t=1
k=1
k=1 ?=1
Free parameter vector s?nt defines the joint assignment probabilities s?ntk? , q(zn,t+1 = ?, znt = k),
so the K 2 non-negative entries of s?nt sum to one. The parameter r?nt defines the marginal probability
PK
r?ntk = q(znt = k), and equals r?ntk = ?=1 s?ntk? . We can find the expected count of transitions
PN PTn ?1
from state k to ? across all sequences via the sufficient statistic Mk? (?
s) , n=1 t=1
s?ntk? .
The truncation level K limits the total number of states to which data is assigned. Under our approximate posterior, only q(zn ) is constrained by this choice; no global factors are truncated. Indeed, if
data is only assigned to the first K states, the conditional independence properties of the HDP-HMM
imply that {?k , uk | k > K} are independent of the data. Their optimal variational posteriors thus
match the prior, and need not be explicitly computed or stored [15, 16]. Simple variational algorithms treat K as a fixed constant [7], but Sec. 4 develops novel algorithms that fit K to data.
Factor q(?). For the starting state (k = 0) and each state k ? 1, 2, . . ., we define q(?k ) as a
Dirichlet distribution: q(?k ) , Dir(??k1 , . . . , ??kK , ??k>K ). Free parameter ??k is a vector of K + 1
positive numbers, with one entry for each of the K active states and a final entry for the aggregate
? ,
mass of all other states. The expected log transition probability between states k and ?, Pk? (?)
P
K+1
Eq [log ?k? ] = ?(??k? ) ? ?( m=1 ??km ), is a key sufficient statistic.
Factor q(?). Emission parameter ?k for state k has factor q(?k ) , H(?
?k ) conjugate to the likelihood F . The supplement provides details for Bernoulli, Gaussian, and auto-regressive F .
We score the approximation q via an objective function L that assigns a scalar value (higher is better)
to each possible input of free parameters, data x, and hyperparameters ?, ?, ?, ??:
L(?) , Eq [log p(x, z, ?, u, ?) ? log q(z, ?, u, ?)] = Ldata + Lentropy + Lhdp-local + Lhdp-global . (5)
3
This function provides a lower bound on the marginal evidence: log p(x|?, ?, ?, ??) ? L. Improving
this bound is equivalent to minimizing KL(q || p). Its four component terms are defined as follows:
i
h
Ldata (x, r?, ??) , Eq log p(x | z, ?) + log p(?)
Lentropy (?
s) , ?Eq [log q(z)] ,
q(?) ,
i
h
h
i
(6)
.
Lhdp-global (?
?, ?
? ) , Eq log p(u)
? ??, ?
Lhdp-local (?
s, ?,
? ) , Eq log p(z | ?) + log p(?)
,
q(u)
q(?)
Detailed analytic expansions for each term are available in the supplement.
3.2 Tractable Posterior Inference for Global State Probabilities
Previous variational methods for the HDP-HMM [7], and for HDP topic models [16] and HDP
grammars [17], used a zero-variance point estimate for the top-level state probabilities ?. While
this approximation simplifies inference, the variational objective no longer bounds the marginal evidence. Such pseudo-bounds are unsuitable for model selection and can favor models with redundant
states that do not explain any data, but nevertheless increase computational and storage costs [14].
Because we seek to learn compact and interpretable models, and automatically adapt the truncation
level K to each dataset, we instead place a proper beta distribution on uk , k ? 1, 2, . . . K:
q(uk ) , Beta(?
?k ?
? k , (1??
?k )?
?k ), where ??k ? (0, 1), ?
? k > 0.
(7)
Qk
Here ??k = Eq(u) [uk ], Eq(u) [?k ] = ??k E[?>k?1 ], and Eq(u) [?>k ] = ?=1 (1??
?? ). The scalar ?
?k
controls the variance, where the zero-variance point estimate is recovered as ?
? k ? ?.
The beta factorization in Eq. (7) complicates evaluation of the marginal likelihood bound in Eq. (6):
PK
? ??, ?
Lhdp-local (?
s, ?,
? ) = Eq(u) [cD (?0 ?)] + k=1 Eq(u) [cD (?? + ??k )]
PK
PK PK+1
?
?
cD (??k ) +
(Mk? (?
s) + ?k Eq(u) [?? ] + ??k (?) ? ??k? )Pk? (?).
(8)
k=0
k=0
?=1
The Dirichlet cumulant function cD maps K +1 positive parameters to a log-normalization constant.
For a non-sticky HDP-HMM where ? = 0, previous work [14] established the following bound:
P
PK+1
cD (??) , log ?(?) ? K+1
(9)
k=1 log ?(??k ) ? K log ? +
?=1 log ?? .
Direct evaluation of Eq(u) [cD (??)] is problematic because the expectations of log-gamma functions
have no closed form, but the lower bound has a simple expectation given beta distributed q(uk ).
Developing a similar bound for sticky models with ? > 0 requires a novel contribution. To begin,
in the supplement we establish the following bound for any ? > 0, ? > 0:
PK+1
cD (?? + ??k ) ? K log ? ? log(? + ?) + log(??k + ?) + ?=1 ?6=k log(?? ).
(10)
To handle the intractable term Eq(u) [log(??k + ?)], we leverage the concavity of the logarithm:
log(??k + ?) ? ?k log(? + ?) + (1 ? ?k ) log ?.
(11)
Combining Eqs. (10) and (11) and taking expectations, we can evaluate a lower bound on Eq. (8) in
closed form, and thereby efficiently optimize its parameters. As illustrated in Fig. 2, this rigorous
lower bound on the marginal evidence log p(x) is quite accurate for practical hyperparameters.
3.3 Batch and Stochastic Variational Inference
Most variational inference algorithms maximize L via coordinate ascent optimization, where the
best value of each parameter is found given fixed values for other variational factors. For the HDPHMM this leads to the following updates, which when iterated converge to some local maximum.
Local update to q(zn ). The assignments for each sequence zn can be updated independently via
dynamic programming [18]. The forward-backward algorithm takes as input a Tn ? K matrix of
log-likelihoods Eq [log p(xn | ?k )] given the current ??, and log transition probabilities Pjk given the
? It outputs the optimal marginal state probabilities s?n , r?n under objective L. This step has
current ?.
cost O(Tn K 2 ) for sequence n, and we can process multiple sequences in parallel for efficiency.
Global update to q(?). Conjugate priors lead to simple closed-form updates ??k = ?? + Sk , where
P
PTn
sufficient statistic Sk summarizes the data assigned to state k: Sk , N
?ntk sF (xnt ).
n=1
t=1 r
Global update to q(?). For each state k ? {0, 1, 2 . . . K}, the positive vector ??k defining the
optimal Dirichlet posterior on transition probabilities from state k is ??k? = Mk? (?
s) + ??? + ??k (?).
Statistic Mk? (?
s) counts the expected number of transitions from state k to ? across all sequences.
4
Global update to q(u). Due to non-conjugacy, our surrogate objective L has no closed-form update to q(u). Instead, we employ numerical optimization to update vectors ??, ?
? simultaneously:
?
arg max Lhdp-local (?
?, ?
? , ?, s?) + Lhdp-global (?
?, ?
? ) subject to ?
? k > 0, ??k ? (0, 1) for k = 1, 2 . . . K.
?,?
??
Details are in the supplement. The update to q(u) requires expectations under q(?), and vice versa,
so it can be useful to iteratively optimize q(?) and q(u) several times given fixed local statistics.
To handle large datasets, we can adapt these updates to perform stochastic variational inference
(SVI) [19]. Stochastic algorithms perform local updates on random subsets of sequences (batches),
and then perturb global parameters by following a noisy estimate of the natural gradient, which
has a simple closed form. SVI has previously been applied to non-sticky HDP-HMMs with pointestimated ? [7], and can be easily adapted to our more principled objective. One drawback of SVI
is the requirement of a learning rate schedule, which must typically be tuned to each dataset.
3.4 Memoized Variational Inference
We now outline a memoized algorithm [13] for our sticky HDP-HMM variational objective. Before
execution, each sequence is randomly assigned to one of B batches. The algorithm repeatedly visits
batches one at a time in random order; we call each full pass through the complete set of B batches a
lap. At each visit to batch b, we perform a local step for all sequences n in batch b and then a global
step. With B = 1 batches, memoized inference reduces to the standard full-dataset algorithm, while
with larger B we have more affordable local steps and faster overall convergence. With just one lap,
memoized inference is equivalent to the synchronous version of streaming variational inference,
presented in Alg. 3 of Broderick et al. [20]. We focus on regimes where dozens of laps are feasible,
which we demonstrate dramatically improves performance.
Affordable, but exact, batch optimization of L is possible by exploiting the additivity of statistics
M , S. For each statistic we track a batch-specific quantity M b , and a whole-dataset summary
P
b
M, B
?b , r?b , we update M b (?
sb ) and S b (?
rb ), increment
b=1 M . After a local step at batch b yields s
each whole-dataset statistic by adding the new batch summary and subtracting the summary stored
in memory from the previous visit, and store (or memoize) the new statistics for future iterations.
This update cycle makes M and S consistent with the most recent assignments for all sequences.
Memoization does require O(BK 2 ) more storage than SVI. However, this cost does not scale with
the number of sequences N or length T . Sparsity in transition counts M may make storage cheaper.
At any point during memoized execution, we can evaluate L exactly for all data seen thus far. This
? ??
is possible because nearly all terms in Eq. (6) are functions of only global parameters ??, ?
? , ?,
and sufficient statistics M, S. The one exception that requires local values s?, r? is the entropy term
Lentropy . To compute it, we track a (K + 1) ? K matrix H b at each batch b:
P PTn ?1
P
b
b
= ? n t=1
s?ntk? log s?r?ntk?
H0?
= ? n r?n1? log r?n1? , Hk?
,
(12)
ntk
where the sums aggregate sequences n that belong to batch b. Each entry of H b is non-negative, and
P
PK PK
b
given the whole-dataset entropy matrix H = B
b=1 H , we have Lentropy =
k=0
?=1 Hk? .
4 State Space Adaptation via Birth, Merge, and Delete Proposals
Reliable nonparametric inference algorithms must quickly identify and create missing states. Splitmerge samplers for HDP topic models [10, 11] are limited because proposals can only split an
existing state into two new states, require expensive traversal of all data points to evaluate an acceptance ratio, and often have low acceptance rates [12]. Some variational methods for HDP topic
models also dynamically create new topics [16, 21], but do not guarantee improvement of the global
objective and can be unstable. We instead interleave stochastic birth proposals with delete and merge
proposals, and use memoization to efficiently verify proposals via the exact full-dataset objective.
Birth proposals. Birth moves can create many new states at once while maintaining the monotonic
increase of the whole-dataset objective, L. Each proposal happens within the local step by trying
to improve q(zn ) for a single sequence n. Given current assignments s?n , r?n with truncation K, the
move proposes new assignments s??n , r?n? that include the K existing states and some new states with
index k > K. If L improves under the proposal, we accept and use the expanded set of states for
all remaining updates in the current lap. To compute L, we require candidate global parameters
??? , ?
? ? , ??? , ??? . These are found via a global step from candidate summaries M ? , S ? , which combine
5
0
?15
50
Hamming dist.
num topics K
15
25
*8
?30
stoch
memo
delete,merge
birth,delete,merge
Non-stick, kappa=0
Sticky, kappa=50
0.6
0.4
0.2
0.0
?30 ?15 0
15 30
10
1
100 1000
1
num pass thru data
stoch: K=47 after 2000 laps in 359 min.
0
sampler
0.8
30
200
400
600
800
sampler: K=10 after 2000 laps in 74 min.
0
200
400
10
100 1000
num pass thru data
600
800
delete,merge: K=8 after 100 laps in 5 min.
0
200
400
600
800
Figure 3: Toy data experiments (Sec. 5). Top left: Data sequences contain 2D points from 8 well-separated
Gaussians with sticky transitions. Top center: Trace plots from initialization with 50 redundant states. Our
state-adaptation algorithms (red/purple) reach ideal K = 8 states and zero Hamming distance regardless of
whether a sticky (solid) or non-sticky (dashed) model is used. Competitors converge slower, especially in the
non-sticky case because non-adaptive methods are more sensitive to hyperparameters. Bottom: Segmentations
of 4 sequences by SVI, the Gibbs sampler, and our method under the non-sticky model (? = 0). Top half shows
true state assignments; bottom shows aligned estimated states. Competitors are polluted by extra states (black).
?
?
the new batch statistics Mb? , Sb? and memoized statistics of other batches M\b
, S\b
expanded by zeros
for states k > K. See the supplement for details on handling multiple sequences within a batch.
The proposal for expanding s?? , r?? with new states can flexibly take any form, from very na??ve to very
data-driven. For data with ?sticky? state persistence, we recommend randomly choosing one interval
[t, t + ?] of the current sequence to reassign when creating s?? , r?? , leaving other timesteps fixed. We
split this interval into two contiguous blocks (one may be empty), each completely assigned to a
new state. In the supplement, we detail a linear-time search that finds the cut point that maximizes
the objective Ldata . Other proposals such as sub-cluster splits [11] could be easily incorporated in
our variational algorithm, but we find this simple interval-based proposal to be fast and effective.
Merge proposals. Merge proposals try to find a less redundant but equally expressive model. Each
proposal takes a pair of existing states i < j and constructs a candidate model where data from state
j is reassigned to state i. Conceptually this reassignment gives a new value s?? , but instead statistics
M ? , S ? can be directly computed and used in a global update for candidate parameters ??? , ??? , ??? .
Si? = Si + Sj ,
M:i? = M:i + M:j ,
Mi:? = Mi: + Mj: ,
Mii? = Mii + Mjj + Mji + Mij .
While most terms in L are linear functions of our cached sufficient statistics, the entropy Lentropy
is not. Thus for each candidate merge pair (i, j), we use O(K) storage and computation to track
column H:i? and row Hi:? of the corresponding merged entropy matrix H ? . Because all terms in the
H ? matrix of Eq. (12) are non-negative, we can lower-bound Lentropy by summing a subset of H ? . As
detailed in the supplement, this allows us to rigorously bound the objective L? for accepting multiple
merges of distinct state pairs. Because many entries of H ? are near-zero, this bound is very tight,
and in practice enables us to scalably merge many redundant state pairs in each lap through the data.
To identify candidate merge pairs i, j, we examine all pairs of states and keep those that satisfy
L?data +L?hdp-local +L?hdp-global > Ldata +Lhdp-local +Lhdp-global . Because entropy must decrease after any
merge (L?entropy < Lentropy ), this test is guaranteed to find all possibly useful merges. It is much more
efficient than the heuristic correlation score used in prior work on HDP topic models [14].
Deletes. Our proposal to delete a rarely-used state j begins by dropping row j and column j from
M to create M ? , and dropping Sj from S to create S ? . Using a target dataset of sequences with
PTn
non-trivial mass on state j, x? = {xn :
?ntj > 0.01}, we run global and local parameter
t=1 r
updates to reassign observations from former state j in a data-driven way. Rather than verifying on
only the target dataset as in [14], we accept or reject the delete proposal via the whole-dataset bound
L. To control computation, we only propose deleting states used in 10 or fewer sequences.
6
-3.5
-3.6
1
10
100
num pass thru data
100
1200
75
50
25
0
0.1
1
10 100
num pass thru data
1000
speedup
-3.4
time (sec)
num states K
objective (x100)
stoch
memo
birth,del,merge
K=50
K=100
800
600
400
200
0
64x
32x
16x
8x
4x
2x
1x
1 2 4 8 16 32 64
1 2 4 8 16 32 64
num parallel workers
num parallel workers
Figure 4: Segmentation of human epigenome: 15 million observations across 173 sequences (Sec. 5). Left:
Adaptive runs started at 1 state grow to 70 states within one lap and reach better L scores than 100-state nonadaptive methods. Each run takes several days. Right: Wallclock times and speedup factors for a parallelized
local step on 1/3 of this dataset. 64 workers complete a local step with K = 50 states in under one minute.
5 Experiments
We compare our proposed birth-merge-delete memoized algorithm to memoized with delete and
merge moves only, and without any moves. We further run a blocked Gibbs sampler [6] that was
previously shown to mix faster than slice samplers [22], and our own implementation of SVI for
objective L. These baselines maintain a fixed number of states K, though some states may have
usage fall to zero. We start all fixed-K methods (including the sampler) from matched initializations.
See the supplement for futher discussion and all details needed to reproduce these experiments.
Toy data. In Fig. 3, we study 32 toy data sequences generated from 8 Gaussian states with sticky
transitions [8]. From an abundant initialization with 50 states, the sampler and non-adaptive variational methods require hundreds of laps to remove redundant states, especially under a non-sticky
model (? = 0). In contrast, our adaptive methods reach the ideal of zero Hamming distance within
a few dozen laps regardless of stickiness, suggesting less sensitivity to hyperparameters.
Speaker diarization. We study 21 unrelated audio recordings of meetings with an unknown number of speakers from the NIST 2007 speaker diarization challenge [23]. The sticky HDP-HMM
previously achieved state-of-the-art diarization performance [6] using a sampler that required hours
of computation. We ran methods from 10 matched initializations with 25 states and ? = 100, computing Hamming distance on non-speech segments as in the standard DER metric. Fig. 5 shows that
within minutes, our algorithms consistently find segmentations better aligned to true speaker labels.
Labelled N = 6 motion capture. Fox et al. [12] introduced a 6 sequence dataset with labels for
12 exercise types, illustrated in Fig. 1. Each sequence has 12 joint angles (wrist, knee, etc.) captured
at 0.1 second intervals. Fig. 6 shows that non-adaptive methods struggle even when initialized
abundantly with 30 (dashed lines) or 60 (solid) states, while our adaptive methods reach better
values of the objective L and cleaner many-to-one alignment to true exercises.
Large N = 124 motion capture. Next, we apply scalable methods to the 124 sequence dataset of
[12]. We lack ground truth here, but Fig. 7 shows deletes and merges making consistent reductions
from abundant initializations and births growing from K = 1. Fig. 7 also shows estimated segmentations for 10 representative sequences, along with skeleton illustrations for the 10 most-used states
in this subset. These segmentations align well with held-out text descriptions.
Chromatin segmentation. Finally, we study segmenting the human genome by the appearance
patterns of regulatory proteins [24]. We observe 41 binary signals from [3] at 200bp intervals
throughout a white blood cell line (CD4T). Each binary value indicates the presence or absence
of an acetylation or methylation that controls gene expression. We divide the whole epigenome into
173 sequences (one per batch) with total size T = 15.4 million. Fig. 4 shows our method can grow
from 1 state to 70 states and compete favorably with non-adaptive competitors. We also demonstrate
that our parallelized local step leads to big 25x speedups in processing such large datasets.
6 Conclusion
Our new variational algorithms adapt HMM state spaces to find clean segmentations driven by
Bayesian model selection. Relative to prior work [14], our contributions include a new bound for the
sticky HDP-HMM, births with guaranteed improvement, local step parallelization, and better merge
selection rules. Our multiprocessing-based Python code is targeted at genome-scale applications.
Acknowledgments This research supported in part by NSF CAREER Award No. IIS-1349774. M. Hughes
supported in part by an NSF Graduate Research Fellowship under Grant No. DGE0228243.
7
0.3
0.2
0.1
0.1
0.2
0.3
0.4
0.5
?2.70
?2.75
?2.80
0.6
delete-merge Hamming
1
10
?2.60
?2.65
?2.70
1
100 1000
Hamming dist.
sampler
memo
delete,merge
birth,delete,merge
10
0.8
0.6
0.4
0.2
0.0
10
?2.45
?2.50
?2.55
100 1000
0.8
0.6
0.4
0.2
0.0
1
?2.40
1
elapsed time (sec)
elapsed time (sec)
Hamming dist.
0.0
0.0
?2.65
train objective
0.4
?2.55
?2.60
Meeting 21 (worst)
?2.55
100 1000
100 1000
0.8
0.6
0.4
0.2
0.0
1
elapsed time (sec)
10
elapsed time (sec)
Hamming dist.
0.5
Meeting 16 (avg.)
train objective
train objective
sampler Hamming
Meeting 11 (best)
?2.50
0.6
10
100 1000
elapsed time (sec)
1
10
100 1000
elapsed time (sec)
Figure 5: Method comparison on speaker diarization from common K = 25 initializations (Sec. 5). Left: Scatterplot of final Hamming distance for our adaptive method and the sampler. Across 21 meetings (each with 10
initializations shown as individual dots) our method finds segmentations closer to ground truth. Right: Traces
of objective L and Hamming distance for meetings representative of good, average, and poor performance.
?2.2
?2.4
?2.6
40
20
?2.8
1
10
0
100 1000
birth: Hdist=0.34 K=28 @ 100 laps
50
0.8
stoch
sampler
0.6
memo
delete,merge
birth,delete,merge
0.4
0.2
0.0
1
10
100 1000
num pass thru data
num pass thru data
0
Hamming dist.
num states K
train objective
60
1
10
100 1000
num pass thru data
del/merge: Hdist=0.30 K=13 @ 100 laps sampler: Hdist=0.49 K=29 @ 1000 laps
100 150 200 250 300 350 400
0
50
100 150 200 250 300 350 400
0
50
100 150 200 250 300 350 400
?2.4
num states K
train objective
Figure 6: Comparison on 6 motion capture streams (Sec. 5). Top: Our adaptive methods reach better L values
and lower distance from true exercise labels. Bottom: Segmentations from the best runs of birth/merge/delete
(left), only deletes and merges from 30 initial states (middle), and the sampler (right). Each sequence shows
true labels (top half) and estimates (bottom half) colored by the true state with highest overlap (many-to-one).
?2.5
?2.6
1
10
100 1000
num pass thru data
Walk
200
1-1: playground jump
150
1-2: playground climb
1-3: playground climb
2-7: swordplay
100
5-3: dance
5-4: dance
50
5-5: dance
6-3: basketball dribble
0
1
10 100 1000
6-4: basketball dribble
6-5: basketball dribble
num pass thru data
Climb
!
Sword
"!
Arms
#!
$!
%!
&!!
Swing
Dribble
Jump
Balance
Ballet Leap
Ballet Pose
Figure 7: Study of 124 motion capture sequences (Sec. 5). Top Left: Objective L and state count K as more data
is seen. Solid lines have 200 initial states; dashed 100. Top Right: Final segmentation of 10 select sequences
by our method, with id numbers and descriptions from mocap.cs.cmu.edu. The 10 most used states are
shown in color, the rest with gray. Bottom: Time-lapse skeletons assigned to each highlighted state.
8
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9
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5,150 | 5,662 | Robust Feature-Sample Linear Discriminant Analysis
for Brain Disorders Diagnosis
Ehsan Adeli-Mosabbeb, Kim-Han Thung, Le An, Feng Shi, Dinggang Shen, for the ADNI?
Department of Radiology and BRIC
University of North Carolina at Chapel Hill, NC, 27599, USA
{eadeli,khthung,le_an,fengshi,dgshen}@med.unc.edu
Abstract
A wide spectrum of discriminative methods is increasingly used in diverse applications for classification or regression tasks. However, many existing discriminative methods assume that the input data is nearly noise-free, which limits their
applications to solve real-world problems. Particularly for disease diagnosis, the
data acquired by the neuroimaging devices are always prone to different sources
of noise. Robust discriminative models are somewhat scarce and only a few attempts have been made to make them robust against noise or outliers. These
methods focus on detecting either the sample-outliers or feature-noises. Moreover, they usually use unsupervised de-noising procedures, or separately de-noise
the training and the testing data. All these factors may induce biases in the learning process, and thus limit its performance. In this paper, we propose a classification method based on the least-squares formulation of linear discriminant analysis, which simultaneously detects the sample-outliers and feature-noises. The
proposed method operates under a semi-supervised setting, in which both labeled
training and unlabeled testing data are incorporated to form the intrinsic geometry
of the sample space. Therefore, the violating samples or feature values are identified as sample-outliers or feature-noises, respectively. We test our algorithm on
one synthetic and two brain neurodegenerative databases (particularly for Parkinson?s disease and Alzheimer?s disease). The results demonstrate that our method
outperforms all baseline and state-of-the-art methods, in terms of both accuracy
and the area under the ROC curve.
1
Introduction
Discriminative methods pursue a direct mapping from the input to the output space for a classification or a regression task. As an example, linear discriminant analysis (LDA) aims to find the
mapping that reduces the input dimensionality, while preserving the most class discriminatory information. Discriminative methods usually achieve good classification results compared to the generative models, when there are enough number of training samples. But they are limited when there
are small number of labeled data, as well as when the data is noisy. Various efforts have been made
to add robustness to these methods. For instance, [17] and [9] proposed robust Fisher/linear discriminant analysis methods, and [19] introduced a worst-case LDA, by minimizing the upper bound of
the LDA cost function. These methods are all robust to sample-outliers. On the other hand, some
methods were proposed to deal with the intra-sample-outliers (or feature-noises), such as [12, 15].
?
Parts of the data used in preparation of this article were obtained from the Alzheimer?s Disease Neuroimaging Initiative (ADNI) database (http://adni.loni.ucla.edu). The investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or
writing of this paper. A complete listing of ADNI investigators can be found at: http://adni.loni.
ucla.edu/wp-content/uploads/howtoapply/ADNIAcknowledgementList.pdf.
1
As in many previous works, de-noising the training and the testing data are often conducted separately. This might induce a bias or inconsistency to the whole learning process. Besides, for many
real-world applications, it is a cumbersome task to acquire enough training samples to perform a
proper discriminative analysis. Hence, we propose to take advantage of the unlabeled testing data
available, to build a more robust classifier. To this end, we introduce a semi-supervised discriminative classification model, which, unlike previous works, jointly estimates the noise model (both
sample-outliers and feature-noises) on the whole labeled training and unlabeled testing data and
simultaneously builds a discriminative model upon the de-noised training data.
In this paper, we introduce a novel classification model based on LDA, which is robust against
both sample-outliers and feature-noises, and hence, here, it is called robust feature-sample linear
discriminant analysis (RFS-LDA). LDA finds the mapping between the sample space and the label
space through a linear transformation matrix, maximizing a so-called Fisher discriminant ratio [17].
In practice, the major drawback of the original LDA is the small sample size problem, which arises
when the number of available training samples is less than the dimensionality of the feature space
[18]. A reformulation of LDA based on the reduced-rank least-squares problem (known LS-LDA)
[10] tackles this problem. LS-LDA finds the mapping ? ? Rl?d by solving the following problem1 :
min kH(Ytr ? ? Xtr )k2F ,
?
(1)
where Ytr ? Rl?Ntr is a binary class label indicator matrix, for l different classes (or labels), and
Xtr ? Rd?Ntr is the matrix containing Ntr d-dimensional training samples. H is a normalization
> ?1/2
factor defined as H = (Ytr Ytr
)
that compensates for the different number of samples in each
class [10]. As a result, the mapping ? is a reduced rank transformation matrix [10, 15], which could
be used to project a test data xtst ? Rd?1 onto a l dimensional space. The class label could therefore
be simply determined using a k-NN strategy.
To make LDA robust against noisy data, Fidler et al. [12] proposed to construct a basis, which contains complete discriminative information for classification. In the testing phase, the estimated basis
identifies the outliers in samples (images in their case) and then is used to calculate the coefficients
using a subsampling approach. On the other hand, Huang et al. [15] proposed a general formulation
for robust regression (RR) and classification (robust LDA or RLDA). In the training stage, they denoise the feature values using a strategy similar to robust principle component analysis (RPCA) [7]
and build the above LS-LDA model using the de-noised data. In the testing stage, they de-noise the
data by performing a locally compact representation of the testing samples from the de-noised training data. This separate de-noising procedure could not effectively form the underlying geometry of
sample space to de-noise the data. Huang et al. [15] only account for feature-noise by imposing a
sparse noise model constraint on the features matrix. On the other hand, the data fitting term in (1)
is vulnerable to large sample-outliers. Recently, in robust statistics, it is found that `1 loss functions
are able to make more reliable estimations [2] than `2 least-squares fitting functions. This has been
adopted in many applications, including robust face recognition [28] and robust dictionary learning
[22]. Reformulating the objective in (1), using this idea, would yield to this problem:
min kH(Ytr ? ? Xtr )k1 .
?
(2)
We incorporate this fitting function in our formulation to deal with the sample-outliers by iteratively
re-weighting each single sample, while simultaneously de-noising the data from feature-noises. This
is done through a semi-supervised setting to take advantage of all labeled and unlabeled data to build
the structure of the sample space more robustly. Semi-supervised learning [8, 34] has long been of
great interest in different fields, because it can make use of unlabeled or poorly labeled data. For
instance, Joulin and Bach [16] introduced a convex relaxation and use the model in different semisupervised learning scenarios. In another work, Cai et al. [5] proposed a semi-supervised discriminant analysis, where the separation between different classes is maximized using the labeled data
points, while the unlabeled data points estimate the structure of the data. In contrast, we incorporate
the unlabeled testing data to form the intrinsic geometry of the sample space and de-noise the data,
whilst building the discriminative model.
1
Bold capital letters denote matrices (e.g., D). All non-bold letters denote scalar variables. dij is the scalar
2
in the row i and column j of D. hd1 , d2 i denotes the inner product between d1 and d2 . kdkP
2 and kdk1
2
>
represent the squared Euclidean Norm and the `1 norm of d, respectively. kDkF = tr(D D) = ij dij and
kDk? designate the squared Frobenius Norm and the nuclear norm (sum of singular values) of D, respectively.
2
X = [Xtr Xtst ] ? Rd?N
x11 x12
? x
?
21 x22
?
?
? x31 x32
?
?
?
.
.
?
.
.
?
.
.
?
xd1 xd2
?
. . . x1N
x1N +1 . . . x1N
tr
tr
. . . x2N
x2N +1 . . . x2N
tr
tr
. . . x3N
x3N +1 . . . x3N
tr
tr
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. . . xdN
xdN +1 . . . xdN
tr
tr
D = [Dtr Dtst ] ? Rd?N
?
?
d11 d12 . . .
? ?
? ? d21 d22 . . .
? ?
? ? d
? ? 31 d32 . . .
?=?
? ?
? ?
.
.
.
? ?
.
.
.
? ?
.
? ?
.
.
dd1 dd2 . . .
? ?
d1N
d
. . . d1N
tr 1Ntr +1
? ?
d2N
d2N +1 . . . d2N ? ?
tr
tr
? ?
?
d3N
d3N +1 . . . d3N ?
? ?
tr
tr
?+?
? ?
? ?
.
.
.
.
?
?
.
.
.
.
? ?
.
? ?
.
.
.
ddN
ddN +1 . . . ddN
tr
tr
E ? Rd?N
e11 e12 . . . e1N
e21 e22 . . . e2N
e31 e32 . . . e3N
.
.
.
.
.
.
.
.
.
.
.
.
ed1 ed2 . . . edN
?
?
?
?
?
?
?
?
?
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?
?
Mapping ?
?
y11 y12 . . . y1N
tr
?
?
?
?
?
?
.
.
.
.
.
.
.
.
.
.
.
.
yl1 yl2 . . . ylN
tr
?
?
?
?
?
Ytr ? Rl?Ntr
Figure 1: Outline of the proposed method: The original data matrix, X, is composed of both labeled
training and unlabeled testing data. Our method decomposes this matrix to a de-noised data matrix,
D, and an error matrix, E, to account for feature-noises. Simultaneously, we learn a mapping from
the de-noised training samples in D (Dtr ) through a robust `1 fitting function, dealing with the
sample-outliers. The same learned mapping on the testing data, Dtst , leads to the test labels.
We apply our method for the diagnosis of neurodegenerative brain disorders. The term neurodegenerative disease is an umbrella term for debilitating and incurable conditions related to progressive
degeneration or death of the cells in the brain nervous system. Although neurodegenerative diseases
manifest with diverse pathological features, the cellular level processes resemble similar structures.
For instance, Parkinson?s disease (PD) mainly affects the basal ganglia region and the substansia
nigra sub-region of the brain, leading to decline in generation of a chemical messenger, dopamine.
Lack of dopamine yields loss of ability to control body movements, along with some non-motor
problems (e.g., depression, anxiety) [35]. In Alzheimer?s disease (AD), deposits of tiny protein
plaques yield into brain damage and progressive loss of memory [26]. These diseases are often
incurable and thus, early diagnosis and treatment are crucial to slow down the progression of the
disease in its initial stages. In this study, we use two popular databases: PPMI and ADNI. The
former aims at investigating PD and its related disorders, while the latter is designed for diagnosing
AD and its prodormal stage, known as mild cognitive impairment (MCI).
Contributions: The contribution of this paper would therefore be multi-fold: (1) We propose an
approach to deal with the sample-outliers and feature-noises simultaneously, and build a robust discriminative classification model. The sample-outliers are penalized through an `1 fitting function,
by re-weighing the samples based on their prediction power, while discarding the feature-noises.
(2) Our proposed model operates under a semi-supervised setting, where the whole data (labeled
training and unlabeled testing samples) are incorporated to build the intrinsic geometry of the sample space, which leads to better de-noising the data. (3) We further select the most discriminative
features for the learning process through regularizing the weights matrix with an `1 norm. This is
specifically of great interest for the neurodegenerative disease diagnosis, where the features from
different regions of the brain are extracted, but not all the regions are associated with a certain disease. Therefore, the most discriminative regions in the brain that utmost affect the disease would be
identified, leading to a more reliable diagnosis model.
2
Robust Feature-Sample Linear Discriminant Analysis (RFS-LDA)
Let?s assume we have Ntr training and Ntst testing samples, each with a d-dimensional feature
vector, which leads to a set of N = Ntr + Ntst total samples. Let X ? Rd?N denote the set of all
samples (both training and testing), in which each column indicates a single sample, and yi ? R1?N
their corresponding ith labels. In general, with l different labels, we can define Y ? Rl?N . Thus,
X and Y are composed by stacking up the training and testing data as: X = [Xtr Xtst ] and
Y = [Ytr Ytst ]. Our goal is to determine the labels of the test samples, Ytst ? Rl?Ntst .
Formulation: An illustration of the proposed method is depicted in Fig 1. First, all the samples
(labeled or unlabeled) are arranged into a matrix, X. We are interested in de-noising this matrix.
Following [14, 21], this could be done by assuming that X can be spanned on a low-rank subspace
and therefore should be rank-deficient. This assumption supports the fact that samples from same
classes should be more correlated [14, 15]. Therefore, the original matrix X is decomposed into two
3
counterparts, D and E, which represent the de-noised data matrix and the error matrix, respectively,
similar to RPCA [7]. The de-noised data matrix shall hold the low-rank assumption and the error
matrix is considered to be sparse. But, this process of de-noising does not incorporate the label
information and is therefore unsupervised. Nevertheless, note that we also seek a mapping between
the de-noised training samples and their respective labels. So, matrix D should be spanned on a
low-rank subspace, which also leads to a good classification model of its sub-matrix, Dtr .
To ensure the rank-deficiency of the matrix D, like in many previous works [7, 14, 21], we approximate the rank function using the nuclear norm (the sum of the singular values of the matrix). The
noise is modeled using the `1 norm of the matrix, which ensures a sparse noise model on the feature
values. Accordingly, the objective function for RFS-LDA under a semi-supervised setting would be:
?
? 1 + kDk? + ?1 kEk1 + ?2 R(?
? ),
min
kH(Ytr ? ? D)k
?
2
? ,D,D,E
(3)
>
?
s.t. D = X + E, D = [Dtr ; 1 ],
where the first term is the `1 regression model introduced in (2). This term only operates on the denoised training samples from matrix D with a row of all 1s is added to it, to ensure an appropriate
linear classification model. The second and the third terms together with the first constraint are
similar to the RPCA formulation [7]. They de-noise the labeled training and unlabeled testing data
together. In combination with the first term, we ensure that the de-noised data also provides a
favorable regression/classification model. The last term is a regularization on the learned mapping
coefficients to ensure the coefficients do not get trivial or unexpectedly large values. The parameters
?, ?1 and ?2 are constant regularization parameters, which are discussed in more details later.
The regularization on the coefficients could be posed as a simple norm of the ? matrix. But, in many
applications like ours (disease diagnosis) many of the features in the feature vectors are redundant. In
practice, features from different brain regions are often extracted, but not all the regions contribute
to a certain disease. Therefore, it is desirable to determine which features (regions) are the most
relevant and the most discriminative to use. Following [11, 26, 28], we are looking for a sparse set
of weights that ensures incorporating the least and the most discriminative features. We propose a
regularization on the weights vector as a combination of the `1 and Frobenius norms:
? ) = k?
? k1 + ?k?
? kF .
R(?
(4)
Evidently, the solution to the objective function in (3) is not easy to achieve, since the first term
contains a quadratic term and minimization of the `1 fitting function is not straightforward (because
of its indifferentiability). To this end, we formalize the solution with a similar strategy as in iteratively re-weighted least squares (IRLS) [2]. The `1 minimization problem is approximated by a
? matrix are weighted with the
conventional `2 least-squares, in which each of the samples in the D
reverse of their regression residual. Therefore the new problem would be:
?
? ? k2 + kDk? + ?1 kEk1 + ?2 R(?
? ),
kH(Ytr ? ? D)?
min
F
?
2
? ,D,D,E
(5)
? = [Dtr ; 1> ].
s.t. D = X + E, D
? is a diagonal matrix, the ith diagonal element of which is the ith sample?s weight:
where ?
q
? i )2 + ? ,
(yi ? ? d
? ii = 1/
?
? ij = 0,
? i, j ? {0, . . . , Ntr }, i 6= j, ?
(6)
where ? is a very small positive number (equal to 0.0001 in our experiments). In the next subsection,
we introduce an algorithm to solve this optimization problem.
Our work is closely related to the RR and RLDA formulations in [15], where the authors impose a
low-rank assumption on the training data feature values and an `1 assumption on the noise model.
The discriminant model is learned similar to LS-LDA, as illustrated in (1), while a sample-weighting
strategy is employed to achieve a more robust model. On the other hand, our model operates under a
semi-supervised learning setting, where both the labeled training and the unlabeled testing samples
are de-noised simultaneously. Therefore, the geometry of the sample space is better modeled on the
low-dimensional subspace, by interweaving both labeled training and unlabeled testing data. In addition, our model further selects the most discriminative features to learn the regression/classification
model, by regularizing the mapping weights vector and enforcing an sparsity condition on them.
4
Algorithm 1 RFS-LDA optimization algorithm.
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
Input: X = [Xtr Xtst ], Ytr , parameters ?, ?1 , ?2 , ? and ?.
? 0 = [Xtr ; 1> ], ? 0 = Ytr (D
? 0 )> (D
? 0 (D
? 0 )> + ?I), E0 = 0, L10 =
Initialization: D0 = [Xtr Xtst ], D
dNtr
dN
0
X/kXk , L 0 = Xtr/kX k , L 0 = ? 0/k?
? 0 k1 .
? k2 , ?1 =
kXk1 , ?2 = 4 kXtr k1 , ?3 = dc
k?
tr 2
2
2
3
4
4
k?0
repeat
. Main optimization loop
0
t ? 0, ?? = ? k
. Update ?
repeat
q
? ij ? 0 and ?
? ii ? 1/ (yk ? ?? t d? ki )2 + 0.0001
? i, j ? {0, . . . , Ntr ? 1}, i 6= j, ?
k > k i>
t+1
> ?k >
k
k
?
? ?
? ) + ?I , t ? t + 1
??
? (D ) + ?3 (B ? L3 ) D
??
? (D
?
? Ytr ?
until k?? t?1 ? ?? t kF/(k?? t?1 kF ? k?? t kF ) < 0.001 or t > 100
t
? k+1 ? ?? .
?1
? k+1 ? ??
?
? > (?
? k+1 )>? k+1?
? + ?k2 I
? > (?
? k+1 )> Ytr ? L2k + ?k2 [Dktr ; 1> ]
D
??
. Update D
k+1
k
k
k
k
k ? k+1
D
? D1/(?k1 + ?k2 ) L1 + ?1 (X ? E ) + [L2 + ?2 D
](1:Ntr ,:) 0
. Update D
Ek+1 ? S?1/?k1 (X ? Dk+1 + L1k/?k1 )
. Update E
? k+1 + L3k )
Bk+1 ? S?2/?k3 (?
. Update B
13:
L1k+1 ? L1k + ?k1 (X ? Dk+1 ? Ek+1 )
. Update multipliers and parameters
k+1
>
? ? [Dk+1
? ? B)
14:
L2k+1 ? L2k + ?k2 (D
? L3k + ?k3 (?
tr ; 1 ]), L3
15:
?k+1
? min(??k1 , 109 ), ?k+1
? min(??k2 , 109 ), ?k+1
? min(??k3 , 109 )
1
2
3
16:
k ?k+1
?8
? k ? [Dk ; 1> ]kF/kD
? k kF < 10?8 and k?
? k ? Bk kF/k?
? k kF < 10
17: until kX ? Dk ? Ek kF/kXkF < 10?8 and kD
tr
Output: ? , D, E and Ytst = ? Xtst .
Optimization: Problem (5) could be efficiently solved using the augmented Lagrangian multipliers
(ALM) approach. Hence, we introduce the Lagrangian multipliers, L1 ? Rd?N , L2 ? R(d+1)?Ntr
and L3 ? Rl?(d+1) , an auxiliary variable, B ? Rl?(d+1) , and write the Lagrangian function as:
? E) = ? kH(Ytr ? ? D)?
? ? k2 + kDk? + ?1 kEk1 + ?2 (kBk1 + ?k?
? , B, D, D,
? kF )
L(?
F
2
?1
? ? [Dtr ; 1> ]i (7)
+ hL1 , X ? D ? Ei +
kX ? D ? Ek2F + hL2 , D
2
?2 ?
?3
? ? Bk2F ,
+
kD ? [Dtr ; 1> ]k2F + hL3 , ? ? Bi +
k?
2
2
? and E) contribut? , B, D, D
where ?1 , ?2 and ?3 are penalty parameters. There are five variables (?
ing to the problem. We alternatively optimize for each variable, while fixing the others. Except for
the matrix ? , all the variables have straightforward or closed-form solutions. ? is calculated through
? and solving the conventional least-squares
IRLS [2], by iteratively calculating the weights in ?
problem, until convergence.
The detailed optimization steps are given in Algorithm 1. The normalization factor H is omitted
in this algorithm, for easier readability. In this algorithm, I is the identity matrix and the operators
?)V? applies singular value threshD? (.) and S? (.) are defined in the following. D? (A) = UD? (?
?) = diag({(?i ? ? )+ }), where U?
? V?
olding algorithm [6] on the intermediate matrix ? , as D? (?
is the singular values decomposition (SVD) of A and ?i s are the singular values. Additionally,
S? (a) = (a ? ?)+ ? (?a ? ?)+ is the soft thresholding operator or the proximal operator for the
`1 norm [3]. Note that s+ is the positive part of s, defined as s+ = max(0, s).
? E is a convex function, while
Algorithm analysis: The solution for each of the matrices B, D, D,
?
all the other variables are fixed. For , the solution is achieved via the IRLS approach, in an iterative
manner. Both the `1 fitting function and the approximated re-weighted least-squares functions are
convex. We only need to ensure that the minimization of the latter is numerically better tractable
than the minimization of the former. This is discussed in depth and the convergence is proved in [2].
To estimate the computational complexity of the algorithm, we need to investigate the complexity of
the sub-procedures of the algorithm. The two most computationally expensive steps in the loop are
the iterative update of ? (Algorithm 1, Steps 4-7) and the SVT operation (Algorithm 1, Step 10). The
former includes solving a least-squares iteratively, which is O(d2 N ) in each iteration and the latter
has the SVD operation as the most computational intensive operation, which is of O(d2 N + N 3 ).
5
By considering the maximum number of iterations for the first sub-procedure equal to tmax = 100,
the overall computational complexity of the algorithm in each iteration would be O(100d2 N + N 3 ).
The number of iterations of the whole algorithm until convergence is dependent on the choice of
{?}s. If ? penalty parameters are increasing smoothly in each iteration (as in Step 15, Algorithm 1),
the overall algorithm would be Q-linearly convergent. A reasonable choice for the sequence of all
{?}s yields in a decrease in the number of required SVD operations [1, 21].
3
Experiments
We compare our method with several baseline and state-of-the-art methods in three different scenarios. The first experiment is on synthetic data, which highlights how the proposed method is robust
against sample-outliers or feature-noises, separately or when they occur at the same time. The next
two experiments are conducted for neurodegenerative brain disorders diagnosis. We use two popular
databases, one for Parkinson?s disease (PD) and the other for Alzheimer?s disease (AD).
We compare our results with different baseline methods, including: Conventional LS-LDA [10],
RLDA [15], RPCA on the X matrix separately to de-noise and then LS-LDA for the classification (denoted as RPCA+LS-LDA) [15], linear support vector machines (SVM), and sparse feature
selection with SVM (SFS+SVM) or with RLDA (SFS+RLDA). Except for RPCA+LDA, the other
methods in comparison do not incorporate the testing data. In order to have a fair set of comparisons,
we also compare against the transductive matrix completion (MC) approach [14]. Additionally, to
also evaluate the effect of the regularization on matrix ? , we report results for RFS-LDA when regu? kF (denoted as RFS-LDA? ), instead of the term introduced in (4). Moreover, we
larized by only ?k?
also train our proposed RFS-LDA in a fully supervised setting, i.e., not involving any testing data
in the training process, to show the effect of the established semi-supervised learning framework in
our proposed method. This is simply done by replacing variable X in (3) with Xtr and solving the
problem correspondingly. This method, referred to as S-RFS-LDA, only uses the training data to
form the geometry of the sample space and, therefore, only cleans the training feature-noises.
For the choice of parameters, the best parameters are selected through an inner 10-fold cross validation on the training data, for all the competing methods. For the proposed method, the parameters are
p
?
? k k2 ,
set with a same strategy as in [15]: ?1 = ?1/( min(d, N )), ?2 = ?2/ d, ? k = ?3 kXk?/kYtr ? ? k D
F
and ? (controlling the {?}s in the algorithm) is set to 1.01. We have set ?1 , ?2 , ?3 and ? through
inner cross validation, and found that all set to 1 yields to reasonable results across all datasets.
Synthetic Data: We construct two independent 100-dimensional subspaces, with bases U1 and U2
(same as described in [21]). U1 ? R100?100 is a random orthogonal matrix and U2 = TU1 , in
which T is a random rotation matrix. Then, 500 vectors are sampled from each subspace through
Xi = Ui Qi , i = {1, 2}, with Qi , a 100 ? 500 matrix, independent and identically distributed
(i.i.d.) from N (0, 1). This leads to a binary classification problem. We gradually add additional
noisy samples and features to the data, drawn i.i.d from N (0, 1), and evaluate our proposed method.
The accuracy means and standard deviations of three different runs are illustrated in Fig. 2. This
experiment is conducted under three settings: (1) First, we analyze the behavior of the method
against gradually added noise to some of the features (feature-noises), illustrated in Fig. 2a. (2)
We randomly add some noisy samples to the aforementioned noise-free samples and evaluate the
methods in the sole presence of sample-outliers. Results are depicted in Fig. 2b. (3) Finally, we
simultaneously add noisy features and samples. Fig. 2c shows the mean?std accuracy as a function
of the additional number of noisy features and samples. Note that all the reported results are obtained
through 10-fold cross-validation. As can be seen, our method is able to select a better subset of
features and samples and achieve superior results compared to RLDA and conventional LS-LDA
approaches. Furthermore, our method behaves more robust against the increase in the noise factor.
Brain neurodegenrative disease diagnosis databases: The first set of data used in this paper is
obtained from the Parkinson?s progression markers initiative (PPMI) database2 [23]. PPMI is the
first substantial study for identifying the PD progression biomarkers to advance the understanding
of the disease. In this research, we use the MRI data acquired by the PPMI study, in which a T1weighted, 3D sequence (e.g., MPRAGE or SPGR) is acquired for each subject using 3T SIEMENS
MAGNETOM TrioTim syngo scanners. We use subjects scanned using MPRAGE sequence to
2
http://www.ppmi-info.org/data
6
Accuracy (%)
100
100
100
80
80
90
80
RFS-LDA
RLDA
70
LS-LDA
60
60
0
100
200
# of added noisy features
(a) Only added noisy features
0
100
200
0
100
200
# of added noisy samples
# of added noisy samples and features
(b) Only added noisy samples
(c) Added noisy samples & features
Figure 2: Results comparisons on synthetic data, for three different runs (mean?std).
Table 1: The accuracy (ACC) and area under ROC curve (AUC) of the PD/NC classification on
PPMI database, compared to the baseline methods.
Method
ACC
AUC
RFS-LDA
RFS-LDA?
S-RFS-LDA
RLDA
SFS+RLDA
RPCA+LS-LDA
LS-LDA
SVM
SFS+SVM
MC
84.1
0.87
78.3
0.81
75.8
0.80
71.0
0.79
73.4
0.80
59.4
0.64
56.6
0.59
55.2
0.56
61.5
0.59
61.5
68.8
minimize the effect of different scanning protocols. The T1-weighted images were acquired for
176 sagittal slices with the following parameters: repetition time = 2300 ms, echo time = 2.98 ms,
flip angle = 9? , and voxel size = 1 ? 1 ? 1 mm3 . All the MR images were preprocessed by skull
stripping [29], cerebellum removal, and then segmented into white matter (WM), gray matter (GM),
and cerebrospinal fluid (CSF) tissues [20]. The anatomical automatic labeling atlas [27], parcellated
with 90 predefined regions of interest (ROI), was registered using HAMMER3 [25, 30] to each
subject?s native space. We further added 8 more ROIs in basal ganglia and brainstem regions, which
are clinically important ROIs for PD. We then computed WM, GM and CSF tissue volumes in each
of the 98 ROIs as features. 56 PD and 56 normal control (NC) subjects are used in our experiments.
The second dataset is from Alzheimer?s disease neuroimaging initiative (ADNI) study4 , including
MRI and FDG-PET data. For this experiment, we used 93 AD patients, 202 MCI patients and 101
NC subjects. To process the data, same tools employed in [29] and [32] are used, including spatial
distortion, skull-stripping, and cerebellum removal. The FSL package [33] was used to segment
each MR image into three different tissues, i.e., GM, WM, and CSF. Then, 93 ROIs are parcellated
for each subject [25] with atlas warping. The volume of GM tissue in each ROI was calculated as
the image feature. For FDG-PET images, a rigid transformation was employed to align it to the
corresponding MR image and the mean intensity of each ROI was calculated as the feature. All
these features were further normalized in a similar way, as in [32].
Results: The first experiment is set up on the PPMI database. Table 1 shows the diagnosis accuracy
of the proposed technique (RFS-LDA) in comparisons with different baseline and state-of-the-art
methods, using a 10-fold cross-validation strategy. As can be seen, the proposed method outperforms
all others. This could be because our method deals with both feature-noises and sample-outliers.
Note that, subjects and their corresponding feature vectors extracted from MRI data are quite prone
to noise, because of many possible sources of noise (e.g. the patient?s body movements, RF emission
due to thermal motion, overall MR scanner measurement chain, or preprocessing artifacts). Therefore, some samples might not be useful (sample-outliers) and some might be contaminated by some
amounts of noise (feature-noises). Our method deals with both types and achieves good results.
The goal for the experiments on ADNI database is to discriminate both MCI and AD patients from
NC subjects, separately. Therefore, NC subjects form our negative class, while the positive class
is defined as AD in one experiment and MCI in the other. The diagnosis results of the AD vs. NC
and MCI vs. NC experiments are reported in Tables 2. As it could be seen, in comparisons with
the state-of-the-art, our method achieves good results in terms of both accuracy and the area under
curve. This is because we successfully discard the sample-outliers and detect the feature-noises.
3
4
Could be downloaded at http://www.nitrc.org/projects/hammerwml
http://www.loni.ucla.edu/ADNI
7
Table 2: The accuracy (ACC) and the area under ROC curve (AUC) of the Alzheimer?s disease
classification on ADNI database, compared to the baseline methods.
Method
ACC
AD/NC
AUC
ACC
MCI/NC
AUC
RFS-LDA
RFS-LDA?
S-RFS-LDA
RLDA
SFS+RLDA
RPCA+LS-LDA
LS-LDA
SVM
SFS+SVM
MC
91.8
0.98
89.8
0.93
89.1
0.96
85.6
0.90
86.3
0.95
84.5
0.90
88.7
0.96
85.0
0.87
90.1
0.98
88.1
0.92
87.6
0.93
84.5
0.87
70.9
0.81
68.9
0.75
72.1
0.80
70.1
0.79
76.3
0.83
76.1
0.80
78.2
0.82
74.3
0.78
Figure 3: The top selected ROIs for AD vs. NC (left) and MCI vs. NC (right) classification problems.
Discussions: In medical imaging applications, many sources of noise (e.g. patient?s movement,
radiations and limitation of imaging devices, preprocessing artifacts) contribute to the acquired data
[13], and therefore methods that deal with noise and outliers are of great interest. Our method
enjoys from a single optimization objective that can simultaneously suppress sample-outliers and
feature-noises, which compared to the competing methods, exhibits a good performance. One of
the interesting functions of the proposed method is the regularization on the mapping coefficients
with the `1 norm, which would select a compact set of features to contribute to the learned mapping.
The magnitude of the coefficients would show the level of contribution of that specific feature to the
learned model. In our application, the features from the whole brain regions are extracted, but only a
small number of regions are associated with the disease (e.g., AD, MCI or PD). Using this strategy,
we can determine which brain regions are highly associated with a certain disease.
Fig. 3 shows the top regions selected by our algorithm in AD vs. NC and MCI vs. NC classification
scenarios. These regions, including middle temporal gyrus, medial front-orbital gyrus, postcentral
gyrus, caudate nucleus, cuneus, and amygdala have been reported to be associated with AD and
MCI in the literature [24, 26]. The figures show the union of regions selected for both MRI and
FDG-PET features. The most frequently used regions for the PD/NC experiment are the substantial
nigra (left and right), putamen (right), middle frontal gyrus (right), superior temporal gyrus (left),
which are also consistent with the literature [4, 31]. This selection of brain regions could be further
incorporated for future clinical analysis.
The semi-supervised setting of the proposed method is also of great interest in the diagnosis of
patients. When new patients first arrive and are to be diagnosed, the previous set of the patients with
no certain diagnosis so far (not labeled yet), could still be used to build a more reliable classifier.
In other words, the current testing samples could contribute the diagnosis of future subjects, as
unlabeled samples.
4
Conclusion
In this paper, we proposed an approach for discriminative classification, which is robust against
both sample-outliers and feature-noises. Our method enjoys a semi-supervised setting, where all
the labeled training and the unlabeled testing data are used to detect outliers and are de-noised,
simultaneously. We have applied our method to the interesting problem of neurodegenerative brain
disease diagnosis and directly applied it for the diagnosis of Parkinson?s and Alzheimer?s diseases.
The results show that our method outperforms all competing methods. As a direction for the future
work, one can develop a multi-task learning reformulation of the proposed method to incorporate
multiple modalities for the subjects, or extend the method for the incomplete data case.
8
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9
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5,151 | 5,663 | Learning spatiotemporal trajectories from
manifold-valued longitudinal data
Jean-Baptiste Schiratti2,1 , St?ephanie Allassonni`ere2 , Olivier Colliot1 , Stanley Durrleman1
1
ARAMIS Lab, INRIA Paris, Inserm U1127, CNRS UMR 7225, Sorbonne Universit?es,
UPMC Univ Paris 06 UMR S 1127, Institut du Cerveau et de la Moelle e? pini`ere,
ICM, F-75013, Paris, France
2
CMAP, Ecole Polytechnique, Palaiseau, France
[email protected],
[email protected],
[email protected],[email protected]
Abstract
We propose a Bayesian mixed-effects model to learn typical scenarios of changes
from longitudinal manifold-valued data, namely repeated measurements of the
same objects or individuals at several points in time. The model allows to estimate
a group-average trajectory in the space of measurements. Random variations of
this trajectory result from spatiotemporal transformations, which allow changes in
the direction of the trajectory and in the pace at which trajectories are followed.
The use of the tools of Riemannian geometry allows to derive a generic algorithm
for any kind of data with smooth constraints, which lie therefore on a Riemannian
manifold. Stochastic approximations of the Expectation-Maximization algorithm
is used to estimate the model parameters in this highly non-linear setting. The
method is used to estimate a data-driven model of the progressive impairments of
cognitive functions during the onset of Alzheimer?s disease. Experimental results
show that the model correctly put into correspondence the age at which each individual was diagnosed with the disease, thus validating the fact that it effectively
estimated a normative scenario of disease progression. Random effects provide
unique insights into the variations in the ordering and timing of the succession of
cognitive impairments across different individuals.
1
Introduction
Age-related brain diseases, such as Parkinson?s or Alzheimer?s disease (AD) are complex diseases
with multiple effects on the metabolism, structure and function of the brain. Models of disease progression showing the sequence and timing of these effects during the course of the disease remain
largely hypothetical [3, 13]. Large databases have been collected recently in the hope to give experimental evidence of the patterns of disease progression based on the estimation of data-driven
models. These databases are longitudinal, in the sense that they contain repeated measurements of
several subjects at multiple time-points, but which do not necessarily correspond across subjects.
Learning models of disease progression from such databases raises great methodological challenges.
The main difficulty lies in the fact that the age of a given individual gives no information about the
stage of disease progression of this individual. The onset of clinical symptoms of AD may vary from
forty and eighty years of age, and the duration of the disease from few years to decades. Moreover,
the onset of the disease does not correspond with the onset of the symptoms: according to recent
studies, symptoms are likely to be preceded by a silent phase of the disease, for which little is
known. As a consequence, statistical models based on the regression of measurements with age are
inadequate to model disease progression.
1
The set of the measurements of a given individual at a specific time-point belongs to a highdimensional space. Building a model of disease progression amounts to estimating continuous
subject-specific trajectories in this space and average those trajectories among a group of individuals. Trajectories need to be registered in space, to account for the fact that individuals follow
different trajectories, and in time, to account for the fact that individuals, even if they follow the
same trajectory, may be at a different position on this trajectory at the same age.
The framework of mixed-effects models seems to be well suited to deal with this hierarchical problem. Mixed-effects models for longitudinal measurements were introduced in the seminal paper of
Laird and Ware [15] and have been widely developed since then (see [6], [16] for instance). However, this kind of models suffers from two main drawbacks regarding our problem. These models
are built on the estimation of the distribution of the measurements at a given time point. In many
situations, this reference time is given by the experimental set up: date at which treatment begins,
date of seeding in studies of plant growth, etc. In studies of ageing, using these models would require to register the data of each individual to a common stage of disease progression before being
compared. Unfortunately, this stage is unknown and such a temporal registration is actually what
we wish to estimate. Another limitation of usual mixed-effects models is that they are defined for
data lying in Euclidean spaces. However, measurements with smooth constraints usually cannot be
summed up or scaled, such as normalized scores of neurospychological tests, positive definite symmetric matrices, shapes encoded as images or meshes. These data are naturally modeled as points on
Riemannian manifolds. Although the development of statistical models for manifold-valued data is
a blooming topic, the construction of statistical models for longitudinal data on a manifold remains
an open problem.
The concept of ?time-warp? was introduced in [8] to allow for temporal registration of trajectories
of shape changes. Nevertheless, the combination of the time-warps with the intrinsic variability of
shapes across individuals is done at the expense of a simplifying approximation: the variance of
shapes does not depend on time whereas it should adapt with the average scenario of shape changes.
Moreover, the estimation of the parameters of the statistical model is made by minimizing a sum of
squares which results from an uncontrolled likelihood approximation. In [18], time-warps are used
to define a metric between curves that are invariant under time reparameterization. This invariance,
by definition, prevents the estimation of correspondences across trajectories, and therefore the estimation of distribution of trajectories in the spatiotemporal domain. In [17], the authors proposed a
model for longitudinal image data but the model is not built on the inference of a statistical model
and does not include a time reparametrization of the estimated trajectories.
In this paper, we propose a generic statistical framework for the definition and estimation of mixedeffects models for longitudinal manifold-valued data. Using the tools of geometry allows us to derive
a method that makes little assumptions about the data and problem to deal with. Modeling choices
boil down to the definition of the metric on the manifold. This geometrical modeling also allows
us to introduce the concept of parallel curves on a manifold, which is key to uniquely decompose
differences seen in the data in a spatial and a temporal component. Because of the non-linearity
of the model, the estimation of the parameters should be based on an adequate maximization of the
observed likelihood. To address this issue, we propose to use a stochastic version of the ExpectationMaximization algorithm [5], namely the MCMC SAEM [2], for which theoretical results regarding
the convergence have been proved in [4], [2].
Experimental results on neuropsychological tests scores and estimates of scenarios of AD progression are given in section 4.
2
2.1
Spatiotemporal mixed-effects model for manifold-valued data
Riemannian geometry setting
The observed data consists in repeated multivariate measurements of p individuals. For a given
individual, the measurements are obtained at time points ti,1 < . . . < ti,ni . The j-th measurement
of the i-th individual is denoted by yi,j . We assume that each observation yi,j is a point on a
N -dimensional Riemannian manifold M embedded in RP (with P ? N ) and equipped with a
Riemannian metric g M . We denote ?M the covariant derivative. We assume that the manifold is
geodesically complete, meaning that geodesics are defined for all time.
2
We recall that a geodesic is a curve drawn on the manifold ? : R ? M, which has no acceleration:
? = 0. For a point p ? M and a vector v ? Tp M, the mapping ExpM
?M
p (v) denotes the
?? ?
Riemannian exponential, namely the point that is reached at time 1 by the geodesic starting at p
with velocity v. The parallel transport of a vector X0 ? T?(t0 ) M in the tangent space at point ?(t0 )
on a curve ? is a time-indexed family of vectors X(t) ? T?(t) M which satisfies ?M
? X(t) = 0
?(t)
and X(t0 ) = X0 . We denote P?,t0 ,t (X0 ) the isometry that maps X0 to X(t).
In order to describe our model, we need to introduce the notion of ?parallel curves? on the manifold:
Definition 1. Let ? be a curve on M defined for all time, a time-point t0 ? R and a vector w ?
T?(t0 ) M, w 6= 0. One defines the curve s ? ? w (?, s), called parallel to the curve ?, as:
? w (?, s) = ExpM
?(s) P?,t0 ,s (w) , s ? R.
The idea is illustrated in Fig. 1. One uses the parallel transport to move the vector w from ?(t0 ) to
?(s) along ?. At the point ?(s), a new point on M is obtained by taking the Riemannian exponential
of P?,t0 ,s (w). This new point is denoted by ? w (?, s). As s varies, one describes a curve ? w (?, ?)
on M, which can be understood as a ?parallel? to the curve ?. It should be pointed out that, even
if ? is a geodesic, ? w (?, ?) is, in general, not a geodesic of M. In the Euclidean case (i.e. a flat
manifold), the curve ? w (?, ?) is the translation of the curve ?: ? w (?, s) = ?(s) + w.
Figure a)
Figure b)
Figure c)
Figure 1: Model description on a schematic manifold. Figure a) (left) : a non-zero vector wi is
choosen in T?(t0 ) M. Figure b) (middle) : the tangent vector wi is transported along the geodesic
? and a point ? wi (?, s) is constructed at time s by use of the Riemannian exponential. Figure c)
(right) : The curve ? wi (?, ?) is the parallel resulting from the construction.
2.2
Generic spatiotemporal model for longitudinal data
Our model is built in a hierarchical manner: data points are seen as samples along individual trajectories, and these trajectories derive from a group-average trajectory. The model writes
yi,j = ? wi (?, ?i (ti,j )) + ?i,j , where we assume the group-average trajectory to be a geodesic,
denoted ? from now on. Individual trajectories derive from the group average by spatiotemporal
transformations. They are defined as a time re-parameterization of a trajectory that is parallel to the
group-average: t ? ? wi (?, ?i (t)). For the ith individual, wi denotes a non-zero tangent vector in
T?(t0 ) M, for some specific time point t0 that needs to be estimated, and which is orthogonal to the
M
? 0 ) for the inner product given by the metric (h?, ?i?(t0 ) = g?(t
tangent vector ?(t
). The time-warp
0)
function ?i is defined as: ?i (t) = ?i (t ? t0 ? ?i ) + t0 . The parameter ?i is an acceleration factor
which encodes whether the i-th individual is progressing faster or slower than the average, ?i is a
time-shift which characterizes the advance or delay of the ith individual with respect to the average
and wi is a space-shift which encodes the variability in the measurements across individuals at the
same stage of progression.
The normal tubular neighborhood theorem ([11]) ensures that parallel shifting defines a spatiotemporal coordinate system as long as the vectors wi are choosen orthogonal and sufficently small.
The orthogonality condition on the tangent vectors wi is necessary to ensure the identifiability of
the model. Indeed, if a vector wi was not choosen orthogonal, its orthogonal projection would play
the same role as the acceleration factor.The spatial and temporal transformations commute, in the
sense that one may re-parameterize the average trajectory before building the parallel curve, or vice
versa. Mathematically, this writes ? wi (? ? ?i , s) = ? wi (?, ?i (s)). This relation also explains the
particular form of the affine time-warp ?i . The geodesic ? is characterized by the fact that it passes
3
at time-point t0 by point p0 = ?(t0 ) with velocity v0 = ?(t
? 0 ). Then, ? ? ?i is the same trajectory,
except that it passes by point p0 at time t0 + ?i with velocity ?i v0 .
The fixed effects of the model are the parameters of the average geodesic: the point p0 on the
manifold, the time-point t0 and the velocity v0 . The random effects are the acceleration factors
?i , time-shifts ?i and space-shifts wi . The first two random effects are scalars. One assumes the
acceleration factors to follow a log-normal distribution (they need to be positive in order not to
reverse time), and time-shifts to follow a zero-mean Gaussian distribution. Space-shifts are vectors
? 0 )? in T?(t0 ) M. In the spirit of independent component
of dimension N ? 1 in the hyperplane ?(t
analysis [12], we assume that wi ?s result from the superposition of Ns < N statistically independent
components. This writes wi = Asi where A is a N ? Ns matrix of rank Ns , si a vector of
Ns independent sources following a heavy tail Laplace distribution with fixed parameter, and each
? 0 )i?(t0 ) = 0.
column cj (A) (1 ? j ? Ns ) of A satisfies the orthogonality condition hcj (A), ?(t
For the dataset (ti,j , yi,j ) (1 ? i ? p, 1 ? j ? ni ), the model may be summarized as:
yi,j = ? wi (?, ?i (ti,j )) + ?i,j .
(1)
with ?i (t) = ?i (t ? t0 ? ?i ) + t0 , ?i = exp(?i ), wi = Asi and
i.i.d.
i.i.d.
i.i.d.
i.i.d.
?i ? N (0, ??2 ), ?i ? N (0, ??2 ), ?i,j ? N (0, ? 2 IN ), si,l ? Laplace(1/2).
Eventually, the parameters of the model one needs to estimate are the fixed effects and the variance
of the random effects, namely ? = (p0 , t0 , v0 , ?? , ?? , ?, vec(A)).
2.3
Propagation model in a product manifold
We wish to use these developments to study the temporal progression of a family of biomarkers.
We assume that each component of yi,j is a scalar measurement of a given biomarker and belongs
to a geodesically complete one-dimensional manifold (M, g). Therefore, each measurement yi,j is
a point in the product manifold M = M N , which we assume to be equipped with the Riemannian
product metric g M = g + . . . + g. We denote ? 0 the geodesic of the one-dimensional manifold
M which goes through the point p0 ? M at time t0 with velocity v0 ? Tp0 M . In order to determine relative progression of the biomarkers among themselves,
we consider a parametric family of
geodesics of M : ? ? (t) = ?0 (t), ?0 (t+?1 ), . . . , ?0 (t+?N ?1 ) . We assume here that all biomarkers
have on average the same dynamics but shifted in time. This hypothesis allows to model a temporal
succession of effects during the course of thedisease. The relative timing in biomarker changes is
measured by the vector ? = 0, ?1 , . . . , ?N ?1 , which becomes a fixed effect of the model.
In this setting, a curve that is parallel to a geodesic ? is given by the following lemma :
Lemma 1. Let ? be a geodesic of the product manifold M = M N and let t0 ? R. If
? w (?, ?) denotes a parallel to the geodesic ? with w = (w1 , . . . , wN ?T?(t0 ) M and ?(t) =
wN
w1
(?1 (t), . . . , ?N (t)), we have ? w (?, s) = ?1 ?(t
? 0 ) + s , . . . , ?N ?(t
? 0 ) + s , s ? R.
As a consequence, a parallel to the average trajectory ? ? has the same form as the geodesic but with
randomly perturbed delays. The model (1) writes : for all k ? {1, . . . , N },
wk,i
yi,j,k = ?0
+ ?i (ti,j ? t0 ? ?i ) + t0 + ?k?1 + ?i,j,k .
(2)
??0 (t0 + ?k?1 )
where wk,i denotes the k-th component of the space-shift wi and yi,j,k , the measurement of the k-th
biomarker, at the j-th time point, for the i-th individual.
2.4
Multivariate logistic curves model
The propagation model given in (2) is now described for normalized biomarkers, such as scores of
neuropsychological tests. In this case, we assume the manifold to be M =]0, 1[ and equipped with
the Riemannian metric g given by : for p ?]0, 1[, (u, v) ? Tp M ? Tp M , gp (u, v) = uG(p)v with
G(p) = 1/(p2 (1 ? p)2 ). The geodesics given by this metric in the one-dimensional Riemannian
?1
v0
(t ? t0 )
manifold M are logistic curves of the form : ?0 (t) = 1 + ( p10 ? 1) exp ? p0 (1?p
0)
and leads to the multivariate logistic curves model in M. We can notice the quite unusual paramaterization of the logistic curve. This parametrization naturally arise because ?0 satisfies : ?0 (t0 ) = p0
and ??0 (t0 ) = v0 . In this case, the model (1) writes:
4
yi,j,k =
1
1+
? 1 exp
p0
?
i )k
v0 ?i (ti,j ? t0 ? ?i ) + v0 ?k + v0 ??0(As
(t0 +?k )
p0 (1 ? p0 )
!!?1
+ ?i,j,k , (3)
where (Asi )k denotes the k-th component of the vector Asi . Note that (3) is not equivalent to a
linear model on the logit of the observations. The logit transform corresponds to the Riemannian
logarithm at p0 = 0.5. In our framework, p0 is not fixed, but estimated as a parameter of our
model. Even with a fixed p0 = 0.5, the model is still non-linear due to the multiplication between
random-effects ?i and ?i , and therefore does not boil down to the usual linear model [15].
3
Parameters estimation
In this section, we explain how to use a stochastic version of the Expectation-Maximization (EM)
algorithm [5] to produce estimates of the parameters ? = (p0 , t0 , v0 , ?, ?? , ?? , ?, vec(A)) of the
model. The algorithm detailed in this section is essentially the same as in [2]. Its scope of application is not limited to statistical models on product manifolds and the MCMC-SAEM algorithm can
actually be used for the inference of a very large family of statistical models.
The random effects z = (?i , ?i , sj,i ) (1 ? i ? p and 1 ? j ? Ns ) are considered as hidden variables.
With the observed data y = (yi,j,k )i,j,k , (y, z) form the complete data of the model. In this context,
the Expectation-Maximization (EM) algorithm proposed in [5] is very efficient to compute the maximum likelihood estimate of ?. Due to the nonlinearity and complexity of the model, the E step is
intractable. As a consequence, we considered a stochastic version of the EM algorithm, namely the
Monte-Carlo Markov Chain Stochastic Approximation Expectation-Maximization (MCMC-SAEM)
algorithm [2], based on [4]. This algorithm is an EM-like algorithm which alternates between three
steps: simulation, stochastic approximation and maximization. If ? (t) denotes the current parameter
estimates of the algorithm, in the simulation step, a sample z(t) of the missing data is obtained from
the transition kernel of an ergodic Markov chain whose stationary distribution is the conditional
distribution of the missing data z knowing y and ? (t) , denoted by q(z | y, ? (t) ). This simulation
step is achieved using Hasting-Metropolis within Gibbs sampler. Note that the high complexity of
our model prevents us from resorting to sampling methods as in [10] as they would require heavy
computations, such as the Fisher information matrix. The stochastic approximation step consists in
a stochastic approximation on the complete log-likelihood log q(y, z | ?) summarized as follows :
Qt (?) = Qt?1 (?) + ?t [log q(y, zP
| ?) ? Qt?1 (?)], where
P (?t )t is a decreasing sequence of positive
step-sizes in ]0, 1] which satisfies t ?t = +? and t ?2t < +?. Finally, the parameter estimates
are updated in the maximization step according to: ? (t+1) = argmax??? Qt (?).
The theoretical convergence of the MCMC SAEM algorithm is proved only if the model belong to the curved exponential family. Or equivalently, if the complete log-likelihood of the
model may be written : log q(y, z | ?) = ??(?) + S(y, z)> ?(?), where S(y, z) is a sufficent
statistic of the model. In this case, the stochastic approximation on the complete log-likelihood
can be replaced with a stochastic approximation on the sufficent statistics of the model. Note
that the multivariate logistic curves model does not belong to the curved exponential family. A
usual workaround consists in regarding the parameters of the model as realizations of independents Gaussian random variables ([14]) : ? ? N (?, D) where D is a diagonal matrix with
very small diagonal entries and the estimation now targets ?. This yields: p0 ? N (p0 , ?p20 ),
t0 ? N (t0 , ?t20 ), v0 ? N (v0 , ?v20 ) and, for all k, ?k ? N (? k , ??2 ). To ensure the orthogonality condition on the columns of A, we assumed that A follows a normal distribution on the
N s
space ? = {A = (c1 (A), . . . , cNs (A)) ? T?? (t0 ) M
; ?j, hcj (A), ??? (t0 )i?? (t0 ) = 0}.
P(N ?1)Ns
Equivalently, we assume that the matrix A writes : A =
?k Bk where, for all k,
k=1
i.i.d.
?k ? N (? k , ??2 ) and (B1 , . . . , B(N ?1)Ns ) is an orthonormal basis of ? obtained by application of the Gram-Schmidt process to a basis of ?. The random variables ?1 , . . . , ?(N ?1)Ns
are considered as new hidden variables of the model. The parameters of the model are ? =
(p0 , t0 , v0 , (? k )1?k?N ?1 , (? k )1?k?(N ?1)Ns , ?? , ?? , ?) whereas the hidden variables of the model
are z = (p0 , t0 , v0 , (?k )1?k?N ?1 , (?k )1?k?(N ?1)Ns , (?i )1?i?p , (?i )1?i?p , (sj,i )1?j?Ns , 1?i?p ).
The algorithm (1) given below summarizes the SAEM algorithm for this model.
The MCMC-SAEM algorithm 1 was tested on synthetic data generated according to (3). The
MCMC-SAEM allowed to recover the parameters used to generate the synthetic dataset.
5
Algorithm 1 Overview of the MCMC SAEM algorithm for the multivariate logistic curves model.
(k) (k)
(k)
If z(k) = p0 , t0 , . . . , (sj,i denotes the vector of hidden variables obtained in the simulation step of the k-th iteration of the MCMC SAEM, let fi,j = [fi,j,l ] ? RN and fi,j,l
(k)
(k)
(k)
(k)
(k)
be the l-th component of ?w(k) (? ? )(k) , exp(?i )(ti,j ? t0 ? ?i ) + t0
and wi
=
i
P(N ?1)Ns (k)
?l Bl .
l=1
Initialization :
? ? ? (0) ; z(0) ? random ; S ? 0 ; (?k )k?0 .
repeat
(k) (k)
(k)
Simulation step : z(k) = p0 , t0 , . . . , (sj,i )j,i ? Gibbs Sampler(z(k?1) , y, ? (k?1) )
(k)
>
fi,j i,j ? RK ; S(k)
Compute the sufficent statistics : S1 ? yi,j
? kfi,j k2 i,j ? RK
2
i
h
Pp
(k)
(k)
(k) 2
? Rp ; S4 ?
with (1 ? i ? p ; 1 ? j ? ni ) and K =
n
;
S
=
(?
)
i
3
i=1
i
i
hi
i
h
(k)
(k)
(k)
(k)
(k)
(k)
(k)
(k)
(k)
? RN ?1 ;
(?i )2 ? Rp ; S5 ? p0 ; S6 ? t0 ; S7 ? v0 ; S8 ? ?j
j
hi i
(k)
S9 ? ?j(k) ? R(N ?1)Ns .
j
(k+1)
Stochastic approximation step : Sj
(k)
Maximization step : p0 (k+1) ? S5
(k+1)
(k)
? Sj
; t0
(k+1)
(k)
(k)
+ ?k (S(y, z(k) ) ? Sj ) for j ? {1, . . . , 9}.
(k)
? S6
(k)
; v0 (k+1) ? S7
(k+1)
; ?j
(k+1)
(k)
? (S8 )j
(k)
? (S9 )j for all 1 ? j ? (N ? 1)Ns ; ??
? p1 (S3 )> 1p
1/2
P
(k) >
(k) >
(k+1)
(k)
2
? p1 (S4 )> 1p ; ? (k+1) ? ?N1 K
.
; ??
i,j,k yi,j,k ? 2(S1 ) 1K + (S2 ) 1K
until convergence.
return ?.
for all 1 ? j ? N ? 1 ; ? j
4
4.1
Experiments
Data
We use the neuropsychological assessment test ?ADAS-Cog 13? from the ADNI1, ADNIGO or
ADNI2 cohorts of the Alzheimer?s Disease Neuroimaging Initiative (ADNI) [1]. The ?ADAS-Cog
13? consists of 13 questions, which allow to test the impairment of several cognitive functions.
For the purpose of our analysis, these items are grouped into four categories: memory (5 items),
language (5 items), praxis (2 items) and concentration (1 item). Scores within each category are
added and normalized by the maximum possible score. Consequently, each data point consists in
four normalized scores, which can be seen as a point on the manifold M =]0, 1[4 .
We included 248 individuals in the study, who were diagnosed with mild cognitive impairment
(MCI) at their first visit and whose diagnosis changed to AD before their last visit. There is an average of 6 visits per subjects (min: 3, max: 11), with an average duration of 6 or 12 months between
consecutive visits. The multivariate logistic curves model was used to analyze this longitudinal data.
4.2
Experimental results
The model was applied with Ns = 1, 2 or 3 independent sources. In each experiment, the MCMC
SAEM was run five times with different initial parameter values. The experiment which returned
the smallest residual variance ? 2 was kept. The maximum number of iterations was arbitrarily set
to 5000 and the number of burn-in iterations was set to 3000 iterations. The limit of 5000 iterations
is enough to observe the convergence of the sequences of parameters estimates. As a result, two
and three sources allowed to decrease the residual variance better than one source (? 2 = 0.012 for
one source, ? 2 = 0.08 for two sources and ? 2 = 0.084 for three sources). The residual variance
? 2 = 0.012 (resp. ? 2 = 0.08, ? 2 = 0.084) mean that the model allowed to explain 79% (resp.
84%, 85%) of the total variance. We implemented our algorithm in MATLAB without any particular
optimization scheme. The 5000 iterations require approximately one day.
The number of parameters to be estimated is equal to 9 + 3Ns . Therefore, the number of sources
do not dramatically impact the runtime. Simulation is the most computationally expensive part of
6
our algorithm. For each run of the Hasting-Metropolis algorithm, the proposal distribution is the
prior distribution. As a consequence, the acceptation ratio simplifies [2] and one computation of
the acceptation ratio requires two computations of the likelihood of the observations, conditionally
on different vectors of latent variables and the vector of current parameters estimates. The runtime
could be improved by parallelizing the sampling per individuals.
For a matter of clarity and because the results obtained with three sources were similar to the results
with two sources, we report here the experimental results obtained with two independent sources.
The average model of disease progression ? ? is plotted in Fig. 2. The estimated fixed effects are
p0 = 0.3, t0 = 72 years, v0 = 0.04 unit per year, and ? = [0; ?15; ?13; ?5] years. This means
that, on average, the memory score (first coordinate) reaches the value p0 = 0.3 at t0 = 72 years,
followed by concentration which reaches the same value at t0 + 5 = 77 years, and then by praxis
and language at age 85 and 87 years respectively.
Random effects show the variability of this average trajectory within the studied population. The
standard deviation of the time-shift equals ?? = 7.5 years, meaning that the disease progression
model in Fig. 2 is shifted by ?7.5 years to account for the variability in the age of disease onset.
The effects of the variance of the acceleration factors, and the two independent components of the
space-shifts are illustrated in Fig. 4. The acceleration factors shows the variability in the pace of
disease progression, which ranges between 7 times faster and 7 times slower than the average. The
first independent component shows variability in the relative timing of the cognitive impairments:
in one direction, memory and concentration are impaired nearly at the same time, followed by
language and praxis; in the other direction, memory is followed by concentration and then language
and praxis are nearly superimposed. The second independent component keeps almost fixed the
timing of memory and concentration, and shows a great variability in the relative timing of praxis
and language impairment. It shows that the ordering of the last two may be inverted in different
individuals. Overall, these space-shift components show that the onset of cognitive impairment
tends to occur by pairs: memory & concentration followed by language & praxis.
Individual estimates of the random effects are obtained from the simulation step of the last iteration
of the algorithm and are plotted in Fig. 5. The figure shows that the estimated individual time-shifts
correspond well to the age at which individuals were diagnosed with AD. This means that the value
p0 estimated by the model is a good threshold to determine diagnosis (a fact that has occurred by
chance), and more importantly that the time-warp correctly register the dynamics of the individual
trajectories so that the normalized age correspond to the same stage of disease progression across
individuals. This fact is corroborated by Fig. 3 which shows that the normalized age of conversion
to AD is picked at 77 years old with a small variance compared to the real distribution of age of
conversion.
Figure 2: The four curves represent the estimated average trajectory. A vertical line is
drawn at t0 = 72 years old and an horizontal
line is drawn at p0 = 0.3.
Figure 3: In blue (resp. red) : histogram
of the ages of conversion to AD (tdiag
i )
(resp. normalized ages of conversion to AD
(?i (tdiag
i ))), with ?i time-warp as in (1).
7
Acceleration factor ??
Independent component ?1
Independent component ?2
-?
+?
Figure 4: Variability in disease progression superimposed with the average trajectory
? ? (dotted
lines): effects of the acceleration factor with plots of ? ? exp(??? )(t ? t0 ) + t0 (first column),
first and second independent component of space-shift with plots of ? ??si ci (A) (? ? , ?) for i = 1 or
2 (second and third column respectively).
Figure 5: Plots of individual random effects:
log-acceleration factor ?i = log(?i ) against
time-shifts t0 + ?i . Color corresponds to the
age of conversion to AD.
4.3
Discussion and perspectives
We proposed a generic spatiotemporal model to analyze longitudinal manifold-valued measurements. The fixed effects define a group-average trajectory, which is a geodesic on the data manifold.
Random effects are subject-specific acceleration factor, time-shift and space-shift which provide insightful information about the variations in the direction of the individual trajectories and the relative
pace at which they are followed.
This model was used to estimate a normative scenario of Alzheimer?s disease progression from
neuropsychological tests. We validated the estimates of the spatiotemporal registration between
individual trajectories by the fact that they put into correspondence the same event on individual
trajectories, namely the age at diagnosis. Alternatives to estimate model of disease progression
include the event-based model [9], which estimates the ordering of categorical variables. Our model
may be seen as a generalization of this model for continuous variables, which do not only estimate
the ordering of the events but also the relative timing between them. Practical solutions to combine
spatial and temporal sources of variations in longitudinal data are given in [7]. Our goal was here to
propose theoretical and algorithmic foundations for the systematic treatment of such questions.
8
References
[1] The Alzheimer?s Disease Neuroimaging Initiative, https://ida.loni.usc.edu/
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approximation algorithm: a convergence study. Bernoulli 16(3), 641?678 (2010)
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aging 16(3), 271?278 (1995)
[4] Delyon, B., Lavielle, M., Moulines, E.: Convergence of a stochastic approximation version of the em
algorithm. Annals of statistics pp. 94?128 (1999)
[5] Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the em algorithm. Journal of the royal statistical society. Series B (methodological) pp. 1?38 (1977)
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(2002)
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[8] Durrleman, S., Pennec, X., Trouv?e, A., Braga, J., Gerig, G., Ayache, N.: Toward a comprehensive framework for the spatiotemporal statistical analysis of longitudinal shape data. International Journal of Computer Vision 103(1), 22?59 (2013)
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[10] Girolami, M., Calderhead, B.: Riemann manifold langevin and hamiltonian monte carlo methods. Journal
of the Royal Statistical Society: Series B (Statistical Methodology) 73(2), 123?214 (2011)
[11] Hirsch, M.W.: Differential topology. Springer Science & Business Media (2012)
[12] Hyv?arinen, A., Karhunen, J., Oja, E.: Independent component analysis, vol. 46. John Wiley & Sons
(2004)
[13] Jack, C.R., Knopman, D.S., Jagust, W.J., Shaw, L.M., Aisen, P.S., Weiner, M.W., Petersen, R.C., Trojanowski, J.Q.: Hypothetical model of dynamic biomarkers of the alzheimer?s pathological cascade. The
Lancet Neurology 9(1), 119?128 (2010)
[14] Kuhn, E., Lavielle, M.: Maximum likelihood estimation in nonlinear mixed effects models. Computational Statistics & Data Analysis 49(4), 1020?1038 (2005)
[15] Laird, N.M., Ware, J.H.: Random-effects models for longitudinal data. Biometrics pp. 963?974 (1982)
[16] Singer, J.D., Willett, J.B.: Applied longitudinal data analysis: Modeling change and event occurrence.
Oxford university press (2003)
[17] Singh, N., Hinkle, J., Joshi, S., Fletcher, P.T.: A hierarchical geodesic model for diffeomorphic longitudinal shape analysis. In: Information Processing in Medical Imaging. pp. 560?571. Springer (2013)
[18] Su, J., Kurtek, S., Klassen, E., Srivastava, A., et al.: Statistical analysis of trajectories on riemannian
manifolds: Bird migration, hurricane tracking and video surveillance. The Annals of Applied Statistics
8(1), 530?552 (2014)
9
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5,152 | 5,664 | Hessian-free Optimization for Learning
Deep Multidimensional Recurrent Neural Networks
Minhyung Cho
Chandra Shekhar Dhir
Jaehyung Lee
Applied Research Korea, Gracenote Inc.
{mhyung.cho,shekhardhir}@gmail.com
[email protected]
Abstract
Multidimensional recurrent neural networks (MDRNNs) have shown a remarkable performance in the area of speech and handwriting recognition. The performance of an MDRNN is improved by further increasing its depth, and the difficulty of learning the deeper network is overcome by using Hessian-free (HF)
optimization. Given that connectionist temporal classification (CTC) is utilized as
an objective of learning an MDRNN for sequence labeling, the non-convexity of
CTC poses a problem when applying HF to the network. As a solution, a convex
approximation of CTC is formulated and its relationship with the EM algorithm
and the Fisher information matrix is discussed. An MDRNN up to a depth of 15
layers is successfully trained using HF, resulting in an improved performance for
sequence labeling.
1
Introduction
Multidimensional recurrent neural networks (MDRNNs) constitute an efficient architecture for
building a multidimensional context into recurrent neural networks [1]. End-to-end training of
MDRNNs in conjunction with connectionist temporal classification (CTC) has been shown to
achieve a state-of-the-art performance in on/off-line handwriting and speech recognition [2, 3, 4].
In previous approaches, the performance of MDRNNs having a depth of up to five layers, which
is limited as compared to the recent progress in feedforward networks [5], was demonstrated. The
effectiveness of MDRNNs deeper than five layers has thus far been unknown.
Training a deep architecture has always been a challenging topic in machine learning. A notable
breakthrough was achieved when deep feedforward neural networks were initialized using layerwise pre-training [6]. Recently, approaches have been proposed in which supervision is added to
intermediate layers to train deep networks [5, 7]. To the best of our knowledge, no such pre-training
or bootstrapping method has been developed for MDRNNs.
Alternatively, Hesssian-free (HF) optimization is an appealing approach to training deep neural
networks because of its ability to overcome pathological curvature of the objective function [8].
Furthermore, it can be applied to any connectionist model provided that its objective function is
differentiable. The recent success of HF for deep feedforward and recurrent neural networks [8, 9]
supports its application to MDRNNs.
In this paper, we claim that an MDRNN can benefit from a deeper architecture, and the application of
second order optimization such as HF allows its successful learning. First, we offer details of the development of HF optimization for MDRNNs. Then, to apply HF optimization for sequence labeling
tasks, we address the problem of the non-convexity of CTC, and formulate a convex approximation.
In addition, its relationship with the EM algorithm and the Fisher information matrix is discussed.
Experimental results for offline handwriting and phoneme recognition show that an MDRNN with
HF optimization performs better as the depth of the network increases up to 15 layers.
1
2
Multidimensional recurrent neural networks
MDRNNs constitute a generalization of RNNs to process multidimensional data by replacing the
single recurrent connection with as many connections as the dimensions of the data [1]. The network
can access the contextual information from 2N directions, allowing a collective decision to be made
based on rich context information. To enhance its ability to exploit context information, long shortterm memory (LSTM) [10] cells are usually utilized as hidden units. In addition, stacking MDRNNs
to construct deeper networks further improves the performance as the depth increases, achieving the
state-of-the-art performance in phoneme recognition [4]. For sequence labeling, CTC is applied as
a loss function of the MDRNN. The important advantage of using CTC is that no pre-segmented
sequences are required, and the entire transcription of the input sample is sufficient.
2.1
Learning MDRNNs
A d-dimensional MDRNN with M inputs and K outputs is regarded as a mapping from an input
sequence x ? RM ?T1 ?????Td to an output sequence a ? (RK )T of length T , where the input data for
M input neurons are given by the vectorization of d-dimensional data, and T1 , . . . , Td is the length
of the sequence in each dimension. All learnable weights and biases are concatenated to obtain a
parameter vector ? ? RN . In the learning phase with fixed training data, the MDRNN is formalized
as a mapping N : RN ? (RK )T from the parameters ? to the output sequence a, i.e., a = N (?).
The scalar loss function is defined over the output sequence as L : (RK )T ? R. Learning an
MDRNN is viewed as an optimization of the objective L(N (?)) = L ? N (?) with respect to ?.
The Jacobian JF of a function F : Rm ? Rn is the n ? m matrix where each element is a partial
derivative of an element of output with respect to an element of input. The Hessian HF of a scalar
function F : Rm ? R is the m ? m matrix of second-order partial derivatives of the output with
respect to its inputs. Throughout this paper, a vector sequence is denoted by boldface a, a vector at
time t in a is denoted by at , and the k-th element of at is denoted by atk .
3
Hessian-free optimization for MDRNNs
The application of HF optimization to an MDRNN is straightforward if the matching loss function [11] for its output layer is adopted. However, this is not the case for CTC, which is necessarily
adopted for sequence labeling. Before developing an appropriate approximation to CTC that is compatible with HF optimization, we discuss two considerations related to the approximation. The first
is obtaining a quadratic approximation of the loss function, and the second is the efficient calculation
of the matrix-vector product used at each iteration of the conjugate gradient (CG) method.
HF optimization minimizes an objective by constructing a local quadratic approximation for the
objective function and minimizing the approximate function instead of the original one. The loss
function L(?) needs to be approximated at each point ?n of the n-th iteration:
1 >
Qn (?) = L(?n ) + ?? L|>
(1)
?n ?n + ?n G?n ,
2
where ?n = ? ? ?n is the search direction, i.e., the parameters of the optimization, and G is a
local approximation to the curvature of L(?) at ?n , which is typically obtained by the generalized
Gauss-Newton (GGN) matrix as an approximation of the Hessian.
HF optimization uses the CG method in a subroutine to minimize the quadratic objective above for
utilizing the complete curvature information and achieving computational efficiency. CG requires
the computation of Gv for an arbitrary vector v, but not the explicit evaluation of G. For neural
networks, an efficient way to compute Gv was proposed in [11], extending the study in [12]. In
section 3.2, we provide the details of the efficient computation of Gv for MDRNNs.
3.1
Quadratic approximation of loss function
The Hessian matrix, HL?N , of the objective L (N (?)) is written as
>
HL?N = JN
HL JN +
KT
X
i=1
2
[JL ]i H[N ]i ,
(2)
where JN ? RKT ?N , HL ? RKT ?KT , and [q]i denotes the i-th component of the vector q.
An indefinite Hessian matrix is problematic for second-order optimization, because it defines an
unbounded local quadratic approximation [13]. For nonlinear systems, the Hessian is not necessarily
positive semidefinite, and thus, the GGN matrix is used as an approximation of the Hessian [11, 8].
The GGN matrix is obtained by ignoring the second term in Eq. (2), as given by
>
GL?N = JN
HL J N .
(3)
The sufficient condition for the GGN approximation to be exact is that the network makes a perfect
prediction for every given sample, that is, JL = 0, or [N ]i stays in the linear region for all i, that is,
H[N ]i = 0.
GL?N has less rank than KT and is positive semidefinite provided that HL is. Thus, L is chosen to
be a convex function so that HL is positive semidefinite. In principle, it is best to define L and N
such that L performs as much of the computation as possible, with the positive semidefiniteness of
HL as a minimum requirement [13]. In practice, a nonlinear output layer together with its matching
loss function [11], such as the softmax function with cross-entropy loss, is widely used.
3.2
Computation of matrix-vector product for MDRNN
>
HL JN v, amounts to the seThe product of an arbitrary vector v by the GGN matrix, Gv = JN
quential multiplication of v by three matrices. First, the product JN v is a Jacobian times vector
and is therefore equal to the directional derivative of N (?) along the direction of v. Thus, JN v can
be written using a differential operator JN v = Rv (N (?)) [12] and the properties of the operator
can be utilized for efficient computation. Because an MDRNN is a composition of differentiable
components, the computation of Rv (N (?)) throughout the whole network can be accomplished by
repeatedly applying the sum, product, and chain rules starting from the input layer. The detailed
derivation of the R operator to LSTM, normally used as a hidden unit in MDRNNs, is provided in
appendix A.
Next, the multiplication of JN v by HL can be performed by direct computation. The dimension
of HL could at first appear problematic, since the dimension of the output vector used by the loss
function L can be as high as KT , in particular, if CTC is adopted as an objective for the MDRNN.
If the loss function can be expressed as the sum of individual loss functions with a domain restricted
in time, the computation can be reduced significantly. For example, with the commonly used crossentropy loss function, the KT ? KT matrix HL can be transformed into a block diagonal matrix
with T blocks of a K ? K Hessian matrix. Let HL,t be the t-th block in HL . Then, the GGN matrix
can be written as
X
>
GL?N =
JN
HL,t JNt ,
(4)
t
t
where JNt is the Jacobian of the network at time t.
>
is calculated using the backFinally, the multiplication of a vector u = HL JN v by the matrix JN
propagation through time algorithm by propagating u instead of the error at the output layer.
4
Convex approximation of CTC for application to HF optimization
Connectioninst temporal classification (CTC) [14] provides an objective function of learning an
MDRNN for sequence labeling. In this section, we derive a convex approximation of CTC inspired
by the GGN approximation according to the following steps. First, the non-convex part of the
original objective is separated out by reformulating the softmax part. Next, the remaining convex
part is approximated without altering its Hessian, making it well matched to the non-convex part.
Finally, the convex approximation is obtained by reuniting the convex and non-convex parts.
4.1
Connectionist temporal classification
CTC is formulated as the mapping from an output sequence of the recurrent network, a ? (RK )T ,
to a scalar loss. The output activations at time t are normalized using the softmax function
ykt = P
exp(atk )
t ,
k0 exp(ak0 )
3
(5)
where ykt is the probability of label k given a at time t.
The conditional probability of the path ? is calculated by the multiplication of the label probabilities
at each timestep, as given by
T
Y
p(?|a) =
y?t t ,
(6)
t=1
where ?t is the label observed at time t along the path ?. The path ? of length T is mapped to a
label sequence of length M ? T by an operator B, which removes the repeated labels and then
the blanks. Several mutually exclusive paths can map to the same label sequence. Let S be a set
containing every possible sequence mapped by B, that is, S = {s|s ? B(?) for some ?} is the
image of B, and let |S| denote the cardinality of the set.
The conditional probability of a label sequence l is given by
X
p(l|a) =
p(?|a),
(7)
??B?1 (l)
which is the sum of probabilities of all the paths mapped to a label sequence l by B.
The cross-entropy loss assigns a negative log probability to the correct answer. Given a target
sequence z, the loss function of CTC for the sample is written as
L(a) = ? log p(z|a).
(8)
From the description above, CTC is composed of the sum of the product of softmax components.
The function ? log(ykt ), corresponding to the softmax with cross-entropy loss, is convex [11].
Therefore, ykt is log-concave. Whereas log-concavity is closed under multiplication, the sum of
log-concave functions is not log-concave in general [15]. As a result, the CTC objective is not
convex in general because it contains the sum of softmax components in Eq. (7).
4.2
Reformulation of CTC objective function
We reformulate the CTC objective Eq. (8) to separate out the terms that are responsible for the nonconvexity of the function. By reformulation, the softmax function is defined over the categorical
label sequences.
By substituting Eq. (5) into Eq. (6), it follows that
exp(b? )
,
0
? 0 ?all exp(b? )
p(?|a) = P
where b? =
as
P
t
(9)
at?t . By substituting Eq. (9) into Eq. (7) and setting l = z, p(z|a) can be re-written
P
exp(b? )
exp(fz )
,
(10)
exp(fz0 )
P
where S is the set of every possible label sequence and fz = log
??B?1 (z) exp(b? ) is the logp(z|a) =
??B?1 (z)
P
??all exp(b? )
=P
z0 ?S
sum-exp function1 , which is proportional to the probability of observing the label sequence z among
all the other label sequences.
With the reformulation above, the CTC objective can be regarded as the cross-entropy loss with the
softmax output, which is defined over all the possible label sequences. Because the cross-entropy
loss function matches the softmax output layer [11], the CTC objective is convex, except the part
that computes fz for each of the label sequences. At this point, an obvious candidate for the convex
approximation of CTC is the GGN matrix separating the convex and non-convex parts.
Let the non-convex part be Nc and the convex part be Lc . The mapping Nc : (RK )T ? R|S| is
defined by
Nc (a) = F = [fz1 , . . . , fz|S| ]> ,
(11)
1
f (x1 , . . . , xn ) = log(ex1 + ? ? ? + exn ) is the log-sum-exp function defined on Rn
4
where fz is given above, and |S| is the number of all the possible label sequences. For given F as
above, the mapping Lc : R|S| ? R is defined by
!
X
exp(fz )
Lc (F ) = ? log P
= ?fz + log
exp(fz0 ) ,
(12)
0
z0 ?S exp(fz )
0
z ?S
where z is the label sequence corresponding to a. The final reformulation for the loss function of
CTC is given by
L(a) = Lc ? Nc (a).
(13)
4.3
Convex approximation of CTC loss function
The GGN approximation of Eq. (13) immediately gives a convex approximation of the Hessian for
>
CTC as GLc ?Nc = JN
HLc JNc . Although HLc has the form of a diagonal matrix plus a rank-1
c
matrix, i.e., diag(Y ) ? Y Y > , the dimension of HLc is |S| ? |S|, where |S| becomes exponentially
large as the length of the sequence increases. This makes the practical calculation of HLc difficult.
On the other hand, removing the linear team
P ?fz from Lc(F ) in Eq. (12) does not alter its Hessian.
0
The resulting formula is Lp (F ) = log
z0 ?S exp(fz ) . The GGN matrices of L = Lc ? Nc
and M = Lp ? Nc are the same, i.e., GLc ?Nc = GLp ?Nc . Therefore, their Hessian matrices are
approximations of each other. The condition that the two Hessian matrices, HL and HM , converges
to the same matrix is discussed below.
P
P
Interestingly, M is given as a compact formula M(a) = Lp ? Nc (a) = t log k exp(atk ), where
atk is the output unit k at time t. Its Hessian HM can be directly computed, resulting in a block
diagonal matrix. Each block is restricted in time, and the t-th block is given by
>
HM,t = diag(Y t ) ? Y t Y t ,
(14)
t >
] and ykt is given in Eq. (5). Because the Hessian of each block is positive
where Y t = [y1t , . . . , yK
semidefinite, HM is positive semidefinite. A convex approximation of the Hessian of an MDRNN
using the CTC objective can be obtained by substituting HM for HL in Eq. (3). Note that the
resulting matrix is block diagonal and Eq. (4) can be utilized for efficient computation.
Our derivation can be summarized as follows:
1. HL = HLc ?Nc is not positive semidefinite.
2. GLc ?Nc = GLp ?Nc is positive semidefinite, but not computationally tractable.
3. HLp ?Nc is positive semidefinite and computationally tractable.
4.4
Sufficient condition for the proposed approximation to be exact
PKT
From Eq. (2), the condition HLc ?Nc = HLp ?Nc holds if and only if
i=1 [JLc ]i H[Nc ]i =
PKT
i=1 [JLp ]i H[Nc ]i . Since JLc 6= JLp in general, we consider only the case of H[Nc ]i = 0 for
all i, which corresponds to the case where Nc is a linear mapping.
[Nc ]i contains a log-sum-exp function mapping from paths to a label sequence. Let l be the label
sequence corresponding to [Nc ]i ; then, [Nc ]i = fl (. . . , b? , . . . ) for ? ? B ?1 (l). If the probability
of one path ? 0 is sufficiently large to ignore all the other paths, that is, exp(b?0 ) exp(b? ) for
? ? {B ?1 (l)\? 0 }, it follows that fl (. . . , b?0 , . . . ) = b?0 . This is a linear mapping, which results in
H[Nc ]i = 0.
In conclusion, the condition HLc ?Nc = HLp ?Nc holds if one dominant path ? ? B ?1 (l) exists such
that fl (. . . , b? , . . . ) = b? for each label sequence l.
4.5
Derivation of the proposed approximation from the Fisher information matrix
The identity of the GGN and the Fisher information matrix [16] has been shown for the network
using the softmax with cross-entropy loss [17, 18]. Thus, it follows that the GGN matrix of Eq. (13)
is identical to the Fisher information matrix. Now, we show that the proposed matrix in Eq. (14)
5
is derived from the Fisher information matrix under the condition given in section 4.4. The Fisher
information matrix of an MDRNN using CTC is written as
"
"
#
>
#
? log p(l|a)
? log p(l|a)
>
F = Ex JN El?p(l|a)
JN ,
(15)
?a
?a
where a = a(x, ?) is the KT -dimensional output of the network N . CTC assumes output probabilities at each timestep to be independent of those at other timesteps [1], and therefore, its Fisher
information matrix is given as the sum of every timestep. It follows that
#
"
"
>
#
X
? log p(l|a)
? log p(l|a)
>
JNt .
(16)
F = Ex
JNt El?p(l|a)
?at
?at
t
Under the condition in section 4.4, the Fisher information matrix is given by
"
#
X
>
t
t t>
F = Ex
JNt (diag(Y ) ? Y Y )JNt ,
(17)
t
which is the same form as Eqs. (4) and (14) combined. See appendix B for the detailed derivation.
4.6
EM interpretation of the proposed approximation
The goal of the Expectation-Maximization (EM) algorithm is to find the maximum likelihood solution for models having latent variables [19]. Given an input sequence x, and
P its corresponding
target label sequence z, the log likelihood of z is given by log p(z|x, ?) = log ??B?1 (z) p(?|x, ?),
where ? represents the model parameters. For each observation x, we have a corresponding latent
variable q which is a 1-of-k binary vector where k is the number
Pof all the paths mapped to z. The
log likelihood can be written in terms of q as log p(z, q|x, ?) = ??B?1 (z) q?|x,z log p(?|x, ?). The
? and repeats the following process until convergence.
EM algorithm starts with an initial parameter ?,
Expectation step calculates: ??|x,z =
P
?
p(?|x,?)
? .
p(?|x,?)
??B?1 (z)
Maximization step updates: ?? = argmax? Q(?),
where Q(?) =
P
??B?1 (z)
??|x,z log p(?|x, ?).
In the context of CTC and RNN, p(?|x, ?) is given as p(?|a(x, ?)) as in Eq. (6), where a(x, ?) is
the KT -dimensional output of the neural network. Taking the second-order derivative of log p(?|a)
t
t
t t>
t
with respect
P to a gives diag(Y )?Y Y , with Y as in Eq. (14). tBecause this term is independent
of ? and ??B?1 (z) ??|x,z = 1, the Hessian of Q with respect to a is given by
>
HQ,t = diag(Y t ) ? Y t Y t ,
(18)
which is the same as the convex approximation in Eq. (14).
5
Experiments
In this section, we present the experimental results for two different sequence labeling tasks, offline
handwriting recognition and phoneme recognition. The performance of Hessian-free optimization
for MDRNNs with the proposed matrix is compared with that of stochastic gradient descent (SGD)
optimization on the same settings.
5.1
Database and preprocessing
The IFN/ENIT Database [20] is a database of handwritten Arabic words, which consists of 32,492
images. The entire dataset has five subsets (a, b, c, d, e). The 25,955 images corresponding to the
subsets (b ? e) were used for training. The validation set consisted of 3,269 images corresponding
to the first half of the sorted list in alphabetical order (ae07 001.tif ? ai54 028.tif) in set a. The
remaining images in set a, amounting to 3,268, were used for the test. The intensity of pixels was
centered and scaled using the mean and standard deviation calculated from the training set.
6
The TIMIT corpus [21] is a benchmark database for evaluating speech recognition performance.
The standard training, validation, and core datasets were used. Each set contains 3,696 sentences,
400 sentences, and 192 sentences, respectively. A mel spectrum with 26 coefficients was used as
a feature vector with a pre-emphasis filter, 25 ms window size, and 10 ms shift size. Each input
feature was centered and scaled using the mean and standard deviation of the training set.
5.2
Experimental setup
For handwriting recognition, the basic architecture was adopted from that proposed in [3]. Deeper
networks were constructed by replacing the top layer with more layers. The number of LSTM cells
in the augmented layer was chosen such that the total number of weights between the different
networks was similar. The detailed architectures are described in Table 1, together with the results.
For phoneme recognition, the deep bidirectional LSTM and CTC in [4] was adopted as the basic
architecture. In addition, the memory cell block [10], in which the cells share the gates, was applied
for efficient information sharing. Each LSTM block was constrained to have 10 memory cells.
According to the results, using a large value of bias for input/output gates is beneficial for training
deep MDRNNs. A possible explanation is that the activation of neurons is exponentially decayed
by input/output gates during the propagation. Thus, setting large bias values for these gates may
facilitate the transmission of information through many layers at the beginning of the learning. For
this reason, the biases of the input and output gates were initialized to 2, whereas those of the forget
gates and memory cells were initialized to 0. All the other weight parameters of the MDRNN were
initialized randomly from a uniform distribution in the range [?0.1, 0.1].
The label error rate was used as the metric for performance evaluation, together with the average
loss of CTC in Eq. (8). It is defined by the edit distance, which sums the total number of insertions,
deletions, and substitutions required to match two given sequences. The final performance, shown
in Tables 1 and 2, was evaluated using the weight parameters that gave the best label error rate on
the validation set. To map output probabilities to a label sequence, best path decoding [1] was used
for handwriting recognition and beam search decoding [4, 22] with a beam width of 100 was used
for phoneme recognition. For phoneme recognition, 61 phoneme labels were used during training
and decoding, and then, mapped to 39 classes for calculating the phoneme error rate (PER) [4, 23].
For phoneme recognition, the regularization method suggested in [24] was used. We applied Gaussian weight noise of standard deviation ? = {0.03, 0.04, 0.05} together with L2 regularization of
strength 0.001. The network was first trained without noise, and then, it was initialized to the weights
that gave the lowest CTC loss on the validation set. Then, the network was retrained with Gaussian
weight noise [4]. Table 2 presents the best result for different values of ?.
5.2.1
Parameters
For HF optimization, we followed the basic setup described in [8], but different parameters were
utilized. Tikhonov damping was used together with Levenberg-Marquardt heuristics. The value of
the damping parameter ? was initialized to 0.1, and adjusted according to the reduction ratio ? (multiplied by 0.9 if ? > 0.75, divided by 0.9 if ? < 0.25, and unchanged otherwise). The initial search
direction for each run of CG was set to the CG direction found by the previous HF optimization
iteration decayed by 0.7. To ensure that CG followed the descent direction, we continued to perform
a minimum 5 and maximum 30 of additional CG iterations after it found the first descent direction.
We terminated CG at iteration i before reaching the maximum iteration if the following condition
was satisfied: (?(xi ) ? ?(xi?5 ))/?(xi ) < 0.005 , where ? is the quadratic objective of CG without offset. The training data were divided into 100 and 50 mini-batches for the handwriting and
phoneme recognition experiments, respectively, and used for both the gradient and matrix-vector
product calculation. The learning was stopped if any of two criteria did not improve for 20 epochs
and 10 epochs in handwriting and phoneme recognition, respectively.
For SGD optimization, the learning rate was chosen from {10?4 , 10?5 , 10?6 }, and the momentum
? from {0.9, 0.95, 0.99}. For handwriting recognition, the best performance obtained using all the
possible combinations of parameters is presented in Table 1. For phoneme recognition, the best
parameters out of nine candidates for each network were selected after training without weight noise
based on the CTC loss. Additionally, the backpropagated error in LSTM layer was clipped to remain
7
in the range [?1, 1] for stable learning [25]. The learning was stopped after 1000 epochs had been
processed, and the final performance was evaluated using the weight parameters that showed the best
label error rate on the validation set. It should be noted that in order to guarantee the convergence,
we selected a conservative criterion as compared to the study where the network converged after 85
epochs in handwriting recognition [3] and after 55-150 epochs in phoneme recognition [4].
5.3
Results
Table 1 presents the label error rate on the test set for handwriting recognition. In all cases, the
networks trained using HF optimization outperformed those using SGD. The advantage of using HF
is more pronounced as the depth increases. The improvements resulting from the deeper architecture
can be seen with the error rate dropping from 6.1% to 4.5% as the depth increases from 3 to 13.
Table 2 shows the phoneme error rate (PER) on the core set for phoneme recognition. The improved
performance according to the depth can be observed for both optimization methods. The best PER
for HF optimization is 18.54% at 15 layers and that for SGD is 18.46% at 10 layers, which are
comparable to that reported in [4], where the reported results are a PER of 18.6% from a network
with 3 layers having 3.8 million weights and a PER of 18.4% from a network with 5 layers having
6.8 million weights. The benefit of a deeper network is obvious in terms of the number of weight
parameters, although this is not intended to be a definitive performance comparison because of
the different preprocessing. The advantage of HF optimization is not prominent in the result of
the experiments using the TIMIT database. One explanation is that the networks tend to overfit
to a relatively small number of the training data samples, which removes the advantage of using
advanced optimization techniques.
Table 1: Experimental results for Arabic offline handwriting recognition. The label error rate is
presented with the different network depths. AB denotes a stack of B layers having A hidden
LSTM cells in each layer. ?Epochs? is the number of epochs required by the network using HF
optimization so that the stopping criteria are fulfilled. is the learning rate and ? is the momentum.
NETWORKS
2-10-50
2-10-213
2-10-146
2-10-128
2-10-1011
2-10-913
DEPTH
WEIGHTS
3
5
8
10
13
15
159,369
157,681
154,209
154,153
150,169
145,417
HF (%)
6.10
5.85
4.98
4.95
4.50
5.69
EPOCHS
77
90
140
109
84
84
SGD (%)
9.57
9.19
9.67
9.25
10.63
12.29
{, ?}
{10?4 ,0.9}
{10?5 ,0.99}
{10?4 ,0.95}
{10?4 ,0.95}
{10?4 ,0.9}
{10?5 ,0.99}
Table 2: Experimental results for phoneme recognition using the TIMIT corpus. PER is presented
with the different MDRNN architectures (depth ? block ? cell/block). ? is the standard deviation
of Gaussian weight noise. The remaining parameters are the same as in Table 1.
NETWORKS
WEIGHTS
3 ? 20 ? 10
5 ? 15 ? 10
8 ? 11 ? 10
10 ? 10 ? 10
13 ? 9 ? 10
15 ? 8 ? 10
3 ? 250 ? 1?
5 ? 250 ? 1?
771,542
795,752
720,826
755,822
806,588
741,230
3.8M
6.8M
HF (%)
20.14
19.18
19.09
18.79
18.59
18.54
EPOCHS
22
30
29
60
93
50
{?}
{0.03}
{0.05}
{0.05}
{0.04}
{0.05}
{0.04}
SGD (%)
20.96
20.82
19.68
18.46
18.49
19.09
18.6
18.4
{, ?, ?}
{10?5 , 0.99, 0.05 }
{10?4 , 0.9, 0.04 }
{10?4 , 0.9, 0.04 }
{10?5 , 0.95, 0.04 }
{10?5 , 0.95, 0.04 }
{10?5 , 0.95, 0.03 }
{10?4 , 0.9, 0.075 }
{10?4 , 0.9, 0.075 }
? The results were reported by Graves in 2013 [4].
6
Conclusion
Hessian-free optimization as an approach for successful learning of deep MDRNNs, in conjunction
with CTC, was presented. To apply HF optimization to CTC, a convex approximation of its objective
function was explored. In experiments, improvements in performance were seen as the depth of the
network increased for both HF and SGD. HF optimization showed a significantly better performance
for handwriting recognition than did SGD, and a comparable performance for speech recognition.
8
References
[1] Alex Graves. Supervised sequence labelling with recurrent neural networks, volume 385. Springer, 2012.
[2] Alex Graves, Marcus Liwicki, Horst Bunke, J?urgen Schmidhuber, and Santiago Fern?andez. Unconstrained on-line handwriting recognition with recurrent neural networks. In Advances in Neural Information Processing Systems, pages 577?584, 2008.
[3] Alex Graves and J?urgen Schmidhuber. Offline handwriting recognition with multidimensional recurrent
neural networks. In Advances in Neural Information Processing Systems, pages 545?552, 2009.
[4] Alex Graves, Abdel-ranhman Mohamed, and Geoffrey Hinton. Speech recognition with deep recurrent
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9
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5,153 | 5,665 | Scalable Inference for Gaussian Process Models with
Black-Box Likelihoods
Edwin V. Bonilla
The University of New South Wales
[email protected]
Amir Dezfouli
The University of New South Wales
[email protected]
Abstract
We propose a sparse method for scalable automated variational inference (AVI) in
a large class of models with Gaussian process (GP) priors, multiple latent functions, multiple outputs and non-linear likelihoods. Our approach maintains the
statistical efficiency property of the original AVI method, requiring only expectations over univariate Gaussian distributions to approximate the posterior with a
mixture of Gaussians. Experiments on small datasets for various problems including regression, classification, Log Gaussian Cox processes, and warped GPs show
that our method can perform as well as the full method under high sparsity levels.
On larger experiments using the MNIST and the SARCOS datasets we show that
our method can provide superior performance to previously published scalable
approaches that have been handcrafted to specific likelihood models.
1
Introduction
Developing automated yet practical approaches to Bayesian inference is a problem that has attracted
considerable attention within the probabilisitic machine learning community (see e.g. [1, 2, 3, 4]).
In the case of models with Gaussian process (GP) priors, the main challenge is that of dealing
with a large number of highly-coupled latent variables. Although promising directions within the
sampling community such as Elliptical Slice Sampling (ESS, [5]) have been proposed, they have
been shown to be particularly slow compared to variational methods. In particular, [6] showed
that their automated variational inference (AVI) method can provide posterior distributions that are
practically indistinguishable from those obtained by ESS, while running orders of magnitude faster.
One of the fundamental properties of the method proposed in [6] is its statistical efficiency, which
means that, in order to approximate a posterior distribution via the maximization of the evidence
lower bound (ELBO), it only requires expectations over univariate Gaussian distributions regardless
of the likelihood model. Remarkably, this property holds for a large class of models involving
multiple latent functions and multiple outputs. However, this method is still impractical for large
datasets as it inherits the cubic computational cost of GP models on the number of observations (N ).
While there have been several approaches to large scale inference in GP models [7, 8, 9, 10, 11],
these have been focused on regression and classification problems. The main obstacle to apply these
approaches to inference with general likelihood models is that it is unclear how they can be extended
to frameworks such as those in [6], while maintaining that desirable property of statistical efficiency.
In this paper we build upon the inducing-point approach underpinning most sparse approximations
to GPs [12, 13] in order to scale up the automated inference method of [6]. In particular, for models
with multiple latent functions, multiple outputs and non-linear likelihoods (such as in multi-class
classification and Gaussian process regression networks [14]) we propose a sparse approximation
whose computational complexity is O(M 3 ) in time, where M N is the number of inducing
points. This approximation maintains the statistical efficiency property of the original AVI method.
As the resulting ELBO decomposes over the training data points, our method can scale up to a very
1
large number of observations and is amenable to stochastic optimization and parallel computation.
Moreover, it can, in principle, approximate arbitrary posterior distributions as it uses a Mixture-ofGaussians (MoG) as the family of approximate posteriors. We refer to our method as SAVIGP, which
stands for scalable automated variational inference for Gaussian process models.
Our experiments on small datasets for problems including regression, classification, Log Gaussian
Cox processes, and warped GPs [15] show that SAVIGP can perform as well as the full method under high levels of sparsity. On a larger experiment on the MNIST dataset, our approach outperforms
the distributed variational inference method in [9], who used a class-conditional density modeling
approach. Our method, unlike [9], uses a single discriminative multi-class framework. Finally, we
use SAVIGP to do inference for the Gaussian process regression network model [14] on the SAR COS dataset concerning an inverse robot dynamics problem [16]. We show that we can outperform
previously published scalable approaches that used likelihood-specific inference algorithms.
2
Related work
There has been a long-standing interest in the GP community to overcome the cubic scaling of
inference in standard GP models [17, 18, 12, 13, 8]. However, none of these approaches actually
dealt with the harder tasks of developing scalable inference methods for multi-output problems and
general likelihood models. The former (multiple output problem) has been addressed, notably, by
[19] and [20] using the convolution process formalism. Nevertheless, such approaches were specific
to regression problems. The latter problem (general likelihood models) has been tackled from a
sampling perspective [5] and within an optimization framework using variational inference [21].
In particular, the work of [21] proposes an efficient full Gaussian posterior approximation for GP
models with iid observations. Our work pushes this breakthrough further by allowing multiple latent
functions, multiple outputs, and more importantly, scalability to large datasets.
A related area of research is that of modeling complex data with deep belief networks based on
Gaussian process mappings [22]. Unlike our approach, these models target the unsupervised problem of discovering structure in high-dimensional data, do not deal with black-box likelihoods, and
focus on small-data applications. Finally, very recent developments in speeding-up probabilistic kernel machines [9, 23, 24] show that the types of problems we are addressing here are highly relevant
to the machine learning community. In particular, [23] has proposed efficient inference methods for
large scale GP classification and [9] has developed a distributed variational approach for GP models,
with a focus on regression and classification problems. Our work, unlike these approaches, allows
practitioners and researchers to investigate new models with GP priors and complex likelihoods for
which currently there is no machinery that can scale to very large datasets.
3
Gaussian Process priors and multiple-output nonlinear likelihoods
We are given a dataset D = {xn , yn }N
n=1 , where xn is a D-dimensional input vector and yn is
a P -dimensional output. Our goal is to learn the mapping from inputs to outputs, which can be
established via Q underlying latent functions {fj }Q
j=1 . A sensible modeling approach to the above
problem is to assume that the Q latent functions {fj } are uncorrelated a priori and that they are
drawn from Q zero-mean Gaussian processes [25]:
p(f ) =
Q
Y
Q
Y
p(f?j ) =
j=1
N (f?j ; 0, Kj ),
(1)
j=1
where f is the set of all latent function values; f?j = {fj (xn )}N
n=1 denotes the values of latent
function j; and Kj is the covariance matrix induced by the covariance function ?j (?, ?), evaluated
at every pair of inputs. Along with the prior in Equation (1), we can also assume that our multidimensional observations {yn } are iid given the corresponding set of latent functions {fn }:
p(y|f ) =
N
Y
p(yn |fn? ),
(2)
n=1
where y is the set of all output observations; yn is the nth output observation; and fn? =
{fj (xn )}Q
j=1 is the set of latent function values which yn depends upon. In short, we are inter2
ested in models for which the following criteria are satisfied: (i) factorization of the prior over the
latent functions; and (ii) factorization of the conditional likelihood over the observations given the
latent functions. Interestingly, a large class of problems can be well modeled with the above assumptions: binary classification [7, 26], warped GPs [15], log Gaussian Cox processes [27], multi-class
classification [26], and multi-output regression [14] all belong to this family of models.
3.1
Automated variational inference
One of the key inference challenges in the above models is that of computing the posterior distribution over the latent functions p(f |y). Ideally, we would like an efficient method that does not need to
know the details of the likelihood in order to carry out posterior inference. This is exactly the main
result in [6], which approximates the posterior with a mixture-of-Gaussians within a variational inference framework. This entails the optimization of an evidence lower bound, which decomposes
as a KL-divergence term and an expected log likelihood (ELL) term. As the KL-divergence term is
relatively straightforward to deal with, we focus on their main result regarding the ELL term:
[6], Th. 1: ?The expected log likelihood and its gradients can be approximated using samples from
univariate Gaussian distributions?. More generally, we say that the ELL term and its gradients can
be estimated using expectations over univariate Gaussian distributions. We refer to this result as
that of statistical efficiency. One of the main limitations of this method is its poor scalability to
large datasets, as it has a cubic time complexity on the number of data points, i.e. O(N 3 ). In the
next section we describe our inference method that scales up to large datasets while maintaining the
statistical efficiency property of the original model.
4
Scalable inference
In order to make inference scalable we redefine our prior to be sparse by conditioning the latent
processes on a set of inducing variables {u?j }Q
j=1 , which lie in the same space as {f?j } and are
drawn from the same zero-mean GP priors. As before, we assume factorization of the prior across
the Q latent functions. Hence the resulting sparse prior is given by:
p(u) =
Q
Y
N (u?j ; 0, ?(Zj , Zj )),
p(f |u) =
j=1
Q
Y
e j ),
?j, K
N (f?j ; ?
(3)
j=1
? j = ?(X, Zj )?(Zj , Zj )?1 u?j ,
?
(4)
e j = ?j (X, X) ? Aj ?(Zj , X) with Aj = ?(X, Zj )?(Zj , Zj )?1 ,
K
(5)
where u?j are the inducing variables for latent process j; u is the set of all the inducing variables; Zj
are all the inducing inputs (i.e. locations) for latent process j; X is the matrix of all input locations
{xi }; and ?(U, V) is the covariance matrix induced by evaluating the covariance function ?j (?, ?) at
all pairwise vectors of matrices U and V. We note that while each of the inducing variables in u?j
lies in the same space as the elements in f?j , each of the M inducing inputs in Zj lies in the same
space as each input data point xn . Given the latent function values fn? , the conditional likelihood
factorizes across data points and is given by Equation (2).
4.1
Approximate posterior
We will approximate the posterior using variational inference. Motivated by the fact that the true
joint posterior is given by p(f , u|y) = p(f |u, y)p(u|y), our approximate posterior has the form:
q(f , u|y) = p(f |u)q(u),
(6)
where p(f |u) is the conditional prior given in Equation (3) and q(u) is our approximate (variational)
posterior. This decomposition has proved effective in problems with a single latent process and a
single output (see e.g. [13]).
Our variational distribution is a mixture of Gaussians (MoG):
q(u|?) =
K
X
?k qk (u|mk , Sk ) =
k=1
K
X
k=1
3
?k
Q
Y
j=1
N (u?j ; mkj , Skj ),
(7)
where ? = {?k , mkj , Skj } are the variational parameters: the mixture proportions {?k }, the posterior means {mkj } and posterior covariances {Skj } of the inducing variables corresponding to
mixture component k and latent function j. We also note that each of the mixture components
qk (u|mk , Sk ) is a Gaussian with mean mk and block-diagonal covariance Sk .
5
Posterior approximation via optimization of the evidence lower bound
Following variational inference principles, the log marginal likelihood log p(y) (or evidence) is
lower bounded by the variational objective:
Z
log p(y) ? Lelbo = q(u|?)p(f |u) log p(y|f )df du ?KL(q(u|?)kp(u)) ,
(8)
|
{z
}
|
{z
}
Lkl
Lell
where the evidence lower bound (Lelbo ) decomposes as the sum of an expected log likelihood term
(Lell ) and a KL-divergence term (Lkl ). Our goal is to estimate our posterior distribution q(u|?) via
maximization of Lelbo . We consider first the Lell term, as it is the most difficult to deal with since
we do not know the details of the implementation of the conditional likelihood p(y|f ).
5.1
Expected log likelihood term
Here we need to compute the expectation of the log conditional likelihood log p(y|f ) over the joint
approximate posterior given in Equation (6). Our goal is to obtain expressions for the Lell term and
its gradients wrt the variational parameters while maintaining the statistical efficiency property of
needing only expectations from univariate Gaussians. For this we first introduce an intermediate
distribution q(f |?) that is obtained by integrating out u from the joint approximate posterior:
Z Z
Z
Z
Lell (?) =
q(u|?)p(f |u) log p(y|f )df du = log p(y|f ) p(f |u)q(u|?)du df . (9)
f u
f
|u
{z
}
q(f |?)
Given our approximate posterior in Equation (7), q(f |?) can be obtained analytically:
q(f |?) =
K
X
?k qk (f |?k ) =
k=1
bkj = Aj mkj ,
K
X
k=1
?k
Q
Y
N (f?j ; bkj , ?kj ), with
(10)
j=1
e j + Aj Skj AT ,
?kj = K
j
(11)
e j and Aj are given in Equation (5). Now we can rewrite Equation (9) as:
where K
Lell (?) =
K
X
?k Eqk (f |?k ) [log p(y|f )] =
N X
K
X
?k Eqk(n) (fn? ) [log p(yn? |fn? )],
(12)
n=1 k=1
k=1
where Eq(x) [g(x)] denotes the expectation of function g(x) over the distribution q(x). Here we have
used the mixture decomposition of q(f |?) in Equation (10) and the factorization of the likelihood
over the data points in Equation (2). Now we are ready to state formally our main result.
Theorem 1 For the sparse GP model with prior defined in Equations (3) to (5), and likelihood
defined in Equation (2), the expected log likelihood over the variational distribution in Equation (7)
and its gradients can be estimated using expectations over univariate Gaussian distributions.
Given the result in Equation (12), the proof is trivial for the computation of Lell as we only need
to realize that qk (f |?k ) = N (f ; bk , ?k ) given in Equation (10) has a block-diagonal covariance
structure. Consequently, qk(n) (fn? ) is a Q-dimensional Gaussian with diagonal covariance. For the
gradients of Lell wrt the variational parameters, we use the following identity:
??k Eqk(n) (fn? ) [log p(yn |fn? )] = Eqk(n) (fn? ) ??k log qk(n) (fn? ) log p(yn |fn? ),
for ?k ? {mk , Sk }, and the result for {?k } is straightforward.
4
(13)
Explicit computation of Lell
We now provide explicit expressions for the computation of Lell . We know that qk(n) (fn? ) is a
Q-dimensional Gaussian with :
qk(n) (fn? ) = N (fn? ; bk(n) , ?k(n) ),
(14)
where ?k(n) is a diagonal matrix. The jth element of the mean and the (j, j)th entry of the covariance are given by:
[bk(n) ]j = [Aj ]n,: mkj ,
e j ]n,n + [Aj ]n,: Skj [AT ]:,n ,
[?k(n) ]j,j = [K
j
(15)
where [A]n,: and [A]:,n denote the nth row and nth column of matrix A respectively. Hence we
can compute Lell as follows:
n
oS
(k,i)
fn?
? N (fn? ; bk(n) , ?k(n) ), k = 1, . . . , K,
(16)
i=1
N K
S
1 XX X
(k,i)
Lbell =
?k
log p(yn? |fn? ).
S n=1
i=1
(17)
k=1
The gradients of Lell wrt variational parameters are given in the supplementary material.
5.2
KL-divergence term
We turn now our attention to the KL-divergence term, which can be decomposed as follows:
?KL(q(u|?)kp(u)) = Eq [? log q(u|?)] + Eq [log p(u)] ,
{z
} |
{z
}
|
Lent
(18)
Lcross
where the entropy term (Lent ) can be lower bounded using Jensen?s inequality:
Lent ? ?
K
X
k=1
?k log
K
X
def
?` N (mk ; m` , Sk + S` ) = L?ent .
(19)
`=1
The negative cross-entropy term (Lcross ) can be computed exactly:
Lcross = ?
Q
K
1X X
?k
[M log 2? + log |?(Zj , Zj )| + mTkj ?(Zj , Zj )?1 mkj + tr ?(Zj , Zj )?1 Skj ].
2
j=1
k=1
(20)
The gradients of the above terms wrt the variational parameters are given in the supplementary
material.
5.3
Hyperparameter learning and scalability to large datasets
For simplicity in the notation we have omitted the parameters of the covariance functions and the
likelihood parameters from the ELBO. However, in our experiments we optimize these along with
the variational parameters in a variational-EM alternating optimization framework. The gradients of
the ELBO wrt these parameters are given in the supplementary material.
The original framework of [6] is completely unfeasible for large datasets, as its complexity is dominated by the inversion of the Gram matrix on all the training data, which is an O(N 3 ) operation
where N is the number of training points. Our sparse framework makes automated variational inference practical for large datasets as its complexity is dominated by inversions of the kernel matrix
on the inducing points, which is an O(M 3 ) operation where M is the number of inducing points
per latent process. Furthermore, as the Lell and its gradients decompose over the training points, and
the Lkl term decomposes over the number of latent process, our method is amenable to stochastic
optimization and / or parallel computation, which makes it scalable to very large number of input
observations, output dimensions and latent processes. In our experiments in section 6 we show that
our sparse framework can achieve similar performance to the full method [6] on small datasets under high levels of sparsity. Moreover, we carried out experiments on larger datasets for which is
practically impossible to apply the full (i.e. non-sparse) method.
5
1.00
FG
MoG1
MoG2
FG
4
0.50
NLPD
SSE
0.75
3
MoG1
MoG2
SF
0.1
0.2
2
0.25
1
0.5
1.0
Figure 1: The SSE and NLPD for warped GPs on the Abalone dataset, where lower values on both
measures are better. Three approximate posteriors are used: FG (full Gaussian), MoG 1 (diagonal
Gaussian), and MoG 2 (mixture of two diagonal Gaussians), along with various sparsity factors (SF
= M/N). The smaller the SF the sparser the model, with SF=1 corresponding to no sparsity.
6
Experiments
Our experiments first consider the same six benchmarks with various likelihood models analyzed
by [6]. The number of training points (N ) on these benchmarks ranges from 300 to 1233 and their
input dimensionality (D) ranges from 1 to 256. The goal of this first set of experiments is to show
that SAVIGP can attain as good performance as the full method under high sparsity levels. We also
carried out experiments at a larger scale using the MNIST dataset and the SARCOS dataset [16]. The
application of the original automated variational inference framework on these datasets is unfeasible.
We refer the reader to the supplementary material for the details of our experimental set-up.
We used two performance measures in each experiment: the standardized squared error (SSE) and
the negative log predictive density (NLPD) for continuous-output problems, and the error rate and
the negative log probability (NLP) for discrete-output problems. We use three versions of SAVIGP:
FG , M o G 1, and M o G 2, corresponding to a full Gaussian, a diagonal Gaussian, and mixture of diagonal Gaussians with 2 components, respectively. We refer to the ratio of the number of inducing
points over the number of training points (M/N ) as sparsity factor.
6.1
Small-scale experiments
In this section we describe the results on three (out of six) benchmarks used by [6] and analyze the
performance of SAVIGP. The other three benchmarks are described in the supplementary material.
Warped Gaussian processes (WGP), Abalone dataset [28],
p(yn |fn )
=
?yn t(yn )N (t(yn )|fn , ? 2 ). For this task we used the same neural-net transformation as in
[15] and the results for the Abalone dataset are shown in Figure 1. We see that the performance of
SAVIGP is practically indistinguishable across all sparsity factors for SSE and NLPD. Here we note
that [6] showed that automated variational inference performed competitively when compared to
hand-crafted methods for warped GPs [15].
?yn exp(?? )
Log Gaussian Cox process (LGCP), Coal-mining disasters dataset [29], p(yn |fn ) = n yn ! n .
Here we used the LGCP for modeling the number of coal-mining disasters between years 1851 to
1962. We note that [6] reported that automated variational inference (the focus of this paper) produced practically indistinguishable distributions (but run order of magnitude faster) when compared
to sampling methods such as Elliptical Slice Sampling [5]. The results for our sparse models are
shown in Figure 2, where we see that both models (FG and MoG 1) remain mostly unaffected when
using high levels of sparsity. We also confirm the findings in [6] that the MoG 1 model underestimates
the variance of the predictions.
Binary classification, Wisconsin breast cancer dataset [28], p(yn = 1) = 1/(1 + exp(?fn )).
Classification error rates and the negative log probability (NLP) on the Wisconsin breast cancer
dataset are shown in Figure 3. We see that the error rates are comparable across all models and
sparsity factors. Interestingly, sparser models achieved lower NLP values, suggesting overconfident
predictions by the less sparse models, especially for the mixtures of diagonal Gaussians.
6
1850 1875 1900 1925 1950
1850
1875
1900
1925
1950
0.6
0.4
0.2
1850
1875
1900
1925
1950
0
SF = 1.0
1850
1875
1900
1925
1950
1
SF = 0.5
1850
1875
1900
1925
1950
2
SF = 0.2
MoG1
intensity
3
SF = 0.1
0.6
0.4
0.2
FG
event counts
4
time
Figure 2: Left: the coal-mining disasters data. Right: the posteriors for a Log Gaussian Cox process
on these data when using a full Gaussian (FG) and a diagonal Gaussian (MoG 1), for various sparsity
factors (SF = M/N). The smaller the SF the sparser the model, with SF=1 corresponding to no
sparsity. The solid line is the posterior mean and the shading area includes 90% confidence interval.
FG
MoG1
MoG2
FG
0.2
NLP
error rate
0.02
MoG2
SF
0.04
0.03
MoG1
0.01
0.1
0.2
0.1
0.5
1.0
0.00
Figure 3: Error rates and NLP for binary classification on the Wisconsin breast cancer dataset. Three
approximate posteriors are used: FG (full Gaussian), MoG 1 (diagonal Gaussian), and MoG 2 (mixture
of two diagonal Gaussians), along with various sparsity factors (SF = M/N). The smaller the SF the
sparser the model, with SF=1 corresponding to the original model without sparsity. Error bars on
the left plot indicate 95% confidence interval around the mean.
6.2
Large-scale experiments
In this section we show the results of the experiments carried out on larger datasets with non-linear
non-Gaussian likelihoods.
Multi-class classification on the MNIST dataset. We first considered a multi-class classification
task on the MNIST dataset using the softmax likelihood. This dataset has been extensively used by the
machine learning community and contains 50,000 examples for training, 10,000 for validation and
10,000 for testing, with 784-dimensional input vectors. Unlike most previous approaches, we did not
tune additional parameters using the validation set. Instead we used our variational framework for
learning all the model parameters using all the training and validation data. This setting most likely
provides a lower bound on test accuracy but our goal here is simply to show that we can achieve
competitive performance with highly-sparse models as our inference algorithm does not know the
details of the conditional likelihood. Figure 4 (left and middle) shows error rates and NLPs where we
see that, although the performance decreases with sparsity, the method is able to attain an accuracy
of 97.49%, while using only around 2000 inducing points (SF = 0.04).
To the best of our knowledge, we are the first to train a multi-class Gaussian process classifier using
a single discriminative probabilistic framework on all classes on MNIST. For example, [17] used a
1-vs-rest approach and [23] focused on the binary classification task of distinguishing the odd digits
from the even digits. Finally, [9] trained one model for each digit and used it as a density model,
achieving an error rate of 5.95%. Our experiments show that by having a single discriminative
probabilistic framework, even without exploiting the details of the conditional likelihood, we can
bring this error rate down to 2.51%. As a reference, previous literature reports about 12% error
rate by linear classifiers and less than 1% error rate by sate-of-the-art large/deep convolutional nets.
7
FG
0.08
SF = 0.04
0.009
0.04
0.02
SMSE
0.4
0.06
NLP
error rate
FG
0.5
0.3
0.2
0.1
0.00
0.006
0.003
0.000
0.001 0.004 0.02 0.04
0.001 0.004 0.02
SF
SF
0.04
1
2
output
Figure 4: Left and middle: classification error rates and negative log probabilities (NLP) for the
multi-class problem on MNIST. Here we used the FG (full Gaussian) approximation with various
sparsity factors (SF = M/N). The smaller the SF the sparser the model. Right: the SMSE for a
Gaussian process regression network model on the SARCOS dataset when learning the 4th and 7th
torques (output 1 and output 2) with a FG (full Gaussian) approximation and 0.04 sparsity factor.
Our results show that our method, while solving the harder problem of full posterior estimation, can
reduce the gap between GPs and deep nets.
Gaussian process regression networks on the SARCOS dataset. Here we apply our SAVIGP inference method to the Gaussian process regression networks (GPRNs) model of [14], using the SARCOS
dataset as a test bed. GPRNs are a very flexible regression approach where P outputs are a linear
combination of Q latent Gaussian processes, with the weights of the linear combination also drawn
from Gaussian processes. This yields a non-linear multiple output likelihood model where the correlations between the outputs can be spatially adaptive, i.e. input dependent. The SARCOS dataset
concerns an inverse dynamics problem of a 7-degrees-of-freedom anthropomorphic robot arm [16].
The data consists of 44,484 training examples mapping from a 21-dimensional input space (7 joint
positions, 7 joint velocities, 7 joint accelerations) to the corresponding 7 joint torques. Similarly to
the work in [10], we consider joint learning for the 4th and 7th torques, which we refer to as output
1 and output 2 respectively, and make predictions on 4,449 test points per output.
Figure 4 (right) shows the standardized mean square error (SMSE) with the full Gaussian approximation (FG) using SF=0.04, i.e. less than 2000 inducing points. The results are considerably better
than those reported by [10] (0.2631 and 0.0127 for each output respectively), although their setting
was much sparser than ours on the first output. This also corroborates previous findings that, on this
problem, having more data does help [16]. To the best of our knowledge, we are the first to perform
inference in GPRNs on problems at this scale.
7
Conclusion
We have presented a scalable approximate inference method for models with Gaussian process (GP)
priors, multiple outputs, and nonlinear likelihoods. One of the key properties of this method is its
statistical efficiency in that it requires only expectations over univariate Gaussian distributions to approximate the posterior with a mixture of Gaussians. Extensive experimental evaluation shows that
our approach can attain excellent performance under high sparsity levels and that it can outperform
previous inference methods that have been handcrafted to specific likelihood models. Overall, this
work makes a substantial contribution towards the goal of developing generic yet scalable Bayesian
inference methods for models based on Gaussian processes.
Acknowledgments
This work has been partially supported by UNSW?s Faculty of Engineering Research Grant Program
project # PS37866 and an AWS in Education Research Grant award. AD was also supported by a
grant from the Australian Research Council # DP150104878.
8
References
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9
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5,154 | 5,666 | Variational Dropout and
the Local Reparameterization Trick
?
Diederik P. Kingma? , Tim Salimans? and Max Welling??
?
Machine Learning Group, University of Amsterdam
?
Algoritmica
University of California, Irvine, and the Canadian Institute for Advanced Research (CIFAR)
[email protected], [email protected], [email protected]
Abstract
We investigate a local reparameterizaton technique for greatly reducing the variance of stochastic gradients for variational Bayesian inference (SGVB) of a posterior over model parameters, while retaining parallelizability. This local reparameterization translates uncertainty about global parameters into local noise that
is independent across datapoints in the minibatch. Such parameterizations can be
trivially parallelized and have variance that is inversely proportional to the minibatch size, generally leading to much faster convergence. Additionally, we explore
a connection with dropout: Gaussian dropout objectives correspond to SGVB with
local reparameterization, a scale-invariant prior and proportionally fixed posterior
variance. Our method allows inference of more flexibly parameterized posteriors;
specifically, we propose variational dropout, a generalization of Gaussian dropout
where the dropout rates are learned, often leading to better models. The method
is demonstrated through several experiments.
1
Introduction
Deep neural networks are a flexible family of models that easily scale to millions of parameters and
datapoints, but are still tractable to optimize using minibatch-based stochastic gradient ascent. Due
to their high flexibility, neural networks have the capacity to fit a wide diversity of nonlinear patterns
in the data. This flexbility often leads to overfitting when left unchecked: spurious patterns are found
that happen to fit well to the training data, but are not predictive for new data. Various regularization
techniques for controlling this overfitting are used in practice; a currently popular and empirically
effective technique being dropout [10]. In [22] it was shown that regular (binary) dropout has a
Gaussian approximation called Gaussian dropout with virtually identical regularization performance
but much faster convergence. In section 5 of [22] it is shown that Gaussian dropout optimizes a lower
bound on the marginal likelihood of the data. In this paper we show that a relationship between
dropout and Bayesian inference can be extended and exploited to greatly improve the efficiency of
variational Bayesian inference on the model parameters. This work has a direct interpretation as a
generalization of Gaussian dropout, with the same fast convergence but now with the freedom to
specify more flexibly parameterized posterior distributions.
Bayesian posterior inference over the neural network parameters is a theoretically attractive method
for controlling overfitting; exact inference is computationally intractable, but efficient approximate
schemes can be designed. Markov Chain Monte Carlo (MCMC) is a class of approximate inference
methods with asymptotic guarantees, pioneered by [16] for the application of regularizing neural
networks. Later useful refinements include [23] and [1].
An alternative to MCMC is variational inference [11] or the equivalent minimum description length
(MDL) framework. Modern variants of stochastic variational inference have been applied to neural
1
networks with some succes [8], but have been limited by high variance in the gradients. Despite
their theoretical attractiveness, Bayesian methods for inferring a posterior distribution over neural
network weights have not yet been shown to outperform simpler methods such as dropout. Even a
new crop of efficient variational inference algorithms based on stochastic gradients with minibatches
of data [14, 17, 19] have not yet been shown to significantly improve upon simpler dropout-based
regularization.
In section 2 we explore an as yet unexploited trick for improving the efficiency of stochastic gradientbased variational inference with minibatches of data, by translating uncertainty about global parameters into local noise that is independent across datapoints in the minibatch. The resulting method
has an optimization speed on the same level as fast dropout [22], and indeed has the original Gaussian dropout method as a special case. An advantage of our method is that it allows for full Bayesian
analysis of the model, and that it?s significantly more flexible than standard dropout. The approach
presented here is closely related to several popular methods in the literature that regularize by adding
random noise; these relationships are discussed in section 4.
2
Efficient and Practical Bayesian Inference
We consider Bayesian analysis of a dataset D, containing a set of N i.i.d. observations of tuples
(x, y), where the goal is to learn a model with parameters or weights w of the conditional probability p(y|x, w) (standard classification or regression)1 . Bayesian inference in such a model consists
of updating some initial belief over parameters w in the form of a prior distribution p(w), after
observing data D, into an updated belief over these parameters in the form of (an approximation
to) the posterior distribution p(w|D). Computing the true posterior distribution through Bayes? rule
p(w|D) = p(w)p(D|w)/p(D) involves computationally intractable integrals, so good approximations are necessary. In variational inference, inference is cast as an optimization problem where we
optimize the parameters of some parameterized model q (w) such that q (w) is a close approximation to p(w|D) as measured by the Kullback-Leibler divergence DKL (q (w)||p(w|D)). This
divergence of our posterior q (w) to the true posterior is minimized in practice by maximizing the
so-called variational lower bound L( ) of the marginal likelihood of the data:
L( ) =
where LD ( ) =
DKL (q (w)||p(w)) + LD ( )
X
Eq (w) [log p(y|x, w)]
(1)
(2)
(x,y)2D
We?ll call LD ( ) the expected log-likelihood.
P The bound L( ) plus DKL (q (w)||p(w|D)) equals
the (conditional) marginal log-likelihood (x,y)2D log p(y|x). Since this marginal log-likelihood
is constant w.r.t. , maximizing the bound w.r.t. will minimize DKL (q (w)||p(w|D)).
2.1
Stochastic Gradient Variational Bayes (SGVB)
Various algorithms for gradient-based optimization of the variational bound (eq. (1)) with differentiable q and p exist. See section 4 for an overview. A recently proposed efficient method for
minibatch-based optimization with differentiable models is the stochastic gradient variational Bayes
(SGVB) method introduced in [14] (especially appendix F) and [17]. The basic trick in SGVB is
to parameterize the random parameters w ? q (w) as: w = f (?, ) where f (.) is a differentiable function and ? ? p(?) is a random noise variable. In this new parameterisation, an unbiased
differentiable minibatch-based Monte Carlo estimator of the expected log-likelihood can be formed:
LD ( ) '
LSGVB
(
D
M
N X
)=
log p(yi |xi , w = f (?, )),
M i=1
(3)
i
i
where (xi , yi )M
i=1 is a minibatch of data with M random datapoints (x , y ) ? D, and ? is a noise
vector drawn from the noise distribution p(?). We?ll assume that the remaining term in the variational lower bound, DKL (q (w)||p(w)), can be computed deterministically, but otherwise it may
be approximated similarly. The estimator (3) is differentiable w.r.t. and unbiased, so its gradient
1
Note that the described method is not limited to classification or regression and is straightforward to apply
to other modeling settings like unsupervised models and temporal models.
2
is also unbiased: r LD ( ) ' r LSGVB
( ). We can proceed with variational Bayesian inference
D
by randomly initializing and performing stochastic gradient ascent on L( ) (1).
2.2
Variance of the SGVB estimator
The theory of stochastic approximation tells us that stochastic gradient ascent using (3) will asymptotically converge to a local optimum for an appropriately declining step size and sufficient weight
updates [18], but in practice the performance of stochastic gradient ascent crucially depends on
the variance of the gradients. If this variance is too large, stochastic gradient descent will fail
to make much progress in any reasonable amount of time. Our objective function consists of an
expected log likelihood term that we approximate using Monte Carlo, and a KL divergence term
DKL (q (w)||p(w)) that we assume can be calculated analytically and otherwise be approximated
with Monte Carlo with similar reparameterization.
Assume that we draw minibatches of datapoints with replacement; see appendix F for a similar
analysis for minibatches without replacement. Using Li as shorthand for log p(yi |xi , w = f (?i , )),
the contribution to the likelihood for the i-th datapoint in the minibatch, the Monte Carlo estimator
PM
N
(3) may be rewritten as LSGVB
( )= M
D
i=1 Li , whose variance is given by
M
M X
M
?
X
?
? N2 ? X
Var LSGVB
(
)
=
Var
[L
]
+
2
Cov
[L
,
L
]
(4)
i
i
j
D
M 2 i=1
i=1 j=i+1
? 1
?
M 1
=N 2
Var [Li ] +
Cov [Li , Lj ] ,
(5)
M
M
where the variances and
the data distribution and ? distribution, i.e.
? covariances are w.r.t. both
?
Var [Li ] = Var?,xi ,yi log p(yi |xi , w = f (?, )) , with xi , yi drawn from the empirical distribution defined by the training set. As can be seen from (5), the total contribution to the variance by
Var [Li ] is inversely proportional to the minibatch size M . However, the total contribution by the
covariances does not decrease with M . In practice, this means that the variance of LSGVB
( ) can be
D
dominated by the covariances for even moderately large M .
2.3
Local Reparameterization Trick
We therefore propose an alternative estimator for which we have Cov [Li , Lj ] = 0, so that the variance of our stochastic gradients scales as 1/M . We then make this new estimator computationally
efficient by not sampling ? directly, but only sampling the intermediate variables f (?) through which
? influences LSGVB
( ). By doing so, the global uncertainty in the weights is translated into a form
D
of local uncertainty that is independent across examples and easier to sample. We refer to such a
reparameterization from global noise to local noise as the local reparameterization trick. Whenever
a source of global noise can be translated to local noise in the intermediate states of computation
(? ! f (?)), a local reparameterization can be applied to yield a computationally and statistically
efficient gradient estimator.
Such local reparameterization applies to a fairly large family of models, but is best explained through
a simple example: Consider a standard fully connected neural network containing a hidden layer
consisting of 1000 neurons. This layer receives an M ? 1000 input feature matrix A from the layer
below, which is multiplied by a 1000 ? 1000 weight matrix W, before a nonlinearity is applied,
i.e. B = AW. We then specify the posterior approximation on the weights to be a fully factor2
ized Gaussian, i.e. q (wi,j ) = N (?i,j , i,j
) 8wi,j 2 W, which means the weights are sampled as
wi,j = ?i,j + i,j ?i,j , with ?i,j ? N (0, 1). In this case we could make sure that Cov [Li , Lj ] = 0
by sampling a separate weight matrix W for each example in the minibatch, but this is not computationally efficient: we would need to sample M million random numbers for just a single layer
of the neural network. Even if this could be done efficiently, the computation following this step
would become much harder: Where we originally performed a simple matrix-matrix product of the
form B = AW, this now turns into M separate local vector-matrix products. The theoretical complexity of this computation is higher, but, more importantly, such a computation can usually not be
performed in parallel using fast device-optimized BLAS (Basic Linear Algebra Subprograms). This
also happens with other neural network architectures such as convolutional neural networks, where
optimized libraries for convolution cannot deal with separate filter matrices per example.
3
Fortunately, the weights (and therefore ?) only influence the expected log likelihood through the
neuron activations B, which are of much lower dimension. If we can therefore sample the random
activations B directly, without sampling W or ?, we may obtain an efficient Monte Carlo estimator
at a much lower cost. For a factorized Gaussian posterior on the weights, the posterior for the
activations (conditional on the input A) is also factorized Gaussian:
2
q (wi,j ) = N (?i,j , i,j
) 8wi,j 2 W =) q (bm,j |A) = N ( m,j , m,j ), with
m,j
=
1000
X
am,i ?i,j , and
i=1
m,j
=
1000
X
a2m,i
2
i,j .
(6)
i=1
Rather than sampling the Gaussian weights and then computing the resulting activations, we may
thus
p sample the activations from their implied Gaussian distribution directly, using bm,j = m,j +
m,j ?m,j , with ?m,j ? N (0, 1). Here, ? is an M ? 1000 matrix, so we only need to sample M
thousand random variables instead of M million: a thousand fold savings.
In addition to yielding a gradient estimator that is more computationally efficient than drawing separate weight matrices for each training example, the local reparameterization trick also leads to an
estimator that has lower variance. To see why, consider the stochastic gradient estimate with respect
2
to the posterior parameter i,j
for a minibatch of size M = 1. Drawing random weights W, we get
@LSGVB
@LSGVB
?i,j am,i
D
D
=
.
2
@ i,j
@bm,j 2 i,j
(7)
If, on the other hand, we form the same gradient using the local reparameterization trick, we get
?m,j a2m,i
@LSGVB
@LSGVB
D
D
p
=
.
(8)
2
@ i,j
@bm,j 2 m,j
Here, there are two stochastic terms: The first is the backpropagated gradient @LSGVB
/@bm,j , and
D
2
the second is the sampled random noise (?i,j or ?m,j ). Estimating the gradient with respect to i,j
then basically comes down to estimating the covariance between these two terms. This is much
easier to do for ?m,j as there are much fewer of these: individually they have higher correlation
with the backpropagated gradient @LSGVB
/@bm,j , so the covariance is easier to estimate. In other
D
words, measuring the effect of ?m,j on @LSGVB
/@bm,j is easy as ?m,j is the only random variable
D
directly influencing this gradient via bm,j . On the other hand, when sampling random weights,
there are a thousand ?i,j influencing each gradient term, so their individual effects get lost in the
noise. In appendix D we make this argument more rigorous, and in section 5 we show that it holds
experimentally.
3
Variational Dropout
Dropout is a technique for regularization of neural network parameters, which works by adding
multiplicative noise to the input of each layer of the neural network during optimization. Using the
notation of section 2.3, for a fully connected neural network dropout corresponds to:
B = (A ?)?, with ?i,j ? p(?i,j )
(9)
where A is the M ? K matrix of input features for the current minibatch, ? is a K ? L weight matrix, and B is the M ? L output matrix for the current layer (before a nonlinearity is applied). The
symbol denotes the elementwise (Hadamard) product of the input matrix with a M ? K matrix
of independent noise variables ?. By adding noise to the input during training, the weight parameters ? are less likely to overfit to the training data, as shown empirically by previous publications.
Originally, [10] proposed drawing the elements of ? from a Bernoulli distribution with probability
1 p, with p the dropout rate. Later it was shown that using a continuous distribution with the same
relative mean and variance, such as a Gaussian N (1, ?) with ? = p/(1 p), works as well or better
[20].
Here, we re-interpret dropout with continuous noise as a variational method, and propose a generalization that we call variational dropout. In developing variational dropout we provide a firm
Bayesian justification for dropout training by deriving its implicit prior distribution and variational
objective. This new interpretation allows us to propose several useful extensions to dropout, such as
a principled way of making the normally fixed dropout rates p adaptive to the data.
4
3.1
Variational dropout with independent weight noise
If the elements of the noise matrix ? are drawn independently from a Gaussian N (1, ?), the marginal
distributions of the activations bm,j 2 B are Gaussian as well:
q (bm,j |A) = N (
m,j , m,j ),
with
m,j
=
K
X
am,i ?i,j , and
i=1
m,j
=?
K
X
2
a2m,i ?i,j
.
(10)
i=1
Making use of this fact, [22] proposed Gaussian dropout, a regularization method where, instead
of applying (9), the activations are directly drawn from their (approximate or exact) marginal distributions as given by (10). [22] argued that these marginal distributions are exact for Gaussian noise
?, and for Bernoulli noise still approximately Gaussian because of the central limit theorem. This
ignores the dependencies between the different elements of B, as present using (9), but [22] report
good results nonetheless.
As noted by [22], and explained in appendix B, this Gaussian dropout noise can also be interpreted
as arising from a Bayesian treatment of a neural network with weights W that multiply the input to
give B = AW, where the posterior distribution of the weights is given by a factorized Gaussian with
2
q (wi,j ) = N (?i,j , ??i,j
). From this perspective, the marginal distributions (10) then arise through
the application of the local reparameterization trick, as introduced in section 2.3. The variational
objective corresponding to this interpretation is discussed in section 3.3.
3.2
Variational dropout with correlated weight noise
Instead of ignoring the dependencies of the activation noise, as in section 3.1, we may retain the
dependencies by interpreting dropout (9) as a form of correlated weight noise:
B = (A ?)?, ?i,j ? N (1, ?) () bm = am W, with
0 0
W = (w10 , w20 , . . . , wK
) , and wi = si ?i , with q (si ) = N (1, ?),
(11)
where am is a row of the input matrix and bm a row of the output. The wi are the rows of the
weight matrix, each of which is constructed by multiplying a non-stochastic parameter vector ?i by
a stochastic scale variable si . The distribution on these scale variables we interpret as a Bayesian
posterior distribution. The weight parameters ?i (and the biases) are estimated using maximum
likelihood. The original Gaussian dropout sampling procedure (9) can then be interpreted as arising
from a local reparameterization of our posterior on the weights W.
3.3
Dropout?s scale-invariant prior and variational objective
The posterior distributions q (W) proposed in sections 3.1 and 3.2 have in common that they can
be decomposed into a parameter vector ? that captures the mean, and a multiplicative noise term
determined by parameters ?. Any posterior distribution on W for which the noise enters this multiplicative way, we will call a dropout posterior. Note that many common distributions, such as
univariate Gaussians (with nonzero mean), can be reparameterized to meet this requirement.
During dropout training, ? is adapted to maximize the expected log likelihood Eq? [LD (?)]. For this
to be consistent with the optimization of a variational lower bound of the form in (2), the prior on
the weights p(w) has to be such that DKL (q (w)||p(w)) does not depend on ?. In appendix C we
show that the only prior that meets this requirement is the scale invariant log-uniform prior:
p(log(|wi,j |)) / c,
i.e. a prior that is uniform on the log-scale of the weights (or the weight-scales si for section 3.2). As
explained in appendix A, this prior has an interesting connection with the floating point format for
storing numbers: From an MDL perspective, the floating point format is optimal for communicating
numbers drawn from this prior. Conversely, the KL divergence DKL (q (w)||p(w)) with this prior
has a natural interpretation as regularizing the number of significant digits our posterior q stores
for the weights wi,j in the floating-point format.
Putting the expected log likelihood and KL-divergence penalty together, we see that dropout training
maximizes the following variatonal lower bound w.r.t. ?:
Eq? [LD (?)]
DKL (q? (w)||p(w)),
5
(12)
where we have made the dependence on the ? and ? parameters explicit. The noise parameters ?
(e.g. the dropout rates) are commonly treated as hyperparameters that are kept fixed during training.
For the log-uniform prior this then corresponds to a fixed limit on the number of significant digits
we can learn for each of the weights wi,j . In section 3.4 we discuss the possibility of making this
limit adaptive by also maximizing the lower bound with respect to ?.
2
For the choice of a factorized Gaussian approximate posterior with q (wi,j ) = N (?i,j , ??i,j
), as
discussed in section 3.1, the lower bound (12) is analyzed in detail in appendix C. There, it is shown
that for this particular choice of posterior the negative KL-divergence DKL (q? (w)||p(w)) is not
analytically tractable, but can be approximated extremely accurately using
DKL [q (wi )|p(wi )] ? constant + 0.5 log(?) + c1 ? + c2 ?2 + c3 ?3 ,
with
c1 = 1.16145124,
c2 =
1.50204118,
c3 = 0.58629921.
The same expression may be used to calculate the corresponding term
posterior approximation of section 3.2.
3.4
DKL (q? (s)||p(s)) for the
Adaptive regularization through optimizing the dropout rate
The noise parameters ? used in dropout training (e.g. the dropout rates) are usually treated as fixed
hyperparameters, but now that we have derived dropout?s variational objective (12), making these
parameters adaptive is trivial: simply maximize the variational lower bound with respect to ?. We
can use this to learn a separate dropout rate per layer, per neuron, of even per separate weight. In
section 5 we look at the predictive performance obtained by making ? adaptive.
We found that very large values of ? correspond to local optima from which it is hard to escape due
to large-variance gradients. To avoid such local optima, we found it beneficial to set a constraint
? ? 1 during training, i.e. we maximize the posterior variance at the square of the posterior mean,
which corresponds to a dropout rate of 0.5.
4
Related Work
Pioneering work in practical variational inference for neural networks was done in [8], where a
(biased) variational lower bound estimator was introduced with good results on recurrent neural network models. In later work [14, 17] it was shown that even more practical estimators can be formed
for most types of continuous latent variables or parameters using a (non-local) reparameterization
trick, leading to efficient and unbiased stochastic gradient-based variational inference. These works
focused on an application to latent-variable inference; extensive empirical results on inference of
global model parameters were reported in [6], including succesful application to reinforcement
learning. These earlier works used the relatively high-variance estimator (3), upon which we improve. Variable reparameterizations have a long history in the statistics literature, but have only
recently found use for efficient gradient-based machine learning and inference [4, 13, 19]. Related
is also probabilistic backpropagation [9], an algorithm for inferring marginal posterior probabilities;
however, it requires certain tractabilities in the network making it insuitable for the type of models
under consideration in this paper.
As we show here, regularization by dropout [20, 22] can be interpreted as variational inference.
DropConnect [21] is similar to dropout, but with binary noise on the weights rather than hidden units.
DropConnect thus has a similar interpretation as variational inference, with a uniform prior over the
weights, and a mixture of two Dirac peaks as posterior. In [2], standout was introduced, a variation
of dropout where a binary belief network is learned for producing dropout rates. Recently, [15]
proposed another Bayesian perspective on dropout. In recent work [3], a similar reparameterization
is described and used for variational inference; their focus is on closed-form approximations of the
variational bound, rather than unbiased Monte Carlo estimators. [15] and [7] also investigate a
Bayesian perspective on dropout, but focus on the binary variant. [7] reports various encouraging
results on the utility of dropout?s implied prediction uncertainty.
6
5
Experiments
We compare our method to standard binary dropout and two popular versions of Gaussian dropout,
which we?ll denote with type A and type B. With Gaussian dropout type A we denote the pre-linear
Gaussian dropout from [20]; type B denotes the post-linear Gaussian dropout from [22]. This way,
the method names correspond to the matrix names in section 2 (A or B) where noise is injected.
Models were implemented in Theano [5], and optimization was performed using Adam [12] with
default hyper-parameters and temporal averaging.
Two types of variational dropout were included. Type A is correlated weight noise as introduced
in section 3.2: an adaptive version of Gaussian dropout type A. Variational dropout type B has
independent weight uncertainty as introduced in section 3.1, and corresponds to Gaussian dropout
type B.
A de facto standard benchmark for regularization methods is the task of MNIST hand-written digit
classification. We choose the same architecture as [20]: a fully connected neural network with 3
hidden layers and rectified linear units (ReLUs). We follow the dropout hyper-parameter recommendations from these earlier publications, which is a dropout rate of p = 0.5 for the hidden layers
and p = 0.2 for the input layer. We used early stopping with all methods, where the amount of
epochs to run was determined based on performance on a validation set.
Variance. We start out by empirically comparing the variance of the different available stochastic
estimators of the gradient of our variational objective. To do this we train the neural network described above for either 10 epochs (test error 3%) or 100 epochs (test error 1.3%), using variational
dropout with independent weight noise. After training, we calculate the gradients for the weights of
the top and bottom level of our network on the full training set, and compare against the gradient
estimates per batch of M = 1000 training examples. Appendix E contains the same analysis for the
case of variational dropout with correlated weight noise.
Table 1 shows that the local reparameterization trick yields the lowest variance among all variational
dropout estimators for all conditions, although it is still substantially higher compared to not having any dropout regularization. The 1/M variance scaling achieved by our estimator is especially
important early on in the optimization when it makes the largest difference (compare weight sample
per minibatch and weight sample per data point). The additional variance reduction obtained by our
estimator through drawing fewer random numbers (section 2.3) is about a factor of 2, and this remains relatively stable as training progresses (compare local reparameterization and weight sample
per data point).
stochastic gradient estimator
local reparameterization (ours)
weight sample per data point (slow)
weight sample per minibatch (standard)
no dropout noise (minimal var.)
top layer
10 epochs
7.8 ? 103
1.4 ? 104
4.9 ? 104
2.8 ? 103
top layer
100 epochs
1.2 ? 103
2.6 ? 103
4.3 ? 103
5.9 ? 101
bottom layer
10 epochs
1.9 ? 102
4.3 ? 102
8.5 ? 102
1.3 ? 102
bottom layer
100 epochs
1.1 ? 102
2.5 ? 102
3.3 ? 102
9.0 ? 100
Table 1: Average empirical variance of minibatch stochastic gradient estimates (1000 examples) for
a fully connected neural network, regularized by variational dropout with independent weight noise.
Speed. We compared the regular SGVB estimator, with separate weight samples per datapoint
with the efficient estimator based on local reparameterization, in terms of wall-clock time efficiency.
With our implementation on a modern GPU, optimization with the na??ve estimator took 1635 seconds per epoch, while the efficient estimator took 7.4 seconds: an over 200 fold speedup.
Classification error. Figure 1 shows test-set classification error for the tested regularization methods, for various choices of number of hidden units. Our adaptive variational versions of Gaussian
dropout perform equal or better than their non-adaptive counterparts and standard dropout under all
tested conditions. The difference is especially noticable for the smaller networks. In these smaller
networks, we observe that variational dropout infers dropout rates that are on average far lower than
the dropout rates for larger networks. This adaptivity comes at negligable computational cost.
7
(a) Classification error on the MNIST dataset
(b) Classification error on the CIFAR-10 dataset
Figure 1: Best viewed in color. (a) Comparison of various dropout methods, when applied to fullyconnected neural networks for classification on the MNIST dataset. Shown is the classification
error of networks with 3 hidden layers, averaged over 5 runs. he variational versions of Gaussian
dropout perform equal or better than their non-adaptive counterparts; the difference is especially
large with smaller models, where regular dropout often results in severe underfitting. (b) Comparison of dropout methods when applied to convolutional net a trained on the CIFAR-10 dataset, for
different settings of network size k. The network has two convolutional layers with each 32k and
64k feature maps, respectively, each with stride 2 and followed by a softplus nonlinearity. This is
followed by two fully connected layers with each 128k hidden units.
We found that slightly downscaling the KL divergence part of the variational objective can be beneficial. Variational (A2) in figure 1 denotes performance of type A variational dropout but with a
KL-divergence downscaled with a factor of 3; this small modification seems to prevent underfitting,
and beats all other dropout methods in the tested models.
6
Conclusion
Efficiency of posterior inference using stochastic gradient-based variational Bayes (SGVB) can often
be significantly improved through a local reparameterization where global parameter uncertainty is
translated into local uncertainty per datapoint. By injecting noise locally, instead of globally at the
model parameters, we obtain an efficient estimator that has low computational complexity, can be
trivially parallelized and has low variance. We show how dropout is a special case of SGVB with
local reparameterization, and suggest variational dropout, a straightforward extension of regular
dropout where optimal dropout rates are inferred from the data, rather than fixed in advance. We
report encouraging empirical results.
Acknowledgments
We thank the reviewers and Yarin Gal for valuable feedback. Diederik Kingma is supported by the
Google European Fellowship in Deep Learning, Max Welling is supported by research grants from
Google and Facebook, and the NWO project in Natural AI (NAI.14.108).
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5,155 | 5,667 | Infinite Factorial Dynamical Model
Isabel Valera?
Max Planck Institute for
Software Systems
[email protected]
Francisco J. R. Ruiz?
Department of Computer Science
Columbia University
[email protected]
Fernando Perez-Cruz
Universidad Carlos III de Madrid, and
Bell Labs, Alcatel-Lucent
[email protected]
Lennart Svensson
Department of Signals and Systems
Chalmers University of Technology
[email protected]
Abstract
We propose the infinite factorial dynamic model (iFDM), a general Bayesian nonparametric model for source separation. Our model builds on the Markov Indian buffet process to consider a potentially unbounded number of hidden Markov
chains (sources) that evolve independently according to some dynamics, in which
the state space can be either discrete or continuous. For posterior inference, we
develop an algorithm based on particle Gibbs with ancestor sampling that can be
efficiently applied to a wide range of source separation problems. We evaluate the
performance of our iFDM on four well-known applications: multitarget tracking,
cocktail party, power disaggregation, and multiuser detection. Our experimental
results show that our approach for source separation does not only outperform
previous approaches, but it can also handle problems that were computationally
intractable for existing approaches.
1
Introduction
The central idea behind Bayesian nonparametrics (BNPs) is the replacement of classical finitedimensional prior distributions with general stochastic processes, allowing for an open-ended number of degrees of freedom in a model [8]. They constitute an approach to model selection and
adaptation in which the model complexity is allowed to grow with data size [17]. In the literature,
BNP priors have been applied for time series modeling. For example, the infinite hidden Markov
model [2, 20] considers a potentially infinite cardinality of the state space; and the BNP construction of switching linear dynamical systems (LDS) [4] considers an unbounded number of dynamical
systems with transitions among them occurring at any time during the observation period.
In the context of signal processing, the source separation problem has captured the attention of the
research community for decades due to its wide range of applications [12, 23, 7, 24]. The BNP
literature for source separation includes [10], in which the authors introduce the nonparametric
counterpart of independent component analysis (ICA), referred as infinite ICA (iICA); and [23],
where the authors present the Markov Indian buffet process (mIBP), which places a prior over an
infinite number of parallel Markov chains and is used to build the infinite factorial hidden Markov
model (iFHMM) and the ICA iFHMM. These approaches can effectively adapt the number of hidden
sources to fit the available data. However, they suffer from several limitations: i) the iFHMM is
restricted to binary on/off hidden states, which may lead to hidden chains that do not match the
actual hidden causes, and it is not able to deal with continuous-valued states, and ii) both the iICA
and the ICA iFHMM make independence assumptions between consecutive values of active hidden
states, which significantly restricts their ability to capture the underlying dynamical models. As a
result, we find that existing approaches are not applicable to many well-known source separation
?
Both authors contributed equally.
1
problems, such as multitarget tracking [12], in which each target can be modeled as a Markov chain
with continuous-valued states describing the target trajectory; or multiuser detection [24], in which
the high cardinality of the hidden states makes this problem computationally intractable for the nonbinary extension of the iFHMM. Hence, there is a lack of both a general BNP model for source
separation, and an efficient inference algorithm to address these limitations.
In this paper, we provide a general BNP framework for source separation that can handle a wide
range of dynamics and likelihood models. We assume a potentially infinite number of sources that
are modeled as Markov chains that evolve according to some dynamical system model. We assume
that only the active sources contribute to the observations, and the states of the Markov chains are not
restricted to be discrete but they can also be continuous-valued. Moreover, we let the observations
depend on both the current state of the hidden sequences, and on some previous states. This system
memory is needed when dealing with applications in which the individual source signals propagate
through the air and may thus suffer from some phenomena, such as reverberation, echo, or multipath
propagation. Our approach results in a general and flexible dynamic model that we refer to as infinite
factorial dynamical model (iFDM), and that can be particularized to recover other models previously
proposed in the literature, e.g., the binary iFHMM.
As for most BNP models, one of the main challenges of our iFDM is posterior inference. In discrete
time series models, including the iFHMM, an approximate inference algorithm based on forwardfiltering backward-sampling (FFBS) sweeps is typically used [23, 5]. However, the exact FFBS
algorithm has exponential computational complexity with respect to the memory length. The FFBS
algorithm also becomes computationally intractable when dealing with on/off hidden states that
are continuous-valued when active. In order to overcome these limitations, we develop a suitable
inference algorithm for our iFDM by building a Markov chain Monte Carlo (MCMC) kernel using
particle Gibbs with ancestor sampling (PGAS) [13]. This algorithm presents quadratic complexity
with respect to the memory length and can easily handle a broad range of dynamical models.
The versatility and efficiency of our approach is shown through a comprehensive experimental validation in which we tackle four well-known source separation problems: multitarget tracking [12],
cocktail party [23], power disaggregation [7], and multiuser detection [24].1 Our results show that
our iFDM provides meaningful estimations of the number of sources and their corresponding individual signal traces even in applications that previous approaches cannot handle. It also outperforms,
in terms of accuracy, the iFHMM (extended to account for the actual state space cardinality) combined with FFBS-based inference in the cocktail party and power disaggregation problems.
2
Infinite Factorial Dynamical Model
In this section, we detail our proposed iFDM. We assume that there is a potentially infinite number of
sources contributing to the observed sequence {yt }Tt=1 , and each source is modeled by an underlying
dynamic system model in which the state of the m-th source at time t, denoted by xtm ? X , evolves
over time as a first-order Markov chain. Here, the state space X can be either discrete or continuous.
In addition, we introduce the auxiliary binary variables stm ? {0, 1} to indicate whether the m-th
source is active at time t, such that the observations only depend on the active sources. We assume
that the variables stm follow a first-order Markov chain and let the states xtm evolve according to
p(xtm |stm , x(t?1)m , s(t?1)m ), i.e., the dynamic system model may depend on whether the source
is active or inactive. We assume dummy states stm = 0 for t ? 0. As an example, in the cocktail
party problem, yt denotes a sample of the recorded audio signal, which depends on the individual
voice signals of the active speakers. The latent states xtm in this example are real-valued and the
transition model p(xtm |stm = 1, x(t?1)m , s(t?1)m ) describes the dynamics of the voice signal.
In many real applications, the individual signals propagate though the air until they are mixed and
gathered by the receiver. In such propagation, different phenomena (e.g., refraction or reflexion of
the signal in the walls) may occur, leading to multipath propagation of the signals and, therefore,
to different delayed copies of the individual signals at the receiver. In order to account for this
?memory? effect, we consider that the state of the m-th source at time t, xtm , influences not only the
observation yt , but also the future L ? 1 observations, yt+1 , . . . , yt+L?1 . Therefore, the likelihood
of yt depends on the last L states of all the Markov chains, yielding
p(yt |X, S) = p(yt |{xtm , stm , x(t?1)m , s(t?1)m , . . . , x(t?L+1)m , s(t?L+1)m }M
m=1 ),
1
Code for these applications can be found at https://github.com/franrruiz/iFDM
2
(1)
0,
m = 1, . . . , 1
am
?
1
bm
s0m
x0m
s1m
s2m
s3m
...
sT m
x1m
x2m
x3m
...
xT m
y1
y2
y3
...
yT
0,
m = 1, . . . , 1
am
?
1
bm
(e)
s0m
(a)
s1m
(e)
s2m
(e)
s3m
(e)
...
sT m
y1
y2
y3
...
yT
(e)
(b)
Figure 1: (a) Graphical representation of the iFDM with memory length L = 2. The dashed lines
represent the memory. (b) Equivalent representation using extended states.
where X and S are T ? M matrices containing all the states xtm and stm , respectively. We remark
that the likelihood of yt cannot depend on any hidden state x? m if s? m = 0.
In order to be able to deal with an infinite number of sources, we place a BNP prior over the binary
matrix S that contains all variables stm . In particular, we assume that S ? mIBP(?, ?0 , ?1 ), i.e., S
is distributed as a mIBP [23] with parameters ?, ?0 and ?1 . The mIBP places a prior distribution
over binary matrices with a finite number of rows T and an infinite number of columns M , in which
each row represents a time instant, and each column represents a Markov chain. The mIBP ensures
that, for any finite value of T , only a finite number of columns M+ in S are active almost surely,
whereas the rest of them remain in the all-zero state and do not influence the observations. We
make use of the stick-breaking construction of the mIBP, which is particularly useful to develop
many practical inference algorithms [19, 23]. Under the stick-breaking construction, two hidden
variables for each Markov chain are introduced, representing the transition probabilities between
the active and inactive states. In particular, we define am = p(stm = 1|s(t?1)m = 0) as the
transition probability from inactive to active, and bm = p(stm = 1|s(t?1)m = 1) as the selftransition probability of the active state of the m-th chain. In the stick-breaking representation, the
columns of S are ordered according to their values of am , such that a1 > a2 > a3 > . . ., and
the probability distribution over variables am is given by a1 ? Beta(?, 1), and p(am |am?1 ) ?
(am )??1 I(0 ? am ? am?1 ), being I(?) the indicator function [19]. Finally, we place a beta
distribution over the transition probabilities bm of the form bm ? Beta(?0 , ?1 ).
The resulting iFDM model, particularized for L = 2, is shown in Figure 1a. Note that this model
(e)
can be equivalently represented as shown in Figure 1b, using the extended states stm , with
(e)
stm = xtm , stm , x(t?1)m , s(t?1)m , . . . , x(t?L+1)m , s(t?L+1)m .
(2)
This extended representation allows for an FFBS-based inference algorithm. However, the exponential complexity of the FFBS with the memory parameter L and with continuous-valued hidden states
xtm makes the algorithm intractable in many real scenarios. Hence, we maintain the representation
in Figure 1a because it allows us to derive an efficient inference algorithm.
The proposed iFDM in Figure 1a can be particularized to resemble some other models that have
been proposed in the literature. In particular, we recover: i) the iFHMM in [23] by choosing the
state space X = {0, 1}, xtm = stm and L = 1, ii) the ICA iFHMM in [23] if we set X = R, L = 1
and assume that p(xtm |stm = 1, x(t?1)m , s(t?1)m ) = p(xtm |stm = 1) is a Gaussian distribution,
and iii) a BNP counterpart of the LDS [9] with on/off states by assuming L = 1 and X = R, and
letting the variables xtm be Gaussian distributed with linear relationships among them.
3
Inference Algorithm
We develop an inference algorithm for the proposed iFDM that can handle different dynamic and
likelihood models. Our approach relies on a blocked Gibbs sampling algorithm that alternates between sampling the number of considered chains and the global variables conditioned on the current
value of matrices S and X, and sampling matrices S and X conditioned on the current value of the
remaining variables. In particular, the algorithm proceeds iteratively as follows:
? Step 1: Add Mnew new inactive chains using an auxiliary slice variable and a slice sampling
method. In this step, the number of considered chains is increased from its initial value M+
to M ? = M+ + Mnew (M+ is not updated because stm = 0 for all t for the new chains).
3
270
271
i=1
x1t
i=2
x2t 1
i=3
x3t
1
1
a1t = 1
at 2
=
at 3
=
x1t
1
1
2
1
=
2 x1t+1
a t+
x2t
x3t
a2t+1 = 2
a3t+1 = 3
x2t+1
x3t+1
272
273
274
275
276
277
278
279
t+1
t
t 1
280
(a) Example of the connection of particles
281
in PGAS. We represent P = 3 particles
xi? for ? = {t ? 1, t, t + 1}. The index 282
ai?
i
denotes the ancestor particle of x? . It can
283
be seen that, e.g., the trajectories x11:t+1
284
and x21:t+1 only differ at time instant t+1.
285
286
287
288
Algorithm 1 Particle Gibbs with ancestor sampling
1
2
3
4
5
6
7
8
9
10
11
Input : Reference particle x0t for t = 1, . . . , T , and global variables.
Output: Sample xout
1:T from the PGAS Markov kernel
Draw xi1 ? r1 (x1 ) for i = 1, . . . , P
1 (Eq. 4)
0
Set xP
1 = x1
Compute the weights w1i = W1 (xi1 ) for i = 1, . . . , P (Eq. 5)
for t = 2, . . . , T do
// Resampling and ancestor sampling
Draw ait ? Categorical(wt1 1 , . . . , wtP 1 ) for i = 1, . . . , P
1
Compute w
eti 1|T for i = 1, . . . , P (Eq. 6)
Draw aP
et1 1|T , . . . , w
etP
t ? Categorical(w
// Particle propagation
ai
t
Draw xit ? rt (xt |x1:t
Set
xP
t
=
x0t
1)
1|T )
for i = 1, . . . , P
1 (Eq. 4)
ai
i
t
Set xi1:t = (x1:t
1 , xt ) for i = 1, . . . , P (Eq. 3)
// Weighting
Compute the weights wti = Wt (xi1:t ) for i = 1, . . . , P (Eq. 5)
1
P
12 Draw k ? Categorical(wT
, . . . , wT
)
k
13 return xout
=
x
1:T
1:T
(b) PGAS algorithm.
Figure 2: Particle Gibbs
with ancestor
sampling.
furthermore,
they can
switch on and off (i.e., start or stop transmitting) at any
289
are
allowed
to
switch
on
at any position. We generate synthetic data in which th
? Step 2: Jointly sample the290states xtm and stm of all the considered chains. Compact
move
within
a
region
of
800
?
metres,
where 25 sensors are located on a re
the representation by removing
inactive
in 800
the entire
observation
291 those chains that remain(1)
(2) (1) (2) >
The
state
x
=
[x
,
x
,
v
,
v
]
of
each
target consists of its position an
tm
period, consequently updating
tm
tm tm tm
292 M+ .
dimensional
plane,
and
we
assume
a
linear
Gaussian
dynamic model such that
? Step 3: Sample the global variables
in
the
model,
which
include
the
transition
probabilities
293
evolves
according
to
and the emission parameters,
from
their
posterior
distribution.
294
2 T2
2 the Indian
3
295 scheme for inference in BNP models based on
In Step 1, we follow the slice sampling
s
1 0 Ts 0
2
buffet process (IBP) [19, 23], which296
effectively transforms the model into a finite factorial
model
6
0
1
0
T
6
7
s
xtm =inGsampling
+G
x
+6 0
with M ? = M+ + Mnew parallel chains.
Step 2 consists
elements
of 0the 0matrix x(t 1)mthe
u ut = 4
297
1
0 5 (t 1)m 4 T
s
ces S and X given the current value
of the global variables. Here, we propose to use PGAS,
298
0 0 0an 1
0
algorithm recently developed for inference
in
state-space
models
and
non-Markovian
latent
vari299
able models [13]. Each iteration of this algorithm presents quadratic complexity with respect to
where Ts = 0.5 is the sampling period, and ut ? N (0, I) is a vector that mod
300
the memory length L, avoiding the exponential complexity
of the standard FFBS algorithm when
noise. For each considered target, we sample the initial position uniformly in
301extended
applied over the equivalent model with
states in Figure 1b. Details on the PGAS approach
space,
and assume
thethat
initial
velocity
is Gaussian
302 PGAS,
are given in Section 3.1. After running
we remove
thosethat
chains
remain
inactive
in the distributed with zero m
0.01I.
Similarly
to
[20,
12],
we
assume
the
observation
of sensor j at time t is gi
m
m
303
whole observation period. In Step 3,
we sample the transition probabilities a Pand b , as well? as ?
d0
signal strength
(RSS),needed
i.e., ytjto =
P0 ? dmjt
+ ntj , where ntj ? N
other model-dependent variables such
variables
evaluate
the=1likelihood
304as the observation
m:stm
p(yt |X, S). Further details on the inference
algorithm
can
be
found
in
the
Supplementary
Material.
305
term, dmjt is the distance between target m and sensor j at time t, P0 = 10 is the
and d0 = 100 metres and = 2 are respectively the reference distance and the
3.1 Particle Gibbs with ancestor306
sampling
307
account
for the[1]
radio
We apply our inference algorithm
PGAS [13] is a method within the frameworkwhich
of particle
MCMC
that propagation
combines themodel.
main ideas,
period
of
length
T
=
300.
In
our
inference
algorithm
308
as well as the strengths, of sequential Monte Carlo and MCMC techniques. In contrast to other we sample the noise var
InvGamma(1,1)
as algorithm
its prior distribution.
309 methods
particle Gibbs with backward simulation
[25, 14], this
can also be conveniently
applied to non-Markovian latent variable
i.e.,
models
that
are
not
expressed
on a state-space
310 models,
In Figure 3, we show the true and inferred
trajectories of the targets, and the tem
form. The PGAS algorithm is an MCMC
and
thus
generates
a
new
sample
of
the
hiddentargets
state in a way that the position
311 kernel,
the
position
error.
We
have
sorted
the
inferred
0
0
matrices (X, S) given an initial sample
),
which
is
the
output
of
the
previous
iteration
of
the algorithm is able to detect
312 (X , S
In this figure, we observe that the proposed model and
PGAS (extended to account for the313
Mnew new
inactive
chains).
The
machinery
inside
the
PGAS
their trajectories with an average position error of around 6 metres. We do not co
algorithm resembles an ordinary particle filter,
with two
main differences:
the particles
is are not multitarget trackin
algorithm
because,
to the best ofone
ourofknowledge,
there
314
deterministically set to the reference input sample,
and
the
ancestor
of
each
particle
is
randomly
literature that can deal with targets that may start and stop transmitting at any tim
315
chosen and stored during the algorithm
execution. We briefly describe the PGAS approach below,
316
but we refer to [13] for a rigorous analysis
of the algorithm properties.
Cocktail Party. We now address a blind speech separation task, also known a
317
specifically,
we each
record
multiple people
who are simultaneousl
In the proposed PGAS, we assume a318
set of Pproblem.
particles More
for each
time instant,
representing
the
?
i microphones. Given the recorded signal, the goal is to separate out th
set
of
states {xtm , stm }M
.
We
denote
by
the
vector
x
the
state
of
the
i-th
particle
at
time
t.
We
also
t
m=1
319
Speakers
may the
startparticle
speaking
becomethe
silent at any given time. Si
introduce the ancestor indexes ait ?320{1, . . . signals.
, P } in order
to denote
thatorprecedes
collect
from
speakers
fromofthe
i-th particle at time t. That is, ait corresponds
to thedata
index
of several
the ancestor
particle
xit .PASCAL
Let also ?CHiME? Speech Separati
1
321 i
website.
Thethat
voice
signal for each
speaker
consists of 4 sentences
xi1:t be the ancestral path of particle
xt , i.e.,Challenge
the particle
trajectory
is recursively
defined
as
i
322
with random pauses in between each sentence. We artificially mix the data 10 ti
at
i
i
x1:t = (x1:t?1 , xt ). Figure 2a shows323
an example to clarify the notation.
1
http://spandh.dcs.shef.ac.uk/projects/chime/PCC/datasets.html
4
6
The algorithm is summarized in Figure 2b. For each time instant t, we first generate the ancestor
i
indexes for the first P ? 1 particles according to the importance weights wt?1
. Given these ant
). For
cestors, the particles are then propagated across time according to a distribution rt (xt |xa1:t?1
simplicity, and dropping the global variables from the notation for conciseness, we assume that
?
t
rt (xt |xa1:t?1
)
=
t
p(xt |xat?1
)
=
M
Y
m=1
t
t
t
p(xtm |stm , xa(t?1)m
, sa(t?1)m
)p(stm |sa(t?1)m
),
(3)
i.e., particles are propagated as in Figure 1a using a simple bootstrap proposal kernel,
p(xtm , stm |s(t?1)m , x(t?1)m ). The P -th particle is instead deterministically set to the reference
0
P
i
particle, xP
et?1|T
.
t = xt , whereas the ancestor indexes at are sampled according to some weights w
Indeed, this is a crucial step that vastly improves the mixing properties of the MCMC kernel.
i
We now focus on the computation on the importance weights wti and the ancestor weights w
et?1|T
.
i
i
For the former, the particles are weighted according to wt = Wt (x1:t ), where
Wt (x1:t ) =
p(x1:t |y1:t )
? p(yt |xt?L+1:t ),
p(x1:t?1 |y1:t?1 )rt (xt |x1:t?1 )
(4)
2
. Eq. 4 implies that, in order to obtain the importance
being y?1 :?2 the set of observations {yt }?t=?
1
i
weights, it suffices to evaluate the likelihood at time t. The weights w
et?1|T
are given by
i
i
w
et?1|T
= wt?1
t+L?2
Y
p(xi1:t?1 , x0t:T |y1:T )
i
0 i
p(y? |xi1:t?1 , x0t:T ).
?
w
p(x
|x
)
t?1
t
t?1
p(xi1:t?1 |y1:t?1 )
? =t
(5)
Note that, for memoryless models (i.e., L = 1), Eq. 5 can be simplified, since the product in the last
i
i
term is not present and, therefore, w
et?1|T
? wt?1
p(x0t |xit?1 ). For L > 1, the computation of the
i
weights w
et?1|T
in (5) for i = 1, . . . , P has computational time complexity scaling as O(P M ? L2 ).
Since this computation needs to be performed for each time instant (and this is the most expensive
calculation), the resulting algorithm complexity scales as O(P T M ? L2 ).
4
Experiments
We now evaluate the proposed model and inference algorithm on four different applications, which
are detailed below and summarized in Table 1. For the PGAS kernel, we use P = 3, 000 particles in
all our experiments. Additional details on the experiments are given in the Supplementary Material.
Multitarget Tracking. In the multitarget tracking problem, we aim at locating the position of
several moving targets based on noisy observations. Under a general setup, a varying number of
indistinguishable targets are moving around in a region, appearing at random in space and time.
Multitarget tracking plays an important role in many areas of engineering such as surveillance,
computer vision and signal processing [18, 16, 21, 6, 12]. Here, we focus on a simple synthetic
example to show that our proposed iFDM can handle time-dependent continuous-valued hidden
states. We place three moving targets within a region of 800 ? 800 metres, where 25 sensors are
(1)
(2) (1) (2)
located on a square grid. The state xtm = [xtm , xtm , vtm , vtm ]> of each target consists of its
position and velocity in a two dimensional plane, and we assume a linear Gaussian dynamic model
such that, while active, xtm evolves according to
xtm = Gx x(t?1)m + Gu ut
T2
[ 2s
Ts2
2 ; Ts
(6)
where Gx = [1 0 Ts 0; 0 1 0 Ts ; 0 0 1 0; 0 0 0 1], Gu =
0; 0
0; 0 Ts ], Ts = 0.5 is the sampling period, and ut ? N (0, I) is a vector that models the acceleration noise. For each considered
target, we sample the initial position uniformly in the sensor network space, and assume that the
initial velocity is Gaussian distributed with zero mean and covariance 0.01I. Following [21, 12], we
generate (T = 300) observations based on the received signal
strength
? (RSS), where the measureP
d0
+ ntj . Here, ntj ? N (0, 2)
ment of sensor j at time t is given by ytj = m:stm =1 P0 ? dmjt
is the noise term, dmjt is the distance between target m and sensor j at time t, P0 = 10 is the
transmitted power, and d0 = 100 metres and ? = 2 are, respectively, the reference distance and the
path loss exponent, which account for the radio propagation model. In our inference algorithm, we
sample the noise variance by placing an InvGamma(1,1) distribution as its prior. Here, we compare
5
Application
Model
Multitarget Tracking
Cocktail Party
Power Dissagregation
Multiuser Detection
Infinite factorial LDS
ICA iFHMM
Non-binary iFHMM
?
X
p(xtm |stm = 1, x(t?1)m , s(t?1)m = 1)
N (xtm |Gx x(t?1)m , Gu G>
u)
2
N (xtm |0, ?x
)
am
jk = p(xtm = k|x(t?1)m = j)
U (A)
R4
R
{0, 1, . . . , Q ? 1}
S
A {0}
Table 1: Applications of the iFDM.
L
1
1
1
?N
35
Target 1
Target 2
Target 3
Inferred Target 1
Inferred Target 2
Inferred Target 3
Sensors
600
400
Target 1
Target 2
Target 3
30
25
Error (m)
800
Target 1
Target 2
Target 3
Average
20
15
10
200
5
0
0
0
200
400
600
800
(a) Target trajectories.
0
100
200
300
iFDM
7.0
5.9
6.3
6.4
Genie-aided model
4.8
6.0
5.4
5.9
(c) Average position error.
Time
(b) Position error.
Figure 3: Results for the multitarget tracking problem.
the performance of the iFDM with a ?genie-aided? finite factorial model with perfect knowledge of
the number of targets and noise variance.
In Figures 3a and 3b, we show the true and inferred trajectories of the targets, and the temporal
evolution of the position error of the iFDM. Additionally, Figure 3c shows the average position error
(in absolute value) for our iFDM and the genie-aided method. In these figures, we observe that the
proposed model and algorithm is able to detect the three targets and their trajectories, providing
similar performance to the genie-aided method. In particular, both approaches provide average
position errors of around 6 metres, which is thrice the noise variance.
Cocktail Party. We now address a blind speech separation task, also known as the cocktail party
problem. Given the recorded audio signals from a set of microphones, the goal is to separate out the
individual speech signals of multiple people who are speaking simultaneously. Speakers may start
speaking or become silent at any time. Similarly to [23], we collect data from several speakers from
the PASCAL ?CHiME? Speech Separation and Recognition Challenge website.2 The voice signal
for each speaker consists of 4 sentences, which we append with random pauses in between each sentence. We artificially mix the data 10 times (corresponding to 10 microphones) with mixing weights
sampled from Uniform(0, 1), such that each microphone receives a linear combination of all the considered signals, corrupted by Gaussian noise with standard deviation 0.3. We consider two scenarios,
with 5 and 15 speakers, and subsample the data so that we learn from T = 1, 354 and T = 1, 087
datapoints, respectively. Following [23], our model assumes p(xtm |stm = 1, x(t?1)m , s(t?1)m ) =
N (xtm |0, 2), and xtm = 0 whenever stm = 0. We also model yt as a linear combination of all
PM+
the voice signals under Gaussian noise, i.e., yt = m=1
wm xtm + nt , where nt ? N (0, ?y2 I)
is the noise term, wm ? N (0, I) is the 10-dimensional weighting vector associated to the m-th
speaker, and ?y2 ? InvGamma(1, 1). We compare our iFDM with the ICA iFHMM in [23] using
FFBS sweeps for inference, with (i) p(xtm |stm = 1) = N (xtm |0, 2) (denoted as FFBS-G), and (ii)
p(xtm |stm = 1) = Laplace(xtm |0, 2) (denoted as FFBS-L).
For the scenario with 5 speakers, we show the true and the inferred (after iteration 10, 000) number
of speakers in Figures 4a, 4b, 4c and 4d, along with their activities during the observation period. In
order to quantitatively evaluate the performance of the different algorithms, we show in Figure 4e
(top) the activity detection error rate (ADER), which is computed as the probability of detecting
activity (inactivity) of a speaker while that speaker is actually inactive (active). As the algorithms
are unsupervised, we sort the estimated chains so that the ADER is minimized. If the inferred
number of speakers M+ is smaller (larger) than the true number of speakers, we consider some
extra inferred inactive chains (additional speakers). The PGAS-based approach outperforms the two
FFBS-based methods because it can jointly sample the states of all chains (speakers) for each time
instant, whereas the FFBS requires sampling each chain conditioned on the current states of the
other chains, leading to poor mixing, as discussed in [22]. As a consequence, the FFBS tends to
overestimate the number of speakers, as shown in Figure 4e (bottom).
2
http://spandh.dcs.shef.ac.uk/projects/chime/PCC/datasets.html
6
Method
M+
ADER
PGAS
FFBS-G
FFBS-L
PGAS
FFBS-G
FFBS-L
(a) Ground truth.
(c) FFBS-G.
(b) PGAS.
# of Speakers
5
15
0.08
0.08
0.25
0.14
0.14
0.12
5
15
7
15
8
15
(e) ADER / Inferred M+ .
(d) FFBS-L.
Figure 4: Results for the cocktail party problem.
Algorithm
PGAS
FFBS
H. 1
0.68
0.59
H. 2
0.79
0.78
H. 3
0.60
0.56
H. 4
0.58
0.53
(a) REDD (?H? stands for ?House?).
H. 5
0.55
0.43
Algorithm
PGAS
FFBS
Day 1
0.76
0.67
Day 2
0.82
0.72
(b) AMP.
Table 2: Accuracy for the power disaggregation problem.
Power Disaggregation. Given the aggregate whole-home power consumption signal, the power
disaggregation problem consists in estimating both the number of active devices in the house and
the power draw of each individual device [11, 7]. We validate the performance of the iFDM on
two different real databases: the Reference Energy Disaggregation Data Set (REDD) [11], and the
Almanac of Minutely Power Dataset (AMP) [15]. For the AMP database, we consider two 24-hour
segments and 8 devices. For the REDD database, we consider a 24-hour segment across 5 houses
and 6 devices. Our model assumes that each device can take Q = 4 different states (one inactive
state and three active states with different power consumption), i.e., xtm ? {0, 1, . . . , Q ? 1}, with
xtm = 0 if stm = 0. We place a symmetric Dirichlet prior over the transition probability vectors
m
of the form am
j ? Dirichlet(1), where each element ajk = p(xtm = k|stm = 1, x(t?1)m =
j, s(t?1)m ). When xtm = 0, the power consumption of device m at time t is zero (P0m = 0), and
when xtm ? {1, . . . , Q ? 1} its average power consumption is given by Pxmtm . Thus, the total power
PM+ m
consumption is given by yt =
m=1 Pxtm + nt , where nt ? N (0, 0.5) represents the additive
Gaussian noise. For q ? {1, . . . , Q ? 1}, we assume a prior power consumption Pqm ? N (15, 10).
In this case, the proposed model for the iFDM resembles a non-binary iFHMM and, therefore, we
can also apply the FFBS algorithm to infer the power consumption draws of each device.
In order to evaluate the performance of the different algorithms, we compute the mean accuracy of
the estimated consumption of each device (higher is better), i.e., acc = 1 ?
(m)
(m)
PT
PM
(m)
(m)
|xt ??
xt |
,
PT m=1
P
(m)
2 t=1 M
x
m=1 t
t=1
where xt and x
?t = Pxmtm are, respectively, the true and the estimated power consumption by
device m at time t. In order to compute the accuracy, we assign each estimated chain to a device
so that the accuracy is maximized. If the inferred number of devices M+ is smaller than the true
(m)
number of devices, we use x
?t = 0 for the undetected devices. If M+ is larger than the true number
(unk)
of devices, we group all the extra chains as an ?unknown? device and use xt
= 0. In Table 2 we
show the results provided by both algorithms. The PGAS approach outperforms the FFBS algorithm
in the five houses of the REDD database and the two selected days of the AMP database. This occurs
because the PGAS can simultaneously sample the hidden states of all devices for each time instant,
whereas the FFBS requires conditioning on the current states of all but one device.
Multiuser Detection. We now consider a digital communication system in which users are allowed
to enter or leave the system at any time, and several receivers cooperate to estimate the number of
users, the (digital) symbols they transmit, and the propagation channels they face. Multipath propagation affects the radio signal, thus causing inter-symbol interference. To capture this phenomenon
in our model, we use L ? 1 in this application. We consider a multiuser Wi-Fi communication system, and we use a ray tracing algorithm (WISE software [3]) to design a realistic indoor wireless system in an office located at Bell Labs Crawford Hill. We place 12 receivers and 6 transmitters across
the office, in the positions respectively marked with circles and crosses in Figure 5 (all transmitters
and receivers are placed at a height of 2 metres).
Transmitted symbols belong to a quadrature phase?
? ?1 }, such that, while active, the transmitted symbols
shift keying (QPSK) constellation, A = { ?1?
2
are independent and uniformly distributed in A, i.e., p(xtm |stm = 1, x(t?1)m , s(t?1)m ) = U(A).
7
1
1
2
7
3
8
2
9
10
3
4
11
5
5
12
14m
4
6 6
120m
Figure 5: Plane of the office building at Bell Labs Crawford Hill.
Model
iFDM
iFHMM
1
6/6
3/11
2
6/6
3/11
L
3
6/6
3/8
4
6/6
1/10
5
6/6
?
(a) # Recovered transmitters / Inferred M+ .
Model
iFDM
iFHMM
1
2.58
2.79
2
2.51
1.38
L
3
0.80
5.53
4
0.30
1.90
5
0.16
?
(b) MSE of the channel coefficients (?10?6 ).
Table 3: Results for the multiuser detection problem.
The observations of all the receivers are weighted replicas of the transmitted symbols under noise,
PM+ PL
m
yt =
m=1
`=1 h` x(t?`+1)m + nt , where xtm = 0 for the inactive states, and the chanm
nel coefficients h` and noise variance ?y2 are provided by WISE software. For inference, we assume Rayleigh-fading channels and, therefore, we place a circularly symmetric complex Gaussian
2
2
prior distribution over the channel coefficients, hm
` |?` ? CN (0, ?` I, 0), and over the noise term,
nt ? CN (0, ?y2 I, 0). We place an inverse gamma prior over ?`2 with mean and standard deviation
0.01e?0.5(`?1) . The choice of this particular prior is based on the assumption that the channel coefficients hm
` are a priori expected to decay with the memory index `, since the radio signal suffers
more attenuation as it propagates through the walls or bounces off them. We use an observation
period T = 2, 000, and vary L from 1 to 5. Five channel taps correspond to the radio signal travelling a distance of 750 m, which should be enough given the dimensions of this office space. We
compare our iFDM with a non-binary iFHMM model with state space cardinality |X | = 5L using
FFBS sweeps for inference (we do not run the FFBS algorithm for L = 5 due to its computational
complexity).
We show in Table 3a the number of recovered transmitters (i.e., the number of transmitters for which
we recover all the transmitted symbols with no error) found after running the inference algorithms,
together with the inferred value of M+ . We see that the iFHMM tends to overestimate the number
of transmitters, which deteriorates the overall symbol estimates and, as a consequence, not all the
transmitted symbols are recovered. We additionally report in Table 3b the MSE of the first channel
P
1
m bm 2
bm
tap, i.e., 6?12
m ||h1 ?h1 || , being h` the inferred channel coefficients. We sort the transmitters
so that the MSE is minimized, and ignore the extra inferred transmitters. In general, the iFDM
outperforms the iFHMM approach, as discussed above. Under our iFDM, the MSE decreases as we
consider a larger value of L, since the model better fits the actual radio propagation model.
5
Conclusions
We have proposed a general BNP approach to solve source separation problems in which the number
of sources is unknown. Our model builds on the mIBP to consider a potentially unbounded number
of hidden Markov chains that evolve independently according to some dynamics, in which the state
space can be either discrete or continuous. For posterior inference, we have developed an algorithm
based on PGAS that solves the intractable complexity that the FFBS presents in many scenarios,
enabling the application of our iFDM in problems such as multitarget tracking or multiuser detection. In addition, we have shown empirically that our PGAS approach outperforms the FFBS-based
algorithm (in terms of accuracy) in the cocktail party and power disaggregation problems, since
the FFBS gets more easily trapped in local modes of the posterior in which several Markov chains
correspond to a single hidden source.
Acknowledgments
I. Valera is currently supported by the Humboldt research fellowship for postdoctoral researchers
program and acknowledges the support of Plan Regional-Programas I+D of Comunidad de Madrid
(AGES-CM S2010/BMD-2422). F. J. R. Ruiz is supported by an FPU fellowship from the Spanish Ministry of Education (AP2010-5333). This work is also partially supported by Ministerio de
Econom??a of Spain (projects COMPREHENSION, id. TEC2012-38883-C02-01, and ALCIT, id.
TEC2012-38800-C03-01), by Comunidad de Madrid (project CASI-CAM-CM, id. S2013/ICE2845), by the Office of Naval Research (ONR N00014-11-1-0651), and by the European Union
7th Framework Programme through the Marie Curie Initial Training Network ?Machine Learning
for Personalized Medicine? (MLPM2012, Grant No. 316861).
8
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9
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5,156 | 5,668 | Variational Information Maximisation for
Intrinsically Motivated Reinforcement Learning
Shakir Mohamed and Danilo J. Rezende
Google DeepMind, London
{shakir, danilor}@google.com
Abstract
The mutual information is a core statistical quantity that has applications in all areas of machine learning, whether this is in training of density models over multiple
data modalities, in maximising the efficiency of noisy transmission channels, or
when learning behaviour policies for exploration by artificial agents. Most learning algorithms that involve optimisation of the mutual information rely on the
Blahut-Arimoto algorithm ? an enumerative algorithm with exponential complexity that is not suitable for modern machine learning applications. This paper
provides a new approach for scalable optimisation of the mutual information by
merging techniques from variational inference and deep learning. We develop our
approach by focusing on the problem of intrinsically-motivated learning, where
the mutual information forms the definition of a well-known internal drive known
as empowerment. Using a variational lower bound on the mutual information,
combined with convolutional networks for handling visual input streams, we develop a stochastic optimisation algorithm that allows for scalable information
maximisation and empowerment-based reasoning directly from pixels to actions.
1 Introduction
The problem of measuring and harnessing dependence between random variables is an inescapable
statistical problem that forms the basis of a large number of applications in machine learning, including rate distortion theory [4], information bottleneck methods [28], population coding [1], curiositydriven exploration [26, 21], model selection [3], and intrinsically-motivated reinforcement learning
[22]. In all these problems the core quantity that must be reasoned about is the mutual information.
In general, the mutual information (MI) is intractable to compute and few existing algorithms are
useful for realistic applications. The received algorithm for estimating mutual information is the
Blahut-Arimoto algorithm [31] that effectively solves for the MI by enumeration ? an approach
with exponential complexity that is not suitable for modern machine learning applications. By combining the best current practice from variational inference with that of deep learning, we bring the
generality and scalability seen in other problem domains to information maximisation problems.
We provide a new approach for maximisation of the mutual information that has significantly lower
complexity, allows for computation with high-dimensional sensory inputs, and that allows us to
exploit modern computational resources.
The technique we derive is generally applicable, but we shall describe and develop our approach
by focussing on one popular and increasingly topical application of the mutual information: as a
measure of ?empowerment? in intrinsically-motivated reinforcement learning. Reinforcement learning (RL) has seen a number of successes in recent years that has now established it as a practical,
scalable solution for realistic agent-based planning and decision making [16, 13]. A limitation of
the standard RL approach is that an agent is only able to learn using external rewards obtained from
its environment; truly autonomous agents will often exist in environments that lack such external
rewards or in environments where rewards are sparsely distributed. Intrinsically-motivated reinforcement learning [25] attempts to address this shortcoming by equipping an agent with a number
of internal drives or intrinsic reward signals, such as hunger, boredom or curiosity that allows the
agent to continue to explore, learn and act meaningfully in a reward-sparse world. There are many
1
State Representation
External Environment
Action
Decoder
Observation
z
q(a1 , . . . , aK |s, s0 )
z
Environment
z
?
Agent
x?
Internal Environment
a1
a2
aK
s?
Source ?(A|s)
Option KB
Critic
h(a1 , . . . , aK |s)
State Repr.
z
State
Embedding
Option
z
z
?
x
Planner
s
a1
a2
aK
?(s)
Figure 1: Perception-action loop separating environment into internal and external facets.
Figure 2: Computational graph for variational information maximisation.
ways in which to formally define internal drives, but what all such definitions have in common is
that they, in some unsupervised fashion, allow an agent to reason about the value of information in
the action-observation sequences it experiences. The mutual information allows for exactly this type
of reasoning and forms the basis of one popular intrinsic reward measure, known as empowerment.
Our paper begins by describing the framework we use for online and self-motivated learning (section 2) and then describes the general problem associated with mutual information estimation and
empowerment (section 3). We then make the following contributions:
? We develop stochastic variational information maximisation, a new algorithm for scalable estimation of the mutual information and channel capacity that is applicable to both discrete and
continuous settings.
? We combine variational information optimisation and tools from deep learning to develop a scalable algorithm for intrinsically-motivated reinforcement learning, demonstrating a new application of the variational theory for problems in reinforcement learning and decision making.
? We demonstrate that empowerment-based behaviours obtained using variational information maximisation match those using the exact computation. We then apply our algorithms to a broad range
of high-dimensional problems for which it is not possible to compute the exact solution, but for
which we are able to act according to empowerment ? learning directly from pixel information.
2 Intrinsically-motivated Reinforcement Learning
Intrinsically- or self-motivated learning attempts to address the question of where rewards come
from and how they are used by an autonomous agent. Consider an online learning system that
must model and reason about its incoming data streams and interact with its environment. This
perception-action loop is common to many areas such as active learning, process control, black-box
optimisation, and reinforcement learning. An extended view of this framework was presented by
Singh et al. [25], who describe the environment as factored into external and internal components
(figure 1). An agent receives observations and takes actions in the external environment. Importantly, the source and nature of any reward signals are not assumed to be provided by an oracle in
the external environment, but is moved to an internal environment that is part of the agent?s decisionmaking system; the internal environment handles the efficient processing of all input data and the
choice and computation of an appropriate internal reward signal.
There are two important components of this framework: the state representation and the critic. We
are principally interested in vision-based self-motivated systems, for which there are no solutions
currently developed. To achieve this, our state representation system is a convolutional neural network [14]. The critic in figure 1 is responsible for providing intrinsic rewards that allow the agent to
act under different types of internal motivations, and is where information maximisation enters the
intrinsically-motivated learning problem.
The nature of the critic and in particular, the reward signal it provides is the main focus of this
paper. A wide variety of reward functions have been proposed, and include: missing information or
Bayesian surprise, which uses the KL divergence to measure the change in an agents internal belief
after the observation of new data [8, 24]; measures based on prediction errors of future states such
predicted L1 change, predicted mode change or probability gain [17], or salient event prediction
[25]; and measures based on information-theoretic quantities such as predicted information gain
(PIG) [15], causal entropic forces [30] or empowerment [23]. The paper by Oudeyer & Kaplan [19]
2
currently provides the widest singular discussion of the breadth of intrinsic motivation measures.
Although we have a wide choice of intrinsic reward measures, none of the available informationtheoretic approaches are efficient to compute or scalable to high-dimensional problems: they require
either knowledge of the true transition probability or summation over all configurations of the state
space, which is not tractable for complex environments or when the states are large images.
3 Mutual Information and Empowerment
The mutual information is a core information-theoretic quantity that acts as a general measure of
dependence between two random variables x and y, defined as:
? ?
?
p(x, y)
I(x, y) = Ep(y|x)p(x) log
,
(1)
p(x)p(y)
where the p(x, y) is a joint distribution over the random variables, and p(x) and p(y) are the corresponding marginal distributions. x and y can be many quantities of interest: in computational
neuroscience they are the sensory inputs and the spiking population code; in telecommunications
they are the input signal to a channel and the received transmission; when learning exploration
policies in RL, they are the current state and the action at some time in the future, respectively.
For intrinsic motivation, we use an internal reward measure referred to as empowerment [12, 23]
that is obtained by searching for the maximal mutual information I(?, ?), conditioned on a starting
state s, between a sequence of K actions a and the final state reached s0 :
? ?
?
p(a, s0 |s)
!
0
E(s) = max I (a, s |s) = max Ep(s0 |a,s)!(a|s) log
,
(2)
!
!
!(a|s)p(s0 |s)
where a = {a1 , . . . , aK } is a sequence of K primitive actions ak leading to a final state s0 , and
p(s0 |a, s) is the K-step transition probability of the environment. p(a, s0 |s) is the joint distribution
of action sequences and the final state, !(a|s) is a distribution over K-step action sequences, and
p(s0 |s) is the joint probability marginalised over the action sequence.
Equation (2) is the definition of the channel capacity in information theory and is a measure of the
amount of information contained in the action sequences a about the future state s0 . This measure
is compelling since it provides a well-grounded, task-independent measure for intrinsic motivation
that fits naturally within the framework for intrinsically motivated learning described by figure 1.
Furthermore, empowerment, like the state- or action-value function in reinforcement learning, assigns a value E(s) to each state s in an environment. An agent that seeks to maximise this value will
move towards states from which it can reach the largest number of future states within its planning
horizon K. It is this intuition that has led authors to describe empowerment as a measure of agent
?preparedness?, or as a means by which an agent may quantify the extent to which it can reliably
influence its environment ? motivating an agent to move to states of maximum influence [23].
An empowerment-based agent generates an open-loop sequence of actions K steps into the future
? this is only used by the agent for its internal planning using !(a|s). When optimised using (2),
the distribution !(a|s) becomes an efficient exploration policy that allows for uniform exploration
of the state space reachable at horizon K, and is another compelling aspect of empowerment (we
provide more intuition for this in appendix A). But this policy is not what is used by the agent for
acting: when an agent must act in the world, it follows a closed-loop policy obtained by a planning
algorithm using the empowerment value (e.g., Q-learning); we expand on this in sect. 4.3. A further
consequence is that while acting, the agent is only ?curious? about parts of its environment that can
be reached within its internal planning horizon K. We shall not explore the effect of the horizon in
this work, but this has been widely-explored and we defer to the insights of Salge et al. [23].
4 Scalable Information Maximisation
The mutual information (MI) as we have described it thus far, whether it be for problems in empowerment, channel capacity or rate distortion, hides two difficult statistical problems. Firstly, computing the MI involves expectations over the unknown state transition probability. This can be seen by
rewriting the MI in terms of the difference between conditional entropies H(?) as:
I(a, s0 |s) = H(a|s)
H(a|s0 , s),
(3)
where H(a|s) = E!(a|s) [log !(a|s)] and H(a|s , s) = Ep(s0 |a,s)!(a|s) [log p(a|s , s)]. This computation requires marginalisation over the K-step transition dynamics of the environment p(s0 |a, s),
0
3
0
which is unknown in general. We could estimate this distribution by building a generative model
of the environment, and then use this model to compute the MI. Since learning accurate generative
models remains a challenging task, a solution that avoids this is preferred (and we also describe one
approach for model-based empowerment in appendix B).
Secondly, we currently lack an efficient algorithm for MI computation. There exists no scalable
algorithm for computing the mutual information that allows us to apply empowerment to highdimensional problems and that allow us to easily exploit modern computing systems. The current
solution is to use the Blahut-Arimoto algorithm [31], which essentially enumerates over all states,
thus being limited to small-scale problems and not being applicable to the continuous domain. More
scalable non-parametric estimators have been developed [7, 6]: these have a high memory footprint
or require a very large number of observations, any approximation may not be a bound on the
MI making reasoning about correctness harder, and they cannot easily be composed with existing
(gradient-based) systems that allow us to design a unified (end-to-end) system. In the continuous
domain, Monte Carlo integration has been proposed [10], but applications of Monte Carlo estimators
can require a large number of draws to obtain accurate solutions and manageable variance. We
have also explored Monte Carlo estimators for empowerment and describe an alternative importance
sampling-based estimator for the MI and channel capacity in appendix B.1.
4.1 Variational Information Lower Bound
The MI can be made more tractable by deriving a lower bound to it and maximising this instead ?
here we present the bound derived by Barber & Agakov [1]. Using the entropy formulation of the MI
(3) reveals that bounding the conditional entropy component is sufficient to bound the entire mutual
information. By using the non-negativity property of the KL divergence, we obtain the bound:
KL[p(x|y)kq(x|y)] 0 ) H(x|y) ? Ep(x|y) [log q? (x|y)]
I ! (s) = H(a|s)
H(a|s0 , s)
H(a) +Ep(s0 |a,s)!? (a|s) [log q? (a|s0 , s)] = I !,q (s)
(4)
where we have introduced a variational distribution q? (?) with parameters ?; the distribution !? (?)
has parameters ?. This bound becomes exact when q? (a|s0 , s) is equal to the true action posterior
distribution p(a|s0 , s). Other lower bounds for the mutual information are also possible: Jaakkola &
Jordan [9] present a lower bound by using the convexity bound for the logarithm; Brunel & Nadal
[2] use a Gaussian assumption and appeal to the Cramer-Rao lower bound.
The bound (4) is highly convenient (especially when compared to other bounds) since the transition
probability p(s0 |a, s) appears linearly in the expectation and we never need to evaluate its probability
? we can thus evaluate the expectation directly by Monte Carlo using data obtained by interaction
with the environment. The bound is also intuitive since we operate using the marginal distribution on
action sequences !? (a|s), which acts as a source (exploration distribution), the transition distribution
p(s0 |a, s) acts as an encoder (transition distribution) from a to s0 , and the variational distribution
q? (a|s0 , s) conveniently acts as a decoder (planning distribution) taking us from s0 to a.
4.2 Variational Information Maximisation
A straightforward optimisation procedure based on (4) is an alternating optimisation for the parameters of the distributions q? (?) and !? (?). Barber & Agakov [1] made the connection between this
approach and the generalised EM algorithm and refer to it as the IM (information maximisation)
algorithm and we follow the same optimisation principle. From an optimisation perspective, the
maximisation of the bound I !,q (s) in (4) w.r.t. !(a|s) can be ill-posed (e.g., in Gaussian models,
the variances can diverge). We avoid such divergent solutions by adding a constraint on the value of
the entropy H(a), which results in the constrained optimisation problem:
? = max I !,q (s) s.t. H(a|s) < ?, E(s)
? = max Ep(s0|a,s)!(a|s) [ 1 ln !(a|s)+ln q? (a|s0, s)] (5)
E(s)
!,q
!,q
where a is the action sequence performed by the agent when moving from s to s0 and
temperature (which is a function of the constraint ?).
is an inverse
At all times we use very general source and decoder distributions formed by complex non-linear
functions using deep networks, and use stochastic gradient ascent for optimisation. We refer to
our approach as stochastic variational information maximisation to highlight that we do all our
computation on a mini-batch of recent experience from the agent. The optimisation for the decoder
q? (?) becomes a maximum likelihood problem, and the optimisation for the source !? (?) requires
computation of an unnormalised energy-based model, which we describe next. We summmarise the
overall procedure in algorithm 1.
4
4.2.1 Maximum Likelihood Decoder
The first step of the alternating optimisation is the optimisation of equation (5) w.r.t. the decoder q,
and is a supervised maximum likelihood problem. Given a set of data from past interactions with
the environment, we learn a distribution from the start and termination states s, s0 , respectively, to
the action sequences a that have been taken. We parameterise the decoder as an auto-regressive
distribution over the K-step action sequence:
q? (a|s0 , s) = q(a1 |s, s0 )
K
Y
k=2
q(ak |f? (ak
1 , s, s
0
(6)
)),
We are free to choose the distributions q(ak ) for each action in the sequence, which we choose as
categorical distributions whose mean parameters are the result of the function f? (?) with parameters
?. f is a non-linear function that we specify using a two-layer neural network with rectified-linear
activation functions. By maximising this log-likelihood, we are able to make stochastic updates to
the variational parameters ? of this distribution. The neural network models used are expanded upon
in appendix D.
4.2.2 Estimating the Source Distribution
Given a current estimate of the decoder q, the variational solution for the distribution !(a|s) computed
I ! (s)/ !(a|s) = 0 under
P by solving the functional derivative
1
?
the constraint that
u(s, a)) , where
a !(a|s) = 1, is given by ! (a|s) =
Z(s) exp (?
P
0
u
?
(s,a)
u(s, a) = Ep(s0 |s,a) [ln q? (a|s, s )], u
?(s, a) = u(s, a) and Z(s) =
is a normalisation
ae
term. By substituting this optimal distribution into the original objective (5) we find that it can be
expressed in terms of the normalisation function Z(s) only, E(s) = 1 log Z(s).
The distribution ! ? (a|s) is implicitly defined as an unnormalised distribution ? there are no direct
mechanisms for sampling actions or computing the normalising function Z(s) for such distributions.
We could use Gibbs or importance sampling, but these solutions are not satisfactory as they would
require several evaluations of the unknown function u(s, a) per decision per state. We obtain a
more convenient problem by approximating the unnormalised distribution ! ? (a|s) by a normalised
(directed) distribution h? (a|s). This is equivalent to approximating the energy term u
?(s, a) by a
function of the log-likelihood of the directed model, r? :
! ? (a|s) ? h? (a|s) ) u
?(s, a) ? r? (s, a);
r? (s, a) = ln h? (a|s) +
? (s).
(7)
We introduced a scalar function ? (s) into the approximation, but since this is not dependent on
the action sequence a it does not change the approximation (7), and can be verified by substituting
(7) into ! ? (a|s). Since h? (a|s) is a normalised distribution, this leaves ? (s) to account for the
normalisation term log Z(s), verified by substituting ! ? (a|s) and (7) into (5). We therefore obtain a
cheap estimator of empowerment E(s) ? 1 ? (s).
To optimise the parameters ? of the directed model h? and the scalar function ? we can minimise
any measure of discrepancy between the two sides of the approximation (7). We minimise the
squared error, giving the loss function L(h? , ? ) for optimisation as:
?
?
L(h? , ? ) = Ep(s0 |s,A) ( ln q? (a|s,s0 ) r? (s, a))2 .
(8)
At convergence of the optimisation, we obtain a compact function with which to compute the empowerment that only requires forward evaluation of the function . h? (a|s) is parameterised using
an auto-regressive distribution similar to (18), with conditional distributions specified by deep networks. The scalar function ? is also parameterised using a deep network. Further details of these
networks are provided in appendix D.
4.3 Empowerment-based Behaviour policies
Using empowerment as an intrinsic reward measure, an agent will seek out states of maximal empowerment. We can treat the empowerment value E(s) as a state-dependent reward and can then
utilise any standard planning algorithm, e.g., Q-learning, policy gradients or Monte Carlo search.
We use the simplest planning strategy by using a one-step greedy empowerment maximisation. This
amounts to choosing actions a = arg maxa C(s, a), where C(s, a) = Ep(s0 |s,a) [E(s)]. This policy
does not account for the effect of actions beyond the planning horizon K. A natural enhancement is
to use value iteration [27] to allow the agent to take actions by maximising its long term (potentially
5
Bound MI
True MI
Approximate
(nats)
Approximate Empowerment
Empowerment (nats)
3.0
True MI
Parameters: ? variational, convolutional, ?
source
while not converged do
x
{Read current state}
s = ConvNet (x) {Compute state repr.}
A ? !(a|s) {Draw action sequence.}
Obtain data (x, a, x0 ) {Acting in env. }
s0 = ConvNet (x0 ) {Compute state repr.}
? / r? log q? (a|s, s0 ) (18)
? / r? L(h? , ? ) (8)
/ r log q? (a|s, s0 ) + r L(h? , ? )
end while
E(s) = 1 ? (s)
{Empowerment}
y = ?0.083 + 1 ? x, r 2 = 0.903
2.5
1.5
?
?
?
?
?
?
?
?
??
? ?
? ? ?
2.0
Bound MI
Algorithm 1: Stochastic Variational Information
Maximisation for Empowerment
?
?
?
? ?
?
?
?
?
?
?
?
?
?
?
1.5
2.0
2.5
Empowerment(nats)
(nats)
TrueTrue
Empowerment
3.0
Figure 3: Comparing exact vs approximate empowerment. Heat maps: empowerment in 3 environments: two
rooms, cross room, two-rooms; Scatter
plot: agreement for two-rooms.
discounted) empowerment. A third approach would be to use empowerment as a potential function
and the difference between the current and previous state?s empowerment as a shaping function with
in the planning [18]. A fourth approach is one where the agent uses the source distribution !(a|s)
as its behaviour policy. The source distribution has similar properties to the greedy behaviour policy and can also be used, but since it effectively acts as an empowered agents internal exploration
mechanism, it has a large variance (it is designed to allow uniform exploration of the state space).
Understanding this choice of behaviour policy is an important line of ongoing research.
4.4
Algorithm Summary and Complexity
The system we have described is a scalable and general purpose algorithm for mutual information
maximisation and we summarise the core components using the computational graph in figure 2 and
in algorithm 1. The state representation mechanism used throughout is obtained by transforming raw
observations x, x0 to produce the start and final states s, s0 , respectively. When the raw observations
are pixels from vision, the state representation is a convolutional neural network [14, 16], while for
other observations (such as continuous measurements) we use a fully-connected neural network ?
we indicate the parameters of these models using . Since we use a unified loss function, we can
apply gradient descent and backpropagate stochastic gradients through the entire model allowing for
joint optimisation of both the information and representation parameters. For optimisation we use a
preconditioned optimisation algorithm such as Adagrad [5].
The computational complexity of empowerment estimators involves the planning horizon K, the
number of actions N , and the number of states S. For the exact computation we must enumerate over
the number of states, which for grid-worlds is S / D2 (for D?D grids), or for binary images is S =
2
2D . The complexity of using the Blahut-Arimoto (BA) algorithm is O(N K S 2 ) = O(N K D4 ) for
2
grid worlds or O(N K 22D ) for binary images. The BA algorithm, even in environments with a small
number of interacting objects becomes quickly intractable, since the state space grows exponentially
with the number of possible interactions, and is also exponential in the planning horizon. In contrast,
our approach deals directly on the image dimensions. Using visual inputs, the convolutional network
produces a vector of size P , upon which all subsequent computation is based, consisting of an Llayer neural network. This gives a complexity for state representation of O(D2 P + LP 2 ). The
autoregressive distributions have complexity of O(H 2 KN ), where H is the size of the hidden layer.
Thus, our approach has at most quadratic complexity in the size of the hidden layers used and linear
in other quantities, and matches the complexity of any currently employed large-scale vision-based
models. In addition, since we use gradient descent throughout, we are able to leverage the power of
GPUs and distributed gradient computations.
5 Results
We demonstrate the use of empowerment and the effectiveness of variational information maximisation in two types of environments. Static environments consists of rooms and mazes in different
configurations in which there are no objects with which the agent can interact, or other moving ob6
Figure 5: Left: empowerment landscape for
Figure 4: Empowerment for a room environagent and key scenario. Yellow is the key and
ment, showing a) an empty room, b) room with
green is the door. Right: Agent in a corridor
an obstacle c) room with a moveable box, d)
with flowing lava. The agent places a bricks to
room with row of moveable boxes.
stem the flow of lava.
jects. The number of states in these settings is equal to the number of locations in the environment,
so is still manageable for approaches that rely on state enumeration. In dynamic environments, aspects of the environment change, such as flowing lava that causes the agent to reset, or a predator
that chases the agent. For the most part, we consider discrete action settings in which the agent has
five actions (up, down, left, right, do nothing). The agent may have other actions, such as picking
up a key or laying down a brick. There are no external rewards available and the agent must reason
purely using visual (pixel) information. For all these experiments we used a horizon of K = 5.
5.1
Effectiveness of the MI Bound
We first establish that the use of the variational information lower bound results in the same behaviour as that obtained using the exact mutual information in a set of static environments. We
consider environments that have at most 400 discrete states and compute the true mutual information using the Blahut-Arimoto algorithm. We compute the variational information bound on the
same environment using pixel information (on 20 ? 20 images). To compare the two approaches we
look at the empowerment landscape obtained by computing the empowerment at every location in
the environment and show these as heatmaps. For action selection, what matters is the location of the
maximum empowerment, and by comparing the heatmaps in figure 3, we see that the empowerment
landscape matches between the exact and the variational solution, and hence, will lead to the same
agent-behaviour.
In each image in figure 3, we show a heat-map of the empowerment for each location in the environment. We then analyze the point of highest empowerment: for the large room it is in the centre of
the room; for the cross-shaped room it is at the centre of the cross, and in a two-rooms environment,
it is located near both doors. In addition, we show that the empowerment values obtained by our
method constitute a close approximation to the true empowerment for the two-rooms environment
(correlation coeff = 1.00, R2 =0.90). These results match those by authors such as Klyubin et al.
[12] (using empowerment) and Wissner-Gross & Freer [30] (using a different information-theoretic
measure ? the causal entropic force). The advantage of the variational approach is clear from this
discussion: we are able to obtain solutions of the same quality as the exact computation, we have far
more favourable computational scaling (one that is not exponential in the size of the state space and
planning horizon), and we are able to plan directly from pixel information.
5.2
Dynamic Environments
Having established the usefulness of the bound and some further understanding of empowerment,
we now examine the empowerment behaviour in environments with dynamic characteristics. Even
in small environments, the number of states becomes extremely large if there are objects that can
be moved, or added and removed from the environment, making enumerative algorithms (such as
BA) quickly infeasible, since we have an exponential explosion in the number of states. We first
reproduce an experiment from Salge et al. [23, ?4.5.3] that considers the empowered behaviour of
an agent in a room-environment, a room that: is empty, has a fixed box, has a moveable box, has
a row of moveable boxes. Salge et al. [23] explore this setup to discuss the choice of the state
representation, and that not including the existence of the box severely limits the planning ability of
the agent. In our approach, we do not face this problem of choosing the state representation, since
the agent will reason about all objects that appear within its visual observations, obviating the need
for hand-designed state representations. Figure 4 shows that in an empty room, the empowerment is
uniform almost everywhere except close to the walls; in a room with a fixed box, the fixed box limits
the set of future reachable states, and as expected, empowerment is low around the box; in a room
where the box can be moved, the box can now be seen as a tool and we have high empowerment
near the box; similarly, when we have four boxes in a row, the empowerment is highest around the
7
Box+Up
Box+Down Box+Left
Box+Right
Up
Down
Left
Right
Stay
1
1
2
3
4
5
6
6
t=4
t=3
t=2
t=1
C(s, a)
t=5
Figure 7: Predator (red) and agent (blue) scenario. Panels 1, 6 show the 3D simulation.
Other panels show a trace of the path that the
Figure 6: Empowerment planning in a lava-filled predator and prey take at points on its trajectory.
maze environment. Black panels show the path The blue/red shows path history; cyan shows the
direction to the maximum empowerment.
taken by the agent.
boxes. These results match those of Salge et al. [23] and show the effectiveness of reasoning from
pixel information directly.
Figure 6 shows how planning with empowerment works in a dynamic maze environment, where lava
flows from a source at the bottom that eventually engulfs the maze. The only way the agent is able to
safeguard itself, is to stem the flow of lava by building a wall at the entrance to one of the corridors.
At every point in time t, the agent decides its next action by computing the expected empowerment
after taking one action. In this environment, we show the planning for all 9 available actions and a
bar graph with the empowerment values for each resulting state. The action that leads to the highest
empowerment is taken and is indicated by the black panels1 .
Figure 5(left) shows two-rooms separated by a door. The agent is able to collect a key that allows
it to open the door. Before collecting the key, the maximum empowerment is in the region around
the key, once the agent has collected the key, the region of maximum empowerment is close to
the door2 . Figure 5(right) shows an agent in a corridor and must protect itself by building a wall of
bricks, which it is able to do successfully using the same empowerment planning approach described
for the maze setting.
5.3 Predator-Prey Scenario
We demonstrate the applicability of our approach to continuous settings, by studying a simple 3D
physics simulation [29], shown in figure 7. Here, the agent (blue) is followed by a predator (red) and
is randomly reset to a new location in the environment if caught by the predator. Both the agent and
the predator are represented as spheres in the environment that roll on a surface with friction. The
state is the position, velocity and angular momentum of the agent and the predator, and the action is
a 2D force vector. As expected, the maximum empowerment lies in regions away from the predator,
which results in the agent learning to escape the predator3 .
6 Conclusion
We have developed a new approach for scalable estimation of the mutual information by exploiting
recent advances in deep learning and variational inference. We focussed specifically on intrinsic
motivation with a reward measure known as empowerment, which requires at its core the efficient
computation of the mutual information. By using a variational lower bound on the mutual information, we developed a scalable model and efficient algorithm that expands the applicability of
empowerment to high-dimensional problems, with the complexity of our approach being extremely
favourable when compared to the complexity of the Blahut-Arimoto algorithm that is currently the
standard. The overall system does not require a generative model of the environment to be built,
learns using only interactions with the environment, and allows the agent to learn directly from visual information or in continuous state-action spaces. While we chose to develop the algorithm in
terms of intrinsic motivation, the mutual information has wide applications in other domains, all
which stand to benefit from a scalable algorithm that allows them to exploit the abundance of data
and be applied to large-scale problems.
Acknowledgements: We thank Daniel Polani for invaluable guidance and feedback.
1
2
Video:
http://youtu.be/eA9jVDa7O38
Video:
http://youtu.be/eSAIJ0isc3Y
http://youtu.be/tMiiKXPirAQ; http://youtu.be/LV5jYY-JFpE
8
3
Videos:
References
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5,157 | 5,669 | Copula variational inference
Dustin Tran
Harvard University
David M. Blei
Columbia University
Edoardo M. Airoldi
Harvard University
Abstract
We develop a general variational inference method that preserves dependency
among the latent variables. Our method uses copulas to augment the families of
distributions used in mean-field and structured approximations. Copulas model the
dependency that is not captured by the original variational distribution, and thus
the augmented variational family guarantees better approximations to the posterior.
With stochastic optimization, inference on the augmented distribution is scalable.
Furthermore, our strategy is generic: it can be applied to any inference procedure
that currently uses the mean-field or structured approach. Copula variational inference has many advantages: it reduces bias; it is less sensitive to local optima;
it is less sensitive to hyperparameters; and it helps characterize and interpret the
dependency among the latent variables.
1
Introduction
Variational inference is a computationally efficient approach for approximating posterior distributions. The idea is to specify a tractable family of distributions of the latent variables and then to minimize the Kullback-Leibler divergence from it to the posterior. Combined with stochastic optimization, variational inference can scale complex statistical models to massive data sets [9, 23, 24].
Both the computational complexity and accuracy of variational inference are controlled by the factorization of the variational family. To keep optimization tractable, most algorithms use the fullyfactorized family, also known as the mean-field family, where each latent variable is assumed independent. Less common, structured mean-field methods slightly relax this assumption by preserving
some of the original structure among the latent variables [19]. Factorized distributions enable efficient variational inference but they sacrifice accuracy. In the exact posterior, many latent variables
are dependent and mean-field methods, by construction, fail to capture this dependency.
To this end, we develop copula variational inference (copula vi). Copula vi augments the traditional
variational distribution with a copula, which is a flexible construction for learning dependencies in
factorized distributions [3]. This strategy has many advantages over traditional vi: it reduces bias;
it is less sensitive to local optima; it is less sensitive to hyperparameters; and it helps characterize
and interpret the dependency among the latent variables. Variational inference has previously been
restricted to either generic inference on simple models?where dependency does not make a significant difference?or writing model-specific variational updates. Copula vi widens its applicability,
providing generic inference that finds meaningful dependencies between latent variables.
In more detail, our contributions are the following.
A generalization of the original procedure in variational inference. Copula vi generalizes variational inference for mean-field and structured factorizations: traditional vi corresponds to running
only one step of our method. It uses coordinate descent, which monotonically decreases the KL
divergence to the posterior by alternating between fitting the mean-field parameters and the copula
parameters. Figure 1 illustrates copula vi on a toy example of fitting a bivariate Gaussian.
Improving generic inference. Copula vi can be applied to any inference procedure that currently
uses the mean-field or structured approach. Further, because it does not require specific knowledge
1
Figure 1: Approximations to an elliptical Gaussian. The mean-field (red) is restricted to fitting
independent one-dimensional Gaussians, which is the first step in our algorithm. The second step
(blue) fits a copula which models the dependency. More iterations alternate: the third refits the meanfield (green) and the fourth refits the copula (cyan), demonstrating convergence to the true posterior.
of the model, it falls into the framework of black box variational inference [15]. An investigator
need only write down a function to evaluate the model log-likelihood. The rest of the algorithm?s
calculations, such as sampling and evaluating gradients, can be placed in a library.
Richer variational approximations. In experiments, we demonstrate copula vi on the standard
example of Gaussian mixture models. We found it consistently estimates the parameters, reduces
sensitivity to local optima, and reduces sensitivity to hyperparameters. We also examine how well
copula vi captures dependencies on the latent space model [7]. Copula vi outperforms competing
methods and significantly improves upon the mean-field approximation.
2
2.1
Background
Variational inference
Let x be a set of observations, z be latent variables, and ? be the free parameters of a variational
distribution q(z; ?). We aim to find the best approximation of the posterior p(z | x) using the variational distribution q(z; ?), where the quality of the approximation is measured by KL divergence.
This is equivalent to maximizing the quantity
L (?) = Eq(z;?) [log p(x, z)] ? Eq(z;?) [log q(z; ?)].
L(?) is the evidence lower bound (elbo), or the variational free energy [25]. For simpler computation, a standard choice of the variational family is a mean-field approximation
q(z; ?) =
d
Y
qi (zi ; ?i ),
i=1
where z = (z1 , . . . , zd ). Note this is a strong independence assumption. More sophisticated approaches, known as structured variational inference [19], attempt to restore some of the dependencies
among the latent variables.
In this work, we restore dependencies using copulas. Structured vi is typically tailored to individual
models and is difficult to work with mathematically. Copulas learn general posterior dependencies
during inference, and they do not require the investigator to know such structure in advance. Further,
copulas can augment a structured factorization in order to introduce dependencies that were not
considered before; thus it generalizes the procedure. We next review copulas.
2.2
Copulas
We will augment the mean-field distribution with a copula. We consider the variational family
" d
#
Y
q(z) =
q(zi ) c(Q(z1 ), . . . , Q(zd )).
i=1
2
1, 3
1
2, 3
3
2
3, 4
1, 2|3
2, 3
4
(T1 )
3, 4
(T2 )
1, 4|3
1, 3
2, 4|13
1, 2|3
1, 4|3
(T3 )
Figure 2: Example of a vine V which factorizes a copula density of four random variables
c(u1 , u2 , u3 , u4 ) into a product of 6 pair copulas. Edges in the tree Tj are the nodes of the lower level
tree Tj+1 , and each edge determines a bivariate copula which is conditioned on all random variables
that its two connected nodes share.
Here Q(zi ) is the marginal cumulative distribution function (CDF) of the random variable zi , and
c is a joint distribution of [0, 1] random variables.1 The distribution c is called a copula of z: it
is a joint multivariate density of Q(z1 ), . . . , Q(zd ) with uniform marginal distributions [21]. For
any distribution, a factorization into a product of marginal densities and a copula always exists and
integrates to one [14].
Intuitively, the copula captures the information about the multivariate random variable after eliminating the marginal information, i.e., by applying the probability integral transform on each variable.
The copula captures only and all of the dependencies among the zi ?s. Recall that, for all random variables, Q(zi ) is uniform distributed. Thus the marginals of the copula give no information.
For example, the bivariate Gaussian copula is defined as
c(u1 , u2 ; ?) = ?? (??1 (u1 ), ??1 (u2 )).
If u1 , u2 are independent uniform distributed, the inverse CDF ??1 of the standard normal transforms (u1 , u2 ) to independent normals. The CDF ?? of the bivariate Gaussian distribution, with
mean zero and Pearson correlation ?, squashes the transformed values back to the unit square. Thus
the Gaussian copula directly correlates u1 and u2 with the Pearson correlation parameter ?.
2.2.1
Vine copulas
It is difficult to specify a copula. We must find a family of distributions that is easy to compute with
and able to express a broad range of dependencies. Much work focuses on two-dimensional copulas,
such as the Student-t, Clayton, Gumbel, Frank, and Joe copulas [14]. However, their multivariate extensions do not flexibly model dependencies in higher dimensions [4]. Rather, a successful approach
in recent literature has been by combining sets of conditional bivariate copulas; the resulting joint is
called a vine [10, 13].
A vine V factorizes a copula density c(u1 , . . . , ud ) into a product of conditional bivariate copulas,
also called pair copulas. This makes it easy to specify a high-dimensional copula. One need only express the dependence for each pair of random variables conditioned on a subset of the others.
Figure 2 is an example of a vine which factorizes a 4-dimensional copula into the product of 6 pair
copulas. The first tree T1 has nodes 1, 2, 3, 4 representing the random variables u1 , u2 , u3 , u4 respectively. An edge corresponds to a pair copula, e.g., 1, 4 symbolizes c(u1 , u4 ). Edges in T1 collapse
into nodes in the next tree T2 , and edges in T2 correspond to conditional bivariate copulas, e.g.,
1, 2|3 symbolizes c(u1 , u2 |u3 ). This proceeds to the last nested tree T3 , where 2, 4|13 symbolizes
1
We overload the notation for the marginal CDF Q to depend on the names of the argument, though we occasionally use Qi (zi ) when more clarity is needed. This is analogous to the standard convention of overloading
the probability density function q(?).
3
c(u2 , u4 |u1 , u3 ). The vine structure specifies a complete factorization of the multivariate copula,
and each pair copula can be of a different family with its own set of parameters:
h
ih
ih
i
c(u1 , u2 , u3 , u4 ) = c(u1 , u3 )c(u2 , u3 )c(u3 , u4 ) c(u1 , u2 |u3 )c(u1 , u4 |u3 ) c(u2 , u4 |u1 , u3 ) .
Formally, a vine is a nested set of trees V = {T1 , . . . , Td?1 } with the following properties:
1. Tree Tj = {Nj , Ej } has d + 1 ? j nodes and d ? j edges.
2. Edges in the j th tree Ej are the nodes in the (j + 1)th tree Nj+1 .
3. Two nodes in tree Tj+1 are joined by an edge only if the corresponding edges in tree Tj
share a node.
Each edge e in the nested set of trees {T1 , . . . , Td?1 } specifies a different pair copula, and the product
of all edges comprise of a factorization of the copula density. Since there are a total of d(d ? 1)/2
edges, V factorizes c(u1 , . . . , ud ) as the product of d(d ? 1)/2 pair copulas.
Each edge e(i, k) ? Tj has a conditioning set D(e), which is a set of variable indices 1, . . . , d. We
define cik|D(e) to be the bivariate copula density for ui and uk given its conditioning set:
cik|D(e) = c Q(ui |uj : j ? D(e)), Q(ui |uj : j ? D(e)) uj : j ? D(e) .
(1)
Both the copula and the CDF?s in its arguments are conditional on D(e). A vine specifies a factorization of the copula, which is a product over all edges in the d ? 1 levels:
c(u1 , . . . , ud ; ?) =
d?1
Y
Y
cik|D(e) .
(2)
j=1 e(i,k)?Ej
We highlight that c depends on ?, the set of all parameters to the pair copulas. The vine construction
provides us with the flexibility to model dependencies in high dimensions using a decomposition of
pair copulas which are easier to estimate. As we shall see, the construction also leads to efficient
stochastic gradients by taking individual (and thus easy) gradients on each pair copula.
3
Copula variational inference
We now introduce copula variational inference (copula vi), our method for performing accurate and
scalable variational inference. For simplicity, consider the mean-field factorization augmented with
a copula (we later extend to structured factorizations). The copula-augmented variational family is
" d
#
Y
q(z; ?, ?) =
q(zi ; ?) c(Q(z1 ; ?), . . . , Q(zd ; ?); ?),
(3)
{z
}
|
i=1
copula
|
{z
}
mean-field
where ? denotes the mean-field parameters and ? the copula parameters. With this family, we maximize the augmented elbo,
L (?, ?) = Eq(z;?,?) [log p(x, z)] ? Eq(z;?,?) [log q(z; ?, ?)].
Copula vi alternates between two steps: 1) fix the copula parameters ? and solve for the mean-field
parameters ?; and 2) fix the mean-field parameters ? and solve for the copula parameters ?. This
generalizes the mean-field approximation, which is the special case of initializing the copula to be
uniform and stopping after the first step. We apply stochastic approximations [18] for each step with
gradients derived
P?in the next section.
P? We set the learning rate ?t ? R to satisfy a Robbins-Monro
schedule, i.e., t=1 ?t = ?, t=1 ?2t < ?. A summary is outlined in Algorithm 1.
This alternating set of optimizations falls in the class of minorize-maximization methods, which
includes many procedures such as the EM algorithm [1], the alternating least squares algorithm, and
the iterative procedure for the generalized method of moments. Each step of copula vi monotonically
increases the objective function and therefore better approximates the posterior distribution.
4
Algorithm 1: Copula variational inference (copula vi)
Input: Data x, Model p(x, z), Variational family q.
Initialize ? randomly, ? so that c is uniform.
while change in elbo is above some threshold do
// Fix ?, maximize over ?.
Set iteration counter t = 1.
while not converged do
Draw sample u ? Unif([0, 1]d ).
Update ? = ? + ?t ?? L. (Eq.5, Eq.6)
Increment t.
end
// Fix ?, maximize over ?.
Set iteration counter t = 1.
while not converged do
Draw sample u ? Unif([0, 1]d ).
Update ? = ? + ?t ?? L. (Eq.7)
Increment t.
end
end
Output: Variational parameters (?, ?).
Copula vi has the same generic input requirements as black-box variational inference [15]?the user
need only specify the joint model p(x, z) in order to perform inference. Further, copula variational inference easily extends to the case when the original variational family uses a structured factorization.
By the vine construction, one simply fixes pair copulas corresponding to pre-existent dependence in
the factorization to be the independence copula. This enables the copula to only model dependence
where it does not already exist.
Throughout the optimization, we will assume that the tree structure and copula families are given
and fixed. We note, however, that these can be learned. In our study, we learn the tree structure using
sequential tree selection [2] and learn the families, among a choice of 16 bivariate families, through
Bayesian model selection [6] (see supplement). In preliminary studies, we?ve found that re-selection
of the tree structure and copula families do not significantly change in future iterations.
3.1
Stochastic gradients of the elbo
To perform stochastic optimization, we require stochastic gradients of the elbo with respect to both
the mean-field and copula parameters. The copula vi objective leads to efficient stochastic gradients
and with low variance.
We first derive the gradient with respect to the mean-field parameters. In general, we can apply the
score function estimator [15], which leads to the gradient
?? L = Eq(z;?,?) [?? log q(z; ?, ?) ? (log p(x, z) ? log q(z; ?, ?))].
(4)
We follow noisy unbiased estimates of this gradient by sampling from q(?) and evaluating the inner
expression. We apply this gradient for discrete latent variables.
When the latent variables z are differentiable, we use the reparameterization trick [17] to take advantage of first-order information from the model, i.e.,?z log p(x, z). Specifically, we rewrite the
expectation in terms of a random variable u such that its distribution s(u) does not depend on the
variational parameters and such that the latent variables are a deterministic function of u and the
mean-field parameters, z = z(u; ?). Following this reparameterization, the gradients propagate
5
inside the expectation,
?? L = Es(u) [(?z log p(x, z) ? ?z log q(z; ?, ?))?? z(u; ?)].
(5)
This estimator reduces the variance of the stochastic gradients [17]. Furthermore, with a copula variational family, this type of reparameterization using a uniform random variable u and a deterministic
function z = z(u; ?, ?) is always possible. (See the supplement.)
The reparameterized gradient (Eq.5) requires calculation of the terms ?zi log q(z; ?, ?) and
??i z(u; ?, ?) for each i. The latter is tractable and derived in the supplement; the former decomposes as
?zi log q(z; ?, ?) = ?zi log q(zi ; ?i ) + ?Q(zi ;?i ) log c(Q(z1 ; ?1 ), . . . , Q(zd ; ?d ); ?)?zi Q(zi ; ?i )
= ?zi log q(zi ; ?i ) + q(zi ; ?i )
d?1
X
X
?Q(zi ;?i ) log ck`|D(e) .
(6)
j=1 e(k,`)?Ej :
i?{k,`}
The summation in Eq.6 is over all pair copulas which contain Q(zi ; ?i ) as an argument. In other
words, the gradient of a latent variable zi is evaluated over both the marginal q(zi ) and all pair
copulas which model correlation between zi and any other latent variable zj . A similar derivation
holds for calculating terms in the score function estimator.
We now turn to the gradient with respect to the copula parameters. We consider copulas which are
differentiable with respect to their parameters. This enables an efficient reparameterized gradient
?? L = Es(u) [(?z log p(x, z) ? ?z log q(z; ?, ?))?? z(u; ?, ?)].
The requirements are the same as for the mean-field parameters.
(7)
Finally, we note that the only requirement on the model is the gradient ?z log p(x, z). This can
be calculated using automatic differentiation tools [22]. Thus Copula vi can be implemented in a
library and applied without requiring any manual derivations from the user.
3.2
Computational complexity
In the vine factorization of the copula, there are d(d ? 1)/2 pair copulas, where d is the number of
latent variables. Thus stochastic gradients of the mean-field parameters ? and copula parameters ?
require O(d2 ) complexity. More generally, one can apply a low rank approximation to the copula by
truncating the number of levels in the vine (see Figure 2). This reduces the number of pair copulas
to be Kd for some K > 0, and leads to a computational complexity of O(Kd).
Using sequential tree selection for learning the vine structure [2], the most correlated variables are at
the highest level of the vines. Thus a truncated low rank copula only forgets the weakest correlations.
This generalizes low rank Gaussian approximations, which also have O(Kd) complexity [20]: it is
the special case when the mean-field distribution is the product of independent Gaussians, and each
pair copula is a Gaussian copula.
3.3
Related work
Preserving structure in variational inference was first studied by Saul and Jordan [19] in the case of
probabilistic neural networks. It has been revisited recently for the case of conditionally conjugate
exponential familes [8]. Our work differs from this line in that we learn the dependency structure
during inference, and thus we do not require explicit knowledge of the model. Further, our augmentation strategy works more broadly to any posterior distribution and any factorized variational family,
and thus it generalizes these approaches.
A similar augmentation strategy is higher-order mean-field methods, which are a Taylor series correction based on the difference between the posterior and its mean-field approximation [11]. Recently,
Giordano et al. [5] consider a covariance correction from the mean-field estimates. All these methods
assume the mean-field approximation is reliable for the Taylor series expansion to make sense, which
is not true in general and thus is not robust in a black box framework. Our approach alternates the
estimation of the mean-field and copula, which we find empirically leads to more robust estimates
than estimating them simultaneously, and which is less sensitive to the quality of the mean-field
approximation.
6
Lambda
method
CVI
LRVB
All off?diagonal covariances
MF
method
CVI
LRVB
0.3
Estimated sd
Estimated sd
0.01
0.2
0.1
0.00
?0.01
0.0
0.0
0.1
0.2
0.3
?0.01
Gibbs standard deviation
0.00
0.01
Gibbs standard deviation
Figure 3: Covariance estimates from copula variational inference (copula vi), mean-field (mf), and
linear response variational Bayes (lrvb) to the ground truth (Gibbs samples). copula vi and lrvb
effectively capture dependence while mf underestimates variance and forgets covariances.
4
Experiments
We study copula vi with two models: Gaussian mixtures and the latent space model [7]. The Gaussian mixture is a classical example of a model for which it is difficult to capture posterior dependencies. The latent space model is a modern Bayesian model for which the mean-field approximation
gives poor estimates of the posterior, and where modeling posterior dependencies is crucial for uncovering patterns in the data.
There are several implementation details of copula vi. At each iteration, we form a stochastic gradient by generating m samples from the variational distribution and taking the average gradient. We
set m = 1024 and follow asynchronous updates [16]. We set the step-size using ADAM [12].
4.1
Mixture of Gaussians
We follow the goal of Giordano et al. [5], which is to estimate the posterior covariance for a Gaussian
mixture. The hidden variables are a K-vector of mixture proportions ? and a set of K P -dimensional
multivariate normals N (?k , ??1
k ), each with unknown mean ?k (a P -vector) and P ? P precision
matrix ?k . In a mixture of Gaussians, the joint probability is
p(x, z, ?, ??1 , ?) = p(?)
K
Y
p(?k , ??1
k )
N
Y
p(xn | zn , ?zn , ??1
zn )p(zn | ?),
n=1
k=1
with a Dirichlet prior p(?) and a normal-Wishart prior p(?k , ??1
k ).
We first apply the mean-field approximation (mf), which assigns independent factors to ?, ?, ?, and
z. We then perform copula vi over the copula-augmented mean-field distribution, i.e., one which
includes pair copulas over the latent variables. We also compare our results to linear response variational Bayes (lrvb) [5], which is a posthoc correction technique for covariance estimation in variational inference. Higher-order mean-field methods demonstrate similar behavior as lrvb. Comparisons to structured approximations are omitted as they require explicit factorizations and are not black
box. Standard black box variational inference [15] corresponds to the mf approximation.
We simulate 10, 000 samples with K = 2 components and P = 2 dimensional Gaussians. Figure
3 displays estimates for the standard deviations of ? for 100 simulations, and plots them against the
ground truth using 500 effective Gibb samples. The second plot displays all off-diagonal covariance
estimates. Estimates for ? and ? indicate the same pattern and are given in the supplement.
When initializing at the true mean-field parameters, both copula vi and lrvb achieve consistent
estimates of the posterior variance. mf underestimates the variance, which is a well-known limitation [25]. Note that because the mf estimates are initialized at the truth, copula vi converges to the
true posterior upon one step of fitting the copula. It does not require alternating more steps.
7
Variational inference methods
Predictive Likelihood
Runtime
Mean-field
lrvb
copula vi (2 steps)
copula vi (5 steps)
copula vi (converged)
-383.2
-330.5
-303.2
-80.2
-50.5
15 min.
38 min.
32 min.
1 hr. 17 min.
2 hr.
Table 1: Predictive likelihood on the latent space model. Each copula vi step either refits the meanfield or the copula. copula vi converges in roughly 10 steps and already significantly outperforms
both mean-field and lrvb upon fitting the copula once (2 steps).
Copula vi is more robust than lrvb. As a toy demonstration, we analyze the MNIST data set of
handwritten digits, using 12,665 training examples and 2,115 test examples of 0?s and 1?s. We perform "unsupervised" classification, i.e., classify without using training labels: we apply a mixture of
Gaussians to cluster, and then classify a digit based on its membership assignment. copula vi reports
a test set error rate of 0.06, whereas lrvb ranges between 0.06 and 0.32 depending on the mean-field
estimates. lrvb and similar higher order mean-field methods correct an existing mf solution?it is
thus sensitive to local optima and the general quality of that solution. On the other hand, copula vi
re-adjusts both the mf and copula parameters as it fits, making it more robust to initialization.
4.2
Latent space model
We next study inference on the latent space model [7], a Bernoulli latent factor model for network
analysis. Each node in an N -node network is associated with a P -dimensional latent variable z ?
N (?, ??1 ). Edges between pairs of nodes are observed with high probability if the nodes are close
to each other in the latent space. Formally, an edge for each pair (i, j) is observed with probability
logit(p) = ? ? |zi ? zj |, where ? is a model parameter.
We generate an N = 100, 000 node network with latent node attributes from a P = 10 dimensional
Gaussian. We learn the posterior of the latent attributes in order to predict the likelihood of held-out
edges. mf applies independent factors on ?, ?, ? and z, lrvb applies a correction, and copula vi
uses the fully dependent variational distribution. Table 1 displays the likelihood of held-out edges and
runtime. We also attempted Hamiltonian Monte Carlo but it did not converge after five hours.
Copula vi dominates other methods in accuracy upon convergence, and the copula estimation without refitting (2 steps) already dominates lrvb in both runtime and accuracy. We note however that
lrvb requires one to invert a O(N K 3 ) ? O(N K 3 ) matrix. We can better scale the method and
achieve faster estimates than copula vi if we applied stochastic approximations for the inversion.
However, copula vi always outperforms lrvb and is still fast on this 100,000 node network.
5
Conclusion
We developed copula variational inference (copula vi). copula vi is a new variational inference
algorithm that augments the mean-field variational distribution with a copula; it captures posterior
dependencies among the latent variables. We derived a scalable and generic algorithm for performing
inference with this expressive variational distribution. We found that copula vi significantly reduces
the bias of the mean-field approximation, better estimates the posterior variance, and is more accurate
than other forms of capturing posterior dependency in variational approximations.
Acknowledgments
We thank Luke Bornn, Robin Gong, and Alp Kucukelbir for their insightful comments. This work
is supported by NSF IIS-0745520, IIS-1247664, IIS-1009542, ONR N00014-11-1-0651, DARPA
FA8750-14-2-0009, N66001-15-C-4032, Facebook, Adobe, Amazon, and the John Templeton Foundation.
8
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9
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5,158 | 567 | Merging Constrained Optimisation with
Deterministic Annealing to "Solve"
Combinatorially Hard Problems
Paul Stolorz?
Santa Fe Institute
1660 Old Pecos Trail, Suite A
Santa Fe, NM 87501
ABSTRACT
Several parallel analogue algorithms, based upon mean field theory (MFT)
approximations to an underlying statistical mechanics formulation, and requiring an externally prescribed annealing schedule, now exist for finding
approximate solutions to difficult combinatorial optimisation problems.
They have been applied to the Travelling Salesman Problem (TSP), as
well as to various issues in computational vision and cluster analysis. I
show here that any given MFT algorithm can be combined in a natural
way with notions from the areas of constrained optimisation and adaptive
simulated annealing to yield a single homogenous and efficient parallel relaxation technique, for which an externally prescribed annealing schedule
is no longer required. The results of numerical simulations on 50-city and
100-city TSP problems are presented, which show that the ensuing algorithms are typically an order of magnitude faster than the MFT algorithms
alone, and which also show, on occasion, superior solutions as well.
1
INTRODUCTION
Several promising parallel analogue algorithms, which can be loosely described by
the term "deterministic annealing" , or "mean field theory (MFT) annealing", have
*also at Theoretical Division and Center for Nonlinear Studies, MSB213, Los Alamos
National Laboratory, Los Alamos, NM 87545.
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Stolorz
recently been proposed as heuristics for tackling difficult combinatorial optimisation
problems [1, 2, 3, 4, 5, 6, 7] . However, the annealing schedules must be imposed
externally in a somewhat ad hoc manner in these procedures (although they can be
made adaptive to a limited degree [8]). As a result, a number of authors [9, 10, 11]
have considered the alternative analogue approach of Lagrangian relaxation, a form
of constrained optimisation due originally to Arrow [12], as a different means of
tackling these problems. The various alternatives require the introduction of a new
set of variables, the Lagrange multipliers. Unfortunately, these usually lead in turn
to either the inclusion of expensive penalty terms, or the consideration of restricted
classes of problem constraints. The penalty terms also tend to introduce unwanted
local minima in the objective function, and they must be included even when the
algorithms are exact [13, 10]. These drawbacks prevent their easy application to
large-scale combinatorial problems, containing 100 or more variables.
In this paper I show that the technical features of analogue mean field approximations can be merged with both Lagrangian relaxation methods, and with the
broad philosophy of adaptive annealing without, importantly, requiring the large
computational resources that typically accompany the Lagrangian methods. The
result is a systematic procedure for crafting from any given MFT algorithm a single parallel homogeneous relaxation technique which needs no externally prescribed
annealing schedule. In this way the computational power of the analogue heuristics is greatly enhanced. In particular, the Lagrangian framework can be used to
construct an efficient adaptation of the elastic net algorithm [2], which is perhaps
the most promising of the analogue heuristics. The results of numerical experiments are presented which display both increased computational efficiency, and on
occasion, better solutions (avoidance of some local minima) over deterministic annealing. Also, the qualitative mechanism at the root of this behaviour is described.
Finally, I note that the apparatus can be generalised to a procedure that uses several
multipliers, in a manner that roughly parallels the notion of different temperatures
at different physical locations in the simulated annealing heuristic.
2
DETERMINISTIC ANNEALING
The deterministic annealing procedures consist of tracking the local minimum of an
objective function of the form
(1)
where x represents the analogue variables used to describe the particular problem at
hand, and T ~ 0 (initially chosen large) is an adjustable annealing, or temperature,
parameter. As T is lowered, the objective function undergoes a qualitative change
from a convex to a distinctly non-convex function. Provided the annealing shedule
is slow enough, however, it is hoped that the local minimum near T = 0 is a close
approximation to the global solution of the problem.
The function S( x) represents an analogue approximation [5, 4, 7] to the entropy of
an underlying discrete statistical physics system, while F(l;,.) approximates its free
energy. The underlying discrete system forms the basis of the simulated annealing
heuristic [14]. Although a general and powerful technique, this heuristic is an inherently stochastic procedure which must consider many individual discrete tours at
Merging Constrained Optimisation with Deterministic Annealing
each and every temperature T. The deterministic annealing approximations have
the advantage of being deterministic, so that an approximate solution at a given
temperature can be found with much less computational effort. In both cases,
however, the complexity of the problem under consideration shows up in the need
to determine with great care an annealing schedule for lowering the temperature
parameter.
The primary contribution of this paper consists in pursuing the relationship between
deterministic annealing and statistical physics one step further, by making explicit
use of the fact that due to the statistical physics embedding of the deterministic
annealing procedures,
(2)
where Xmin is the local minimum obtained for the parameter value T. This deceptively simple observation allows the consideration of the somewhat different approach of Lagrange multiplier methods to automatically determine a dynamics for
T in the analogue heuristics, using as a constraint the vanishing of the entropy
function at zero temperature. This particular fact has not been explicitly used in
any previous optimisation procedures based upon Lagrange multipliers, although it
is implicit in the work of [9]. Most authors have focussed instead on the syntactic
constraints contained in the function U(~) when incorporating Lagrange multipliers. As a result the issue of eliminating an external annealing schedule has not been
directly confronted.
3
LAGRANGE MULTIPLIERS
Multiplier methods seek the critical points of a "Lagrangian" function
F(~,;\)
= U(x) -
;\S(x)
(3)
where the notation of (1) has been retained, in accordance with the philosophy
discussed above. The only difference is that the parameter T has been replaced by
a variable ;\ (the Lagrange multiplier), which is to be treated on the same basis
as the variables~. By definition, the critical points of F(x,;\) obey the so-called
Kuhn-Tucker conditions
\1LF(~,;\)
\1>..F(x,;\)
= 0 = \1-rU(~) - ;\ \1-rS(~)
=0=
(4)
-Sex)
Thus, at any critical point of this function, the constraint S(~) = 0 is satisfied. This
corresponds to a vanishing entropy estimate in (1). Hopefully, in addition, U(~) is
minimised, subject to the constraint.
The difficulty with this approach when used in isolation is that finding the critical
points of F(~,;\) entails, in general, the minimisation of a transformed "unconstrained" function, whose set of local minima contains the critical points of F as
a subset. This transformed function is required in order to ensure an algorithm
which is convergent, because the critical points of F(~,;\) are saddle points, not
local minima. One well-known way to do this is to add a term S2(~) to (3), giving
an augmented Lagrangian with the same fixed points as (3), but hopefully with
better convergence properties. Unfortunately, the transformed function is invariably more complicated than F(~, ;\), typically containing extra quadratic penalty
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Stolorz
terms (as in the above case), which tend to convert harmless saddle points into
unwanted local minima. It also leads to greater computational overhead, usually
in the form of either second derivatives of the functions U(L) and S(L) , or of matrix inversions [13, 10] (although see [11] for an approach which minimises this
overhead). For large-scale combinatorial problems such as the TSP these disadvantages become prohibitive. In addition, the entropic constraint functions occurring
in deterministic annealing tend to be quite complicated nonlinear functions of the
variables involved, often with peculiar behaviour near the constraint condition. In
these cases (the Hopfield /Tank method is an example) a term quadratic in the entropy cannot simply be added to (3) in a straightforward way to produce a suitable
augmented Lagrangian (of course, such a procedure is possible with several of the
terms in the internal energy U (.~?.
4
COMBINING BOTH METHODS
The best features of each of the two approaches outlined above may be retained by
using the following modification of the original first-order Arrow technique:
Xi
>.
= -"Vr,F(x,>.) = -"Vr,U(x) + >'''Vx,S(x)
=+"V>.F(x,>.)
(5)
=-S(x)+c/>'
where F(x, >.) is a slightly modified "free energy" function given by
F(x, >.) = U(x) - >'S(x)
+ c In >.
(6)
In these expressions, c > 0 is a constant, chosen small on the scale of the other parameters, and characterises the sole, inexpensive, penalty requirement. It is needed
purely in order to ensure that>. remain positive. In fact, in the numerical experiment that I will present, this penalty term for>. was not even used - the algorithm
was simply terminated at a suitably small value of >..
The reason for insisting upon>. > 0, in contrast to most first-order relaxation methods, is that it ensures that the free energy objective function is bounded below with
respect to the X variables. This in turn allows (5) to be proven locally convergent
[15] using techniques discussed in [13]. Furthermore, the methods described by (5)
are found empirically to be globally convergent as well. This feature is in fact the
key to their computational efficiency, as it means that they need not be grafted
onto more sophisticated and inefficient methods in order to ensure convergence.
This behaviour can be traced to the fact that the ''free energy" functions, while
non-convex overall with respect to L, are nevertheless convex over large volumes of
the solution space. The point can be illustrated by the construction of an energy
function similar to that used by Platt and Barr [9], which also displays the mechanism by which some of the unwanted local minima in deterministic annealing may
be avoided. These issues are discussed further in Section 6.
The algorithms described above have several features which distinguish them from
previous work. Firstly, the entropy estimate Sex) has been chosen explicitly as
the appropriate constraint function, a fact which has previously been unexploited
in the optimisation context (although a related piecewise linear function has been
used by [9]). Further, since this estimate is usually positive for the mean field theory
Merging Constrained Optimisation with Deterministic Annealing
heuristics, A (the only new variable) decreases monotonically in a manner roughly
similar to the temperature decrease schedule used in simulated and deterministic
annealing, but with the ad hoc drawback now removed. Moreover, there is no
requirement that the system be at or near a fixed point each time A is altered there is simply one homogeneous dynamical system which must approach a fixed
point only once at the very end of the simulation, and furthermore A appears linearly
except near the end of the procedure (a major reason for its efficiency). Finally,
the algorithms do not require computationally cumbersome extra structure in the
form of quadratic penalty terms, second derivatives or inverses, in contrast to the
usual Lagrangian relaxation techniques. All of these features can be seen to be due
to the statistical physics setting of the annealing "Lagrangian", and the use of an
entropic constraint instead of the more usual syntactic constraints.
The apparatus outlined above can immediately be used to adapt the Hopfield/Tank
heuristic for the Travelling Salesman Problem (TSP) [1], which can easily be written
in the form (1). However, the elastic net method [2] is known to be a somewhat
superior method, and is therefore a better candidate for modification. There is
an impediment to the procedure here: the objective function for the elastic net is
actually of the form
F(z., A) = U(x) - AS(X, A)
(7)
which precludes the use of a true Lagrange muliplier, since A now appears nontrivially in the constraint function itself! However, I find surprisingly that the
algorithm obtained by applying the Lagrangian relaxation apparatus in a straightforward way as before still leads to a coherent algorithm. The equations are
Xi
= -V'1',F(x,A) = -V'1',U(z.) + AV'1',S(X)
A = +(V'>.F(z.,A)
= -([S(.~,A)
(8)
+ AV'>.S(X,A)]
The parameter ( > 0 is chosen so that an explicit barrier term for A can be avoided.
It is the only remaining externally prescribed part of the former annealing schedule,
and is fixed just once at the begining of the algorithm.
It can be shown that the global convergence of (8) is highly plausible in general (and
seems to always occur in practice), as in the simpler case described by (5). Secondly,
and most importantly, it can be shown that the constraints that are obeyed at
the new fixed points satisfy the syntax of the original discrete problem [15]. The
procedure is not limited to the elastic net method for the TSP. The mean field
approximations discussed in [3, 4, 5] all behave in a similar way, and can therefore
be adapted successfully to Lagrangian relaxation methods. The form of the elastic
net entropy function suggests a further natural generalisation of the procedure.
A different "multiplier" Aa can be assigned to each city a, each variable being
responsible for satisfying a different additive component of the entropy constraint.
The idea has an obvious parallel to the notion in simulated annealing of lowering
the temperature in different geographical regions at different rates in response to
the behaviour of the system. The number of extra variables required is a modest
computational investment, since there are typically many more tour points than
city points for a given implementation.
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Stolorz
5
RESULTS FOR THE TSP
Numerical simulations were performed on various TSP instances using the elastic
net method, the Lagrangian adaptation with a single global Lagrange multiplier,
and the modification discussed above involving one Lagrange multiplier for each city.
The results are shown in Table 1. The tours for the Lagrangian relaxation methods
are about 0.5% shorter than those for the elastic net, although these differences
are not yet at a statistically significant level. The differences in the computational
requirements are, however, much more dramatic. No attempt has been made to
optimise any of the techniques by using sophisticated descent procedures, although
the size of the update step has been chosen to seperately optimise each method.
Table 1: Performance of heuristics described in the text on a set of 40 randomly
distibuted 50-city instances of the TSP in the unit square. CPU times quoted are
for a SUN SPARC Station 1+. Q' and j3 are the standard tuning parameters [4].
METHOD
Elastic net
Global multiplier
Local multipliers
Q'
0.2
0.4
0.4
j3
TOUR LENGTH
CPU(SEC)
2.5
2.5
2.5
5.95 ? 0.10
5.92 ? 0.09
5.92 ? 0.08
260 ? 33
49 ? 5
82 ? 12
I have also been able to obtain a superior solution to the 100-city problem analysed
by Durbin and Willshaw [2], namely a solution of length 7.746 [15] (c.f. length
7.783 for the elastic net) in a fraction of the time taken by elastic net annealing.
This represents an improvement of roughly 0.5%. Although still about 0.5% longer
than the best tour found by simulated annealing, this result is quite encouraging,
because it was obtained with far less CPU time than simulated annealing, and in
substantially less time than the elastic net: improvements upon solutions within
about 1% of optimality typically require a substantial increase in CPU investment.
6
HOW IT WORKS - VALLEY ASCENT
Inspection of the solutions obtained by the various methods indicates that the multiplier schemes can sometimes exchange enough "inertial" energy to overcome the
energy barriers which trap the annealing methods, thus offering better solutions as
well as much-improved computational efficiency. This point is illustrated in Figure
l(a), which displays the evolution of the following function during the algorithm for
a typical set of parameters:
1~ .
E
= "2 ~ Xi
2
1. 2
+ 2"\
(9)
I
The two terms can be thought of as different components of an overall kinetic energy
E. During the procedure, energy can be exchanged between these two components,
so the function E(t) does not decrease monotonically with time. This allows the
system to occasionally escape from local minima. Nevertheless, after a long enough
Merging Constrained Optimisation with Deterministic Annealing
(a)
6
(I)
Q)
.~
~4
o
'j
c:
22
..
I~?,
\
o
.....,.
.... -"'..._......... .
... :'": ....
_.,-_....._._._.-. .
...
.....:.~
L-LJ......L....J..............L-&...L..I...I...I....J'-..I...L...J....L..J....L..L...L=.::J
o
20
40
60
80
100
Iteration number
Figure I: (a) Evolution of variables for a typical 50-city TSP. The solid curve shows
the total kinetic energy E given by (9). The dotted curve shows the A component
of this energy, and the dash-dotted curve shows the x component. (b) Trajectories
taken by various algorithms on a schematic free energy surface. The two dashdotted curves show possible paths for elastic net annealing, each ascending a valley
floor. The dotted curve shows a Lagrangian relaxation, which displays oscillations
about the valley floor leading to the superior solution.
time the function does decrease smoothly, ensuring convergence to a valid solution
to the problem.
The basic mechanism can also be understood by plotting schematically the free
energy "surface" F(ll, A), as shown in Figure l(b) . This surface has a single valley
in the foreground, where A is large. Bifurcations occur as A becomes smaller, with
a series of saddles, each a valid problem solution, being reached in the background
at A O. Deterministic annealing can be viewed as the ascent of just one of these
valleys along the valley floor. It is hoped that the broadest and deepest minimum
is chosen at each valley bifurcation, leading eventually to the lowest background
saddle point as the optimal solution. A typical trajectory for one of the Lagrangian
modifications also consists roughly of the ascent of one of these valleys. However,
oscillations about the valley floor now occur on the way to the final saddle point,
due to the interplay between the different kinetic components displayed in Figure
lea). It is hoped that the extra degrees of freedom allow valleys to be explored more
fully near bifurcation points, thus biasing the larger valleys more than deterministic
annealing. Notice that in order to generate the A dynamics, computational significance is now assigned to the actual value of the free energy in the new schemes, in
contrast to the situation in regular annealing.
=
7
CONCLUSION
In summary, a simple yet effective framework has been developed for systematically
generalising any algorithm described by a mean field theory approximation procedure to a Lagrangian method which replaces annealing by the relaxation of a single
dynamical system. Even in the case of the elastic net, which has a slightly awkward
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Stolorz
form, the resulting method can be shown to be sensible, and I find in fact that it
substantially improves the speed (and accuracy) of that method. The adaptations
depend crucially upon the vanishing of the analogue entropy at zero temperature.
This allows the entropy to be used as a powerful constraint function, even though
it is a highly nonlinear function and might be expected at first sight to be unsuitable for the task. In fact, this observation can also be applied in a wider context
to design objective functions and architectures for neural networks which seek to
improve generalisation ability by limiting the number of network parameters [16].
References
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Alamos National Laboratory, 1991.
[16] P.Stolorz. Analogue entropy as a constraint in adaptive learning and optimisation. Technical report, in preparation, Santa Fe Institute, 1992.
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5,159 | 5,670 | Fast Second-Order Stochastic Backpropagation for
Variational Inference
Kai Fan
Duke University
[email protected]
Ziteng Wang?
HKUST?
[email protected]
Jeffrey Beck
Duke University
[email protected]
Katherine Heller
Duke University
[email protected]
James T. Kwok
HKUST
[email protected]
Abstract
We propose a second-order (Hessian or Hessian-free) based optimization method
for variational inference inspired by Gaussian backpropagation, and argue that
quasi-Newton optimization can be developed as well. This is accomplished
by generalizing the gradient computation in stochastic backpropagation via a
reparametrization trick with lower complexity. As an illustrative example, we apply this approach to the problems of Bayesian logistic regression and variational
auto-encoder (VAE). Additionally, we compute bounds on the estimator variance
of intractable expectations for the family of Lipschitz continuous function. Our
method is practical, scalable and model free. We demonstrate our method on several real-world datasets and provide comparisons with other stochastic gradient
methods to show substantial enhancement in convergence rates.
1
Introduction
Generative models have become ubiquitous in machine learning and statistics and are now widely
used in fields such as bioinformatics, computer vision, or natural language processing. These models
benefit from being highly interpretable and easily extended. Unfortunately, inference and learning
with generative models is often intractable, especially for models that employ continuous latent
variables, and so fast approximate methods are needed. Variational Bayesian (VB) methods [1] deal
with this problem by approximating the true posterior that has a tractable parametric form and then
identifying the set of parameters that maximize a variational lower bound on the marginal likelihood.
That is, VB methods turn an inference problem into an optimization problem that can be solved, for
example, by gradient ascent.
Indeed, efficient stochastic gradient variational Bayesian (SGVB) estimators have been developed for auto-encoder models [17] and a number of papers have followed up on this approach
[28, 25, 19, 16, 15, 26, 10]. Recently, [25] provided a complementary perspective by using stochastic
backpropagation that is equivalent to SGVB and applied it to deep latent gaussian models. Stochastic backpropagation overcomes many limitations of traditional inference methods such as the meanfield or wake-sleep algorithms [12] due to the existence of efficient computations of an unbiased
estimate of the gradient of the variational lower bound. The resulting gradients can be used for parameter estimation via stochastic optimization methods such as stochastic gradient decent(SGD) or
adaptive version (Adagrad) [6].
?
?
Equal Contribution to this work
Refer to Hong Kong University of Science and Technology
1
Unfortunately, methods such as SGD or Adagrad converge slowly for some difficult-to-train models,
such as untied-weights auto-encoders or recurrent neural networks. The common experience is that
gradient decent always gets stuck near saddle points or local extrema. Meanwhile the learning rate
is difficult to tune. [18] gave a clear explanation on why Newton?s method is preferred over gradient
decent, which often encounters under-fitting problem if the optimizing function manifests pathological curvature. Newton?s method is invariant to affine transformations so it can take advantage of
curvature information, but has higher computational cost due to its reliance on the inverse of the
Hessian matrix. This issue was partially addressed in [18] where the authors introduced Hessian
free (HF) optimization and demonstrated its suitability for problems in machine learning.
In this paper, we continue this line of research into 2nd order variational inference algorithms. Inspired by the property of location scale families [8], we show how to reduce the computational
cost of the Hessian or Hessian-vector product, thus allowing for a 2nd order stochastic optimization scheme for variational inference under Gaussian approximation. In conjunction with the HF
optimization, we propose an efficient and scalable 2nd order stochastic Gaussian backpropagation
for variational inference called HFSGVI. Alternately, L-BFGS [3] version, a quasi-Newton method
merely using the gradient information, is a natural generalization of 1st order variational inference.
The most immediate application would be to look at obtaining better optimization algorithms for
variational inference. As to our knowledge, the model currently applying 2nd order information is
LDA [2, 14], where the Hessian is easy to compute [11]. In general, for non-linear factor models
like non-linear factor analysis or the deep latent Gaussian models this is not the case. Indeed,
to our knowledge, there has not been any systematic investigation into the properties of various
optimization algorithms and how they might impact the solutions to optimization problem arising
from variational approximations.
The main contributions of this paper are to fill such gap for variational inference by introducing a
novel 2nd order optimization scheme. First, we describe a clever approach to obtain curvature information with low computational cost, thus making the Newton?s method both scalable and efficient.
Second, we show that the variance of the lower bound estimator can be bounded by a dimension-free
constant, extending the work of [25] that discussed a specific bound for univariate function. Third,
we demonstrate the performance of our method for Bayesian logistic regression and the VAE model
in comparison to commonly used algorithms. Convergence rate is shown to be competitive or faster.
2
Stochastic Backpropagation
In this section, we extend the Bonnet and Price theorem [4, 24] to develop 2nd order Gaussian
backpropagation. Specifically, we consider how to optimize an expectation of the form Eq? [f (z|x)],
where z and x refer to latent variables and observed variables respectively, and expectation is taken
w.r.t distribution q? and f is some smooth loss function (e.g. it can be derived from a standard
variational lower bound [1]). Sometimes we abuse notation and refer to f (z) by omitting x when no
ambiguity exists. To optimize such expectation, gradient decent methods require the 1st derivatives,
while Newton?s methods require both the gradients and Hessian involving 2nd order derivatives.
2.1
Second Order Gaussian Backpropagation
If the distribution q is a dz -dimensional Gaussian N (z|?, C), the required partial derivative is easily
computed with a lower algorithmic cost O(d2z ) [25]. By using the property of Gaussian distribution,
we can compute the 2nd order partial derivative of EN (z|?,C) [f (z)] as follows:
?2?i ,?j EN (z|?,C) [f (z)]
= EN (z|?,C) [?2zi ,zj f (z)] = 2?Cij EN (z|?,C) [f (z)],
(1)
1
?2Ci,j ,Ck,l EN (z|?,C) [f (z)] =
EN (z|?,C) [?4zi ,zj ,zk ,zl f (z)],
(2)
4
1
?2?i ,Ck,l EN (z|?,C) [f (z)] =
EN (z|?,C) ?3zi ,zk ,zl f (z) .
(3)
2
Eq. (1), (2), (3) (proof in supplementary) have the nice property that a limited number of samples
from q are sufficient to obtain unbiased gradient estimates. However, note that Eq. (2), (3) needs
to calculate the third and fourth derivatives of f (z), which is highly computationally inefficient. To
avoid the calculation of high order derivatives, we use a co-ordinate transformation.
2
2.2
Covariance Parameterization for Optimization
By constructing the linear transformation (a.k.a. reparameterization) z = ? + R, where ?
N (0, Idz ), we can generate samples from any Gaussian distribution N (?, C) by simulating data
from a standard normal distribution, provided the decomposition C = RR> holds. This fact allows
us to derive the following theorem indicating that the computation of 2nd order derivatives can be
scalable and programmed to run in parallel.
Theorem 1 (Fast Derivative). If f is a twice differentiable function and z follows Gaussian distribution N (?, C), C = RR> , where both the mean ? and R depend on a d-dimensional parameter ? = (?l )dl=1 , i.e. ?(?), R(?), we have ?2?,R EN (?,C) [f (z)] = E?N (0,Idz ) [> ? H], and
?2R EN (?,C) [f (z)] = E?N (0,Idz ) [(T ) ? H]. This then implies
> ?(? + R)
,
(4)
??l EN (?,C) [f (z)] = E?N (0,I) g
??l
"
#
>
?(? + R)
?(? + R)
? 2 (? + R)
?2?l ?l EN (?,C) [f (z)] = E?N (0,I)
H
+ g>
, (5)
1 2
??l1
??l2
??l1 ?l2
where ? is Kronecker product, and gradient g, Hessian H are evaluated at ? + R in terms of f (z).
If we consider the mean and covariance matrix as the variational parameters in variational inference,
the first two results w.r.t ?, R make parallelization possible, and reduce computational cost of the
Hessian-vector multiplication due to the fact that (A> ? B)vec(V ) = vec(AV B). If the model has
few parameters or a large resource budget (e.g. GPU) is allowed, Theorem 1 launches the foundation
for exact 2nd order derivative computation in parallel. In addition, note that the 2nd order gradient
computation on model parameter ? only involves matrix-vector or vector-vector multiplication, thus
leading to an algorithmic complexity that is O(d2z ) for 2nd order derivative of ?, which is the same
as 1st order gradient [25]. The derivative computation at function f is up to 2nd order, avoiding to
calculate 3rd or 4th order derivatives. One practical parametrization assumes a diagonal covariance
matrix C = diag{?12 , ..., ?d2z }. This reduces the actual computational cost compared with Theorem
1, albeit the same order of the complexity (O(d2z )) (see supplementary material). Theorem 1 holds
for a large class of distributions in addition to Gaussian distributions, such as student t-distribution.
If the dimensionality d of embedded parameter ? is large, computation of the gradient G? and
Hessian H? (differ from g, H above) will be linear and quadratic w.r.t d, which may be unacceptable.
Therefore, in the next section we attempt to reduce the computational complexity w.r.t d.
2.3
Apply Reparameterization on Second Order Algorithm
In standard Newton?s method, we need to compute the Hessian matrix and its inverse, which is
intractable for limited computing resources. [18] applied Hessian-free (HF) optimization method in
deep learning effectively and efficiently. This work largely relied on the technique of fast Hessian
matrix-vector multiplication [23]. We combine reparameterization trick with Hessian-free or quasiNewton to circumvent matrix inverse problem.
Hessian-free Unlike quasi-Newton methods HF doesn?t make any approximation on the Hessian.
HF needs to compute H? v, where v is any vector that has the matched dimension to H? , and then
uses conjugate gradient algorithm to solve the linear system H? v = ??F (?)> v, for any objective
function F . [18] gives a reasonable explanation for Hessian free optimization. In short, unlike a
pre-training method that places the parameters in a search region to regularize[7], HF solves issues
of pathological curvature in the objective by taking the advantage of rescaling property of Newton?s
(?)
method. By definition H? v = lim??0 ?F (?+?v)??F
indicating that H? v can be numerically
?
computed by using finite differences at ?. However, this numerical method is unstable for small ?.
In this section, we focus on the calculation of H? v by leveraging a reparameterization trick.
Specifically, we apply an R-operator technique [23] for computing the product H? v exactly. Let
F = EN (?,C) [f (z)] and reparametrize z again as Sec. 2.2, we do variable substitution ? ? ? + ?v
after gradient Eq. (4) is obtained, and then take derivative on ?. Thus we have the following analyt3
Algorithm 1 Hessian-free Algorithm on Stochastic Gaussian Variational Inference (HFSGVI)
Parameters: Minibatch Size B, Number of samples to estimate the expectation M (= 1 as default),
Input: Observation X (and Y if required), Lower bound function L = EN (?,C) [fL ]
Output: Parameter ? after having converged.
1: for t = 1, 2, . . . do
2:
xB
b=1 ? Randomly draw B datapoints from full data set X;
3:
M
mb =1 ? sample M times from N (0, I) for each xb ;
P P
1
> ?(?+Rmb )
4:
Define gradient G(?) = M
, gb,m = ?z (fL (z|xb ))|z=?+Rmb ;
b
mb gb,m
??
5:
Define function B(?, v) = ?? G(? + ?v)|?=0 , where v is a d-dimensional vector;
6:
Using Conjugate Gradient algorithm to solve linear system: B(?t , pt ) = ?G(?t );
7:
?t+1 = ?t + pt ;
8: end for
ical expression for Hessian-vector multiplication:
#
"
?
?
?
(?(?)
+
R(?))
>
H? v =
=
?F (? + ?v)
EN (0,I) g
??
??
??
?=0
???+?v ?=0
"
!#
?
? (?(?) + R(?))
= EN (0,I)
g>
.
(6)
??
??
???+?v
?=0
Eq. (6) is appealing since it does not need to store the dense matrix and provides an unbiased H? v
estimator with a small sample size. In order to conduct the 2nd order optimization for variational inference, if the computation of the gradient for variational lower bound is completed, we only need to
add one extra step for gradient evaluation via Eq. (6) which has the same computational complexity
as Eq. (4). This leads to a Hessian-free variational inference method described in Algorithm 1.
For the worst case of HF, the conjugate gradient (CG) algorithm requires at most d iterations to
terminate, meaning that it requires d evaluations of H? v product. However, the good news is that
CG leads to good convergence after a reasonable number of iterations. In practice we found that it
may not necessary to wait CG to converge. In other words, even if we set the maximum iteration K
in CG to a small fixed number (e.g., 10 in our experiments, though with thousands of parameters),
the performance does not deteriorate. The early stoping strategy may have the similar effect of
Wolfe condition to avoid excessive step size in Newton?s method. Therefore we successfully reduce
the complexity of each iteration to O(Kdd2z ), whereas O(dd2z ) is for one SGD iteration.
L-BFGS Limited memory BFGS utilizes the information gleaned from the gradient vector to approximate the Hessian matrix without explicit computation, and we can readily utilize it within
our framework. The basic idea of BFGS approximates Hessian by an iterative algorithm Bt+1 =
>
>
>
Bt + ?Gt ?G>
t /??t ??t ? Bt ??t ??t Bt /??t Bt ??t , where ?Gt = Gt ? Gt?1 and
??t = ?t ? ?t?1 . By Eq. (4), the gradient Gt at each iteration can be obtained without any
difficulty. However, even if this low rank approximation to the Hessian is easy to invert analytically due to the Sherman-Morrison formula, we still need to store the matrix. L-BFGS will further
implicitly approximate this dense Bt or B?1
by tracking only a few gradient vectors and a short
t
history of parameters and therefore has a linear memory requirement. In general, L-BFGS can perform a sequence of inner products with the K most recent ??t and ?Gt , where K is a predefined
constant (10 or 15 in our experiments). Due to the space limitations, we omit the details here but
none-the-less will present this algorithm in experiments section.
2.4
Estimator Variance
The framework of stochastic backpropagation [16, 17, 19, 25] extensively uses the mean of very
few samples (often just one) to approximate the expectation. Similarly we approximate the left side
of Eq. (4), (5), (6) by sampling few points from the standard normal distribution. However, the
magnitude of the variance of such an estimator is not seriously discussed. [25] simply explored the
variance quantitatively for separable functions.[19] merely borrowed the variance reduction technique from reinforcement learning by centering the learning signal in expectation and performing
variance normalization. Here, we will generalize the treatment of variance to a broader family,
Lipschitz continuous function.
4
Theorem 2 (Variance Bound). If f is an L-Lipschitz differentiable function and ? N (0, Idz ),
2 2
then E[(f () ? E[f ()])2 ] ? L 4? .
The proof of Theorem 2 (see supplementary) employs the properties of sub-Gaussian distributions
and the duplication trick that are commonly used in learning theory. Significantly, the result implies
a variance bound independent
the dimensionality
of Gaussian variable. Note that from the proof,
?(fof()?E[f
()])
L2 ?2 ? 2 /8
we can only obtain the E e
?e
for ? > 0. Though this result is enough
to illustrate the variance independence of dz , we can in fact tighten it to a sharper upper bound by
2 2
a constant scalar, i.e. e? L /2 , thus leading to the result of Theorem 2 with Var(f ()) ? L2 . If
all the results above hold for smooth (twice continuous and differentiable) functions with Lipschitz
constant L then it holds for all Lipschitz functions by a standard approximation argument. This
means the condition can be relaxed to Lipschitz continuous function.
P
t2
1
M
? 2M
? 2 L2 .
Corollary 3 (Bias Bound). P M
m=1 f (m ) ? E[f ()] ? t ? 2e
It is also worth mentioning that the significant corollary of Theorem 2 is probabilistic inequality
to measure the convergence rate of Monte Carlo approximation in our setting. This tail bound,
together with variance bound, provides the theoretical guarantee for stochastic backpropagation on
Gaussian variables and provides an explanation for why a unique realization (M = 1) is enough
in practice. By reparametrization, Eq. (4), (5, (6) can be formulated as the expectation w.r.t the
isotropic Gaussian distribution with identity covariance matrix leading to Algorithm 1. Thus we
can rein in the number of samples for Monte Carlo integration regardless dimensionality of latent
variables z. This seems counter-intuitive. However, we notice that larger L may require more
samples, and Lipschitz constants of different models vary greatly.
3
Application on Variational Auto-encoder
Note that our method is model free. If the loss function has the form of the expectation of a function
w.r.t latent Gaussian variables, we can directly use Algorithm 1. In this section, we put the emphasis
on a standard framework VAE model [17] that has been intensively researched; in particular, the
function endows the logarithm form, thus bridging the gap between Hessian and fisher information
matrix by expectation (see a survey [22] and reference therein).
3.1
Model Description
(i)
Suppose we have N i.i.d. observations X = {x(i) }N
? RD is a data vector that can
i=1 , where x
take either continuous or discrete values. In contrast to a standard auto-encoder model constructed
by a neural network with a bottleneck structure, VAE describes the embedding process from the
prospective of a Gaussian latent variable model. Specifically, each data point x follows a generative
model p? (x|z), where this process is actually a decoder that is usually constructed by a non-linear
transformation with unknown parameters ? and a prior distribution p? (z). The encoder or recognition model q? (z|x) is used to approximate the true posterior p? (z|x), where ? is similar to the
parameter of variational distribution. As suggested in [16, 17, 25], multi-layered perceptron (MLP)
is commonly considered as both the probabilistic encoder and decoder. We will later see that this
construction is equivalent to a variant deep neural networks under the constrain of unique realization
for z. For this model and each datapoint, the variational lower bound on the marginal likelihood is,
log p? (x(i) ) ?
Eq? (z|x(i) ) [log p? (x(i) |z)] ? DKL (q? (z|x(i) )kp? (z)) = L(x(i) ).
(7)
We can actually write the KL divergence into the expectation term and denote (?, ?) as ?.
By the previous
discussion, this means that our objective is to solve the optimization problem
P
arg max? i L(x(i) ) of full dataset variational lower bound. Thus L-BFGS or HF SGVI algorithm
can be implemented straightforwardly to estimate the parameters of both generative and recognition
models. Since the first term of reconstruction error appears in Eq. (7) with an expectation form
on latent variable, [17, 25] used a small finite number M samples as Monte Carlo integration with
reparameterization trick to reduce the variance. This is, in fact, drawing samples from the standard normal distribution. In addition, the second term is the KL divergence between the variational
distribution and the prior distribution, which acts as a regularizer.
5
3.2
Deep Neural Networks with Hybrid Hidden Layers
In the experiments, setting M = 1 can not only achieve excellent performance but also speed up
the program. In this special case, we discuss the relationship between VAE and traditional deep
PD
auto-encoder. For binary inputs, denote the output as y, we have log p? (x|z) = j=1 xj log yj +
(1 ? xj ) log(1 ? yj ), which is exactly the negative cross-entropy. It is also apparent that log p? (x|z)
is equivalent to negative squared error loss for continuous data. This means that maximizing the
lower bound is roughly equal to minimizing the loss function of a deep neural network (see Figure
1 in supplementary), except for different regularizers. In other words, the prior in VAE only imposes a regularizer in encoder or generative model, while L2 penalty for all parameters is always
considered in deep neural nets. From the perspective of deep neural networks with hybrid hidden
nodes, the model consists of two Bernoulli layers and one Gaussian layer. The gradient computation
can simply follow a variant of backpropagation layer by layer (derivation given in supplementary).
To further see the rationale of setting M = 1, we will investigate the upper bound of the Lipschitz
constant under various activation functions in the next lemma. As Theorem 2 implies, the variance
of approximate expectation by finite samples mainly relies on the Lipschitz constant, rather than
dimensionality. According to Lemma 4, imposing a prior or regularization to the parameter can
control both the model complexity and function smoothness. Lemma 4 also implies that we can get
the upper bound of the Lipschitz constant for the designed estimators in our algorithm.
Lemma 4. For a sigmoid activation function g in deep neural networks with one Gaussian layer z,
z ? N (?, C), C = R> R. Let z = ? + R, then the Lipschitz constant of g(Wi, (? + R) + bi )
is bounded by 14 kWi, Rk2 , where Wi, is ith row of weight matrix and bi is the ith element bias.
Similarly, for hyperbolic tangent or softplus function, the Lipschitz constant is bounded by kWi, Rk2 .
4
Experiments
We apply our 2nd order stochastic variational inference to two different non-conjugate models. First,
we consider a simple but widely used Bayesian logistic regression model, and compare with the
most recent 1st order algorithm, doubly stochastic variational inference (DSVI) [28], designed for
sparse variable selection with logistic regression. Then, we compare the performance of VAE model
with our algorithms.
4.1
Bayesian Logistic Regression
D
Given a dataset {xi , yi }N
includes the default feature 1 and
i=1 , where each instance xi ? R
yi ? {?1, 1} is the binary label, the Bayesian logistic regression models the probability of outputs conditional on features and the coefficients ? with an imposed prior. The likelihood and the
QN
prior can usually take the form as i=1 g(yi x>
i ?) and N (0, ?) respectively, where g is sigmoid
function and ? is a diagonal covariance matrix for simplicity. We can propose a variational Gaussian
distribution q(?|?, C) to approximate the posterior of regression parameter. If we further assume a
QD
diagonal C, a factorized form j=1 q(?j |?j , ?j ) is both efficient and practical for inference. Unlike
iteratively optimizing ? and ?, C as in variational EM, [28] noticed that the calculation of the gradient w.r.t the lower bound indicates the updates of ? can be analytically worked out by variational
parameters, thus resulting a new objective function for the representation of lower bound that only
relies on ?, C (details refer to [28]). We apply our algorithm to this variational logistic regression on
three appropriate datasets: DukeBreast and Leukemia are small size but high-dimensional for
sparse logistic regression, and a9a which is large. See Table 1 for additional dataset descriptions.
Fig. 1 shows the convergence of Gaussian variational lower bound for Bayesian logistic regression
in terms of running time. It is worth mentioning that the lower bound of HFSGVI converges within
3 iterations on the small datasets DukeBreast and Leukemia. This is because all data points
are fed to all algorithms and the HFSGVI uses a better approximation of the Hessian matrix to
proceed 2nd order optimization. L-BFGS-SGVI also take less time to converge and yield slightly
larger lower bound than DSVI. In addition, as an SGD-based algorithm, it is clearly seen that DSVI
is less stable for small datasets and fluctuates strongly even at the later optimized stage. For the
large a9a, we observe that HFSGVI also needs 1000 iterations to reach a good lower bound and
becomes less stable than the other two algorithms. However, L-BFGS-SGVI performs the best
6
Table 1: Comparison on number of misclassification
Dataset(size: #train/test/feature)
DukeBreast(38/4/7129)
Leukemia(38/34/7129)
A9a(32561/16281/123)
Duke Breast
DSVI
L-BFGS-SGVI HFSGVI
train test train
test
train test
0
2
0
1
0
0
0
3
0
3
0
3
4948 2455 4936 2427 4931 2468
Leukemia
0
A9a
4
?1
0
x 10
?20
?50
?1.1
?100
?150
DSVI
L?BFGS?SGVI
HFSGVI
?200
?250
0
20
40
time(s)
60
80
Lower Bound
Lower Bound
Lower Bound
?40
?60
?80
?100
?120
?140
0
20
40
time(s)
?1.2
?1.3
DSVI
L?BFGS?SGVI
HFSGVI
?1.4
60
?1.5
0
80
DSVI
L?BFGS?SGVI
HFSGVI
100
200
300
time(s)
400
500
Figure 1: Convergence rate on logistic regression (zoom out or see larger figures in supplementary)
both in terms of convergence rate and the final lower bound. The misclassification report in Table
1 reflects the similar advantages of our approach, indicating a competitive predication ability on
various datasets. Finally, it is worth mentioning that all three algorithms learn a set of very sparse
regression coefficients on the three datasets (see supplement for additional visualizations).
4.2
Variational Auto-encoder
We also apply the 2nd order stochastic variational inference to train a VAE model (setting M = 1 for
Monte Carlo integration to estimate expectation) or the equivalent deep neural networks with hybrid
hidden layers. The datasets we used are images from the Frey Face, Olivetti Face and MNIST. We
mainly learned three tasks by maximizing the variational lower bound: parameter estimation, images
reconstruction and images generation. Meanwhile, we compared the convergence rate (running
time) of three algorithms, where in this section the compared SGD is the Ada version [6] that is
recommended for VAE model in [17, 25]. The experimental setting is as follows. The initial weights
are randomly drawn from N (0, 0.012 I) or N (0, 0.0012 I), while all bias terms are initialized as 0.
The variational lower bound only introduces the regularization on the encoder parameters, so we add
an L2 regularizer on decoder parameters with a shrinkage parameter 0.001 or 0.0001. The number of
hidden nodes for encoder and decoder is the same for all auto-encoder model, which is reasonable
and convenient to construct a symmetric structure. The number is always tuned from 200 to 800
with 100 increment. The mini-batch size is 100 for L-BFGS and Ada, while larger mini-batch is
recommended for HF, meaning it should vary according to the training size.
The detailed results are shown in Fig. 2 and 3. Both Hessian-free and L-BFGS converge faster than
Ada in terms of running time. HFSGVI also performs competitively with respet to generalization
on testing data. Ada takes at least four times as long to achieve similar lower bound. Theoretically,
Newton?s method has a quadratic convergence rate in terms of iteration, but with a cubic algorithmic complexity at each iteration. However, we manage to lower the computation in each iteration
to linear complexity. Thus considering the number of evaluated training data points, the 2nd order
algorithm needs much fewer step than 1st order gradient descent (see visualization in supplementary
on MNIST). The Hessian matrix also replaces manually tuned learning rates, and the affine invariant property allows for automatic learning rate adjustment. Technically, if the program can run in
parallel with GPU, the speed advantages of 2nd order algorithm should be more obvious [21].
Fig. 2(b) and Fig. 3(b) are reconstruction results of input images. From the perspective of deep
neural network, the only difference is the Gaussian distributed latent variables z. By corollary of
Theorem 2, we can roughly tell the mean ? is able to represent the quantity of z, meaning this layer
is actually a linear transformation with noise, which looks like dropout training [5]. Specifically,
Olivetti includes 64?64 pixels faces of various persons, which means more complicated models
or preprocessing [13] (e.g. nearest neighbor interpolation, patch sampling) is needed. However,
even when simply learning a very bottlenecked auto-encoder, our approach can achieve acceptable
results. Note that although we have tuned the hyperparameters of Ada by cross-validation, the
best result is still a bunch of mean faces. For manifold learning, Fig. 2(c) represents how the
7
Frey Face
1600
1400
Lower Bound
1200
1000
Ada train
Ada test
L?BFGS?SGVI train
L?BFGS?SGVI test
HFSGVI train
HFSGVI test
800
600
400
200
0
0.5
1
time(s)
1.5
2
4
x 10
(a) Convergence
(b) Reconstruction
(c) Manifold by Generative Model
Figure 2: (a) shows how lower bound increases w.r.t program running time for different algorithms; (b) illustrates the reconstruction ability of this auto-encoder model when dz = 20 (left 5 columns are randomly
sampled from dataset); (c) is the learned manifold of generative model when dz = 2.
Olivetti Face
6000
Lower Bound
4000
2000
Ada train
Ada test
L?BFGS?SGVI train
L?BFGS?SGVI test
HFSGVI train
HFSGVI test
0
?2000
?4000
0
0.5
1
time (s)
1.5
2
4
x 10
(a) Convergence
(b) HFSGVI v.s L-BFGS-SGVI v.s. Ada-SGVI
Figure 3: (a) shows running time comparison; (b) illustrates reconstruction comparison without patch sampling, where dz = 100: top 5 rows are original faces.
learned generative model can simulate the images by HFSGVI. To visualize the results, we choose
the 2D latent variable z in p? (x|z), where the parameter ? is estimated by the algorithm. The
two coordinates of z take values that were transformed through the inverse CDF of the Gaussian
distribution from equal distance grid (10?10 or 20?20) on the unit square. Then we merely use the
generative model to simulate the images. Besides these learning tasks, denoising, imputation [25]
and even generalizing to semi-supervised learning [16] are possible application of our approach.
5
Conclusions and Discussion
In this paper we proposed a scalable 2nd order stochastic variational method for generative models
with continuous latent variables. By developing Gaussian backpropagation through reparametrization we introduced an efficient unbiased estimator for higher order gradients information. Combining with the efficient technique for computing Hessian-vector multiplication, we derived an efficient
inference algorithm (HFSGVI) that allows for joint optimization of all parameters. The algorithmic
complexity of each parameter update is quadratic w.r.t the dimension of latent variables for both 1st
and 2nd derivatives. Furthermore, the overall computational complexity of our 2nd order SGVI is
linear w.r.t the number of parameters in real applications just like SGD or Ada. However, HFSGVI
may not behave as fast as Ada in some situations, e.g., when the pixel values of images are sparse
due to fast matrix multiplication implementation in most softwares.
Future research will focus on some difficult deep models such as RNNs [10, 27] or Dynamic SBN
[9]. Because of conditional independent structure by giving sampled latent variables, we may construct blocked Hessian matrix to optimize such dynamic models. Another possible area of future
work would be reinforcement learning (RL) [20]. Many RL problems can be reduced to compute
gradients of expectations (e.g., in policy gradient methods) and there has been series of exploration
in this area for natural gradients. However, we would suggest that it might be interesting to consider
where stochastic backpropagation fits in our framework and how 2nd order computations can help.
Acknolwedgement This research was supported in part by the Research Grants Council of the
Hong Kong Special Administrative Region (Grant No. 614513).
8
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9
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5,160 | 5,671 | Rethinking LDA: Moment Matching for Discrete ICA
Anastasia Podosinnikova
Francis Bach
Simon Lacoste-Julien
?
INRIA - Ecole
normale sup?erieure Paris
Abstract
We consider moment matching techniques for estimation in latent Dirichlet allocation (LDA). By drawing explicit links between LDA and discrete versions of
independent component analysis (ICA), we first derive a new set of cumulantbased tensors, with an improved sample complexity. Moreover, we reuse standard
ICA techniques such as joint diagonalization of tensors to improve over existing
methods based on the tensor power method. In an extensive set of experiments on
both synthetic and real datasets, we show that our new combination of tensors and
orthogonal joint diagonalization techniques outperforms existing moment matching methods.
1
Introduction
Topic models have emerged as flexible and important tools for the modelisation of text corpora.
While early work has focused on graphical-model approximate inference techniques such as variational inference [1] or Gibbs sampling [2], tensor-based moment matching techniques have recently
emerged as strong competitors due to their computational speed and theoretical guarantees [3, 4].
In this paper, we draw explicit links with the independent component analysis (ICA) literature
(e.g., [5] and references therein) by showing a strong relationship between latent Dirichlet allocation
(LDA) [1] and ICA [6, 7, 8]. We can then reuse standard ICA techniques and results, and derive new
tensors with better sample complexity and new algorithms based on joint diagonalization.
2
Is LDA discrete PCA or discrete ICA?
Notation. Following the text modeling terminology, we define a corpus X = {x1 , . . . , xN } as a
collection of N documents. Each document is a collection {wn1 , . . . , wnLn } of Ln tokens. It is
convenient to represent the `-th token of the n-th document as a 1-of-M encoding with an indicator
vector wn` ? {0, 1}M with only
where M is the vocabulary size, and each document
P one non-zero,
M
as the count vector P
xn :=
` wn` ? R . In such representation, the length Ln of the n-th
document is Ln = m xnm . We will always use index k ? {1, . . . , K} to refer to topics, index
n ? {1, . . . , N } to refer to documents, index m ? {1, . . . , M } to refer to words from the vocabulary,
and index ` ? {1, . . . , Ln } to refer to tokens of the n-th document. The plate diagrams of the models
from this section are presented in Appendix A.
Latent Dirichlet allocation [1] is a generative probabilistic model for discrete data such as text
corpora. In accordance to this model, the n-th document is modeled as an admixture over the vocabulary of M words with K latent topics. Specifically, the latent variable ?n , which is sampled
from the Dirichlet distribution, represents the topic mixture proportion over K topics for the n-th
document. Given ?n , the topic choice zn` |?n for the `-th token is sampled from the multinomial distribution with the probability vector ?n . The token wn` |zn` , ?n is then sampled from the multinomial
distribution with the probability vector dzn` , or dk if k is the index of the non-zero element in zn` .
This vector dk is the k-th topic, that is a vector of probabilities over the words from the
P vocabulary
subject to the simplex constraint, i.e., dk ? ?M , where ?M := {d ? RM : d 0, m dm = 1}.
This generative process of a document (the index n is omitted for simplicity) can be summarized as
1
? ? Dirichlet(c),
z` |? ? Multinomial(1, ?),
w` |z` , ? ? Multinomial(1, dz` ).
(1)
One can think of the latent variables z` as auxiliary variables which were introduced for convenience
of inference, but can in fact be marginalized out [9], which leads to the following model
? ? Dirichlet(c),
x|? ? Multinomial(L, D?),
LDA model (2)
where D ? RM ?K is the topic matrix with the k-th column equal to the k-th topic dk , and c ? RK
++
is the vector of parameters for the Dirichlet distribution. While a document is represented as a set
of tokens w` in the formulation (1), the formulation (2) instead compactly represents a document as
the count vector x. Although the two representations are equivalent, we focus on the second one in
this paper and therefore refer to it as the LDA model.
Importantly, the LDA model does not model the length of documents. Indeed, although the original
paper [1] proposes to model the document length as L|? ? Poisson(?), this is never used in practice
and, in particular, the parameter ? is not learned. Therefore, in the way that the LDA model is
typically used, it does not provide a complete generative process of a document as there is no rule to
sample L|?. In this paper, this fact is important, as we need to model the document length in order
to make the link with discrete ICA.
Discrete PCA. The LDA model (2) can be seen as a discretization of principal component analysis (PCA) via replacement of the normal likelihood with the multinomial one and adjusting the
prior [9] in the following probabilistic PCA model [10, 11]: ? ? Normal(0, IK ) and x|? ?
Normal(D?, ? 2 IM ), where D ? RM ?K is a transformation matrix and ? is a parameter.
Discrete ICA (DICA). Interestingly, a small extension of the LDA model allows its interpretation as a discrete independent component analysis model. The extension naturally arises when the
document length for the LDA model is modeled as a random variable from the gamma-Poisson
mixture (which is equivalent to a P
negative binomial random variable), i.e., L|? ? Poisson(?) and
? ? Gamma(c0 , b), where c0 := k ck is the shape parameter and b > 0 is the rate parameter. The
LDA model (2) with such document length is equivalent (see Appendix B.1) to
?k ? Gamma(ck , b),
xm |? ? Poisson([D?]m ),
GP model (3)
where all ?1 , ?2 , . . . , ?K are mutually independent, the parameters ck coincide with the ones of the
LDA model in (2), and the free parameter b can be seen (see Appendix B.2) as a scaling parameter
for the document length when c0 is already prescribed.
This model was introduced by Canny [12] and later named as a discrete ICA model [13]. It is more
natural, however, to name model (3) as the gamma-Poisson (GP) model and the model
?1 , . . . , ?K ? mutually independent,
xm |? ? Poisson([D?]m )
DICA model (4)
as the discrete ICA (DICA) model. The only difference between (4) and the standard ICA model [6,
7, 8] (without additive noise) is the presence of the Poisson noise which enforces discrete, instead of
continuous, values of xm . Note also that (a) the discrete ICA model is a semi-parametric model that
can adapt to any distribution on the topic intensities ?k and that (b) the GP model (3) is a particular
case of both the LDA model (2) and the DICA model (4).
Thanks to this close connection between LDA and ICA, we can reuse standard ICA techniques to
derive new efficient algorithms for topic modeling.
3
Moment matching for topic modeling
The method of moments estimates latent parameters of a probabilistic model by matching theoretical
expressions of its moments with their sample estimates. Recently [3, 4], the method of moments
was applied to different latent variable models including LDA, resulting in computationally fast
2
learning algorithms with theoretical guarantees. For LDA, they (a) construct LDA moments with a
particular diagonal structure and (b) develop algorithms for estimating the parameters of the model
by exploiting this diagonal structure. In this paper, we introduce novel GP/DICA cumulants with
a similar to the LDA moments structure. This structure allows to reapply the algorithms of [3, 4]
for the estimation of the model parameters, with the same theoretical guarantees. We also consider
another algorithm applicable to both the LDA moments and the GP/DICA cumulants.
3.1
Cumulants of the GP and DICA models
In this section, we derive and analyze the novel cumulants of the DICA model. As the GP model is
a particular case of the DICA model, all results of this section extend to the GP model.
The first three cumulant tensors for the random vector x can be defined as follows
cum(x) := E(x),
(5)
cum(x, x) := cov(x, x) = E (x ? E(x))(x ? E(x))
>
,
(6)
cum(x, x, x) := E [(x ? E(x)) ? (x ? E(x)) ? (x ? E(x))] ,
(7)
where ? denotes the tensor product (see some properties of cumulants in Appendix C.1). The
essential property of the cumulants (which does not hold for moments) that we use in this paper is
that the cumulant tensor for a random vector with independent components is diagonal.
Let y = D?; then for the Poisson random variable xm |ym ? Poisson(ym ), the expectation is
E(xm |ym ) = ym . Hence, by the law of total expectation and the linearity of expectation, the
expectation in (5) has the following form
E(x) = E(E(x|y)) = E(y) = DE(?).
(8)
Further, the variance of the Poisson random variable xm is var(xm |ym ) = ym and, as x1 ,
x2 , . . . , xM are conditionally independent given y, then their covariance matrix is diagonal, i.e.,
cov(x, x|y) = diag(y). Therefore, by the law of total covariance, the covariance in (6) has the form
cov(x, x) = E [cov(x, x|y)] + cov [E(x|y), E(x|y)]
= diag [E(y)] + cov(y, y) = diag [E(x)] + Dcov(?, ?)D> ,
(9)
where the last equality follows by the multilinearity property of cumulants (see Appendix C.1).
Moving the first term from the RHS of (9) to the LHS, we define
S := cov(x, x) ? diag [E(x)] .
DICA S-cum. (10)
From (9) and by the independence of ?1 , . . . , ?K (see Appendix C.3), S has the following diagonal
structure
X
>
S=
var(?k )dk d>
(11)
k = Ddiag [var(?)] D .
k
By analogy with the second order case, using the law of total cumulance, the multilinearity property
of cumulants, and the independence of ?1 , . . . , ?K , we derive in Appendix C.2 expression (24),
similar to (9), for the third cumulant (7). Moving the terms in this expression, we define a tensor T
with the following element
[T ]m1 m2 m3 := cum(xm1 , xm2 , xm3 ) + 2?(m1 , m2 , m3 )E(xm1 )
DICA T-cum. (12)
? ?(m2 , m3 )cov(xm1 , xm2 ) ? ?(m1 , m3 )cov(xm1 , xm2 ) ? ?(m1 , m2 )cov(xm1 , xm3 ),
where ? is the Kronecker delta. By analogy with (11) (Appendix C.3), the diagonal structure of
tensor T :
X
T =
cum(?k , ?k , ?k )dk ? dk ? dk .
(13)
k
In Appendix E.1, we recall (in our notation) the matrix S (39) and the tensor T (40) for the LDA
model [3], which are analogues of the matrix S (10) and the tensor T (12) for the GP/DICA models. Slightly abusing terminology, we refer to the matrix S (39) and the tensor T (40) as the LDA
moments and to the matrix S (10) and the tensor T (12) as the GP/DICA cumulants. The diagonal
structure (41) & (42) of the LDA moments is similar to the diagonal structure (11) & (13) of the
GP/DICA cumulants, though arising through a slightly different argument, as discussed at the end of
3
Appendix E.1. Importantly, due to this similarity, the algorithmic frameworks for both the GP/DICA
cumulants and the LDA moments coincide.
The following sample complexity results apply to the sample estimates of the GP cumulants:1
Proposition 3.1. Under the GP model, the expected error for the sample estimator Sb (29) for the
GP cumulant S (10) is:
h
i r h
i
1
2
2
b
b
?
?
E kS ? SkF ? E kS ? SkF ? O ? max ?L , c?0 L ,
(14)
N
? := E(L).
where ? := max k kdk k2 , c?0 := min(1, c0 ) and L
2
A high probability bound could be derived using concentration inequalities for Poisson random
variables [14]; but the expectation already gives the right order of magnitude for the error (for
example via Markov?s inequality). The expression (29) for an unbiased finite sample estimate Sb of S
and the expression (30) for an unbiased finite sample estimate Tb of T are defined2 in Appendix C.4.
A sketch of a proof for Proposition 3.1 can be found in Appendix D.
By following a similar analysis as in [15], we can rephrase the topic recovery error in term of the
error on the GP cumulant. Importantly, the whitening transformation (introduced in Section 4) redi? 2 , which is the scale of S (see Appendix D.5 for details). This means
vides the error on S (14) by L
?
?
that the contribution from S? to the recovery error will scale as O(1/ N max{?, c?0 /L}),
where
? are smaller than 1 and can be very small. We do not present the exact expression
both ? and c?0 /L
for the expected squared error for the estimator of T , but due to a similar structure in the derivation,
?
? 3/2 }.
? 3 , c?3/2 L
we expect the analogous bound of E[kTb ? T kF ] ? 1/ N max{?3/2 L
0
?
Current sample complexity results of the LDA moments [3] can be summarized as O(1/ N ). However, the proof (which can be found in the supplementary material [15]) analyzes only the case when
finite sample estimates of the LDA moments are constructed from one triple per document, i.e.,
w1 ? w2 ? w3 only, and not from the U-statistics that average multiple (dependent) triples per
document as in the practical expressions (43) and (44). Moreover, one has to be careful when comparing upper bounds. Nevertheless, comparing the bound (14) with the current theoretical results
for the LDA moments, we see that the GP/DICA cumulants sample complexity contains the `2 norm of the columns of the topic matrix D in the numerator, as opposed to the O(1) coefficient
for the LDA moments. This norm can be significantly smaller than 1 for vectors in the simplex
(e.g., ? = O(1/kdk k0 ) for sparse topics). This suggests that the GP/DICA cumulants may have
better finite sample convergence properties than the LDA moments and our experimental results in
Section 5.2 are indeed consistent with this statement.
The GP/DICA cumulants have a somewhat more intuitive derivation than the LDA moments as
they are expressed via the count vectors x (which are the sufficient statistics for the model) and
not the tokens w` ?s. Note also that the construction of the LDA moments depend on the unknown
parameter c0 . Given that we are in an unsupervised setting and that moreover the evaluation of
LDA is a difficult task [16], setting this parameter is non-trivial. In Appendix G.4, we observe
experimentally that the LDA moments are somewhat sensitive to the choice of c0 .
4
Diagonalization algorithms
How is the diagonal structure (11) of S and (13) of T going to be helpful for the estimation of the
model parameters? This question has already been thoroughly investigated in the signal processing
(see, e.g., [17, 18, 19, 20, 21, 5] and references therein) and machine learning (see [3, 4] and references therein) literature. We review the approach in this section. Due to similar diagonal structure,
the algorithms of this section apply to both the LDA moments and the GP/DICA cumulants.
For simplicity, let us rewrite expressions (11) and (13) for S and T as follows
X
X
S=
sk dk d>
T =
tk dk ? dk ? dk ,
k,
k
k
1
(15)
Note that the expected squared error for the DICA cumulants is similar, but the expressions are less compact
and, in general, depend on the prior on ?k .
2
For completeness, we also present the finite sample estimates Sb (43) and Tb (44) of S (39) and T (40) for
the LDA moments (which are consistent with the ones suggested in [4]) in Appendix F.4.
4
?
where sk := var(?k ) and tk := cum(?k , ?k , ?k ). Introducing the rescaled topics dek := sk dk ,
eD
e > . Following the same assumption from [3] that the topic vectors are
we can also rewrite S = D
e is full rank), we can compute a whitening matrix W ? RK?M of S, i.e.,
linearly independent (D
a matrix such that W SW > = IK where IK is the K-by-K identity matrix (see Appendix F.1 for
more details). As a result, the vectors zk := W dek form an orthonormal set of vectors.
Further, let us define a projection T (v) ? RK?K of a tensor T ? RK?K?K onto a vector u ? RK :
X
Tk1 k2 k3 uk3 .
(16)
T (u)k1 k2 :=
k3
Applying the multilinear transformation (see, e.g., [4] for the definition) with W > to the tensor T
from (15) and projecting the resulting tensor T := T (W > , W > , W > ) onto some vector u ? RK ,
we obtain
X
e
tk hzk , uizk zk> ,
(17)
T (u) =
k
3/2
where e
tk := tk /sk is due to the rescaling of topics and h?, ?i stands for the inner product. As the
vectors zk are orthonormal, the pairs zk and ?k := e
tk hzk , ui are eigenpairs of the matrix T (u),
which are uniquely defined if the eigenvalues ?k are all different. If they are unique, we can recover
the GP/DICA (as well as LDA) model parameters via dek = W ? zk and e
tk = ?k /hzk , ui.
This procedure was referred to as the spectral algorithm for LDA [3] and the fourth-order3 blind
identification algorithm for ICA [17, 18]. Indeed, one can expect that the finite sample estimates
Sb (29) and Tb (30) possess approximately the diagonal structure (11) and (13) and, therefore, the reasoning from above can be applied, assuming that the effect of the sampling error is controlled.
This spectral algorithm, however, is known to be quite unstable in practice (see, e.g., [22]). To overcome this problem, other algorithms were proposed. For ICA, the most notable ones are probably
the FastICA algorithm [20] and the JADE algorithm [21]. The FastICA algorithm, with appropriate
choice of a contrast function, estimates iteratively the topics, making use of the orthonormal structure (17), and performs the deflation procedure at every step. The recently introduced tensor power
method (TPM) for the LDA model [4] is close to the FastICA algorithm. Alternatively, the JADE algorithm modifies the spectral algorithm by performing multiple projections for (17) and then jointly
diagonalizing the resulting matrices with an orthogonal matrix. The spectral algorithm is a special
case of this orthogonal joint diagonalization algorithm when only one projection is chosen. Importantly, a fast implementation [23] of the orthogonal joint diagonalization algorithm from [24] was
proposed, which is based on closed-form iterative Jacobi updates (see, e.g., [25] for the later).
In practice, the orthogonal joint diagonalization (JD) algorithm is more robust than FastICA (see,
e.g., [26, p. 30]) or the spectral algorithm. Moreover, although the application of the JD algorithm
for the learning of topic models was mentioned in the literature [4, 27], it was never implemented in
practice. In this paper, we apply the JD algorithm for the diagonalization of the GP/DICA cumulants
as well as the LDA moments, which is described in Algorithm 1. Note that the choice of a projection
c > up for some vector up ? RK is important and corresponds to
vector vp ? RM obtained as vp = W
c > along the third mode. Importantly, in Algorithm 1, the
the multilinear transformation of Tb with W
joint diagonalization routine is performed over (P + 1) matrices of size K?K, where the number of
topics K is usually not too big. This makes the algorithm computationally fast (see Appendix G.1).
The same is true for the spectral algorithm, but not for TPM.
In Section 5.1, we compare experimentally the performance of the spectral, JD, and TPM algorithms
for the estimation of the parameters of the GP/DICA as well as LDA models. We are not aware of
any experimental comparison of these algorithms in the LDA context. While already working on
this manuscript, the JD algorithm was also independently analyzed by [27] in the context of tensor
factorization for general latent variable models. However, [27] focused mostly on the comparison
of approaches for tensor factorization and their stability properties, with brief experiments using a
latent variable model related but not equivalent to LDA for community detection. In contrast, we
provide a detailed experimental comparison in the context of LDA in this paper, as well as propose
a novel cumulant-based estimator. Due to the space restriction the estimation of the topic matrix D
and the (gamma/Dirichlet) parameter c are moved to Appendix F.6.
3
See Appendix C.5 for a discussion on the orders.
5
Algorithm 1 Joint diagonalization (JD) algorithm for GP/DICA cumulants (or LDA moments)
1: Input: X ? RM ?N , K, P (number of random projections); (and c0 for LDA moments)
b ? RM ?M ((29) for GP/DICA / (43) for LDA in Appendix F)
2: Compute sample estimate S
c ? RK?M of Sb (see Appendix F.1)
3: Estimate whitening matrix W
option (a): Choose vectors {u1 , u2 , . . . , uP } ? RK uniformly at random from the unit `2 c > up ? RM for all p = 1, . . . , P
sphere and set vp = W
(P = 1 yields the spectral algorithm)
option (b): Choose vectors {u1 , u2 , . . . , uP } ? RK as the canonical basis e1 , e2 , . . . , eK of
c > up ? RM for all p = 1, . . . , K
RK and set vp = W
c Tb(vp )W
c > ? RK?K ((52) for GP/DICA / (54) for LDA; Appendix F)
4: For ?p, compute Bp = W
c SbW
c > = IK , Bp , p = 1, . . . , P }
5: Perform orthogonal joint diagonalization of matrices {W
K?K
(see [24] and [23]) to find an orthogonal matrix V ? R
and vectors {a1 , a2 , . . . , aP } ? RK
such that
c SbW
c > V > = IK , and V Bp V > ? diag(ap ), p = 1, . . . , P
VW
c and values ap , p = 1, . . . , P
6: Estimate joint diagonalization matrix A = V W
7: Output: Estimate of D and c as described in Appendix F.6
5
Experiments
In this section, (a) we compare experimentally the GP/DICA cumulants with the LDA moments and
(b) the spectral algorithm [3], the tensor power method [4] (TPM), the joint diagonalization (JD)
algorithm from Algorithm 1, and variational inference for LDA [1].
Real data: the associated press (AP) dataset, from D. Blei?s web page,4 with N = 2, 243 documents
b = 194; the NIPS papers
and M = 10, 473 vocabulary words and the average document length L
5
b = 1, 321; the KOS dataset,6 from the
dataset [28] of 2, 483 NIPS papers and 14, 036 words, and L
b = 136.
UCI Repository, with 3, 430 documents and 6, 906 words, and L
Semi-synthetic data are constructed by analogy with [29]: (1) the LDA parameters D and c are
learned from the real datasets with variational inference and (2) toy data are sampled from a model
of interest with the given parameters D and c. This provides the ground truth parameters D and c.
For each setting, data are sampled 5 times and the results are averaged. We plot error bars that are
the minimum and maximum values. For the AP data, K ? {10, 50} topics are learned and, for
the NIPS data, K ? {10, 90} topics are learned. For larger K, the obtained topic matrix is illconditioned, which violates the identifiability condition for topic recovery using moment matching
techniques [3]. All the documents with less than 3 tokens are resampled.
Sampling techniques. All the sampling models have the parameter c which is set to c = c0 c?/ k?
ck1 ,
where c? is the learned c from the real dataset with variational LDA, and c0 is a parameter that we
b so that
can vary. The GP data are sampled from the gamma-Poisson model (3) with b = c0 /L
b
the expected document length is L (see Appendix B.2). The LDA-fix(L) data are sampled from the
LDA model (2) with the document length being fixed to a given L. The LDA-fix2(?,L1 ,L2 ) data
are sampled as follows: (1 ? ?)-portion of the documents are sampled from the LDA-fix(L1 ) model
with a given document length L1 and ?-portion of the documents are sampled from the LDA-fix(L2 )
model with a given document length L2 .
Evaluation. Evaluation of topic recovery for semi-synthetic data is performed with the `1 b and true D topic matrices with the best permutation of columns:
error between the recovered D
P b
1
b
err`1 (D, D) := min??PERM 2K
k kd?k ? dk k1 ? [0, 1]. The minimization is over the possible
b and can be efficiently obtained with the Hungarian
permutations ? ? PERM of the columns of D
algorithm for bipartite matching. For the evaluation of topic recovery in the real data case, we use
an approximation of the log-likelihood for held out documents as the metric [16]. See Appendix G.6
for more details.
4
http://www.cs.columbia.edu/?blei/lda-c
http://ai.stanford.edu/?gal/data
6
https://archive.ics.uci.edu/ml/datasets/Bag+of+Words
5
6
1
1
JD
JD(k)
JD(f)
Spec
TPM
0.6
0.4
0.8
?1 -error
?1 -error
0.8
0.2
0
1
0.6
0.4
0.2
10
20
30
40
0
1
50
Number of docs in 1000s
10
20
30
40
50
Number of docs in 1000s
Figure 1: Comparison of the diagonalization algorithms. The topic matrix D and Dirichlet parameter c are
learned for K = 50 from AP; c is scaled to sum up to 0.5 and b is set to fit the expected document length
b = 200. The semi-synthetic dataset is sampled from GP; number of documents N varies from 1, 000 to
L
50, 000. Left: GP/DICA moments. Right: LDA moments. Note: a smaller value of the `1 -error is better.
We use our Matlab implementation of the GP/DICA cumulants, the LDA moments, and the diagonalization algorithms. The datasets and the code for reproducing our experiments are available
online.7 In Appendix G.1, we discuss implementation and complexity of the algorithms. We explain
how we initialize the parameter c0 for the LDA moments in Appendix G.3.
5.1
Comparison of the diagonalization algorithms
In Figure 1, we compare the diagonalization algorithms on the semi-synthetic AP dataset for K = 50
using the GP sampling. We compare the tensor power method (TPM) [4], the spectral algorithm
(Spec), the orthogonal joint diagonalization algorithm (JD) described in Algorithm 1 with different
options to choose the random projections: JD(k) takes P = K vectors up sampled uniformly from
the unit `2 -sphere in RK and selects vp = W > up (option (a) in Algorithm 1); JD selects the full basis
e1 , . . . , eK in RK and sets vp = W > ep (as JADE [21]) (option (b) in Algorithm 1); JD(f ) chooses
the full canonical basis of RM as the projection vectors (computationally expensive).
Both the GP/DICA cumulants and LDA moments are well-specified in this setup. However, the
LDA moments have a slower finite sample convergence and, hence, a larger estimation error for the
same value N . As expected, the spectral algorithm is always slightly inferior to the joint diagonalization algorithms. With the GP/DICA cumulants, where the estimation error is low, all algorithms
demonstrate good performance, which also fulfills our expectations. However, although TPM shows
almost perfect performance in the case of the GP/DICA cumulants (left), it significantly deteriorates
for the LDA moments (right), which can be explained by the larger estimation error of the LDA
moments and lack of robustness of TPM. The running times are discussed in Appendix G.2. Overall, the orthogonal joint diagonalization algorithm with initialization of random projections as W >
multiplied with the canonical basis in RK (JD) is both computationally efficient and fast.
5.2
Comparison of the GP/DICA cumulants and the LDA moments
In Figure 2, when sampling from the GP model (top, left), both the GP/DICA cumulants and LDA
moments are well specified, which implies that the approximation error (i.e., the error for the infinite number of documents) is low for both. The GP/DICA cumulants achieve low values of the
estimation error already for N = 10, 000 documents independently of the number of topics, while
the convergence is slower for the LDA moments. When sampling from the LDA-fix(200) model (top,
right), the GP/DICA cumulants are mis-specified and their approximation error is high, although the
estimation error is low due to the faster finite sample convergence. One reason of poor performance
of the GP/DICA cumulants, in this case, is the absence of variance in document length. Indeed, if
documents with two different lengths are mixed by sampling from the LDA-fix2(0.5,20,200) model
(bottom, left), the GP/DICA cumulants performance improves. Moreover, the experiment with a
changing fraction ? of documents (bottom, right) shows that a non-zero variance on the length improves the performance of the GP/DICA cumulants. As in practice real corpora usually have a
non-zero variance for the document length, this bad scenario for the GP/DICA cumulants is not
likely to happen.
7
https://github.com/anastasia-podosinnikova/dica
7
1
0.8
0.8
?1 -error
?1 -error
1
0.6
0.4
0.2
JD-GP(10)
JD-LDA(10)
JD-GP(90)
JD-LDA(90)
0.6
0.4
0.2
0
1
10
20
30
40
0
1
50
10
20
30
40
50
Number of docs in 1000s
1
1
0.8
0.8
?1 -error
?1 -error
Number of docs in 1000s
0.6
0.4
0.6
0.4
0.2
0.2
0
0
1
10
20
30
40
0
50
0.2
0.4
0.6
0.8
1
Fraction of doc lengths ?
Number of docs in 1000s
-11.5
JD-GP
JD-LDA
Spec-GP
Spec-LDA
VI
VI-JD
-12
-12.5
-13
-13.5
10
50
100
150
Log-likelihood (in bits)
Log-likelihood (in bits)
Figure 2: Comparison of the GP/DICA cumulants and LDA moments. Two topic matrices and parameters c1
and c2 are learned from the NIPS dataset for K = 10 and 90; c1 and c2 are scaled to sum up to c0 = 1.
Four corpora of different sizes N from 1, 000 to 50, 000: top, left: b is set to fit the expected document length
b = 1300; sampling from the GP model; top, right: sampling from the LDA-fix(200) model; bottom, left:
L
sampling from the LDA-fix2(0.5,20,200) model. Bottom, right: the number of documents here is fixed to
N = 20, 000; sampling from the LDA-fix2(?,20,200) model varying the values of the fraction ? from 0 to 1
with the step 0.1. Note: a smaller value of the `1 -error is better.
-10.5
-11
-11.5
-12
-12.5
10
Topics K
50
100
150
Topics K
Figure 3: Experiments with real data. Left: the AP dataset. Right: the KOS dataset. Note: a higher value of
the log-likelihood is better.
5.3
Real data experiments
In Figure 3, JD-GP, Spec-GP, JD-LDA, and Spec-LDA are compared with variational inference (VI)
and with variational inference initialized with the output of JD-GP (VI-JD). We measure held out
log-likelihood per token (see Appendix G.7 for details on the experimental setup). The orthogonal joint diagonalization algorithm with the GP/DICA cumulants (JD-GP) demonstrates promising
performance. In particular, the GP/DICA cumulants significantly outperform the LDA moments.
Moreover, although variational inference performs better than the JD-GP algorithm, restarting variational inference with the output of the JD-GP algorithm systematically leads to better results. Similar
behavior has already been observed (see, e.g., [30]).
6
Conclusion
In this paper, we have proposed a new set of tensors for a discrete ICA model related to LDA, where
word counts are directly modeled. These moments make fewer assumptions regarding distributions,
and are theoretically and empirically more robust than previously proposed tensors for LDA, both
on synthetic and real data. Following the ICA literature, we showed that our joint diagonalization
procedure is also more robust. Once the topic matrix has been estimated in a semi-parametric way
where topic intensities are left unspecified, it would be interesting to learn the unknown distributions
of the independent topic intensities.
Acknowledgments. This work was partially supported by the MSR-Inria Joint Center. The authors
would like to thank Christophe Dupuy for helpful discussions.
8
References
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2003.
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9
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5,161 | 5,672 | Model-Based Relative Entropy Stochastic Search
Abbas Abdolmaleki1,2,3 , Rudolf Lioutikov4 , Nuno Lau1 , Luis Paulo Reis2,3 ,
Jan Peters4,6 , and Gerhard Neumann5
1: IEETA, University of Aveiro, Aveiro, Portugal
2: DSI, University of Minho, Braga, Portugal
3: LIACC, University of Porto, Porto, Portugal
4: IAS, 5: CLAS, TU Darmstadt, Darmstadt, Germany
6: Max Planck Institute for Intelligent Systems, Stuttgart, Germany
{Lioutikov,peters,neumann}@ias.tu-darmstadt.de
{abbas.a, nunolau}@ua.pt, [email protected]
Abstract
Stochastic search algorithms are general black-box optimizers. Due to their ease
of use and their generality, they have recently also gained a lot of attention in operations research, machine learning and policy search. Yet, these algorithms require
a lot of evaluations of the objective, scale poorly with the problem dimension, are
affected by highly noisy objective functions and may converge prematurely. To
alleviate these problems, we introduce a new surrogate-based stochastic search
approach. We learn simple, quadratic surrogate models of the objective function.
As the quality of such a quadratic approximation is limited, we do not greedily exploit the learned models. The algorithm can be misled by an inaccurate optimum
introduced by the surrogate. Instead, we use information theoretic constraints to
bound the ?distance? between the new and old data distribution while maximizing
the objective function. Additionally the new method is able to sustain the exploration of the search distribution to avoid premature convergence. We compare our
method with state of art black-box optimization methods on standard uni-modal
and multi-modal optimization functions, on simulated planar robot tasks and a
complex robot ball throwing task. The proposed method considerably outperforms the existing approaches.
1
Introduction
Stochastic search algorithms [1, 2, 3, 4] are black box optimizers of an objective function that is
either unknown or too complex to be modeled explicitly. These algorithms only make weak assumption on the structure of underlying objective function. They only use the objective values and don?t
require gradients or higher derivatives of the objective function. Therefore, they are well suited
for black box optimization problems. Stochastic search algorithms typically maintain a stochastic search distribution over parameters of the objective function, which is typically a multivariate
Gaussian distribution [1, 2, 3]. This policy is used to create samples from the objective function.
Subsequently, a new stochastic search distribution is computed by either computing gradient based
updates [2, 4, 5], evolutionary strategies [1], the cross-entropy method [7], path integrals [3, 8], or
information-theoretic policy updates [9]. Information-theoretic policy updates [10, 9, 2] bound the
relative entropy (also called Kullback Leibler or KL divergence) between two subsequent policies.
Using a KL-bound for the update of the search distribution is a common approach in the stochastic
search. However, such information theoretic bounds could so far only be approximately applied
either by using Taylor-expansions of the KL-divergence resulting in natural evolutionary strategies (NES) [2, 11], or sample-based approximations, resulting in the relative entropy policy search
1
(REPS) [9] algorithm. In this paper, we present a novel stochastic search algorithm which is called
MOdel-based Relative-Entropy stochastic search (MORE). For the first time, our algorithm bounds
the KL divergence of the new and old search distribution in closed form without approximations.
We show that this exact bound performs considerably better than approximated KL bounds.
In order to do so, we locally learn a simple, quadratic surrogate of the objective function. The
quadratic surrogate allows us to compute the new search distribution analytically where the KL
divergence of the new and old distribution is bounded. Therefore, we only exploit the surrogate
model locally which prevents the algorithm to be misled by inaccurate optima introduced by an
inaccurate surrogate model.
However, learning quadratic reward models directly in parameter space comes with the burden of
quadratically many parameters that need to be estimated. We therefore investigate new methods that
rely on dimensionality reduction for learning such surrogate models. In order to avoid over-fitting,
we use a supervised Bayesian dimensionality reduction approach. This dimensionality reduction
technique avoids over fitting, which makes the algorithm applicable also to high dimensional problems. In addition to solving the search distribution update in closed form, we also upper-bound the
entropy of the new search distribution to ensure that exploration is sustained in the search distribution throughout the learning progress, and, hence, premature convergence is avoided. We will show
that this method is more effective than commonly used heuristics that also enforce exploration, for
example, adding a small diagonal matrix to the estimated covariance matrix [3].
We provide a comparison of stochastic search algorithms on standard objective functions used for
benchmarking and in simulated robotics tasks. The results show that MORE considerably outperforms state-of-the-art methods.
1.1
Problem Statement
We want to maximize an objective function R(?) : Rn ? R. The goal is to find one or more
parameter vectors ? ? Rn which have the highest possible objective value. We maintain a search
distribution ?(?) over the parameter space ? of the objective function R(?). The search distribution ?(?) is implemented as a multivariate Gaussian distribution, i.e., ?(?) = N (?|?, ?). In each
iteration, the search distribution ?(?) is used to create samples ? [k] of the parameter vector ?. Subsequently, the (possibly noisy) evaluation R[k] of ? [k] is obtained by querying the objective function.
The samples {? [k] , R[k] }k=1...N are subsequently used to compute a new search distribution. This
process will run iteratively until the algorithm converges to a solution.
1.2
Related Work
Recent information-theoretic (IT) policy search algorithms [9] are based on the relative entropy policy search (REPS) algorithm which was proposed in [10] as a step-based policy search algorithm.
However, in [9] an episode-based version of REPS that is equivalent to stochastic search was presented. The key idea behind episode-based REPS is to control the exploration-exploitation trade-off
by bounding the relative entropy between the old ?data? distribution q(?) and the newly estimated
search distribution ?(?) by a factor . Due to the relative entropy bound, the algorithm achieves a
smooth and stable learning process. However, the episodic REPS algorithm uses a sample based
approximation of the KL-bound, which needs a lot of samples in order to be accurate. Moreover, a
typical problem of REPS is that the entropy of the search distribution decreases too quickly, resulting
in premature convergence.
Taylor approximations of the KL-divergence have also been used very successfully in the area of
stochastic search, resulting in natural evolutionary strategies (NES). NES uses the natural gradient to
optimize the objective [2]. The natural gradient has been shown to outperform the standard gradient
in many applications in machine learning [12]. The intuition of the natural gradient is that we want
to obtain an update direction of the parameters of the search distribution that is most similar to the
standard gradient while the KL-divergence between new and old search distributions is bounded.
To obtain this update direction, a second order approximation of the KL, which is equivalent to the
Fisher information matrix, is used.
2
Surrogate based stochastic search algorithms [6][13] have been shown to be more sample efficient
than direct stochastic search methods and can also smooth out the noise of the objective function. For
example, an individual optimization method is used on the surrogate that is stopped whenever the
KL-divergence between the new and the old distribution exceeds a certain bound [6]. For the first
time, our algorithm uses the surrogate model to compute the new search distribution analytically,
which bounds the KL divergence of the new and old search distribution, in closed form.
Quadratic models have been used successfully in trust region methods for local surrogate approximation [14, 15]. These methods do not maintain a stochastic search distribution but a point estimate
and a trust region around this point. They update the point estimate by optimizing the surrogate and
staying in the trusted region. Subsequently, heuristics are used to increase or decrease the trusted
region. In the MORE algorithm, the trusted region is defined implicitly by the KL-bound.
The Covariance Matrix Adaptation-Evolutionary Strategy (CMA-ES) is considered as the state of
the art in stochastic search optimization. CMA-ES also maintains a Gaussian distribution over the
problem parameter vector and uses well-defined heuristics to update the search distribution.
2
Model-Based Relative Entropy Stochastic Search
Similar to information theoretic policy search algorithms [9], we want to control the explorationexploitation trade-off by bounding the relative entropy of two subsequent search distribution. However, by bounding the KL, the algorithm can adapt the mean and the variance of the algorithm. In
order to maximize the objective for the immediate iteration, the shrinkage in the variance typically
dominates the contribution to the KL-divergence, which often leads to a premature convergence of
these algorithms. Hence, in addition to control the KL-divergence of the update, we also need to
control the shrinkage of the covariance matrix. Such a control mechanism can be implemented by
lower-bounding the entropy of the new distribution. In this paper, we will set the bound always
to a certain percentage of the entropy of the old search distribution such that MORE converges
asymptotically to a point estimate.
2.1
The MORE framework
Similar as in [9], we can formulate an optimization problem to obtain a new search distribution
that maximizes the expected objective value while upper-bounding the KL-divergence and lowerbounding the entropy of the distribution
Z
max
?
?(?)R? d?,
s.t. KL ?(?)||q(?) ? ,
Z
H(?) ? ?,
1=
?(?)d?,
(1)
1
where
R R? denotes the expected objective when evaluating parameter vector ?. The term H(?) =
? ?(?) log ?(?)d? denotes the entropy of the distribution ? and q(?) is the old distribution. The
parameters and ? are user-specified parameters to control the exploration-exploitation trade-off of
the algorithm.
We can obtain a closed form solution for ?(?) by optimizing the Lagrangian for the optimization
problem given above. This solution is given as
R?
?/(?+?)
?(?) ? q(?)
exp
,
(2)
?+?
where ? and ? are the Lagrangian multipliers. As we can see, the new distribution is now a geometric average between the old sampling distribution q(?) and the exponential transformation of the
objective function. Note that, by setting ? = 0, we obtain the standard episodic REPS formulation
[9]. The optimal value for ? and ? can be obtained by minimizing the dual function g(?, ?) such
that ? > 0 and ? > 0, see [16]. The dual function g(?, ?) is given by
Z
?
R?
g(?, ?) = ? ? ?? + (? + ?) log
q(?) ?+? exp
d? .
(3)
?+?
1
Note that we are typically not able to obtain the expected reward but only a noisy estimate of the underlying
reward distribution.
3
As we are dealing with continuous distributions, the entropy can also be negative. We specify ?
such that the relative difference of H(?) to a minimum exploration policy H(?0 ) is decreased for a
certain percentage, i.e., we change the entropy constraint to
H(?) ? H(?0 ) ? ?(H(q) ? H(?0 )) ? ? = ?(H(q) ? H(?0 )) + H(?0 ).
Throughout all our experiments, we use the same ? value of 0.99 and we set minimum entropy
H(?0 ) of search distribution to a small enough value like ?75. We will show that using the additional entropy bound considerably alleviates the premature convergence problem.
2.2
Analytic Solution of the Dual-Function and the Policy
Using a quadratic surrogate model of the objective function, we can compute the integrals in the
dual function analytically, and, hence, we can satisfy the introduced bounds exactly in the MORE
framework. At the same time, we take advantage of surrogate models such as a smoothed estimate
in the case of noisy objective functions and a decrease in the sample complexity2 .
We will for now assume that we are given a quadratic surrogate model
R? ? ? T R? + ? T r + r0
of the objective function R? which we will learn from data in Section 3. Moreover, the search
distribution is Gaussian, i.e., q(?) = N (?|b, Q). In this case the integrals in the dual function given
in Equation 3 can be solved in closed form. The integral inside the log-term in Equation (3) now
represents an integral over an un-normalized Gaussian distribution. Hence, the integral evaluates to
the inverse of the normalization factor of the corresponding Gaussian. After rearranging terms, the
dual can be written as
1 T
g(?, ?) = ? ? ?? +
f F f ? ?bT Q?1 b ? ? log |2?Q| + (? + ?) log |2?(? + ?)F | (4)
2
with F = (?Q?1 ? 2R)?1 and f = ?Q?1 b + r. Hence, the dual function g(?, ?) can be efficiently
evaluated by matrix inversions and matrix products. Note that, for a large enough value of ?, the
matrix F will be positive definite and hence invertible even if R is not. In our optimization, we
always restrict the ? values such that F stays positive definite3 .
Nevertheless, we could always find the ? value with the correct KL-divergence. In contrast to
MORE, Episodic REPS relies on a sample based approximation of the integrals in the dual function
in Equation (3). It uses the sampled rewards R? of the parameters ? to approximate this integral.
We can also obtain the update rule for the new policy ?(?). From Equation (2), we know that the new
policy is the geometric average of the Gaussian sampling distribution q(?) and a squared exponential
given by the exponentially transformed surrogate. After re-arranging terms and completing the
square, the new policy can be written as
?(?) = N (?|F f , F (? + ?)) ,
(5)
where F , f are given in the previous section.
3
Learning Approximate Quadratic Models
In this section, we show how to learn a quadratic surrogate. Note that we use the quadratic surrogate
in each iteration to locally approximate the objective function and not globally. As the search distribution will shrink in each iteration, the model error will also vanish asymptotically. A quadratic
surrogate is also a natural choice if a Gaussian distribution is used, cause the exponent of the Gaussian is also quadratic in the parameters. Hence, even using a more complex surrogate, it can not
be exploited by a Gaussian distribution. A local quadratic surrogate model provides similar secondorder information as the Hessian in standard gradient updates. However, a quadratic surrogate model
also has quadratically many parameters which we have to estimate from a (ideally) very small data
2
The regression performed for learning the quadratic surrogate model estimates the expectation of the objective function from the observed samples.
3
To optimize g, any constrained nonlinear optimization method can be used[13].
4
6
-10
y
0.50 -10
y
-2.00 -10
-1.00
3
1.50
-0.00
2
1
0
-1
-2
-3
-4
0
1
2
3
4
5
Episodes
6
2.00
2.50
MORE
REPS
PoWER
xNES
CMA-ES
3.00
3.50
MORE
REPS
PoWER
xNES
CMA-ES
4.00
7
8
9
? 1500
(a) Rosenbrock
Average Return
4
1.00
Average Return
Average Return
5
0
0.5
1
1.5
2
Episodes
2.5
? 10
1.00
2.00
3.00
4.00
5.00
4
3
y
6.00
(b) Rastrigin
0
MORE
REPS
PoWER
xNES
CMA-ES
1
2
3
4
Episodes
5
6
? 10
7
4
(c) Noisy Function
Figure 1: Comparison of stochastic search methods for optimizing the uni-modal Rosenbrock (a) and the
multi modal (b) Rastrigin function. (c) Comparison for a noisy objective function. All results show that MORE
clearly outperforms other methods.
set. Therefore, already learning a simple local quadratic surrogate is a challenging task. In order
to learn the local quadratic surrogate, we can use linear regression to fit a function of the form
f (?) = ?(?)?, where ?(?) is a feature function that returns a bias term, all linear and all quadratic
terms of ?. Hence, the dimensionality of ?(?) is D = 1 + d + d(d + 1)/2, where d is the dimensionality of the parameter space. To reduce the dimensionality of the regression problem, we
project ? in a lower dimensional space lp?1 = W ? and solve the linear regression problem in this
reduced space4 . The quadratic form of the objective function can then be computed from ? and
W . Still, the question remains how to choose the projection matrix W . We did not achieve good
performance with standard PCA [17] as PCA is unsupervised. Yet, the W matrix is typically quite
high dimensional such that it is hard to obtain the matrix by supervised learning and simultaneously
avoid over-fitting. Inspired by [18], where supervised Bayesian dimensionality reduction are used
for classification, we also use a supervised Bayesian approach where we integrate out the projection
matrix W .
3.1
Bayesian Dimensionality Reduction for Quadratic Functions
In order to integrate out the parameters W , we use the following probabilistic dimensionality reduction model
Z
p(r? |? ? , D) = p(r? |? ? , W )p(W |D)dW ,
(6)
where r? is prediction of the objective at query point ? ? , D is the training data set consisting of
parameters ? [k] and their objective evaluations R[k] . The posterior for W is given by Bayes rule,
i.e., p(W |D) = p(D|W )p(W )/p(D). The likelihood function p(D|W ) is given by
Z
p(D|W ) = p(D|W , ?)p(?)d?,
(7)
where p(D|W , ?) is the likelihood of the linear model ? and p(?) its prior. For the likelihood
of the linear model we use a multiplicative noise model, i.e., the higher the absolute value of the
objective, the higher the variance. The intuition behind this choice is that we are mainly interested
in minimizing the relative error instead of the absolute error5 . Our likelihood and prior is therefore
given by
p(D|W , ?) =
N
Y
N (R[k] |?(W ? [k] )?, ? 2 |R[k] |),
p(?) = N (?|0, ? 2 I),
(8)
k=1
4
W (p?d) is a projection matrix that projects a vector from a d dimension manifold to a p dimension manifold.
5
We observed empirically that such relative error performs better if we have non-smooth objective functions
with a large difference in the objective values. For example, an error of 10 has a huge influence for an objective
value of ?1, while for a value of ?10000, such an error is negligible.
5
Equation 7 is a weighted Bayesian linear regression model in ? where the weight of each sample
is scaled by the absolute value of |R[k] |?1 . Therefore, p(D|W ) can be obtained efficiently in
closed form. However, due to the feature transformation, the output R[k] depends non-linearly on
the projection W . Therefore, the posterior p(W |D) cannot be obtained in closed form any more.
We use a simple sample-based approach in order to approximate the posterior p(W |D). We use K
samples from the prior p(W ) to approximate the integrals in Equation (6) and in p(D). In this case,
the predictive model is given by
1 X
p(D|W i )
p(r? |? ? , D) ?
p(r? |? ? , W i )
,
(9)
K i
p(D)
P
where p(D) ? 1/K i p(D|W i ). The prediction for a single W i can again be obtained by a
standard Bayesian linear regression. Our algorithm is only interested in the expectation R? = E[r|?]
in the form of a quadratic model. Given a certain W i , we can obtain a single quadratic model from
?(W i ?)?? , where ?? is the mean of the posterior distribution p(?|W , D) obtained by Bayesian
linear regression. The expected quadratic model is then obtained by a weighted average over all K
quadratic models with weight p(D|W i )/p(D). Note that with a higher number of projection matrix
samples(K), the better the posterior can be approximated. Generating these samples is typically
inexpensive as it just requires computation time but no evaluation of the objective function. We
also investigated using more sophisticated sampling techniques such as elliptical slice sampling
[19] which achieved a similar performance but considerably increased computation time. Further
optimization of the sampling technique is part of future work.
4
Experiments
We compare MORE with state of the art methods in stochastic search and policy search such as
CMA-ES [1], NES [2], PoWER [20] and episodic REPS [9]. In our first experiments, we use
standard optimization test functions [21], such as the the Rosenbrock (uni modal) and the Rastrigin
(multi modal) functions. We use a 15 dimensional version of these functions.
Furthermore, we use a 5-link planar robot that has to reach a given point in task space as a toy task
for the comparisons. The resulting policy has 25 parameters, but we also test the algorithms in highdimensional parameter spaces by scaling the robot up to 30 links (150 parameters). We subsequently
made the task more difficult by introducing hard obstacles, which results in a discontinuous objective
function. We denote this task hole-reaching task. Finally, we evaluate our algorithm on a physical
simulation of a robot playing beer pong. The used parameters of the algorithms and a detailed
evaluation of the parameters of MORE can be found in the supplement.
4.1
Standard Optimization Test Functions
Pn?1
We chose one uni-modal functions f (x) = i=1 [100(xi+1 ? x2i )2 + (1 ? xi )2 ], also known as
Rosenbrock
Pn function and a multi-modal function which is known as the Rastgirin function f (x) =
10n + i=1 [x2i ? 10 cos(2?xi )]. All these functions have a global minimum equal f (x) = 0. In
our experiments, the mean of the initial distributions has been chosen randomly.
Algorithmic Comparison. We compared our algorithm against CMA-ES, NES, PoWER and REPS.
In each iteration, we generated 15 new samples 6 . For MORE, REPS and PoWER, we always keep
the last L = 150 samples, while for NES and CMA-ES only the 15 current samples are kept7 . As
we can see in the Figure 1, MORE outperforms all the other methods in terms of learning speed and
final performance in all test functions. However, in terms of the computation time, MORE was 5
times slower than the other algorithms. Yet, MORE was sufficiently fast as one policy update took
less than 1s.
Performance on a Noisy Function. We also conducted an experiment on optimizing the Sphere
function where we add multiplicative noise to the reward samples, i.e., y = f (x) + |f (x)|, where
? N (0, 1.0) and f (x) = xM x with a randomly chosen M matrix.
6
We use the heuristics introduced in [1, 2] for CMA-ES and NES
NES and CMA-ES algorithms typically only use the new samples and discard the old samples. We also
tried keeping old samples or getting more new samples which decreased the performance considerably.
7
6
y
3.50 -10
4.00
4.00
Average Return
Average Return
3.50
REPS
MORE
xNES
CMA-ES
4.50
5.00
5.50
6.00
4.50
y
3.00 -10
REPS
MORE
xNES
CMA-ES
5.00
5.50
6.00
6.50
7.00
0
0.5
1
1.5 2 2.5
Episodes
3
(a) Reaching Task
3.5
? 10
4
4
7.50
0
0.5
1
y
3.50
Average Return
3.00 -10
1.5 2 2.5
Episodes
3
3.5
? 10
(b) High-D Reaching Task
4
4
4.00
4.50
Gamma = 0.99
Gamma = 0.96
Gamma = 0.99999
5.00
5.50
0
0.5
1
1.5 2 2.5
Episodes
3
3.5
? 10
4
4
(c) Evaluation of ?
Figure 2: (a) Algorithmic comparison for a planar task (5 joints, 25 parameters). MORE outperforms all the
other methods considerably.(b) Algorithmic comparison for a high-dimensional task (30 joints, 150 parameters). The performance of NES degraded while MORE could still outperform CMA-ES. (c) Evaluation of the
entropy bound ?. For a low ?, the entropy bound is not active and the algorithm converges prematurely. If ? is
close to one, the entropy is reduced too slowly and convergence takes long.
Figure 1(c) shows that MORE successfully smooths out the noise and converges, while other methods diverge. The result shows that MORE can learn highly noisy reward functions.
4.2
Planar Reaching and Hole Reaching
We used a 5-link planar robot with DMPs [22] as the underlying control policy. Each link had a
length of 1m. The robot is modeled as a decoupled linear dynamical system. The end-effector of
the robot has to reach a via-point v 50 = [1, 1] at time step 50 and at the final time step T = 100
the point v 100 = [5, 0] with its end effector. The reward was given by a quadratic cost term for the
two via-points as well as quadratic costs for high accelerations. Note that this objective function
is highly non-quadratic in the parameters as the via-points are defined in end effector space. We
used 5 basis functions per degree of freedom for the DMPs while the goal attractor for reaching the
final state was assumed to be known. Hence, our parameter vector had 25 dimensions. The setup,
including the learned policy is shown in the supplement.
Algorithmic Comparison. We generated 40 new samples. For MORE, REPS, we always keep
the last L = 200 samples, while for NES and CMA-ES only the 40 current samples are kept. We
empirically optimized the open parameters of the algorithms by manually testing 50 parameter sets
for each algorithm. The results shown in Figure 2(a) clearly show that MORE outperforms all other
methods in terms of speed and the final performance.
Entropy Bound. We also evaluated the entropy bound in Figure 2(c). We can see that the entropy
constraint is a crucial component of the algorithm to avoid the premature convergence.
High-Dimensional Parameter Spaces. We also evaluated the same task with a 30-link planar robot,
resulting in a 150 dimensional parameter space. We compared MORE, CMA, REPS and NES. While
NES considerably degraded in performance, CMA and MORE performed well, where MORE found
considerably better policies (average reward of -6571 versus -15460 of CMA-ES), see Figure 2(b).
The setup with the learned policy from MORE is depicted in the supplement.
We use the same robot setup as in the planar reaching task for hole reaching task. For completing
the hole reaching task, the robot?s end effector has to reach the bottom of a hole (35cm wide and 1
m deep) centering at [2, 0] without any collision with the ground or the walls, see Figure 3(c). The
reward was given by a quadratic cost term for the desired final point, quadratic costs for high accelerations and additional punishment for collisions with the walls. Note that this objective function is
discontinuous due to the costs for collisions. The goal attractor of the DMP for reaching the final
state in this task is unknown and is also learned. Hence, our parameter vector had 30 dimensions.
Algorithmic Comparison. We used the same learning parameters as for the planar reaching task.
The results shown in Figure 3(a) show that MORE clearly outperforms all other methods. In this
task, NES could not find any reasonable solution while Power, REPS and CMA-ES could only learn
sub-optimal solutions. MORE could also achieve the same learning speed as REPS and CMA-ES,
but would then also converge to a sub-optimal solution.
7
y
-1.50 -10
-1.00
5.00
Average Return
Average Return
4.50
-0.50
5.50
6.00
5
REPS
PoWER
MORE
CMA-ES
4
-0.00
6.50
7.00
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y
y-axis [m]
4.00 -10
0
0.5
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1.5 2 2.5
Episodes
REPS
PoWER
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xNES
CMA-ES
3 3.5 4 4
(a) Hole Reaching Task
? 10
0.50
2
1
0
1.00
1.50
3
-1
0
50 Episodes100
(b) Beer Pong Task
150
-1.5
-1
-0.5
0
0.5
1
1.5
x-axis [m]
2
2.5
3
(c) Hole Reaching Task Posture
Figure 3: (a) Algorithmic comparison for the hole reaching task. MORE could find policies of much higher
quality. (b) Algorithmic comparison for the beer pong task. Only MORE could reliably learn high-quality
policies while for the other methods, even if some trials found good solutions, other trials got stuck prematurely.
4.3
Beer Pong
In this task, a seven DoF simulated barrett WaM robot arm
had to play beer-pong, i.e., it had to throw a ball such that
it bounces once on the table and falls into a cup. The ball
was placed in a container mounted on the end-effector. The
ball could leave the container by a strong deceleration of the
robot?s end-effector. We again used a DMP as underlying policy representation, where we used the shape parameters (five
(a) Beer Pong Task
per DoF) and the goal attractor (one per DoF) as parameters.
The mean of our search distribution was initialized with imitation learning. The cup was placed at a distance of 2.2m from Figure 4: The Beer Pong Task. The
robot has to throw a ball such that it
the robot and it had a height of 7cm. As reward function, we bounces of the table and ends up in the
computed the point of the ball trajectory after the bounce on cup.
the table, where the ball is passing the plane of the entry of the
cup. The reward was set to be 20 times the negative squared
distance of that point to the center of the cup while punishing the acceleration of the joints. We
evaluated MORE, CMA, PoWER and REPS on this task. The setup is shown in Figure 4 and the
learning curve is shown in Figure 3(b). MORE was able to accurately hit the ball into the cup while
the other algorithms couldn?t find a robust policy.
5
Conclusion
Using KL-bounds to limit the update of the search distribution is a wide-spread idea in the stochastic search community but typically requires approximations. In this paper, we presented a new
model-based stochastic search algorithm that computes the KL-bound analytically. By relying on a
Gaussian search distribution and on locally learned quadratic models of the objective function, we
can obtain a closed form of the information theoretic policy update. We also introduced an additional entropy term in the formulation that is needed to avoid premature shrinkage of the variance of
the search distribution. Our algorithm considerably outperforms competing methods in all the considered scenarios. The main disadvantage of MORE is the number of parameters. However based
on our experiments, these parameters are not problem specific.
Acknowledgment
This project has received funding from the European Unions Horizon 2020 research and innovation
programme under grant agreement No #645582 (RoMaNS) and the first author is supported by FCT
under grant SFRH/BD/81155/2011.
8
References
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9
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5,162 | 5,673 | Supervised Learning for Dynamical System Learning
Ahmed Hefny ?
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Carlton Downey ?
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Geoffrey J. Gordon ?
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Abstract
Recently there has been substantial interest in spectral methods for learning dynamical systems. These methods are popular since they often offer a good tradeoff
between computational and statistical efficiency. Unfortunately, they can be difficult to use and extend in practice: e.g., they can make it difficult to incorporate
prior information such as sparsity or structure. To address this problem, we present
a new view of dynamical system learning: we show how to learn dynamical systems by solving a sequence of ordinary supervised learning problems, thereby
allowing users to incorporate prior knowledge via standard techniques such as
L1 regularization. Many existing spectral methods are special cases of this new
framework, using linear regression as the supervised learner. We demonstrate the
effectiveness of our framework by showing examples where nonlinear regression
or lasso let us learn better state representations than plain linear regression does;
the correctness of these instances follows directly from our general analysis.
1
Introduction
Likelihood-based approaches to learning dynamical systems, such as EM [1] and MCMC [2], can
be slow and suffer from local optima. This difficulty has resulted in the development of so-called
?spectral algorithms? [3], which rely on factorization of a matrix of observable moments; these
algorithms are often fast, simple, and globally optimal.
Despite these advantages, spectral algorithms fall short in one important aspect compared to EM and
MCMC: the latter two methods are meta-algorithms or frameworks that offer a clear template for
developing new instances incorporating various forms of prior knowledge. For spectral algorithms,
by contrast, there is no clear template to go from a set of probabilistic assumptions to an algorithm.
In fact, researchers often relax model assumptions to make the algorithm design process easier,
potentially discarding valuable information in the process.
To address this problem, we propose a new framework for dynamical system learning, using the
idea of instrumental-variable regression [4, 5] to transform dynamical system learning to a sequence
of ordinary supervised learning problems. This transformation allows us to apply the rich literature
on supervised learning to incorporate many types of prior knowledge. Our new methods subsume a
variety of existing spectral algorithms as special cases.
The remainder of this paper is organized as follows: first we formulate the new learning framework
(Sec. 2). We then provide theoretical guarantees for the proposed methods (Sec. 4). Finally, we give
?
This material is based upon work funded and supported by the Department of Defense under Contract No.
FA8721-05-C-0003 with Carnegie Mellon University for the operation of the Software Engineering Institute, a
federally funded research and development center.
?
Supported by a grant from the PNC Center for Financial Services Innovation
?
Supported by NIH grant R01 MH 064537 and ONR contract N000141512365.
1
S1A regression ??[?? |?? ]
future ?? /??
history ??
???1
??
??+??1 ??+?
S2 regression
shifted future ??+1
extended future ?? /??
S1B regression ? ?[?? |?? ]
Condition on ?? (filter) ? ??+1
Marginalize ?? (predict) ? ??+1|??1
Figure 1: A latent-state dynamical system.
Observation ot is determined by latent state
st and noise t .
Figure 2: Learning and applying a dynamical system with instrumental regression. The
predictions from S1 provide training data to
S2. At test time, we filter or predict using the
weights from S2.
two examples of how our techniques let us rapidly design new and useful dynamical system learning
methods by encoding modeling assumptions (Sec. 5).
2
A framework for spectral algorithms
A dynamical system is a stochastic process (i.e., a distribution over sequences of observations) such
that, at any time, the distribution of future observations is fully determined by a vector st called the
latent state. The process is specified by three distributions: the initial state distribution P (s1 ), the
state transition distribution P (st+1 | st ), and the observation distribution P (ot | st ). For later use,
we write the observation ot as a function of the state st and random noise t , as shown in Figure 1.
Given a dynamical system, one of the fundamental tasks is to perform inference, where we predict
future observations given a history of observations. Typically this is accomplished by maintaining
a distribution or belief over states bt|t?1 = P (st | o1:t?1 ) where o1:t?1 denotes the first t ? 1
observations. bt|t?1 represents both our knowledge and our uncertainty about the true state of the
system. Two core inference tasks are filtering and prediction.1 In filtering, given the current belief
bt = bt|t?1 and a new observation ot , we calculate an updated belief bt+1 = bt+1|t that incorporates
ot . In prediction, we project our belief into the future: given a belief bt|t?1 we estimate bt+k|t?1 =
P (st+k | o1:t?1 ) for some k > 0 (without incorporating any intervening observations).
The typical approach for learning a dynamical system is to explicitly learn the initial, transition, and
observation distributions by maximum likelihood. Spectral algorithms offer an alternate approach
to learning: they instead use the method of moments to set up a system of equations that can be
solved in closed form to recover estimates of the desired parameters. In this process, they typically
factorize a matrix or tensor of observed moments?hence the name ?spectral.?
Spectral algorithms often (but not always [6]) avoid explicitly estimating the latent state or the initial,
transition, or observation distributions; instead they recover observable operators that can be used
to perform filtering and prediction directly. To do so, they use an observable representation: instead
of maintaining a belief bt over states st , they maintain the expected value of a sufficient statistic of
future observations. Such a representation is often called a (transformed) predictive state [7].
In more detail, we define qt = qt|t?1 = E[?t | o1:t?1 ], where ?t = ?(ot:t+k?1 ) is a vector of future
features. The features are chosen such that qt determines the distribution of future observations
1
There are other forms of inference in addition to filtering and prediction, such as smoothing and likelihood
evaluation, but they are outside the scope of this paper.
2
P (ot:t+k?1 | o1:t?1 ).2 Filtering then becomes the process of mapping a predictive state qt to qt+1
conditioned on ot , while prediction maps a predictive state qt = qt|t?1 to qt+k|t?1 = E[?t+k |
o1:t?1 ] without intervening observations.
A typical way to derive a spectral method is to select a set of moments involving ?t , work out the
expected values of these moments in terms of the observable operators, then invert this relationship
to get an equation for the observable operators in terms of the moments. We can then plug in an
empirical estimate of the moments to compute estimates of the observable operators.
While effective, this approach can be statistically inefficient (the goal of being able to solve for the
observable operators is in conflict with the goal of maximizing statistical efficiency) and can make
it difficult to incorporate prior information (each new source of information leads to new moments
and a different and possibly harder set of equations to solve). To address these problems, we show
that we can instead learn the observable operators by solving three supervised learning problems.
The main idea is that, just as we can represent a belief about a latent state st as the conditional
expectation of a vector of observable statistics, we can also represent any other distributions needed
for prediction and filtering via their own vectors of observable statistics. Given such a representation,
we can learn to filter and predict by learning how to map these vectors to one another.
In particular, the key intermediate quantity for filtering is the ?extended and marginalized? belief
P (ot , st+1 | o1:t?1 )?or equivalently P (ot:t+k | o1:t?1 ). We represent this distribution via a vector
?t = ?(ot:t+k ) of features of the extended future. The features are chosen such that the extended
state pt = E[?t | o1:t?1 ] determines P (ot:t+k | o1:t?1 ). Given P (ot:t+k | o1:t?1 ), filtering and
prediction reduce respectively to conditioning on and marginalizing over ot .
In many models (including Hidden Markov Models (HMMs) and Kalman filters), the extended state
pt is linearly related to the predictive state qt ?a property we exploit for our framework. That is,
pt = W qt
(1)
for some linear operator W . For example, in a discrete system ?t can be an indicator vector representing the joint assignment of the next k observations, and ?t can be an indicator vector for the next
k + 1 observations. The matrix W is then the conditional probability table P (ot:t+k | ot:t+k?1 ).
Our goal, therefore, is to learn this mapping W . Na??vely, we might try to use linear regression for
this purpose, substituting samples of ?t and ?t in place of qt and pt since we cannot observe qt or
pt directly. Unfortunately, due to the overlap between observation windows, the noise terms on ?t
and ?t are correlated. So, na??ve linear regression will give a biased estimate of W .
To counteract this bias, we employ instrumental regression [4, 5]. Instrumental regression uses instrumental variables that are correlated with the input qt but not with the noise t:t+k . This property
provides a criterion to denoise the inputs and outputs of the original regression problem: we remove
that part of the input/output that is not correlated with the instrumental variables. In our case, since
past observations o1:t?1 do not overlap with future or extended future windows, they are not correlated with the noise t:t+k+1 , as can be seen in Figure 1. Therefore, we can use history features
ht = h(o1:t?1 ) as instrumental variables.
In more detail, by taking the expectation of (1) given ht , we obtain an instrument-based moment
condition: for all t,
E[pt | ht ] = E[W qt | ht ]
E[E[?t | o1:t?1 ] | ht ] = W E[E[?t | o1:t?1 ] | ht ]
E[?t | ht ] = W E[?t | ht ]
(2)
Assuming that there are enough independent dimensions in ht that are correlated with qt , we maintain the rank of the moment condition when moving from (1) to (2), and we can recover W by least
squares regression if we can compute E[?t | ht ] and E[?t | ht ] for sufficiently many examples t.
Fortunately, conditional expectations such as E[?t | ht ] are exactly what supervised learning algorithms are designed to compute. So, we arrive at our learning framework: we first use supervised
2
For convenience we assume that the system is k-observable: that is, the distribution of all future observations is determined by the distribution of the next k observations. (Note: not by the next k observations
themselves.) At the cost of additional notation, this restriction could easily be lifted.
3
Model/Algorithm
future features ?t
Spectral Algorithm
for HMM [3]
U > eot where eot is an indicator vector and U spans the range of qt (typically the top m left singular vectors of
the joint probability table P (ot+1 , ot ))
xt and xt ? xt , where xt =
U > ot:t+k?1 for a matrix U that spans
the range of qt (typically the top m left
singular vectors of the covariance matrix
Cov(ot:t+k?1 , ot?k:t?1 ))
U > ot:t+k?1 (U obtained as above)
SSID for Kalman
filters (time dependent gain)
SSID for stable
Kalman
filters
(constant gain)
Uncontrolled HSEPSR [9]
Evaluation functional ks (ot:t+k?1 , .)
for a characteristic kernel ks
extended future features
?t
U > eot+1 ? eot
ffilter
yt and yt ? yt , where
yt is formed by stacking
U > ot+1:t+k and ot .
pt specifies a Gaussian distribution
where conditioning on ot is straightforward.
ot and U > ot+1:t+k
Estimate steady-state covariance by
solving Riccati equation [8]. pt together with the steady-state covariance
specify a Gaussian distribution where
conditioning on ot is straightforward.
Kernel Bayes rule [10].
ko (ot , .) ? ko (ot , .)
and ?t+1 ? ko (ot , .)
Estimate a state normalizer from S1A
output states.
Table 1: Examples of existing spectral algorithms reformulated as two-stage instrument regression
with linear S1 regression. Here ot1 :t2 is a vector formed by stacking observations ot1 through ot2 and
? denotes the outer product. Details and derivations can be found in the supplementary material.
learning to estimate E[?t | ht ] and E[?t | ht ], effectively denoising the training examples, and then
use these estimates to compute W by finding the least squares solution to (2).
In summary, learning and inference of a dynamical system through instrumental regression can be
described as follows:
? Model Specification: Pick features of history ht = h(o1:t?1 ), future ?t = ?(ot:t+k?1 )
and extended future ?t = ?(ot:t+k ). ?t must be a sufficient statistic for P(ot:t+k?1 |
o1:t?1 ). ?t must satisfy
? E[?t+1 | o1:t?1 ] = fpredict (E[?t | o1:t?1 ]) for a known function fpredict .
? E[?t+1 | o1:t ] = ffilter (E[?t | o1:t?1 ], ot ) for a known function ffilter .
? S1A (Stage 1A) Regression: Learn a (possibly non-linear) regression model to estimate
??t = E[?t | ht ]. The training data for this model are (ht , ?t ) across time steps t.3
? S1B Regression: Learn a (possibly non-linear) regression model to estimate ??t = E[?t |
ht ]. The training data for this model are (ht , ?t ) across time steps t.
? S2 Regression: Use the feature expectations estimated in S1A and S1B to train a model
to predict ??t = W ??t , where W is a linear operator. The training data for this model are
estimates of (??t , ??t ) obtained from S1A and S1B across time steps t.
? Initial State Estimation: Estimate an initial state q1 = E[?1 ] by averaging ?1 across
several example realizations of our time series.4
? Inference: Starting from the initial state q1 , we can maintain the predictive state qt =
E[?t | o1:t?1 ] through filtering: given qt we compute pt = E[?t | o1:t?1 ] = W qt . Then,
given the observation ot , we can compute qt+1 = ffilter (pt , ot ). Or, in the absence of ot ,
we can predict the next state qt+1|t?1 = fpredict (pt ). Finally, by definition, the predictive
state qt is sufficient to compute P(ot:t+k?1 | o1:t?1 ).5
The process of learning and inference is depicted in Figure 2. Modeling assumptions are reflected
in the choice of the statistics ?, ? and h as well as the regression models in stages S1A and S1B.
Table 1 demonstrates that we can recover existing spectral algorithms for dynamical system learning
using linear S1 regression. In addition to providing a unifying view of some successful learning
algorithms, the new framework also paves the way for extending these algorithms in a theoretically
justified manner, as we demonstrate in the experiments below.
3
Our bounds assume that the training time steps t are sufficiently spaced for the underlying process to mix,
but in practice, the error will only get smaller if we consider all time steps t.
4
Assuming ergodicity, we can set the initial state to be the empirical average vector of future features in a
P
single long sequence, T1 Tt=1 ?t .
5
It might seem reasonable to learn qt+1 = fcombined (qt , ot ) directly, thereby avoiding the need to separately
estimate pt and condition on ot . Unfortunately, fcombined is nonlinear for common models such as HMMs.
4
3
Related Work
This work extends predictive state learning algorithms for dynamical systems, which include spectral algorithms for Kalman filters [11], Hidden Markov Models [3, 12], Predictive State Representations (PSRs) [13, 14] and Weighted Automata [15]. It also extends kernel variants such as [9],
which builds on [16]. All of the above work effectively uses linear regression or linear ridge regression (although not always in an obvious way).
One common aspect of predictive state learning algorithms is that they exploit the covariance structure between future and past observation sequences to obtain an unbiased observable state representation. Boots and Gordon [17] note the connection between this covariance and (linear) instrumental
regression in the context of the HSE-HMM. We use this connection to build a general framework for
dynamical system learning where the state space can be identified using arbitrary (possibly nonlinear) supervised learning methods. This generalization lets us incorporate prior knowledge to learn
compact or regularized models; our experiments demonstrate that this flexibility lets us take better
advantage of limited data.
Reducing the problem of learning dynamical systems with latent state to supervised learning bears
similarity to Langford et al.?s sufficient posterior representation (SPR) [18], which encodes the state
by the sufficient statistics of the conditional distribution of the next observation and represents system dynamics by three vector-valued functions that are estimated using supervised learning approaches. While SPR allows all of these functions to be non-linear, it involves a rather complicated
training procedure involving multiple iterations of model refinement and model averaging, whereas
our framework only requires solving three regression problems in sequence. In addition, the theoretical analysis of [18] only establishes the consistency of SPR learning assuming that all regression
steps are solved perfectly. Our work, on the other hand, establishes convergence rates based on the
performance of S1 regression.
4
Theoretical Analysis
In this section we present error bounds for two-stage instrumental regression. These bounds hold
regardless of the particular S1 regression method used, assuming that the S1 predictions converge to
the true conditional expectations. The bounds imply that our overall method is consistent.
Let (xt , yt , zt ) ? (X , Y, Z) be i.i.d. triplets of input, output, and instrumental variables. (Lack of
independence will result in slower convergence in proportion to the mixing time of our process.) Let
? t | zt ] and E[y
? t | zt ] as
x
?t and y?t denote E[xt | zt ] and E[yt | zt ]. And, let x
?t and y?t denote E[x
estimated by the S1A and S1B regression steps. Here x
?t , x
?t ? X and y?t , y?t ? Y.
We want to analyze the convergence of the output of S2 regression?that is, of the weights W given
by ridge regression between S1A outputs and S1B outputs:
! T
!?1
T
X
X
?? =
W
y?t ? x
?t
x
?t ? x
?t + ?IX
(3)
t=1
t=1
Here ? denotes tensor (outer) product, and ? > 0 is a regularization parameter that ensures the
invertibility of the estimated covariance.
Before we state our main theorem we need to quantify the quality of S1 regression in a way that is
independent of the S1 functional form. To do so, we place a bound on the S1 error, and assume that
this bound converges to zero: given the definition below, for each fixed ?, limN ?? ??,N = 0.
Definition 1 (S1 Regression Bound). For any ? > 0 and N ? N+ , the S1 regression bound ??,N > 0
is a number such that, with probability at least (1 ? ?/2), for all 1 ? t ? N :
k?
xt ? x
?t kX < ??,N
k?
yt ? y?t kY < ??,N
In many applications, X , Y and Z will be finite dimensional real vector spaces: Rdx , Rdy and
Rdz . However, for generality we state our results in terms of arbitrary reproducing kernel Hilbert
spaces. In this case S2 uses kernel ridge regression, leading to methods such as HSE-PSRs. For
5
this purpose, let ?x?x? and ?y?y? denote the (uncentered) covariance operators of x
? and y? respectively:
?x?x? = E[?
x?x
?], ?y?y? = E[?
y ? y?]. And, let R(?x?x? ) denote the closure of the range of ?x?x? .
With the above assumptions, Theorem 2 gives a generic error bound on S2 regression in terms of
S1 regression. If X and Y are finite dimensional and ?x?x? has full rank, then using ordinary least
squares (i.e., setting ? = 0) will give the same bound, but with ? in the first two terms replaced by
the minimum eigenvalue of ?x?x? , and the last term dropped.
Theorem 2. Assume that k?
xkX , k?
xkY < c < ? almost surely. Assume W is a Hilbert-Schmidt
? ? be as defined in (3). Then, with probability at least 1 ? ?, for each xtest ?
operator, and let W
? ? xtest ? W xtest kY is bounded by
R(?x?x? ) s.t. kxtest kX ? 1, the error kW
r
?
?
??
q
log(1/?)
1
+
?
?
?1
??
N
log(1/?) 1
1
?
?
?
?
?
+
O
O?
+
+
?
+
O
3
? ?,N ? ?
??
? ? 32
N
?2
{z
}
|
|
{z
}
error from regularization
error
from
finite
samples
|
{z
}
error in S1 regression
We defer the proof to the supplementary material. The supplementary material also provides explicit
finite-sample bounds (including expressions for the constants hidden by O-notation), as well as
concrete examples of S1 regression bounds ??,N for practical regression models.
Theorem 2 assumes that xtest is in R(?x?x? ). For dynamical systems, all valid states satisfy this
property. However, with finite data, estimation errors may cause the estimated state q?t (i.e., xtest ) to
have a non-zero component in R? (?x?x? ). Lemma 3 bounds the effect of such errors: it states that, in
a stable system, this component gets smaller as S1 regression performs better. The main limitation
of Lemma 3 is the assumption that ffilter is L-Lipchitz, which essentially means that the model?s
estimated probability for ot is bounded below. There is no way to guarantee this property in practice;
so, Lemma 3 provides suggestive evidence rather than a guarantee that our learned dynamical system
will predict well.
Lemma 3. For observations o1:T , let q?t be the estimated state given o1:t?1 . Let q?t be the projection
of q?t onto R(?x?x? ). Assume ffilter is L-Lipchitz on pt when evaluated at ot , and ffilter (pt , ot ) ?
R(?x?x? ) for any pt ? R(?y?y?). Given the assumptions of theorem 2 and assuming that k?
qt kX ? R
for all 1 ? t ? T , the following holds for all 1 ? t ? T with probability at least 1 ? ?/2.
??,N
kt kX = k?
qt ? q?t kX = O ?
?
?
Since W? is bounded, the prediction error due to t diminishes at the same rate as kt kX .
5
Experiments and Results
We now demonstrate examples of tweaking the S1 regression to gain advantage. In the first experiment we show that nonlinear regression can be used to reduce the number of parameters needed in
S1, thereby improving statistical performance for learning an HMM. In the second experiment we
show that we can encode prior knowledge as regularization.
5.1
Learning A Knowledge Tracing Model
In this experiment we attempt to model and predict the performance of students learning from an
interactive computer-based tutor. We use the Bayesian knowledge tracing (BKT) model [19], which
is essentially a 2-state HMM: the state st represents whether a student has learned a knowledge
component (KC), and the observation ot represents the success/failure of solving the tth question in
a sequence of questions that cover this KC. Figure 3 summarizes the model. The events denoted by
guessing, slipping, learning and forgetting typically have relatively low probabilities.
5.1.1
Data Description
We evaluate the model using the ?Geometry Area (1996-97)? data available from DataShop [20].
This data was generated by students learning introductory geometry, and contains attempts by 59
6
Correct
Answer
Skill
Known
Skill
Known
Incorrect
Answer
Skill
Unknown
Skill
Unknown
Figure 3: Transitions and observations in BKT. Each node represents a possible value of the state
or observation. Solid arrows represent transitions while dashed arrows represent observations.
students in 12 knowledge components. As is typical for BKT, we consider a student?s attempt at a
question to be correct iff the student entered the correct answer on the first try, without requesting
any hints from the help system. Each training sequence consists of a sequence of first attempts for a
student/KC pair. We discard sequences of length less than 5, resulting in a total of 325 sequences.
5.1.2
Models and Evaluation
Under the (reasonable) assumption that the two states have distinct observation probabilities, this
model is 1-observable. Hence we define the predictive state to be the expected next observation,
which results in the following statistics: ?t = ot and ?t = ot ?k ot+1 , where ot is represented by
a 2 dimensional indicator vector and ?k denotes the Kronecker product. Given these statistics, the
extended state pt = E[?t | o1:t?1 ] is a joint probability table of ot:t+1 .
We compare three models that differ by history features and S1 regression method:
Spec-HMM: This baseline uses ht = ot?1 and linear S1 regression, making it equivalent to the
spectral HMM method of [3], as detailed in the supplementary material.
Feat-HMM: This baseline represents ht by an indicator vector of the joint assignment of the previous b observations (we set b to 4) and uses linear S1 regression. This is essentially a feature-based
spectral HMM [12]. It thus incorporates more history information compared to Spec-HMM at the
expense of increasing the number of S1 parameters by O(2b ).
LR-HMM: This model represents ht by a binary vector of length b encoding the previous b observations and uses logistic regression as the S1 model. Thus, it uses the same history information as
Feat-HMM but reduces the number of parameters to O(b) at the expense of inductive bias.
We evaluated the above models using 1000 random splits of the 325 sequences into 200 training
and 125 testing. For each testing observation ot we compute the absolute error between actual and
expected value (i.e. |?ot =1 ? P? (ot = 1 | o1:t?1 )|). We report the mean absolute error for each split.
The results are displayed in Figure 4.6 We see that, while incorporating more history information
increases accuracy (Feat-HMM vs. Spec-HMM), being able to incorporate the same information
using a more compact model gives an additional gain in accuracy (LR-HMM vs. Feat-HMM). We
also compared the LR-HMM method to an HMM trained using expectation maximization (EM). We
found that the LR-HMM model is much faster to train than EM while being on par with it in terms
of prediction error.7
5.2
Modeling Independent Subsystems Using Lasso Regression
Spectral algorithms for Kalman filters typically use the left singular vectors of the covariance between history and future features as a basis for the state space. However, this basis hides any sparsity
that might be present in our original basis. In this experiment, we show that we can instead use
lasso (without dimensionality reduction) as our S1 regression algorithm to discover sparsity. This is
useful, for example, when the system consists of multiple independent subsystems, each of which
affects a subset of the observation coordinates.
6
7
The differences have similar sign but smaller magnitude if we use RMSE instead of MAE.
We used MATLAB?s built-in logistic regression and EM functions.
7
0.34
0.3
0.28
0.26
0.26
0.3
0.34
Spec-HMM
0.32
0.3
0.28
0.26
0.26
0.3
0.34
0.32
0.3
0.28
0.26
0.26
0.3
0.34
0.32
0.3
0.28
0.26
0.26
Feat-HMM
Spec-HMM
Model
Training time (relative to Spec-HMM)
0.34
LR-HMM
0.32
LR-HMM
0.34
LR-HMM
Feat-HMM
0.34
Spec-HMM
1
Feat-HMM
1.02
0.3
0.34
EM
LR-HMM
2.219
EM
14.323
Figure 4: Experimental results: each graph compares the performance of two models (measured
by mean absolute error) on 1000 train/test splits. The black line is x = y. Points below this line
indicate that model y is better than model x. The table shows training time.
Figure 5: Left singular vectors of (left) true linear predictor from ot?1 to ot (i.e. OT O+ ), (middle)
covariance matrix between ot and ot?1 and (right) S1 Sparse regression weights. Each column
corresponds to a singular vector (only absolute values are depicted). Singular vectors are ordered by
their mean coordinate, interpreting absolute values as a probability distribution over coordinates.
To test this idea we generate a sequence of 30-dimensional observations from a Kalman filter. Observation dimensions 1 through 10 and 11 through 20 are generated from two independent subsystems
of state dimension 5. Dimensions 21-30 are generated from white noise. Each subsystem?s transition and observation matrices have random Gaussian coordinates, with the transition matrix scaled
to have a maximum eigenvalue of 0.95. States and observations are perturbed by Gaussian noise
with covariance of 0.01I and 1.0I respectively.
We estimate the state space basis using 1000 examples (assuming 1-observability) and compare the
singular vectors of the past to future regression matrix to those obtained from the Lasso regression
matrix. The result is shown in figure 5. Clearly, using Lasso as stage 1 regression results in a basis
that better matches the structure of the underlying system.
6
Conclusion
In this work we developed a general framework for dynamical system learning using supervised
learning methods. The framework relies on two key principles: first, we extend the idea of predictive
state to include extended state as well, allowing us to represent all of inference in terms of predictions
of observable features. Second, we use past features as instruments in an instrumental regression,
denoising state estimates that then serve as training examples to estimate system dynamics.
We have shown that this framework encompasses and provides a unified view of some previous
successful dynamical system learning algorithms. We have also demostrated that it can be used
to extend existing algorithms to incorporate nonlinearity and regularizers, resulting in better state
estimates. As future work, we would like to apply this framework to leverage additional techniques
such as manifold embedding and transfer learning in stage 1 regression. We would also like to
extend the framework to controlled processes.
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the 26th Annual International Conference on Machine Learning, ICML 2009, Montreal, Quebec, Canada, June 14-18, 2009, pages 961?968, 2009.
9
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5,163 | 5,674 | Expectation Particle Belief Propagation
Thibaut Lienart, Yee Whye Teh, Arnaud Doucet
Department of Statistics
University of Oxford
Oxford, UK
{lienart,teh,doucet}@stats.ox.ac.uk
Abstract
We propose an original particle-based implementation of the Loopy Belief Propagation (LPB) algorithm for pairwise Markov Random Fields (MRF) on a continuous state space. The algorithm constructs adaptively efficient proposal distributions approximating the local beliefs at each note of the MRF. This is achieved
by considering proposal distributions in the exponential family whose parameters
are updated iterately in an Expectation Propagation (EP) framework. The proposed particle scheme provides consistent estimation of the LBP marginals as the
number of particles increases. We demonstrate that it provides more accurate results than the Particle Belief Propagation (PBP) algorithm of [1] at a fraction of
the computational cost and is additionally more robust empirically. The computational complexity of our algorithm at each iteration is quadratic in the number
of particles. We also propose an accelerated implementation with sub-quadratic
computational complexity which still provides consistent estimates of the loopy
BP marginal distributions and performs almost as well as the original procedure.
1
Introduction
Undirected Graphical Models (also known as Markov Random Fields) provide a flexible framework
to represent networks of random variables and have been used in a large variety of applications in
machine learning, statistics, signal processing and related fields [2]. For many applications such as
tracking [3, 4], sensor networks [5, 6] or computer vision [7, 8, 9] it can be beneficial to define MRF
on continuous state-spaces.
Given a pairwise MRF, we are here interested in computing the marginal distributions at the nodes
of the graph. A popular approach to do this is to consider the Loopy Belief Propagation (LBP) algorithm [10, 11, 2]. LBP relies on the transmission of messages between nodes. However when dealing with continuous random variables, computing these messages exactly is generally intractable.
In practice, one must select a way to tractably represent these messages and a way to update these
representations following the LBP algorithm. The Nonparametric Belief Propagation (NBP) algorithm [12] represents the messages with mixtures of Gaussians while the Particle Belief Propagation
(PBP) algorithm [1] uses an importance sampling approach. NBP relies on restrictive integrability
conditions and does not offer consistent estimators of the LBP messages. PBP offers a way to circumvent these two issues but the implementation suggested proposes sampling from the estimated
beliefs which need not be integrable. Moreover, even when they are integrable, sampling from
the estimated beliefs is very expensive computationally. Practically the authors of [1] only sample
approximately from those using short MCMC runs, leading to biased estimators.
In our method, we consider a sequence of proposal distributions at each node from which one can
sample particles at a given iteration of the LBP algorithm. The messages are then computed using
importance sampling. The novelty of the approach is to propose a principled and automated way
of designing a sequence of proposals in a tractable exponential family using the Expectation Prop1
agation (EP) framework [13]. The resulting algorithm, which we call Expectation Particle Belief
Propagation (EPBP), does not suffer from restrictive integrability conditions and sampling is done
exactly which implies that we obtain consistent estimators of the LBP messages. The method is empirically shown to yield better approximations to the LBP beliefs than the implementation suggested
in [1], at a much reduced computational cost, and than EP.
2
2.1
Background
Notations
We consider a pairwise MRF, i.e. a distribution over a set of p random variables indexed by a set
V = {1, . . . , p}, which factorizes according to an undirected graph G = (V, E) with
Y
Y
p(xV ) ?
?u (xu )
?uv (xu , xv ).
(1)
u?V
(u,v)?E
The random variables are assumed to take values on a continuous, possibly unbounded, space X .
The positive functions ?u : X 7? R+ and ?uv : X ? X 7? R+ are respectively known as the node
and edge potentials. The aim is to approximate the marginals pu (xu ) for all u ? V . A popular
approach is the LBP algorithm discussed earlier. This algorithm is a fixed point iteration scheme
yielding approximations called the beliefs at each node [10, 2]. When the underlying graph is a tree,
the resulting beliefs can be shown to be proportional to the exact marginals. This is not the case in
the presence of loops in the graph. However, even in these cases, LBP has been shown to provide
good approximations in a wide range of situations [14, 11]. The LBP fixed-point iteration can be
written as follows at iteration t:
?
Z
Y
?
t
?
m
(x
)
=
?
(x
,
x
)?
(x
)
mt?1
?
v
uv
u
v
u
u
uv
wu (xu )dxu
?
w??u \v
,
(2)
Y
?
?
But (xu ) = ?u (xu )
mtwu (xu )
?
?
w??u
where ?u denotes the neighborhood of u i.e., the set of nodes {w | (w, u) ? E}, muv is known as
the message from node u to node v and Bu is the belief at node u.
2.2
Related work
The crux of any generic implementation of LBP for continuous state spaces is to select a way to represent the messages and design an appropriate method to compute/approximate the message update.
In Nonparametric BP (NBP) [12], the messages are represented by mixtures of Gaussians. In theory,
computing the product of such messages can be done analytically but in practice this is impractical
due to the exponential growth in the number of terms to consider. To circumvent this issue, the
authors suggest an importance sampling approach targeting the beliefs and fitting mixtures of Gaussians to the resulting weighted particles. The computation of the update (2) is then always done over
a constant number of terms.
A restriction of ?vanilla? Nonparametric BP is that the messages must be finitely integrable for the
message representation to make sense. This is the case if the following two conditions hold:
Z
Z
sup
?uv (xu , xv )dxu < ?, and
?u (xu )dxu < ?.
(3)
xv
These conditions do however not hold in a number of important cases as acknowledged in [3]. For
instance, the potential ?u (xu ) is usually proportional to a likelihood of the form p(yu |xu ) which
need not be integrable in xu . Similarly, in imaging applications for example, the edge potential can
encode similarity between pixels which also need not verify the integrability condition as in [15].
Further, NBP does not offer consistent estimators of the LBP messages.
Particle BP (PBP) [1] offers a way to overcome the shortcomings of NBP: the authors also consider
importance sampling to tackle the update of the messages but without fitting a mixture of Gaussians.
2
(i)
For a chosen proposal distribution qu on node u and a draw of N particles {xu }N
i=1 ? qu (xu ), the
messages are represented as mixtures:
m
b PBP
uv (xv ) :=
N
X
(i)
?uv
?uv (x(i)
u , xv ),
(i)
with ?uv
:=
i=1
(i)
1 ?u (xu ) Y
(i)
m
b PBP
wu (xu ).
N qu (xu(i) )
w??u \v
(4)
This algorithm has the advantage that it does not require the conditions (3) to hold. The authors
suggest two possible choices of sampling distributions: sampling from the local potential ?u , or
sampling from the current belief estimate. The first case is only valid if ?u is integrable w.r.t. xu
which, as we have mentioned earlier, might not be the case in general and the second case implies
sampling from a distribution of the form
Y
buPBP (xu ) ? ?u (xu )
B
m
b PBP
(5)
wu (xu )
w??u
which is a product of mixtures. As in NBP, na??ve sampling of the proposal has complexity O(N |?u | )
and is thus in general too expensive to consider. Alternatively, as the authors suggest, one can run
a short MCMC simulation targeting it which reduces the complexity to order O(|?u |N 2 ) since the
buPBP point-wise, is of order O(|?u |N ), and we
cost of each iteration, which requires evaluating B
need O(N ) iterations of the MCMC simulation. The issue with this approach is that it is still computationally expensive, and it is unclear how many iterations are necessary to get N good samples.
2.3
Our contribution
In this paper, we consider the general context where the edge and node-potentials might be nonnormalizable and non-Gaussian. Our proposed method is based on PBP, as PBP is theoretically
better suited than NBP since, as discussed earlier, it does not require the conditions (3) to hold, and,
provided that one samples from the proposals exactly, it yields consistent estimators of the LBP
messages while NBP does not. Further, the development of our method also formally shows that
considering proposals close to the beliefs, as suggested by [1], is a good idea. Our core observation
is that since sampling from a proposal of the form (5) using MCMC simulation is very expensive,
we should consider using a more tractable proposal distribution instead. However it is important that
the proposal distribution is constructed adaptively, taking into account evidence collected through
the message passing itself, and we propose to achieve this by using proposal distributions lying in a
tractable exponential family, and adapted using the Expectation Propagation (EP) framework [13].
3
Expectation Particle Belief Propagation
Our aim is to address the issue of selecting the proposals in the PBP algorithm. We suggest using
exponential family distributions as the proposals on a node for computational efficiency reasons,
with parameters chosen adaptively based on current estimates of beliefs and EP. Each step of our
algorithm involves both a projection onto the exponential family as in EP, as well as a particle
approximation of the LBP message, hence we will refer to our method as Expectation Particle
Belief Propagation or EPBP for short.
For each pair of adjacent nodes u and v, we will use muv (xv ) to denote the exact (but unavailable)
LBP message from u to v, m
b uv (xv ) to denote the particle approximation of muv , and ?uv an exponential family projection of m
b uv . In addition, let ??u denote an exponential family projection of the
node potential ?u . We will consider approximations consisting of N particles. In the following, we
will derive the form of our particle approximated message m
b uv (xv ), along with the choice of the
proposal distribution qu (xu ) used to construct m
b uv . Our starting point is the edge-wise belief over
xu and xv , given the incoming particle approximated messages,
Y
Y
buv (xu , xv ) ? ?uv (xu , xv )?u (xu )?v (xv )
B
m
b wu (xu )
m
b ?v (xv ).
(6)
w??u \v
???v \u
buv (xv ),
The exact LBP message muv (xv ) can be derived by computing the marginal distribution B
and constructing muv (xv ) such that
buv (xv ) ? muv (xv )M
cvu (xv ),
B
3
(7)
cvu (xv ) = ?v (xv ) Q
where M
b ?v (xv ) is the (particle approximated) pre-message from v to
???v \u m
u. It is easy to see that the resulting message is as expected,
Z
Y
muv (xv ) ? ?uv (xu , xv )?u (xu )
m
b wu (xu )dxu .
(8)
w??u \v
Since the above exact LBP belief and message are intractable in our scenario of interest, the idea
buv (xu , xv ) instead. Consider a proposal distribution
is to use an importance sampler targeting B
of the form qu (xu )qv (xv ). Since xu and xv are independent under the proposal, we can draw
(i)
(j) N
N independent samples, say {xu }N
i=1 and {xv }j=1 , from qu and qv respectively. We can then
approximate the belief using a N ? N cross product of the particles,
N
(j)
X
buv (x(i)
B
u , xv )
buv (xu , xv ) ? 1
B
?(x(i) ,x(j) ) (xu , xv )
u
v
N 2 i,j=1 qu (xu(i) )qv (x(j)
v )
(i)
(j)
(i) c
(j) Q
(i)
N
b wu (xu )
vu (xv )
1 X ?uv (xu , xv )?u (xu )M
w??u \v m
? 2
?(x(i) ,x(j) ) (xu , xv )
(i)
(j)
u
v
N i,j=1
qu (xu )qv (xv )
(9)
buv (xv ),
Marginalizing onto xv , we have the following particle approximation to B
N
(j)
(j) c
b uv (xv )M
1 Xm
vu (xv )
b
?x(j) (xv )
Buv (xv ) ?
(j)
v
N j=1
qv (xv )
(10)
where the particle approximated message m
b uv (xv ) from u to v has the form of the message representation in the PBP algorithm (4).
buv .
To determine sensible proposal distributions, we can find qu and qv that are close to the target B
buv kqu qv ) as the measure of closeness, the optimal qu required for the
Using the KL divergence KL(B
u to v message is the node belief,
Y
buv (xu ) ? ?u (xu )
B
m
b wu (xu )
(11)
w??u
thus supporting the claim in [1] that a good proposal to use is the current estimate of the node belief.
As pointed out in Section 2, it is computationally inefficient to use the particle approximated node
belief as the proposal distribution. An idea is to use a tractable exponential family distribution for
qu instead, say
Y
qu (xu ) ? ??u (xu )
?wu (xu )
(12)
w??u
where ??u and ?wu are exponential family approximations of ?u and m
b wu respectively. In Section
4 we use a Gaussian family, but we are not limited to this. Using the framework of expectation
propogation (EP) [13], we can iteratively find good exponential family approximations as follows.
\w
For each w ? ?u , to update the ?wu , we form the cavity distribution qu ? qu /?wu and the corre\w
+
sponding tilted distribution m
b wu qu . The updated ?wu
is the exponential family factor minimising
the KL divergence,
h
i
+
?wu
= arg min KL m
b wu (xu )qu\w (xu )
?(xu )qu\w (xu ) .
(13)
??exp.fam.
Geometrically, the update projects the tilted distribution onto the exponential family manifold.
The optimal solution requires computing the moments of the tilted distribution through numeri\w
cal quadrature, and selecting ?wu so that ?wu qu matches the moments of the tilted distribution. In
our scenario the moment computation can be performed crudely on a small number of evaluation
points since it only concerns the updating of the importance sampling proposal. If an optimal ? in
the exponential family does not exist, e.g. in the Gaussian case that the optimal ? has a negative
variance, we simply revert ?wu to its previous value [13]. An analogous update is used for ??u .
In the above derivation, the expectation propagation steps for each incoming message into u and for
the node potential are performed first, to fit the proposal to the current estimated belief at u, before
4
it is used to draw N particles, which can then be used to form the particle approximated messages
from u to each of its neighbours. Alternatively, once each particle approximated message m
b uv (xv )
is formed, we can update its exponential family projection ?uv (xv ) immediately. This alternative
scheme is described in Algorithm 1.
Algorithm 1 Node update
(i)
1: sample {xu } ? qu ( ? )
(i) Q
(i)
bu (x(i)
2: compute B
b wu (xu )
u ) = ?u (xu )
w??u m
3: for v ? ?u do
cuv (x(i)
b (i) b vu (xu(i) )
4:
compute M
u ) := Bu (xu )/m
(i)
(i)
cuv (x(i)
5:
compute the normalized weights wuv ? M
u )/qu (xu )
PN
(i)
(i)
6:
update the estimator of the outgoing message m
b uv (xv ) = i=1 wuv ?uv (xu , xv )
\?
+
compute the cavity distribution qv ? qv /??v , get ??v
in the exponential family such that
\?
+ \?
+
+
??v qv approximates ?v qv , update qv ? ??v and let ??v ? ??v
\u
+
8:
compute the cavity distribution qv ? qv /?uv , get ?uv in the exponential family such that
\u
+ \u
+
+
qv approximates m
b uv qv , update qv ? ?uv
and let ?uv ? ?uv
?uv
9: end for
7:
3.1
Computational complexity and sub-quadratic implementation
Each EP projection step costs O(N ) computations since the message m
b wu is a mixture of N components (see (4)). Drawing N particles from the exponential family proposal qu costs O(N ). The step
with highest computational complexity is in evaluating the particle weights in (4). Indeed, evaluating
the mixture representation of a message on a single point is O(N ), and we need to compute this for
each of N particles. Similarly, evaluating the estimator of the belief on N sampling points at node
u requires O(|?u |N 2 ). This can be reduced since the algorithm still provides consistent estimators
if we consider the evaluation of unbiased estimators of the messages instead. Since the messages
PN
i
i
have the form m
b uv (xv ) =
i=1 wuv ?uv (xv ), we can follow a method presented in [16] where
? M
i
}N
one draws M indices {i` }`=1 from a multinomial with weights {wuv
i=1 and evaluates the correi?
`
sponding M components ?uv
. This reduces the cost of the evaluation of the beliefs to O(|?u |M N )
which leads to an overall sub-quadratic complexity if M is o(N ). We show in the next section how
it compares to the quadratic implementation when M = O(log N ).
4
Experiments
We investigate the performance of our method on MRFs for two simple graphs. This allows us
to compare the performance of EPBP to the performance of PBP in depth. We also illustrate the
behavior of the sub-quadratic version of EPBP. Finally we show that EPBP provides good results in
a simple denoising application.
4.1
Comparison with PBP
We start by comparing EPBP to PBP as implemented by Ihler et al. on a 3 ? 3 grid (figure 1)
with random variables taking values on R. The node and edge potentials are selected such that the
marginals are multimodal, non-Gaussian and skewed with
?u (xu )
= ?1 N (xu ? yu ; ?2, 1) + ?2 G(xu ? yu ; 2, 1.3)
,
(14)
?uv (xu , xv ) = L(xu ? xv ; 0, 2)
where yu denotes the observation at node u, N (x; ?, ?) ? exp(?x2 /2? 2 ) (density of a Normal
distribution), G(x; ?, ?) ? exp(?(x??)/? +exp(?(x??)/?)) (density of a Gumbel distribution)
and L(x; ?, ?) ? exp(?|x ? ?|/?) (density of a Laplace distribution). The parameters ?1 and ?2
are respectively set to 0.6 and 0.4. We compare the two methods after 20 LBP iterations.1
1
The scheduling used alternates between the classical orderings: top-down-left-right, left-right-top-down,
down-up-right-left and right-left-down-up. One ?LBP iteration? implies that all nodes have been updated once.
5
1
4
7
2
5
8
3
6
9
1
2
4
5
3
6
7
8
Figure 1: Illustration of the grid (left) and tree (right) graphs used in the experiments.
PBP as presented in [1] is implemented using the same parameters than those in an implementation
code provided by the authors: the proposal on each node is the last estimated belief and sampled with
a 20-step MCMC chain, the MH proposal is a normal distribution. For EPBP, the approximation of
the messages are Gaussians. The ground truth is approximated by running LBP on a deterministic
equally spaced mesh with 200 points. All simulations were run with Julia on a Mac with 2.5 GHz
Intel Core i5 processor, our code is available online.2
Figure 2 compares the performances of both methods. The error is computed as the mean L1 error
over all nodes between the estimated beliefs and the ground truth evaluated over the same deterministic mesh. One can observe that not only does PBP perform worse than EPBP but also that the
error plateaus with increasing number of samples. This is because the secondampling within PBP
is done approximately and hence the consistency of the estimators is lost. The speed-up offered by
EPBP is very substantial (figure 4 left). Hence, although it would be possible to use more MCMC
(Metropolis-Hastings) iterations within PBP to improve its performance, it would
? make the method
prohibitively expensive to use. Note that for EPBP, one observes the usual 1/ N convergence of
particle methods.
Figure 3 compares the estimator of the beliefs obtained by the two methods for three arbitrarily
picked nodes (node 1, 5 and 9 as illustrated on figure 1). The figure also illustrates the last proposals
constructed with our approach and one notices that their supports match closely the support of the
true beliefs. Figure 4 left illustrates how the estimated beliefs converge as compared to the true
beliefs with increasing number of iterations. One can observe that PBP converges more slowly and
that the results display more variability which might be due to the MCMC runs being too short.
We repeated the experiments on a tree with 8 nodes (figure 1 right) where we know that, at convergence, the beliefs computed using BP are proportional to the true marginals. The node and edge
potentials are again picked such that the marginals are multimodal with
?u (xu )
= ?1 N (xu ? yu ; ?2, 1) + ?2 N (xu ? yu ; 1, 0.5)
,
(15)
?uv (xu , xv ) = L(xu ? xv ; 0, 1)
with ?1 = 0.3 and ?2 = 0.7. On this example, we also show how ?pure EP? with normal distributions performs. We also try using the distributions obtained with EP as proposals for PBP (referred
to as ?PBP after EP? in figures). Both methods underperform compared to EPBP as illustrated visually in Figure 5. In particular one can observe in Figure 3 that ?PBP after EP? converges slower
than EPBP with increasing number of samples.
4.2
Sub-quadratic implementation and denoising application
As outlined in Section 3.1, in the implementation of EPBP one can use an unbiased estimator of
the edge weights based on a draw of M components from a multinomial. The complexity of the
resulting algorithm is O(M N ). We apply this method to the 3 ? 3 grid example in the case where
M is picked to be roughly of order log(N ): i.e., for N = {10, 20, 50, 100, 200, 500}, we pick
M = {5, 6, 8, 10, 11, 13}. The results are illustrated in Figure 6 where one can see that the N log N
implementation compares very well to the original quadratic implementation at a much reduced
cost. We apply this sub-quadratic method on a simple probabilistic model for an image denoising
problem. The aim of this example is to show that the method can be applied to larger graphs and still
provide good results. The model underlined is chosen to showcase the flexibility and applicability
of our method in particular when the edge-potential is non-integrable. It is not claimed to be an
optimal approach to image denoising.3 The node and edge potentials are defined as follows:
?u (xu )
= N (xu ? yu ; 0, 0.1)
,
(16)
?uv (xu , xv ) = L? (xu ? xv ; 0, 0.03)
2
3
https://github.com/tlienart/EPBP.
In this case in particular, an optimization-based method such as [17] is likely to yield better results.
6
where L? (x; ?, ?) = L(x; ?, ?) if |x| ? ? and L(?; ?, ?) otherwise. In this example we set
? = 0.2. The value assigned to each pixel of the reconstruction is the estimated mean obtained over
the corresponding node (figure 7). The image has size 50 ? 50 and the simulation was run with
N = 30 particles per nodes, M = 5 and 10 BP iterations taking under 2 minutes to complete. We
compare it with the result obtained with EP on the same model.
10 0
10 0
EPBP
PBP after EP
Mean L1 error
Mean L1 error
PBP
EPBP
10 -1
10 -2
10 1
10 2
10 -1
10 -2
10 3
10 1
10 2
Number of samples per node
10 3
Number of samples per node
Figure 2: (left) Comparison of the mean L1 error for PBP and EPBP for the 3 ? 3 grid example.
(right) Comparison of the mean L1 error for ?PBP after EP? and EPBP for the tree example. In both
cases, EPBP is more accurate for the same number of samples.
0.35
0.6
0.3
0.3
0.5
0.25
0.4
0.2
0.3
0.15
0.25
True belief
Estimated belief (EPBP)
Estimated belief (PBP)
Proposal (EPBP)
0.2
0.15
0.1
0.05
0
-5
0
5
10
15
0.2
0.1
0.1
0.05
0
-5
0
5
10
15
0
-5
0
5
10
15
Figure 3: Comparison of the beliefs on node 1, 5 and 9 as obtained by evaluating LBP on a deterministic mesh (true belief ), with PBP and with EPBP for the 3 ? 3 grid example. The proposal used
by EPBP at the last step is also illustrated. The results are obtained with N = 100 samples on each
node and 20 BP iterations. One can observe visually that EPBP outperforms PBP.
4.5
4
10
3
10
2
PBP
EPBP
3.5
Wall-clock time [s]
3
Mean L1 error
10
EPBP
PBP
4
2.5
2
1.5
1
10 1
10 0
0.5
0
0
5
10
15
10 -1
20
Number of BP iterations
10 1
10 2
10 3
Number of samples per node
Figure 4: (left) Comparison of the convergence in L1 error with increasing number of BP iterations
for the 3 ? 3 grid example when using N = 30 particles. (right) Comparison of the wall-clock time
needed to perform PBP and EPBP on the 3 ? 3 grid example.
5
Discussion
We have presented an original way to design adaptively efficient and easy-to-sample-from proposals
for a particle implementation of Loopy Belief Propagation. Our proposal is inspired by the Expectation Propagation framework.
We have demonstrated empirically that the resulting algorithm is significantly faster and more accurate than an implementation of PBP using the estimated beliefs as proposals and sampling from
them using MCMC as proposed in [1]. It is also more accurate than EP due to the nonparametric
nature of the messages and offers consistent estimators of the LBP messages. A sub-quadratic version of the method was also outlined and shown to perform almost as well as the original method on
7
mildly multi-modal models, it was also applied successfully in a simple image denoising example
illustrating that the method can be applied on graphical models with several hundred nodes.
We believe that our method could be applied successfully to a wide range of applications such as
smoothing for Hidden Markov Models [18], tracking or computer vision [19, 20]. In future work,
we will look at considering other divergences than the KL and the ?Power EP? framework [21], we
will also look at encapsulating the present algorithm within a sequential Monte Carlo framework
and the recent work of Naesseth et al. [22].
1.2
0.9
0.5
0.8
0.45
0.7
0.4
0.6
0.35
1
0.8
True belief
Est. bel. (EPBP)
Est. bel. (PBP)
Est. bel. (EP)
Est. bel. (PBP after EP)
0.3
0.5
0.25
0.6
0.4
0.2
0.4
0.3
0.15
0.2
0.1
0.1
0.05
0.2
0
-2
0
2
4
0
-2
6
0
2
4
6
0
-2
0
2
4
6
Figure 5: Comparison of the beliefs on node 1, 3 and 8 as obtained by evaluating LBP on a deterministic mesh, using EPBP, PBP, EP and PBP using the results of EP as proposals. This is for the
tree example with N = 100 samples on each node and 20 LBP iterations. Again, one can observe
visually that EPBP outperforms the other methods.
10
0
10
2
10
10
NlogN implementation
Quadratic implementation
Wall-clock time [s]
Mean L1 error
NlogN implementation
Quadratic implementation
-1
-2
10 1
10 2
10 1
10 0
10
10 3
Number of samples
-1
10 1
10 2
10 3
Number of samples per node
Figure 6: Comparison of the mean L1 error for PBP and EPBP on a 3 ? 3 grid (left). For the
same number of samples, EPBP is more accurate. It is also faster by about two orders of magnitude
(right). The simulations were run several times for the same observations to illustrate the variability
of the results.
Figure 7: From left to right: comparison of the original (first), noisy (second) and recovered image
using the sub-quadratic implementation of EPBP (third) and with EP (fourth).
Acknowledgments
We thank Alexander Ihler and Drew Frank for sharing their implementation of Particle Belief Propagation. TL gratefully acknowledges funding from EPSRC (grant 1379622) and the Scatcherd European scholarship scheme. YWT?s research leading to these results has received funding from
EPSRC (grant EP/K009362/1) and ERC under the EU?s FP7 Programme (grant agreement no.
617411). AD?s research was supported by the EPSRC (grant EP/K000276/1, EP/K009850/1) and
by AFOSR/AOARD (grant AOARD-144042).
8
References
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[2] Martin J. Wainwright and Michael I. Jordan. Graphical models, exponential families, and
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[16] Mark Briers, Arnaud Doucet, and Sumeetpal S. Singh. Sequential auxiliary particle belief
propagation. In Proc. 8th ICIF, volume 1, pages 705?711, 2005.
[17] Leonid I. Rudin, Stanley Osher, and Emad Fatemi. Nonlinear total variation based noise removal algorithms. Physica D, 60(1):259?268, 1992.
[18] M. Briers, A. Doucet, and S. Maskell. Smoothing algorithms for state-space models. Ann. Inst.
Stat. Math., 62(1):61?89, 2010.
[19] Erik B. Sudderth, Michael I. Mandel, William T. Freeman, and Alan S. Willsky. Visual hand
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[21] Thomas P. Minka. Power EP. Technical Report MSR-TR-2004-149, 2004.
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9
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5,164 | 5,675 | Embedding Inference
for Structured Multilabel Prediction
Farzaneh Mirzazadeh Siamak Ravanbakhsh
University of Alberta
Nan Ding
Google
Dale Schuurmans
University of Alberta
{mirzazad,mravanba}@ualberta.ca
[email protected]
[email protected]
Abstract
A key bottleneck in structured output prediction is the need for inference during training and testing, usually requiring some form of dynamic programming.
Rather than using approximate inference or tailoring a specialized inference
method for a particular structure?standard responses to the scaling challenge?
we propose to embed prediction constraints directly into the learned representation. By eliminating the need for explicit inference a more scalable approach to
structured output prediction can be achieved, particularly at test time. We demonstrate the idea for multi-label prediction under subsumption and mutual exclusion
constraints, where a relationship to maximum margin structured output prediction
can be established. Experiments demonstrate that the benefits of structured output
training can still be realized even after inference has been eliminated.
1
Introduction
Structured output prediction has been an important topic in machine learning. Many prediction
problems involve complex structures, such as predicting parse trees for sentences [28], predicting
sequence labellings for language and genomic data [1], or predicting multilabel taggings for documents and images [7, 8, 13, 20]. Initial breakthroughs in this area arose from tractable discriminative training methods?conditional random fields [19, 27] and structured large margin training
[26, 29]?that compare complete output configurations against given target structures, rather than
simply learning to predict each component in isolation. More recently, search based approaches that
exploit sequential prediction methods have also proved effective for structured prediction [4, 21].
Despite these improvements, the need to conduct inference or search over complex outputs both
during the training and testing phase proves to be a significant bottleneck in practice.
In this paper we investigate an alternative approach that eliminates the need for inference or search
at test time. The idea is to shift the burden of coordinating predictions to the training phase, by
embedding constraints in the learned representation that ensure prediction relationships are satisfied.
The primary benefit of this approach is that prediction cost can be significantly reduced without
sacrificing the desired coordination of structured output components.
We demonstrate the proposed approach for the problem of multilabel classification with hierarchical and mutual exclusion constraints on output labels [8]. Multilabel classification is an important
subfield of structured output prediction where multiple labels must be assigned that respect semantic
relationships such as subsumption, mutual exclusion or weak forms of correlation. The problem is of
growing importance as larger tag sets are being used to annotate images and documents on the Web.
Research in this area can be distinguished by whether the relationships between labels are assumed
to be known beforehand or whether such relationships need to be inferred during training. In the latter case, many works have developed tailored training losses for multilabel prediction that penalize
joint prediction behavior [6, 9, 30] without assuming any specific form of prior knowledge. More
recently, several works have focused on coping with large label spaces by using low dimensional
1
projections to label subspaces [3, 17, 22]. Other work has focused on exploiting weak forms of prior
knowledge expressed as similarity information between labels that can be obtained from auxiliary
sources [11]. Unfortunately, none of these approaches strictly enforce prior logical relationships between label predictions. By contrast, other research has sought to exploit known prior relationships
between labels. The most prominent such approaches have been to exploit generative or conditional
graphical model structures over the label set [5, 16]. Unfortunately, the graphical model structures
are either limited to junction trees with small treewidth [5] or require approximation [12]. Other
work, using output kernels, has also been shown able to model complex relationships between labels [15] but is hampered by an intractable pre-image problem at test time.
In this paper, we focus on tractable methods and consider the scenario where a set of logical label
relationships is given a priori; in particular, implication and mutual exclusion relationships. These
relationships have been the subject of extensive work on multilabel prediction, where it is known
that if the implication/subsumption relationships form a tree [25] or a directed acyclic graph [2, 8]
then efficient dynamic programming algorithms can be developed for tractable inference during
training and testing, while for general pairwise models [32] approximate inference is required. Our
main contribution is to show how these relationships can be enforced without the need for dynamic
programming. The idea is to embed label relationships as constraints on the underlying score model
during training so that a trivial labelling algorithm can be employed at test time, a process that can
be viewed as pre-compiling inference during the training phase.
The literature on multivariate prediction has considered many other topics not addressed by this
paper, including learning from incomplete labellings, exploiting hierarchies and embeddings for
multiclass prediction [31], exploiting multimodal data, deriving generalization bounds for structured
and multilabel prediction problems, and investigating the consistency of multilabel losses.
2
Background
We consider a standard prediction model where a score function s : X ? Y ? R with parameters ?
is used to determine the prediction for a given input x via
?
y
=
arg max s(x, y).
y?Y
(1)
Here y is a configuration of assignments over a set of components (that might depend on x). Since
Y is a combinatorial set, (1) cannot usually be solved by enumeration;P
some structure required for
efficient prediction. For example, s might decompose as s(x, y) =
c?C s(x, yc ) over a set of
cliques C that form a junction tree, where yc denotes the portion of y covered by clique c. Y might
also encode constraints to aid tractability, such as y forming a consistent matching in a bipartite
graph, or a consistent parse tree [28]. The key practical requirement is that s and Y allow an efficient
solution to (1). The operation of maximizing or summing over all y ? Y is referred to as inference,
and usually involves a dynamic program tailored to the specific structure encoded by s and Y.
For supervised learning one attempts to infer a useful score function given a set of t training pairs
(x1 , y1 ), (x2 , y2 ), ..., (xt , yt ) that specify the correct output associated with each input. Conditional
random fields [19] and structured large margin training (below with margin scaling) [28, 29] can both
be expressed as optimizations over the score model parameters ? respectively:
min r(?) +
???
min r(?) +
???
t
X
log
X
i=1
t
X
i=1
exp(s? (xi , y)) ? s? (xi , yi )
(2)
y?Y
max ?(y, yi ) + s? (xi , y) ? s? (xi , yi ),
y?Y
(3)
where r(?) is a regularizer over ? ? ?. Equations (1), (2) and (3) suggest that inference over y ? Y
is required at each stage of training and testing, however we show this is not necessarily the case.
Multilabel Prediction To demonstrate how inference might be avoided, consider the special case
of multilabel prediction with label constraints. Multilabel prediction specializes the previous set up
by assuming y is a boolean assignment to a fixed set of variables, where y = (y1 , y2 , ..., y` ) and
yi ? {0, 1}, i.e. each label is assigned 1 (true) or 0 (false). As noted, an extensive literature that
2
has investigated various structural assumptions on the score function to enable tractable prediction.
For simplicity we adopt thePfactored form that has been reconsidered in recent work [8, 11] (and
originally [13]): s(x, y) = k s(x, yk ). This form allows (1) to be simplified to
X
X
? = arg max
y
s(x, yk ) = arg max
yk sk (x)
(4)
y?Y
y?Y
k
k
where sk (x) := s(x, yk = 1) ? s(x, yk = 0) gives the decision function associated with label
yk ? {0, 1}. That is, based on (4), if the constraints in Y were ignored, one would have the
relationship y?k = 1 ? sk (x) ? 0. The constraints in Y play an important role however: it has been
shown in [8] that imposing prior implications and mutual exclusions as constraints in Y yields state
of the art accuracy results for image tagging on the ILSVRC corpus. This result was achieved in [8]
by developing a novel and rather sophisticated dynamic program that can efficiently solve (4) under
these constraints. Here we show how such a dynamic program can be eliminated.
3
Embedding Label Constraints
Consider the two common forms of logical relationships between labels: implication and mutual
exclusion. For implication one would like to enforce relationships of the form y1 ? y2 , meaning
that whenever the label y1 is set to 1 (true) then the label y2 must also be set to 1 (true). For mutual
exclusion one would like to enforce relationships of the form ?y1 ? ?y2 , meaning that at least one
of the labels y1 and y2 must be set to 0 (false) (i.e., not both can be simultaneously true). These
constraints arise naturally in multilabel classification, where label sets are increasingly large and
embody semantic relationships between categories [2, 8, 32]. For example, images can be tagged
with labels ?dog?, ?cat? and ?Siamese? where ?Siamese? implies ?cat?, while ?dog? and ?cat? are
mutually exclusive (but an image could depict neither). These implication and mutual exclusion
constraints constitute the ?HEX? constraints considered in [8].
Our goal is to express the logical relationships between label assignments as constraints on the score
function that hold universally over all x ? X . In particular, using the decomposed representation
(4), the desired label relationships correspond to the following constraints
Implication y1 ? y2 :
Mutual exclusion ?y1 ? ?y2 :
s1 (x) ? ?? ? s2 (x) ? ?
s1 (x) < ?? or s2 (x) < ??
?x ? X
?x ? X
(5)
(6)
where we have introduced the additional margin quantity ? ? 0 for subsequent large margin training.
3.1
Score Model
The first key consideration is representing the score function in a manner that allows the desired
relationships to be expressed. Unfortunately, the standard linear form s(x, y) = h?, f (x, y)i cannot
allow the needed constraints to be enforced over all x ? X without further restricting the form
of the feature representation f ; a constraint we would like to avoid. More specifically, consider
a standard set up where there is a mapping f (x, yk ) that produces a feature representation for an
input-label pair (x, yk ). For clarity, we additionally make the standard assumption that the inputs
and outputs each have independent feature representations [11], hence f (x, yk ) = ?(x) ? ?k for an
input feature map ? and label feature representation ?k . In this case, a bi-linear score function has
the form sk (x) = ?(x)> A?k + b> ?(x) + c> ?k + d for parameters ? = (A, b, c, d). Unfortunately,
such a score function does not allow sk (x) ? ? (e.g., in Condition (5)) to be expressed over all
x ? X without either assuming A = 0 and b = 0, or special structure in ?.
To overcome this restriction we consider a more general scoring model that extends the standard
bi-linear form to a form that is linear in the parameters but quadratic in the feature representations:
?"
?
?
"
#> ?
#
P
A b
P
A b
?(x)
?(x)
? A> Q c ?
?k
?k
?sk (x) =
for ? = ? A> Q c ? . (7)
>
>
1
1
b
c
r
b> c> r
Here ? = ?> and sk is linear in ? for each k. The benefit of a quadratic form in the features is that
it allows constraints over x ? X to be easily imposed on label scores via convex constraints on ?.
3
Lemma 1 If ? 0 then ?sk (x) = kU ?(x) + u ? V ?k k2 for some U , V and u.
Proof: First expand (7), obtaining ?sk (x) = ?(x)> P ?(x) + 2?(x)> A?k + 2b> ?(x) + ?k> Q?k +
2c> ?k + r. Since ? 0 there must exist U , V and u such that ? = [U > , ?V > , u]> [U > , ?V > , u],
where U > U = P , U > V = ?A, U > u = b, V > V = Q, V > u = ?c, and u> u = r. A simple
substitution and rearrangement shows the claim.
The representation (7) generalizes both standard bi-linear and distance-based models. The standard
bi-linear model is achieved by P = 0 and Q = 0. By Lemma 1, the semidefinite assumption ? 0
also yields a model that has a co-embedding [24] interpretation: the feature representations ?(x)
and ?k are both mapped (linearly) into a common Euclidean space where the score is determined
by the squared distance between the embedded vectors (with an additional offset u). To aid the
presentation below we simplify this model a bit further. Set b = 0 and observe that (7) reduces to
>
P A
?(x)
?(x)
sk (x) = ?k ?
(8)
?k
?k
A> Q
where ?k = ?r ? 2c> ?k . In particular, we modify the parameterization to ? = {?k }`k=1 ? {?P AQ }
such that ?P AQ denotes the matrix of parameters in (8). Importantly, (8) remains linear in the new
parameterization. Lemma 1 can then be modified accordingly for a similar convex constraint on ?.
Lemma 2 If ?P AQ 0 then there exist U and V such that for all labels k and l
sk (x)
?k> Q?k
?
?k> Q?l
?
?l> Q?k
+
?l> Q?l
=
=
?k ? kU ?(x) ? V ?k k2
2
kV ?k ? V ?l k .
(9)
(10)
Proof: Similar to Lemma 1, since ?P AQ 0, there exist U and V such that ?P AQ =
[U > , ?V > ]> [U > , ?V > ] where U > U = P , V > V = Q and U > V = ?A. Expanding (8) and substituting gives (9). For (10) note ?k> Q?k ? ?k> Q?l ? ?l> Q?k + ?l> Q?l = (?k ? ?l )> Q(?k ? ?l ).
Expanding Q gives (?k ? ?l )> Q(?k ? ?l ) = (?k ? ?l )> V > V (?k ? ?l ) = kV ?k ? V ?l k2 .
This representation now allows us to embed the desired label relationships as simple convex constraints on the score model parameters ?.
3.2
Embedding Implication Constraints
Theorem 3 Assume the quadratic-linear score model (8) and ?P AQ 0. Then for any ? ? 0 and
? > 0, the implication constraint in (5) is implied for all x ? X by:
?1 + ? + (1 + ?)(?1> Q?1 ? ?1> Q?2 ? ?2> Q?1 + ?2> Q?2 ) ? ?2 ? ?
? 2
(?1> Q?1 ? ?1> Q?2 ? ?2> Q?1 + ?2> Q?2 ) ? ?1 + ?.
2
(11)
(12)
Proof: First, since ?P AQ 0 we have the relationship (10), which implies that there must exist
vectors ?1 = V ?1 and ?2 = V ?2 such that ?1> Q?1 ? ?1> Q?2 ? ?2> Q?1 + ?2> Q?2 = k?1 ? ?2 k2 .
Therefore, the constraints (11) and (12) can be equivalently re-expressed as
?1 + ? + (1 + ?)k?1 ? ?2 k2
? 2
k?1 ? ?2 k2
2
?
?2 ? ?
(13)
?
?1 + ?
(14)
with respect to these vectors. Next let ?(x) := U ?(x) (which exists by (9)) and observe that
k?(x) ? ?2 k2
= k?(x) ? ?1 + ?1 ? ?2 k2
= k?(x) ? ?1 k2 + k?1 ? ?2 k2 + 2h?(x) ? ?1 , ?1 ? ?2 i,
(15)
Consider two cases.
Case 1: 2h?(x) ? ?1 , ?1 ? ?2 i > ?k?1 ? ?2 k2 . In this case, by the Cauchy Schwarz inequality we
have 2k?(x) ? ?1 kk?1 ? ?2 k ? 2h?(x) ? ?1 , ?1 ? ?2 i > ?k?1 ? ?2 k2 , which implies k?(x) ? ?1 k >
?
? 2
2
k?1 ? ?2 k2 ? ?1 + ? by constraint (14). But this implies
2 k?1 ? ?2 k, hence k?(x) ? ?1 k > 2
that s1 (x) < ?? therefore it does not matter what value s2 (x) has.
4
Case 2: 2h?(x) ? ?1 , ?1 ? ?2 i ? ?k?1 ? ?2 k2 . In this case, assume that s1 (x) ? ??, i.e.
k?(x) ? ?1 k2 ? ?1 + ?, otherwise it does not matter what value s2 (x) has. Then from (15) it follows that k?(x) ? ?2 k2 ? k?(x) ? ?1 k2 + (1 + ?)k?1 ? ?2 k2 ? ?1 + ? + (1 + ?)k?1 ? ?2 k2 ? ?2 ? ?
by constraint (13). But this implies that s2 (x) ? ?, hence the implication is enforced.
3.3
Embedding Mutual Exclusion Constraints
Theorem 4 Assume the quadratic-linear score model (8) and ?P AQ 0. Then for any ? ? 0 the
mutual exclusion constraint in (6) is implied for all x ? X by:
>
1
2 (?1 Q?1
? ?1> Q?2 ? ?2> Q?1 + ?2> Q?2 ) >
?1 + ?2 + 2?.
(16)
Proof: As before, since ?P AQ 0 we have the relationship (10), which implies that there must exist
vectors ?1 = V ?1 and ?2 = V ?2 such that ?1> Q?1 ? ?1> Q?2 ? ?2> Q?1 + ?2> Q?2 = k?1 ? ?2 k2 .
Observe that the constraint (16) can then be equivalently expressed as
1
2 k?1
? ?2 k2
>
?1 + ?2 + 2?,
(17)
and observe that
k?1 ? ?2 k2
= k?1 ? ?(x) + ?(x) ? ?2 k2
= k?1 ? ?(x)k2 + k?(x) ? ?2 k2 + 2h?1 ? ?(x), ?(x) ? ?2 i
(18)
using ?(x) := U ?(x) as before (which exists by (9)). Therefore
k?(x) ? ?1 k2 + k?(x) ? ?2 k2 = k?1 ? ?2 k2 ? 2h?1 ? ?(x), ?(x) ? ?2 i
= k(?1 ??(x))+(?(x)??2 )k2 ? 2h?1 ??(x), ?(x)??2 i (19)
?
=
1
2 k(?1 ? ?(x))
2
1
2 k?1 ? ?2 k .
+ (?(x) ? ?2 )k2
(20)
(21)
(To prove the inequality (20) observe that, since 0 ? 12 ka ? bk2 , we must have ha, bi ? 12 kak2 +
1
1
1
1
2
2
2
2
2 kbk , hence 2ha, bi ? 2 kak + 2 kbk + ha, bi = 2 ka + bk , which establishes ?2ha, bi ?
1
2
? 2 ka + bk . The inequality (20) then follows simply by setting a = ?1 ? ?(x) and b = ?(x) ? ?2 .)
Now combining (21) with the constraint (17) implies that k?(x) ? ?1 k2 + k?(x) ? ?2 k2 ?
1
2
2
2
2 k?1 ? ?2 k > ?1 + ?2 + 2?, therefore one of k?(x) ? ?1 k > ?1 + ? or k?(x) ? ?2 k > ?2 + ?
must hold, hence at least one of s1 (x) < ?? or s2 (x) < ?? must hold. Therefore, the mutual
exclusion is enforced.
Importantly, once ?P AQ 0 is imposed, the other constraints in Theorems 3 and 4 are all linear in
the parameters Q and ?.
4
Properties
We now establish that the above constraints on the parameters in (8) achieve the desired properties.
In particular, we show that given the constraints, inference can be removed both from the prediction
problem (4) and from structured large margin training (3).
4.1
Prediction Equivalence
First note that the decision of whether a label yk is associated with x can be determined by
s(x, yk = 1) > s(x, yk = 0) ?
max yk sk (x) > 0 ? 1 = arg max yk sk (x).
yk ?{0,1}
yk ?{0,1}
(22)
Consider joint assignments y = (y1 , ..., yl ) ? {0, 1}l and let Y denote the set of joint assignments
that are consistent with a set of implication and mutual exclusion constraints. (It is assumed the
constraints are satisfiable; that is, Y is not the empty set.) Then the optimal joint assignment for a
Pl
given x can be specified by arg maxy?Y k=1 yk sk (x).
5
Proposition 5 If the constraint set Y imposes the constraints in (5) and (6) (and is nonempty), and
the score function s satisfies the corresponding constraints for some ? > 0, then
max
y?Y
l
X
yk sk (x)
=
k=1
l
X
k=1
max yk sk (x)
(23)
yk
Proof: First observe that
max
y?Y
l
X
yk sk (x) ? max
y
k=1
l
X
l
X
yk sk (x) =
k=1
max yk sk (x)
k=1
yk
(24)
so making local classifications for each label gives an upper bound. However, if the score function
satisfies the constraints, then the concatenation of the local label decisions y = (y1 , ..., yl ) must
be jointly feasible; that is, y ? Y. In particular, for the implication y1 ? y2 the score constraint
(5) ensures that if s1 (x) > 0 ? ?? (implying 1 = arg maxy1 y1 s1 (x)) then it must follow
that s2 (x) ? ?, hence s2 (x) > 0 (implying 1 = arg maxy2 y2 s2 (x)). Similarly, for the mutual
exclusion ?y1 ? ?y2 the score constraint (6) ensures min(s1 (x), s2 (x)) < ?? ? 0, hence if
s1 (x) > 0 ? ?? (implying 1 = arg maxy1 y1 s1 (x)) then it must follow that s2 (x) < ?? ? 0
(implying 0 = arg maxy2 y2 s2 (x)), and vice versa. Therefore, since the maximizer y of (24) is
feasible, we actually have that the leftmost term in (24) is equal to the rightmost.
Since the feasible set Y embodies non-trivial constraints over assignment vectors in (23), interchanging maximization with summation is not normally justified. However, Proposition 5 establishes that,
if the score model also satisfies its respective constraints (e.g., as established in the previous section),
then maximization and summation can be interchanged, and inference over predicted labellings can
be replaced by greedy componentwise labelling, while preserving equivalence.
4.2
Re-expressing Large Margin Structured Output Training
Given a target joint assignment over labels t = (t1 , ..., tl ) ? {0, 1}l , and using the score model (8),
the standard structured output large margin training loss (3) can then be written as
X
max ?(y, ti ) +
y?Y
i
l
X
s(xi , yk ) ? s(xi , tik )
=
X
i
k=1
max ?(y, ti ) +
y?Y
l
X
(yk ? tik )sk (xi ), (25)
k=1
using the simplified score function representation such that tik denotes the k-th label of the i-th
training example. If we furthermore make the standard assumption that ?(y, ti ) decomposes as
Pl
?(y, ti ) = k=1 ?k (yk , tik ), the loss can be simplified to
X
i
max
y?Y
l
X
?k (yk , tik ) + (yk ? tik )sk (xi ).
(26)
k=1
Note also that since yk ? {0, 1} and tik ? {0, 1} the margin functions ?k typically have the form
?k (0, 0) = ?k (1, 1) = 0 and ?k (0, 1) = ?k01 and ?k (1, 0) = ?k10 for constants ?k01 and ?k10 , which
for simplicity we will assume are equal, ?k01 = ?k10 = ? for all k (although label specific margins
might be possible). This is the same ? used in the constraints (5) and (6).
The difficulty in computing this loss is that it apparently requires an exponential search over y. When
this exponential search can be avoided, it is normally avoided by developing a dynamic program.
Instead, we can now see that the search over y can be eliminated.
Proposition 6 If the score function s satisfies the constraints in (5) and (6) for ? > 0, then
X
i
max
y?Y
l
X
?(yk , tik ) + (yk ? tik )sk (xi )
=
l
XX
i
k=1
6
k=1
max ?(yk , tik ) + (yk ? tik )sk (xi ).(27)
yk
Proof: For a given x and t ? Y, let fk (y) = ?(y, tk ) + (y ? tk )sk (x), hence yk =
arg maxy?{0,1} fk (y). It is easy to show that
1 ? arg max fk (y) ??
y?{0,1}
sk (x) ? tk ? ? (1 ? tk )?,
(28)
which can be verified by checking the two cases, tk = 0 and tk = 1. When tk = 0 we have fk (0) =
0 and fk (1) = ? + s(x), therefore 1 = yk ? arg maxy?{0,1} fk (y) iff ? + s(x) ? 0. Similarly,
when tk = 1 we have fk (0) = ? ? s(x) and fk (1) = 0, therefore 1 = yk ? arg maxy?{0,1} fk (y)
iff ? ? s(x) ? 0. Combining these two conditions yields (28).
Next, we verify that if the score constraints hold, then the logical constraints over y are automatically
satisfied even by locally assigning yk , which implies the optimal joint assignment is feasible, i.e.
y ? Y, establishing the claim. In particular, for the implication y1 ? y2 , it is assumed that t1 ? t2
in the target labeling and also that score constraints hold, ensuring s1 (x) ? ?? ? s2 (x) ? ?.
Consider the cases over possible assignments to t1 and t2 :
If t1 = 0 and t2 = 0 then y1 = 1 ? f1 (1) ? f1 (0) ? ? + s1 (x) ? 0 ? s1 (x) ? ?? ? s2 (x) ? ?
(by assumption) ? s2 (x) ? ?? ? ? + s2 (x) ? 0 ? f2 (1) ? f2 (0) ? y2 = 1.
If t1 = 0 and t2 = 1 then y1 = 1 ? f1 (1) ? f1 (0) ? ? + s1 (x) ? 0 ? s1 (x) ? ?? ? s2 (x) ? ?
(by assumption) ? 0 ? ? ? s2 (x) ? f2 (1) ? f2 (0) ? y2 = 1 (tight case).
The case t1 = 1 and t2 = 0 cannot happen by the assumption that t ? Y.
If t1 = 1 and t2 = 1 then y1 = 1 ? f1 (1) ? f1 (0) ? 0 ? ? ? s1 (x) ? s1 (x) ? ?? ? s2 (x) ? ?
(by assumption) ? 0 ? ? ? s2 (x) ? f2 (1) ? f2 (0) ? y2 = 1.
Similarly, for the mutual exclusion ?y1 ? ?y2 , it is assumed that ?t1 ? ?t2 in the target labeling
and also that the score constraints hold, ensuring min(s1 (x), s2 (x)) < ??. Consider the cases over
possible assignments to t1 and t2 :
If t1 = 0 and t2 = 0 then y1 = 1 and y2 = 1 implies that s1 (x) ? ?? and s2 (x) ? ??, which
contradicts the constraint that min(s1 (x), s2 (x)) < ?? (tight case).
If t1 = 0 and t2 = 1 then y1 = 1 and y2 = 1 implies that s1 (x) ? ?? and s2 (x) ? ?, which
contradicts the same constraint.
If t1 = 1 and t2 = 0 then y1 = 1 and y2 = 1 implies that s1 (x) ? ? and s2 (x) ? ??, which again
contradicts the same constraint.
The case t1 = 1 and t2 = 1 cannot happen by the assumption that t ? Y.
Therefore, since the concatenation, y, of the independent maximizers of (27) is feasible, i.e. y ? Y,
we have that the rightmost term in (27) equals the leftmost.
Similar to Section 4.1, Proposition 6 demonstrates that if the constraints (5) and (6) are satisfied
by the score model s, then structured large margin training (3) reduces to independent labelwise
training under the standard hinge loss, while preserving equivalence.
5
Efficient Implementation
Even though Section 3 achieves the primary goal of demonstrating how desired label relationships
can be embedded as convex constraints on score model parameters, the linear-quadratic representation (8) unfortunately does not allow convenient scaling: the number of parameters in ?P AQ (8)
is n+`
(accounting for symmetry), which is quadratic in the number of features, n, in ? and the
2
number of labels, `. Such a large optimization variable is not practical for most applications, where
n and ` can be quite large. The semidefinite constraint ?P AQ 0 can also be costly in practice.
Therefore, to obtain scalable training we require some further refinement.
In our experiments below we obtained a scalable training procudure by exploiting trace norm regularization on ?P AQ to reduce its rank. The key benefit of trace norm regularization is that efficient solution methods exist that work with a low rank factorization of the matrix variable while
automatically ensuring positive semidefiniteness and still guaranteeing global optimality [10, 14].
Therefore, we conducted the main optimization in terms of a smaller matrix variable B such that
BB > = ?P AQ . Second, to cope with the constraints, we employed an augmented Lagrangian
method that increasingly penalizes constraint violations, but otherwise allows simple unconstrained
optimization. All optimizations for smooth problems were performed using LBFGS and nonsmooth
problems were solved using a bundle method [23].
7
Dataset
Enron
WIPO
Reuters
Features
1001
74435
47235
Labels
57
183
103
Depth
4
5
5
# Training
988
1352
3000
# Testing
660
358
3000
Reference
[18]
[25]
[20]
Table 1: Data set properties
% test error
unconstrained
constrained
inference
Enron
12.4
9.8
6.8
WIPO
21.0
2.6
2.7
Reuters
27.1
4.0
29.3
test time (s)
unconstrained
constrained
inference
Enron
0.054
0.054
0.481
WIPO
0.070
0.070
0.389
Reuters
0.60
0.60
5.20
Table 2: (left) test set prediction error (percent); (right) test set prediction time (s)
6
Experimental Evaluation
To evaluate the proposed approach we conducted experiments on multilabel text classification data
that has a natural hierarchy defined over the label set. In particular, we investigated three multilabel text classification data sets, Enron, WIPO and Reuters, obtained from https://sites.
google.com/site/hrsvmproject/datasets-hier (see Table 1 for details). Some preprocessing was performed on the label relations to ensure consistency with our assumptions. In
particular, all implications were added to each instance to ensure consistency with the hierarchy,
while mutual exclusions were defined between siblings whenever this did not create a contradiction.
We conducted experiments to compare the effects of replacing inference with the constraints outlined
in Section 3, using the score model (8). For comparison, we trained using the structured large margin
formulation (3), and trained under a multilabel prediction loss without inference, but both including
then excluding the constraints. For the multilabel training loss we used the smoothed calibrated
separation ranking loss proposed in [24]. In each case, the regularization parameter was simply set
to 1. For inference, we implemented the inference algorithm outlined in [8].
The results are given in Table 2, showing both the test set prediction error (using labelwise prediction
error, i.e. Hamming loss) and the test prediction times. As expected, one can see benefits from
incorporating known relationships between the labels when training a predictor. In each case, the
addition of constraints leads to a significant improvement in test prediction error, versus training
without any constraints or inference added. Training with inference (i.e., classical structured large
margin training) still proves to be an effective training method overall, in one case improving the
results over the constrained approach, but in two other cases falling behind. The key difference
between the approach using constraints versus that using inference is in terms of the time it takes
to produce predictions on test examples. Using inference to make test set predictions clearly takes
significantly longer than applying labelwise predictions from either a constrained or unconstrained
model, as shown in the right subtable of Table 2.
7
Conclusion
We have demonstrated a novel approach to structured multilabel prediction where inference is replaced with constraints on the score model. On multilabel text classification data, the proposed
approach does appear to be able to achieve competitive generalization results, while reducing the
time needed to make predictions at test time. In cases where logical relationships are known to
hold between the labels, using either inference or imposing constraints on the score model appear to
yield benefits over generic training approaches that ignore the prior knowledge. For future work we
are investigating extensions of the proposed approach to more general structured output settings, by
combining the method with search based prediction methods. Other interesting questions include
exploiting learned label relations and coping with missing labels.
8
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encoding. In Proceedings of the International Conference on Machine Learning (ICML), 2014.
[23] M. M?akel?a. Multiobjective proximal bundle method for nonconvex nonsmooth optimization: Fortran
subroutine MPBNGC 2.0. Technical report, U. of Jyv?askyk?a, 2003.
[24] F. Mirzazadeh, Y. Guo, and D. Schuurmans. Convex co-embedding. In Proceedings AAAI, 2014.
[25] J. Rousu, C. Saunders, S. Szedmak, and J. Shawe-Taylor. Kernel-based learning of hierarchical multilabel
classification models. Journal of Machine Learning Research, 7:1601?1626, 2006.
[26] V. Srikumar and C. Manning. Learning distributed representations for structured output prediction. In
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[27] X. Sun. Structure regularization for structured prediction. In Proceedings NIPS, 2014.
[28] B. Taskar. Learning structured prediction models: A large margin approach. PhD thesis, Stanford, 2004.
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[30] G. Tsoumakas, I. Katakis, and I. Vlahavas. Mining multi-label data. In Data Mining and Knowledge
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[31] K. Weinberger and O. Chapelle. Large margin taxonomy embedding for document categorization. In
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9
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5,165 | 5,676 | Tractable Learning for Complex Probability Queries
Jessa Bekker, Jesse Davis
KU Leuven, Belgium
{jessa.bekker,jesse.davis}@cs.kuleuven.be
Arthur Choi, Adnan Darwiche, Guy Van den Broeck
University of California, Los Angeles
{aychoi,darwiche,guyvdb}@cs.ucla.edu
Abstract
Tractable learning aims to learn probabilistic models where inference is guaranteed to be efficient. However, the particular class of queries that is tractable depends on the model and underlying representation. Usually this class is MPE
or conditional probabilities Pr(x|y) for joint assignments x, y. We propose a
tractable learner that guarantees efficient inference for a broader class of queries.
It simultaneously learns a Markov network and its tractable circuit representation,
in order to guarantee and measure tractability. Our approach differs from earlier
work by using Sentential Decision Diagrams (SDD) as the tractable language instead of Arithmetic Circuits (AC). SDDs have desirable properties, which more
general representations such as ACs lack, that enable basic primitives for Boolean
circuit compilation. This allows us to support a broader class of complex probability queries, including counting, threshold, and parity, in polytime.
1
Introduction
Tractable learning [1] is a promising new machine learning paradigm that focuses on learning probability distributions that support efficient querying. It is motivated by the observation that while
classical algorithms for learning Bayesian and Markov networks excel at fitting data, they ignore the
cost of reasoning with the learned model. However, many applications, such as health-monitoring
systems, require efficient and (guaranteed) accurate reasoning capabilities. Hence, new learning
techniques are needed to support applications with these requirements.
Initially, tractable learning focused on the first model class recognized to be tractable: low-treewidth
graphical models [2?5]. Recent advances in probabilistic inference exploit other properties of a
model, including local structure [6] and exchangeability [7], which even scale to models that have
high treewidth. In particular, the discovery of local structure led to arithmetic circuits (ACs) [8],
which are a much more powerful representation of tractable probability distributions. In turn, this
led to new tractable learners that targeted ACs to guarantee efficient inference [9, 10]. In this context, ACs with latent variables are sometimes called sum-product networks (SPNs) [11, 12]. Other
tractable learners target exchangeable models [13, 14] or determinantal point processes [15].
There is a trade-off in tractable learning that is poorly understood and often ignored: tractability
is not absolute, and always relative to a class of queries that the user is interested in. Existing approaches define tractability as the ability to efficiently compute most-probable explanations (MPE)
or conditional probabilities Pr(x|y) where x, y are joint assignments to subsets of random variables.
While these queries are indeed efficient on ACs, many other queries of interest are not. For example,
computing partial MAP remains NP-hard on low-treewidth and AC models [16]. Similarly, various
1
decision [17, 18], monotonicity [19], and utility [20] queries remain (co-)NP-hard.1 Perhaps the
simplest query beyond the reach of tractable AC learners is for probabilities Pr(?|?), where ?, ?
are complex properties, such as counts, thresholds, comparison, and parity of sets of random variables. These properties naturally appear throughout the machine learning literature, for example, in
neural nets [21], and in exchangeable [13] and statistical relational models [22]. We believe they
have not been used to their full potential in the graphical models? world due to their intractability.
We call these types of queries complex probability queries.
This paper pushes the boundaries of tractable learning by supporting more queries efficiently. While
we currently lack any representation tractable for partial MAP, we do have all the machinery available to learn tractable models for complex probability queries. Their tractability is enabled by the
weighted model counting (WMC) [6] encoding of graphical models and recent advances in compilation of Boolean functions into Sentential Decision Diagrams (SDDs) [23]. SDDs can be seen as
a syntactic subset of ACs with more desirable properties, including the ability to (1) incrementally
compile a Markov network, via a conjoin operator, (2) dynamically minimize the size and complexity of the representation, and (3) efficiently perform complex probability queries.
Our first contribution is a tractable learning algorithm for Markov networks with compact SDDs,
following the outer loop of the successful ACMN learner [9] for ACs, that uses SDD primitives to
modify the circuit during the Markov network structure search. Support for the complex queries
listed above also means that these properties can be employed as features in the learned network.
Second, we prove that complex symmetric probability queries over n variables, as well as their
extensions, run in time polynomial in n and linear in the size of the learned SDD. Tighter complexity
bounds are obtained for specific classes of queries. Finally, we illustrate these tractability properties
in an empirical evaluation on four real-world data sets and four types of complex queries.
2
Background
2.1
Markov Networks
A Markov network or Markov random field compactly represents the joint distribution over a set of
variables X = (X1 , X2 , . . . , Xn ) [24]. Markov networks are often represented as log-linear models,
that is,P
an exponentiated weighted sum of features of the state x of variables X: Pr(X = x) =
1
j wj fj (x). The fj (X) are real-valued functions of the state, wj is the weight associated
Z exp
with fj , and Z is the partition function. For discrete models, features are often Boolean functions;
typically a conjunction of tests of the form (Xi = xi ) ? ? ? ? ? (Xj = xj ). One is interested in
performing certain inference tasks, such as computing the posterior marginals or most-likely state
(MPE) given observations. In general, such tasks are intractable (#P- and NP-hard).
Learning Markov networks from data require estimating the weights of the features (parameter
learning), and the features themselves (structure learning). We can learn the parameters by optimizing some convex objective function, which is typically the log-likelihood. Evaluation of this
function and its gradient is in general intractable (#P-complete). Therefore, it is common to optimize an approximate objective, such as the pseudo-log-likelihood. The classical structure learning
approach [24] is a greedy, top-down search. It starts with features over individual variables, and
greedily searches for new features to add to the model from a set of candidate features, found by
conjoining pairs of existing features. Other approaches convert local models into a global one [25].
To prevent overfitting, one puts a penalty on the complexity of the model (e.g., number of features).
2.2
Tractable Circuit Representations and Tractable Learning
Tractable circuit representations overcome the intractability of inference in Markov networks. Although we are not always guaranteed to find a compact tractable representation for every Markov
network, in this paper we will guarantee their existence for the learned models.
AC Arithmetic Circuits (ACs) [8] are directed acyclic graphs whose leafs are inputs representing either indicator variables (to assign values to random variables), parameters (weights wj ) or constants.
Figure 1c shows an example. ACs encode the partition function computation of a Markov network.
1
The literature typically shows hardness for polytrees. Results carry over because these have compact ACs.
2
+
?
?
IA
I?A
+
?
A
weight feature
w1
A?B
w2
?A ? ?B
IB
B
parameter
variable
P1
P2
+
?
?
I?B
IB
+
?
e w1 1
+
?
1 0
?
1
+
?
1
?
I?B
e w1 0
?
?
1 e w2 e w1 0
(a) Markov Network
(b) Sentential Decision Diagram
(c) Arithmetic Circuit
Figure 1: A Markov network over variables A, B, and its tractable SDD and AC representations.
By setting indicators to 1 and evaluating the AC bottom-up, the value of the partition function, Z, is
obtained at the root. Other settings of the indicators encode arbitrary evidence. Moreover, a second,
top-down pass yields all single-variable marginal probabilities; similar procedures exist for MPE.
All these algorithms run in time linear in the size of the AC (number of edges). The tractable learning paradigm for Markov networks is best exemplified by ACMN [9], which concurrently learns a
Markov network and its AC. It employs a complexity penalty based on the inference cost. Moreover,
ACMN efficiently computes the exact log-likelihood (as opposed to pseudo-log-likelihood) and its
gradient on the AC. ACMN uses the standard greedy top-down feature search outlined above.
SDD Sentential Decision Diagrams (SDDs) are a tractable representation of sentences in propositional logic [23]. The supplement2 reviews SDDs in detail; a brief summary is next. SDDs are
directed acyclic graphs, as depicted in Figure 1b. A circle node represents the disjunction of its
children. A pair of boxes denotes the conjunction of the two boxes, and each box can be a (negated)
Boolean variable or a reference to another SDD node. The detailed properties of SDDs yield two
benefits. First, SDDs support an efficient conjoin operator that can incrementally construct new
SDDs from smaller SDDs in linear time. Second, SDDs support dynamic minimization, which
allows us to control the growth of an SDD during incremental construction.
There is a close connection between SDDs for logic and ACs for graphical models, through an
intermediate weighted model counting formulation [6], which is reviewed in the supplement. Given
a graphical model M , one can construct a logical sentence ? whose satisfying assignments are in
one-to-one correspondence with the possible worlds of M . Moreover, each satisfying assignment of
? encodes the weights wj that apply to its possible world in M . For each feature fj of M , this ?
includes a constraint fj ? Pj , meaning that weight wj applies when ?parameter? variable Pj is true;
see Figure 1a. A consequence of this correspondence is that, given an SDD for ?, we can efficiently
construct an AC for the original Markov network M ; see Figure 1. Hence, an SDD corresponding
to M is a tractable representation of M . Different from ACs, SDDs have the following properties:
support for efficient (linear) conjunction allows us to add new features fj and incrementally learn a
Markov network. Moreover, dynamic minimization lets us systematically search for more compact
circuits for the same Markov network, mitigating the increasing complexity of inference as we learn
more features. Such operations are not available for ACs in general.
3
Learning Algorithm
We propose LearnSDD, which employs a greedy, general-to-specific search that simultaneously
learns a Markov network and its underlying SDD which is used for inference. The cost of inference
in the learned model is dictated by the size of its SDD. Conceptually, our approach is similar to
ACMN [9] with the key differences being our use of SDDs instead of ACs, which gives us more
tractability and freedom in the types of features that are considered.
2
https://dtai.cs.kuleuven.be/software/learnsdd
3
Algorithm 1 LearnSDD(T, e, ?)
initialize model M with variables as features
Mbest ? M
while number of edges |SDD M | < e and not timeout
best score = ??
F ? generateFeatures(M, T )
for each feature f in F do
M 0 ? M .add(f )
if score(M 0 , T, ?) > best score
best score = score(M 0 , T, ?)
Mbest ? M 0
M ? Mbest
LearnSDD, outlined in Algorithm 1, receives as input a training set T , a maximum number of edges
e, and a parameter ? to control the relative importance of fitting the data vs. the cost of inference. As
is typical with top-down approaches to structure learning [24], the initial model has one feature Xi =
true for each variable, which corresponds to a fully-factorized Markov network. Next, LearnSDD
iteratively constructs a set of candidate features, where each feature is a logical formula. It scores
each feature by compiling it into an SDD, conjoining the feature to the current model temporarily,
and then evaluating the score of the model that includes the feature. The supplement shows how
a features is added to an SDD. In each iteration, the highest scoring feature is selected and added
to the model permanently. The process terminates when the maximum number of edges is reached
or when it runs out of time. Inference time is dictated by the size of the learned SDD. To control
this cost, we invoke dynamic SDD minimization each time a feature is evaluated, and when we
permanently add a feature to the model.
Performing structure learning with SDDs offers advantages over ACs. First, SDDs support a practical conjoin operation, which greatly simplifies the design of a top-down structure learning algorithm
(ACMN instead relies on a complex special-purpose AC modification algorithm). Second, SDDs
support dynamic minimization, allowing us to search for smaller SDDs, as needed. The following
two sections discuss the score function and feature generation in greater detail.
3.1
Score Function and Weight Learning
Score functions capture a trade-off between fitting the training data and the preference for simpler
models, captured by a regularization term. In tractable learning, the regularization term reflects the
cost of inference in the model. Therefore, we use the following score function:
score(M 0 , T ) = [log Pr(T |M 0 ) ? log Pr(T |M )] ? ? [|SDD M 0 | ? |SDD M |] /|SDD M |
(1)
0
where T is the training data, M is the model extended with feature f , M is the old model, |SDD . |
returns the number of edges in the SDD representation, and ? is a user-defined parameter. The first
term is the improvement in the model?s log-likelihood due to incorporating f . The second term
measures the relative growth of the SDD representation after incorporating f . We use the relative
growth because adding a feature to a larger model adds many more edges than adding a feature to
a smaller model. Section 4 shows that any query?s inference complexity depends on the SDD size.
Finally, ? lets us control the trade-off between fitting the data and the cost of inference.
Scoring a model requires learning the weights associated with each feature. Because we use SDDs,
we can efficiently compute the exact log-likelihood and its gradient using only two passes over the
SDD. Therefore, we learn maximum-likelihood estimates of the weights.
3.2
Generating Features
In each iteration, LearnSDD constructs a set of candidate features using two different feature generators: conjunctive and mutex. The conjunctive generator considers each pair of features f1 , f2 in
the model and proposes four new candidates per pair: f1 ? f2 , ?f1 ? f2 ,f1 ? ?f2 and ?f1 ? ?f2 .
The mutex generator automatically identifies mutually exclusive sets of variables in the data and
proposes a feature to capture this relationship. Mutual exclusivity arises naturally in data. It occurs in tractable learning because existing approaches typically assume Boolean data. Hence,
4
any multi-valued attribute is converted into multiple binary variables. For all variable sets X =
{X1 ,WX2 , ? ? ? , XV
n } that have exactly one ?true? value in each training example, the exactly one fean
ture i=1 (Xi ? j6=i ?Xj ) is added to the candidate set. When at most one variable is ?true?, the
Wn
V
Vn
mutual exclusivity feature i=1 (Xi ? j6=i ?Xj ) ? j=1 ?Xj is added to the candidate set.
4
Complex Queries
Tractable learning focuses on learning models that can efficiently compute the probability of a query
given some evidence, where both the query and evidence are conjunctions of literals. However, many
other important and interesting queries do not conform to this structure, including the following:
? Consider the task of predicting the probability that a legislative bill will pass given that
some lawmakers have already announced how they will vote. Answering this query requires estimating the probability that a count exceeds a given threshold.
? Imagine only observing the first couple of sentences of a long review, and wanting to assess
the probability that the entire document has more positive words than negative words in
it, which could serve as proxy for how positive (negative) the review is. Answering this
requires comparing two groups, in this case positive words and negative words.
Table 1 lists these and other examples of what we call complex queries, which are logical functions
that cannot be written as a conjunction of literals. Unfortunately, tractable models based on ACs
are, in general, unable to answer these types of queries efficiently. We show that using a model
with an SDD as the target tractable representation can permit efficient exact inference for certain
classes of complex queries: symmetric queries and their generalizations. No known algorithm exists
for efficiently answering these types of queries in ACs. For other classes of complex queries, the
complexity is never worse than for ACs, and in many cases SDDs will be more efficient. Note that
SPNs have the same complexity for answering queries as ACs since they are interchangeable [12].
We first discuss how to answer complex queries using ACs and SDDs. We then discuss some classes
of complex queries and when we can guarantee tractable inference in SDDs.
4.1
Answering Complex Queries
Currently, it is only known how to solve conjunctive queries in ACs. Therefore, we will answer
complex queries by asking multiple conjunctive queries. We convert the query into DNF format
W
C consisting of n mutually exclusive clauses C = {C
n }. Now, the probability of the
W1 , . . . , CP
n
query is the sum of the probabilities of the clauses: Pr ( C) = i=1 Pr(Ci ). In the worst case,
m
this construction requires 2 clauses for queries over m variables. The inference complexity for
each clause on the AC is O(|AC |). Hence, the total inference complexity is O(2m ? |AC |).
SDDs can answer complex queries without transforming them into mutually exclusive clauses. Instead, the query Q can directly be conjoined with the weighted model counting formulation ? of
the Markov network M . Given an SDD Sm for the Markov network and an SDD Sq for Q, we
can efficiently compile an SDD Sa for Q ? ?. From Sa , we can compute the partition function
of the Markov network after asserting Q, which gives us the probability of Q. This computation is
performed efficiently on the AC that corresponds to Sa (cf. Section 2.2). The supplement explains
the protocol for answering a query. The size of the SDD Sa is at most |Sq | ? |Sm | [23], and inference
is linear in the circuit size, therefore it is O(|Sq | ? |Sm |). When converting an arbitrary query into
SDD, the size may grow as large as 2m , with m the number of variables in the query. But often
it will be much smaller (see Section 4.2). Thus, the overall complexity is O(2m ? |Sm |), but often
much better, depending on the query class.
4.2
Classes of Complex Queries
A first class of tractable queries are symmetric Boolean functions. These queries do not depend on
the exact input values, but only on how many of them are true.
Definition 1. A Boolean function f (X1 , . . . , Xn ) : {0, 1}n ? {0, 1} is a symmetric query precisely
when f (X1 , . . . , Xn ) = f (X?(1) , . . . , X?(n) ) for every permutation ? of the n indexes.
5
Table 1: Examples of complex queries, with m the SDD size and n the number of query variables.
Query class
Symmetric
Query
Asymmetric
Tractable
Query
Query Type
Parity
k-Threshold
Exactly-k
Modulo-k
Exactly-k
Hamming distance k
Group comparison
Inference Complexity
O(mn)
O(mnk2 )
O(mnk2 )
O(mnk)
O(mnk2 )
O(mnk2 )
O(mn3 )
Example
#(A, B, C)%2 = 0
#(A, B, C) > 1
#(A, B, C) = 2
#(A, B, C)%3 = 0
#(A, B, ?C) = 2
#(A, B, ?C) ? 2
#(A, B, ?C) > #(D, ?E)
Table 1 lists examples of functions that can always be answered in polytime because they have
a compact SDD. Note that the features generated by the mutex generator are types of exactly-k
queries where k = 1, and therefore have a compact SDD. We have the following result.
Theorem 1. Markov networks with compact SDDs support tractable querying of symmetric functions. More specifically, let M be a Markov network with an SDD of size m, and let Q be any
symmetric function of n variables. Then, PrM (Q) can be computed in O(mn3 ) time. Moreover,
when Q is a parity function, querying takes O(mn) time, and when Q is a k-threshold or exactly-k
function, querying takes O(mnk 2 ) time.
The proof shows that any SDD can be conjoined with these queries without increasing the SDD
size by more than a factor polynomial in n. The proof of Theorem 1 is given in the supplement.
This tractability result can be extended to certain non-symmetric functions. For example, negating
the inputs to a symmetric functions still yields a tractable complex query. This allows queries for
the probability that the state is within a given Hamming distance from a desired state. Moreover,
Boolean combinations of a bounded number of tractable function also admit efficient querying. This
allows queries that compare symmetric properties of different groups of variables.
We cannot guarantee tractability for other classes of complex queries, because some queries do not
have a compact SDD representation. An example of such a query is the weighed k?threshold where
each literal has a corresponding weight and the total weight of true literals must be bigger than some
threshold. While the worst-case complexity of using SDDs and ACs to answer such queries is the
same, we show in the supplement that SDDs can still be more efficient in practice.
5
Empirical Evaluation
The goal of this section is to evaluate the merits of using SDDs as a target representation in tractable
learning for complex queries. Specifically, we want to address the following questions:
Q1 Does capturing mutual exclusivity allow LearnSDD to learn more accurate models than ACMN?
Q2 Do SDDs produced by LearnSDD answer complex queries faster than ACs learned by ACMN?
To resolve these questions, we run LearnSDD and ACMN on real-world data and compare their
performance. Our LearnSDD implementation builds on the publicly available SDD package.3
5.1
Data
Table 2 describes the characteristics of each data set.
Table 2: Data Set Characteristics
Data Set
Traffic
Temperature
Voting
Movies
3
Train Set Size
3,311
13,541
1,214
1,600
Tune Set Size
441
1,805
200
150
http://reasoning.cs.ucla.edu/sdd/
6
Test Set Size
662
2,708
350
250
Num. Vars.
128
216
1,359
1000
Mutex features We used the Traffic and Temperature data sets [5] to evaluate the benefit of detecting mutual exclusivity. In the initial version of these data sets, each variable had four values, which
were binarized using a 1-of-n encoding.
Complex queries To evaluate complex queries, we used voting data from GovTrac.us and Pang
and Lee?s Movie Review data set.4 The voting data contains all 1764 votes in the House of Representatives from the 110th Congress. Each bill is an example and the variables are the votes of the
453 congressmen, which can be yes, no, or present. The movie review data contains 1000 positive
and 1000 negative movie reviews. We first applied the Porter stemmer and then used the Scikit
Learn CountVectorizer,5 which counts all 1- and 2-grams, while omitting the standard Scikit Learn
stop words. We selected the 1000 most frequent n-grams in the training data to serve as the features.
5.2
Methodology
For all data sets, we divided the data into a single train, tune, and test partition. All experiments
were run on identically configured machines with 128GB RAM and twelve 2.4GHz cores.
Mutex features Using the training set, we learned models with both LearnSDD and ACMN. For
LearnSDD, we tried setting ? to 1.0, 0.1, 0.01 and 0.001. For ACMN, we did a grid search for
the hyper-parameters (per-split penalty ps and the L1 and L2-norm weights l1 and l2) with ps ?
{2, 5, 10}, l1 ? {0.1, 1, 5} and l2 ? {0.1, 0.5, 1}. For both methods, we stopped learning if the
circuit exceeded two million edges or the algorithm ran for 72 hours. For each approach, we picked
the best learned model according to the tuning set log-likelihood. We evaluated the quality of the
selected model using the log-likelihood on the test set.
Complex queries In this experiment, the goal is to compare the time needed to answer a query in
models learned by LearnSDD and ACMN. In both SDDs and ACs, inference time depends linearly
on the number of edges in the circuit. Therefore, to ensure a fair comparison, the learned models
should have approximately the same number of edges. Hence, we first learned an SDD and then
used the number of edges in the learned SDD to limit the size of the model learned by ACMN.
In the voting data set, we evaluated the threshold query: what is the probability that at least 50%
of the congressmen vote ?yes? on a bill, given as evidence that some lawmakers have already
announced their vote? We vary the percentage of unknown votes from 1 to 100% in intervals
of 1% point. We evaluated several queries on the movie data set. The first two queries mimic
an active sensing setting to predict features of the review without reading it entirely. The evidence for each query are the features that appear in the first 750 characters of the stemmed review. On average, the stemmed reviews have approximately 3,600 characters. The first query is
Pr(#(positive ngrams) > 5) and second is Pr(#(positive ngrams) > #(negative ngrams)),
which correspond to a threshold query and a group comparison query, respectively. For both queries,
we varied the size of the positive and negative ngram sets from 5 to 100 ngrams with an increment
size of 1. We randomly selected which ngrams are positive and negative as we are only interested
in a query?s evaluation time. The third query is the probability that a parity function over a set of
features is even. We vary the number of variables considered by the parity function from 5 to 100.
For each query, we report the average per example inference time for each learned model on the
test set. We impose a 10 minute average time limit and 100 minutes individual time limit for each
query. For completeness, the supplement reports run times for queries that are guaranteed to (not)
be tractable for both ACs and SDDs as well as the conditional log-likelihoods of all queries.
5.3
Results and Discussion
Mutex features Figure 2 shows the test set log-likelihoods as a function of the size of the learned
model. In both data sets, LearnSDD produces smaller models that have the same accuracy as AC.
This is because it can add mutex features without the need to add other features that are needed as
building blocks but are redundant afterwards. These results allow us to affirmatively answer (Q1).
Complex queries Figure 3 shows the inference times for complex queries that are extensions of
symmetric queries. For all queries, we see that LearnSDD?s model results in significantly faster
inference times than ACMN?s model. In fact, ACMN?s model exceeds the ten minute time limit on
4
5
http://www.cs.cornell.edu/people/pabo/movie-review-data/
http://tartarus.org/martin/PorterStemmer/ and http://scikit-learn.org/
7
Log-likelihood
Log-likelihood
-40
-50
-60
LearnSDD
ACMN
-70
-80
0
-20
-25
-30
LearnSDD
ACMN
-35
-40
500000
0
500000
Size
1e+06
Size
(a) Temperature
(b) Traffic
Figure 2: The size and log-likelihood of the models learned by LearnSDD and ACMN. Ideally, the model is
small with high accuracy (upper left corner), which is best approached by the LearnSDD models.
334 out of 388 of the query settings whereas this only happens in 25 settings for LearnSDD. The
SDD can answer all parity questions and positive word queries in less than three hundred milliseconds and the group comparison in less than three seconds. It can answer the voting query with up to
75% of the votes unknown in less than ten minutes. These results demonstrate LearnSDD?s superior
ability to answer complex queries compared to ACMN and allow us to positively answer (Q2).
Timeout
600
500
400
300
200
100
0
Timeout
600
500
400
300
200
100
0
0
20
40
60
% Unknown votes
SDD
AC
Time (s)
Time (s)
SDD
AC
80
100
0
(a) Threshold query (Voting)
Time (s)
SDD
AC
0
40
60
80
#positive words ? 5
100
(b) Threshold query (Movie)
Timeout
600
500
400
300
200
100
0
SDD
AC
Time (s)
Timeout
600
500
400
300
200
100
0
20
20
40
60
80
100
#positive words ? #negative words
0
(c) Group comparison (Movie)
20
40
60
# variables
80
100
(d) Parity (Movie)
Figure 3: The time for SDDs vs. ACs to answer complex queries, varying the number of query variables.
SDDs need less time in all settings, answering nearly all queries. ACs timeout in more than 85% of the cases.
6
Conclusions
This paper highlighted the fact that tractable learning approaches learn models for only a restricted
classes of queries, primarily focusing on the efficient computation of conditional probabilities. We
focused on enabling efficient inference for complex queries. To achieve this, we proposed using
SDDs as the target representation for tractable learning. We provided an algorithm for simultaneously learning a Markov network and its SDD representation. We proved that SDDs support polytime inference for complex symmetric probability queries. Empirically, SDDs enable significantly
faster inference times than ACs for multiple complex queries. Probabilistic SDDs are a closely related representation: they also support complex queries (in structured probability spaces) [26, 27],
but they lack general-purpose structure learning algorithms (a subject of future work).
Acknowledgments
We thank Songbai Yan for prior collaborations on related projects. JB is supported by IWT
(SB/141744). JD is partially supported by the Research Fund KU Leuven (OT/11/051, C22/15/015),
EU FP7 Marie Curie CIG (#294068), IWT (SBO-HYMOP) and FWO-Vlaanderen (G.0356.12). AC
and AD are partially supported by NSF (#IIS-1514253) and ONR (#N00014-12-1-0423).
8
References
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9
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5,166 | 5,677 | Double or Nothing: Multiplicative
Incentive Mechanisms for Crowdsourcing
Nihar B. Shah
University of California, Berkeley
[email protected]
Dengyong Zhou
Microsoft Research
[email protected]
Abstract
Crowdsourcing has gained immense popularity in machine learning applications
for obtaining large amounts of labeled data. Crowdsourcing is cheap and fast, but
suffers from the problem of low-quality data. To address this fundamental challenge in crowdsourcing, we propose a simple payment mechanism to incentivize
workers to answer only the questions that they are sure of and skip the rest. We
show that surprisingly, under a mild and natural ?no-free-lunch? requirement, this
mechanism is the one and only incentive-compatible payment mechanism possible. We also show that among all possible incentive-compatible mechanisms
(that may or may not satisfy no-free-lunch), our mechanism makes the smallest possible payment to spammers. Interestingly, this unique mechanism takes a
?multiplicative? form. The simplicity of the mechanism is an added benefit. In
preliminary experiments involving over several hundred workers, we observe a
significant reduction in the error rates under our unique mechanism for the same
or lower monetary expenditure.
1
Introduction
Complex machine learning tools such as deep learning are gaining increasing popularity and are
being applied to a wide variety of problems. These tools, however, require large amounts of labeled
data [HDY+ 12, RYZ+ 10, DDS+ 09, CBW+ 10]. These large labeling tasks are being performed by
coordinating crowds of semi-skilled workers through the Internet. This is known as crowdsourcing.
Crowdsourcing as a means of collecting labeled training data has now become indispensable to the
engineering of intelligent systems.
Most workers in crowdsourcing are not experts. As a consequence, labels obtained from crowdsourcing typically have a significant amount of error [KKKMF11, VdVE11, WLC+ 10]. Recent
efforts have focused on developing statistical techniques to post-process the noisy labels in order
to improve its quality (e.g., [RYZ+ 10, ZLP+ 15, KOS11, IPSW14]). However, when the inputs to
these algorithms are erroneous, it is difficult to guarantee that the processed labels will be reliable
enough for subsequent use by machine learning or other applications. In order to avoid ?garbage in,
garbage out?, we take a complementary approach to this problem: cleaning the data at the time of
collection.
We consider crowdsourcing settings where the workers are paid for their services, such as in the
popular crowdsourcing platforms of Amazon Mechanical Turk and others. These commercial platforms have gained substantial popularity due to their support for a diverse range of tasks for machine
learning labeling, varying from image annotation and text recognition to speech captioning and machine translation. We consider problems that are objective in nature, that is, have a definite answer.
Figure 1a depicts an example of such a question where the worker is shown a set of images, and for
each image, the worker is required to identify if the image depicts the Golden Gate Bridge.
1
Is this the Golden Gate Bridge?
Is this the Golden Gate Bridge?
Yes!
Yes!
No!
No
I?m not sure
(b)!
(a)!
Figure 1: Different interfaces in a crowdsourcing setup: (a) the conventional interface, and (b) with
an option to skip.
Our approach builds on the simple insight that in typical crowdsourcing setups, workers are simply
paid in proportion to the amount of tasks they complete. As a result, workers attempt to answer
questions that they are not sure of, thereby increasing the error rate of the labels. For the questions
that a worker is not sure of, her answers could be very unreliable [WLC+ 10, KKKMF11, VdVE11,
JSV14]. To ensure acquisition of only high-quality labels, we wish to encourage the worker to
skip the questions about which she is unsure, for instance, by providing an explicit ?I?m not sure?
option for every question (see Figure 1b). Our goal is to develop payment mechanisms to encourage
the worker to select this option when she is unsure. We will term any payment mechanism that
incentivizes the worker to do so as ?incentive compatible?.
In addition to incentive compatibility, preventing spammers is another desirable requirement from
incentive mechanisms in crowdsourcing. Spammers are workers who answer randomly without
regard to the question being asked, in the hope of earning some free money, and are known to exist
in large numbers on crowdsourcing platforms [WLC+ 10, Boh11, KKKMF11, VdVE11]. It is thus
of interest to deter spammers by paying them as low as possible. An intuitive objective, to this end,
is to ensure a zero expenditure on spammers who answer randomly. In this paper, however, we
impose a strictly and significantly weaker condition, and then show that there is one and only one
incentive-compatible mechanism that can satisfy this weak condition. Our requirement, referred to
as the ?no-free-lunch? axiom, says that if all the questions attempted by the worker are answered
incorrectly, then the payment must be zero.
We propose a payment mechanism for the aforementioned setting (?incentive compatibility? plus
?no-free-lunch?), and show that surprisingly, this is the only possible mechanism. We also show that
additionally, our mechanism makes the smallest possible payment to spammers among all possible
incentive compatible mechanisms that may or may not satisfy the no-free-lunch axiom. Our payment
mechanism takes a multiplicative form: the evaluation of the worker?s response to each question is
a certain score, and the final payment is a product of these scores. This mechanism has additional
appealing features in that it is simple to compute, and is also simple to explain to the workers. Our
mechanism is applicable to any type of objective questions, including multiple choice annotation
questions, transcription tasks, etc.
In order to test whether our mechanism is practical, and to assess the quality of the final labels
obtained, we conducted experiments on the Amazon Mechanical Turk crowdsourcing platform. In
our preliminary experiments that involved over several hundred workers, we found that the quality
of data improved by two-fold under our unique mechanism, with the total monetary expenditure
being the same or lower as compared to the conventional baseline.
2
Problem Setting
In the crowdsourcing setting that we consider, one or more workers perform a task, where a task
consists of multiple questions. The questions are objective, by which we mean, each question has
precisely one correct answer. Examples of objective questions include multiple-choice classification
questions such as Figure 1, questions on transcribing text from audio or images, etc.
For any possible answer to any question, we define the worker?s confidence about an answer as the
probability, according to her belief, of this answer being correct. In other words, one can assume
that the worker has (in her mind) a probability distribution over all possible answers to a question,
and the confidence for an answer is the probability of that answer being correct. As a shorthand, we
also define the confidence about a question as the confidence for the answer that the worker is most
2
confident about for that question. We assume that the worker?s confidences for different questions
are independent. Our goal is that for every question, the worker should be incentivized to:
1. skip if the confidence is below a certain pre-defined threshold, otherwise:
2. select the answer that she thinks is most confident about.
More formally, let T 2 (0, 1) be a predefined value. The goal is to design payment mechanisms that
incentivize the worker to skip the questions for which her confidence is lower than T , and attempt
those for which her confidence is higher than T . 1 Moreover, for the questions that she attempts to
answer, she must be incentivized to select the answer that she believes is most likely to be correct.
The threshold T may be chosen based on various factors of the problem at hand, for example, on
the downstream machine learning algorithms using the crowdsourced data, or the knowledge of the
statistics of worker abilities, etc. In this paper we assume that the threshold T is given to us.
Let N denote the total number of questions in the task. Among these, we assume the existence of
some ?gold standard? questions, that is, a set of questions whose answers are known to the requester.
Let G (1 ? G ? N ) denote the number of gold standard questions. The G gold standard questions
are assumed to be distributed uniformly at random in the pool of N questions (of course, the worker
does not know which G of the N questions form the gold standard). The payment to a worker for
a task is computed after receiving her responses to all the questions in the task. The payment is
based on the worker?s performance on the gold standard questions. Since the payment is based on
known answers, the payments to different workers do not depend on each other, thereby allowing us
to consider the presence of only one worker without any loss in generality.
We will employ the following standard notation. For any positive integer K, the set {1, . . . , K} is
denoted by [K]. The indicator function is denoted by 1, i.e., 1{z} = 1 if z is true, and 0 otherwise.
The notation R+ denotes the set of all non-negative real numbers.
Let x1 , . . . , xG 2 { 1, 0, +1} denote the evaluations of the answers that the worker gives to the G
gold standard questions. Here, ?0? denotes that the worker skipped the question, ? 1? denotes that
the worker attempted to answer the question and that answer was incorrect, and ?+1? denotes that
the worker attempted to answer the question and that answer was correct. Let f : { 1, 0, +1}G !
R+ denote the payment function, namely, a function that determines the payment to the worker
based on these evaluations x1 , . . . , xG . Note that the crowdsourcing platforms of today mandate the
payments to be non-negative. We will let ? (> 0) denote the budget, i.e., the maximum amount that
can be paid to any individual worker for this task:
max f (x1 , . . . , xG ) = ?.
x1 ,...,xG
The amount ? is thus the amount of compensation paid to a perfect agent for her work. We will
assume this budget condition of ? throughout the rest of the paper.
We assume that the worker attempts to maximize her overall expected payment. In what follows, the
expression ?the worker?s expected payment? will refer to the expected payment from the worker?s
point of view, and the expectation will be taken with respect to the worker?s confidences about her
answers and the uniformly random choice of the G gold standard questions among the N questions
in the task. For any question i 2 [N ], let yi = 1 if the worker attempts question i, and set yi = 0
otherwise. Further, for every question i 2 [N ] such that yi 6= 0, let pi be the confidence of the
worker for the answer she has selected for question i, and for every question i 2 [N ] such that
yi = 0, let pi 2 (0, 1) be any arbitrary value. Let E = (?1 , . . . , ?G ) 2 { 1, 1}G . Then from the
worker?s perspective, the expected payment for the selected answers and confidence-levels is
!
G
X
X
Y
1+?i
1 ?i
1
f (?1 yj1 , . . . , ?G yjG ) (pji ) 2 (1 pji ) 2
.
N
G
i=1
(j1 ,...,jG ) E2{ 1,1}G
?{1,...,N }
In the expression above, the outermost summation corresponds to the expectation with respect to the
randomness arising from the unknown choice of the gold standard questions. The inner summation
corresponds to the expectation with respect to the worker?s beliefs about the correctness of her
responses.
1
In the event that the confidence about a question is exactly equal to T , the worker may be equally incentivized to answer or skip.
3
We will call any payment function f as an incentive-compatible mechanism if the expected payment
of the worker under this payment function is strictly maximized when the worker responds in the
manner desired.2
3
Main results: Incentive-compatible mechanism and guarantees
In this section, we present the main results of the paper, namely, the design of incentive-compatible
mechanisms with practically useful properties. To this end, we impose the following natural requirement on the payment function f that is motivated by the practical considerations of budget
constraints and discouraging spammers and miscreants [Boh11, KKKMF11, VdVE11, WLC+ 10].
We term this requirement as the ?no-free-lunch axiom?:
Axiom 1 (No-free-lunch axiom). If all the answers attempted by the worker in the gold standard are
wrong, then the payment is zero. More formally, for every set of evaluations (x1 , . . . , xG ) that satisfy
PG
PG
0 < i=1 1{xi 6= 0} = i=1 1{xi = 1}, we require the payment to satisfy f (x1 , . . . , xG ) = 0.
Observe that no-free-lunch is an extremely mild requirement. In fact, it is significantly weaker than
imposing a zero payment on workers who answer randomly. For instance, if the questions are of
binary-choice format, then randomly choosing among the two options for each question would result
in 50% of the answers being correct in expectation, while the no-free-lunch axiom is applicable only
when none of them turns out to be correct.
3.1
Proposed ?Multiplicative? Mechanism
We now present our proposed payment mechanism in Algorithm 1.
Algorithm 1 ?Multiplicative? incentive-compatible mechanism
? Inputs: Threshold T , Budget ?, Evaluations (x1 , . . . , xG ) 2 { 1, 0, +1}G of the worker?s answers to the G gold standard questions
PG
PG
? Let C = i=1 1{xi = 1} and W = i=1 1{xi = 1}
? The payment is
f (x1 , . . . , xG ) = ?T G
C
1{W = 0}.
The proposed mechanism has a multiplicative form: each answer in the gold standard is given a
score based on whether it was correct (score = T1 ), incorrect (score = 0) or skipped (score = 1),
and the final payment is simply a product of these scores (scaled by ?). The mechanism is easy to
describe to workers: For instance, if T = 12 , G = 3 and ? = 80 cents, then the description reads:
?The reward starts at 10 cents. For every correct answer in the 3 gold standard questions,
the reward will double. However, if any of these questions are answered incorrectly, then
the reward will become zero. So please use the ?I?m not sure? option wisely.?
Observe how this payment rule is similar to the popular ?double or nothing? paradigm [Dou14].
The algorithm makes a zero payment if one or more attempted answers in the gold standard are
wrong. Note that this property is significantly stronger than the property of no-free-lunch which
we originally required, where we wanted a zero payment only when all attempted answers were
wrong. Surprisingly, as we prove shortly, Algorithm 1 is the only incentive-compatible mechanism
that satisfies no-free-lunch.
The following theorem shows that the proposed payment mechanism indeed incentivizes a worker
to skip the questions for which her confidence is below T , while answering those for which her
confidence is greater than T . In the latter case, the worker is incentivized to select the answer which
she thinks is most likely to be correct.
Theorem 1. The payment mechanism of Algorithm 1 is incentive-compatible and satisfies the nofree-lunch condition.
2
Such a payment function that is based on gold standard questions is also called a ?strictly proper scoring
rule? [GR07].
4
The proof of Theorem 1 is presented in Appendix A. It is easy to see that the mechanism satisfies nofree-lunch. The proof of incentive compatibility is also not hard: We consider any arbitrary worker
(with arbitrary belief distributions), and compute the expected payment for that worker for the case
when her choices in the task follow the requirements. We then show that any other choice leads to a
strictly smaller expected payment.
While we started out with a very weak condition of no-free-lunch of making a zero payment when
all attempted answers are wrong, the mechanism proposed in Algorithm 1 is significantly more
strict and makes a zero payment when any of the attempted answers is wrong. A natural question
that arises is: can we design an alternative mechanism satisfying incentive compatibility and nofree-lunch that operates somewhere in between?
3.2
Uniqueness of the Mechanism
In the previous section we showed that our proposed multiplicative mechanism is incentive compatible and satisfies the intuitive requirement of no-free-lunch. It turns out, perhaps surprisingly, that
this mechanism is unique in this respect.
Theorem 2. The payment mechanism of Algorithm 1 is the only incentive-compatible mechanism
that satisfies the no-free-lunch condition.
Theorem 2 gives a strong result despite imposing very weak requirements. To see this, recall our earlier discussion on deterring spammers, that is, incurring a low expenditure on workers who answer
randomly. For instance, when the task comprises binary-choice questions, one may wish to design
mechanisms which make a zero payment when the responses to 50% or more of the questions in the
gold standard are incorrect. The no-free-lunch axiom is a much weaker requirement, and the only
mechanism that can satisfy this requirement is the mechanism of Algorithm 1.
The proof of Theorem 2 is available in Appendix B. The proof relies on the following key lemma
that establishes a condition that any incentive-compatible mechanism must necessarily satisfy. The
lemma applies to any incentive-compatible mechanism and not just to those satisfying no-free-lunch.
Lemma. Any incentive-compatible payment mechanism f must satisfy, for every i 2 {1, . . . , G}
and every (y1 , . . . , yi 1 , yi+1 , . . . , yG ) 2 { 1, 0, 1}G 1 ,
T f (y1 , . . . , yi
1 , 1, yi+1 , . . . , yG )
+ (1
T )f (y1 , . . . , yi 1 , 1, yi+1 , . . . , yG )
= f (y1 , . . . , yi 1 , 0, yi+1 , . . . , yG ).
The proof of this lemma is provided in Appendix C. Given this lemma, the proof of Theorem 2 is
then completed via an induction on the number of skipped questions.
3.3
Optimality against Spamming Behavior
As discussed earlier, crowdsouring tasks, especially those with multiple choice questions, often
encounter spammers who answer randomly without heed to the question being asked. For instance,
under a binary-choice setup, a spammer will choose one of the two options uniformly at random for
every question. A highly desirable objective in crowdsourcing settings is to deter spammers. To this
end, one may wish to impose a condition of zero payment when the responses to 50% or more of
the attempted questions in the gold standard are incorrect. A second desirable metric could be to
minimize the expenditure on a worker who simply skips all questions. While the aforementioned
requirements were deterministic functions of the worker?s responses, one may alternatively wish to
impose requirements that depend on the distribution of the worker?s answering process. For instance,
a third desirable feature would be to minimize the expected payment to a worker who answers all
questions uniformly at random. We now show that interestingly, our unique multiplicative payment
mechanism simultaneously satisfies all these requirements. The result is stated assuming a multiplechoice setup, but extends trivially to non-multiple-choice settings.
Theorem 3.A (Distributional). Consider any value A 2 {0, . . . , G}. Among all incentivecompatible mechanisms (that may or may not satisfy no-free-lunch), Algorithm 1 strictly minimizes
the expenditure on a worker who skips some A of the questions in the the gold standard, and chooses
answers to the remaining (G A) questions uniformly at random.
5
Theorem 3.B (Deterministic). Consider any value B 2 (0, 1]. Among all incentive-compatible
mechanisms (that may or may not satisfy no-free-lunch), Algorithm 1 strictly minimizes the expenditure on a worker who gives incorrect answers to a fraction B or more of the questions attempted
in the gold standard.
The proof of Theorem 3 is presented in Appendix D. We see from this result that the multiplicative
payment mechanism of Algorithm 1 thus possesses very useful properties geared to deter spammers,
while ensuring that a good worker will be paid a high enough amount.
To illustrate this point, let us compare the mechanism of Algorithm 1 with the popular additive class
of payment mechanisms.
Example 1. Consider the popular class of ?additive? mechanisms, where the payments to a worker
are added across the gold standard questions. This additive payment mechanism offers a reward of
?
?T
G for every correct answer in the gold standard, G for every question skipped, and 0 for every
incorrect answer. Importantly, the final payment to the worker is the sum of the rewards across the
G gold standard questions. One can verify that this additive mechanism is incentive compatible.
One can also see that that as guaranteed by our theory, this additive payment mechanism does not
satisfy the no-free-lunch axiom.
Suppose each question involves choosing from two options. Let us compute the expenditure that
these two mechanisms make under a spamming behavior of choosing the answer randomly to each
question. Given the 50% likelihood of each question being correct, on can compute that the additive
mechanism makes a payment of ?2 in expectation. On the other hand, our mechanism pays an
expected amount of only ?2 G . The payment to spammers thus reduces exponentially with the
number of gold standard questions under our mechanism, whereas it does not reduce at all in the
additive mechanism.
Now, consider a different means of exploiting the mechanism(s) where the worker simply skips all
questions. To this end, observe that if a worker skips all the questions then the additive payment
mechanism will incur an expenditure of ?T . On the other hand, the proposed payment mechanism
of Algorithm 1 pays an exponentially smaller amount of ?T G (recall that T < 1).
4
Simulations and Experiments
In this section, we present synthetic simulations and real-world experiments to evaluate the effects
of our setting and our mechanism on the final label quality.
4.1
Synthetic Simulations
We employ synthetic simulations to understand the effects of various kinds of labeling errors in
crowdsourcing. We consider binary-choice questions in this set of simulations. Whenever a worker
answers a question, her confidence for the correct answer is drawn from a distribution P independent
of all else. We investigate the effects of the following five choices of the distribution P:
?
?
?
?
?
The uniform distribution on the support [0.5, 1].
A triangular distribution with lower end-point 0.2, upper end-point 1 and a mode of 0.6.
A beta distribution with parameter values ? = 5 and = 1.
The hammer-spammer distribution [KOS11], that is, uniform on the discrete set {0.5, 1}.
A truncated Gaussian distribution: a truncation of N (0.75, 0.5) to the interval [0, 1].
When a worker has a confidence p (drawn from the distribution P) and attempts the question, the
probability of making an error equals (1 p).
We compare (a) the setting where workers attempt every question, with (b) the setting where workers
skip questions for which their confidence is below a certain threshold T . In this set of simulations,
we set T = 0.75. In either setting, we aggregate the labels obtained from the workers for each
question via a majority vote on the two classes. Ties are broken by choosing one of the two options
uniformly at random.
6
Figure 2: Error under different interfaces for synthetic simulations of five distributions of the workers? error probabilities.
Figure 2 depicts the results from these simulations. Each bar represents the fraction of questions that
are labeled incorrectly, and is an average across 50,000 trials. (The standard error of the mean is too
small to be visible.) We see that the skip-based setting consistently outperforms the conventional
setting, and the gains obtained are moderate to high depending on the underlying distribution of the
workers? errors. In particular, the gains are quite striking under the hammer-spammer model: this
result is not surprising since the mechanism (ideally) screens the spammers out and leaves only the
hammers who answer perfectly.
4.2
Experiments on Amazon Mechanical Turk
We conducted preliminary experiments on the Amazon Mechanical Turk commercial crowdsourcing
platform (mturk.com) to evaluate our proposed scheme in real-world scenarios. The complete
data, including the interface presented to the workers in each of the tasks, the results obtained from
the workers, and the ground truth solutions, are available on the website of the first author.
Goal. Before delving into details, we first note certain caveats relating to such a study of mechanism design on crowdsourcing platforms. When a worker encounters a mechanism for only a
small amount of time (a handful of tasks in typical research experiments) and for a small amount of
money (at most a few dollars in typical crowdsourcing tasks), we cannot expect the worker to completely understand the mechanism and act precisely as required. For instance, we wouldn?t expect
our experimental results to change significantly even upon moderate modifications in the promised
amounts, and furthermore, we do expect the outcomes to be noisy. Incentive compatibility kicks
in when the worker encounters a mechanism across a longer term, for example, when a proposed
mechanism is adopted as a standard for a platform, or when higher amounts are involved. This is
when we would expect workers or others (e.g., bloggers or researchers) to design strategies that can
game the mechanism. The theoretical guarantee of incentive compatibility or strict properness then
prevents such gaming in the long run.
We thus regard these experiments as preliminary. Our intentions towards this experimental exercise
were (a) to evaluate the potential of our algorithms to work in practice, and (b) to investigate the
effect of the proposed algorithms on the net error in the collected labelled data.
Experimental setup. We conducted the five following experiments (?tasks?) on Amazon Mechanical Turk: (a) identifying the golden gate bridge from pictures, (b) identifying the breeds of dogs
from pictures, (c) identifying heads of countries, (d) identifying continents to which flags belong,
and (e) identifying the textures in displayed images. Each of these tasks comprised 20 to 126 multi7
Figure 3: Error under different interfaces and mechanisms for five experiments conducted on Mechanical Turk.
ple choice questions.3 For each experiment, we compared (i) a baseline setting (Figure 1a) with an
additive payment mechanism that pays a fixed amount per correct answer, and (ii) our skip-based
setting (Figure 1b) with the multiplicative mechanism of Algorithm 1. For each experiment, and for
each of the two settings, we had 35 workers independently perform the task.
Upon completion of the tasks on Amazon Mechanical Turk, we aggregated the data in the following
manner. For each mechanism in each experiment, we subsampled 3, 5, 7, 9 and 11 workers, and
took a majority vote of their responses. We averaged the accuracy across all questions and across
1, 000 iterations of this subsample-and-aggregate procedure.
Results. Figure 3 reports the error in the aggregate data in the five experiments. We see that in
most cases, our skip-based setting results in a higher quality data, and in many of the instances, the
reduction is two-fold or higher. All in all, in the experiments, we observed a substantial reduction in
the amount of error in the labelled data while expending the same or lower amounts and receiving
no negative comments from the workers. These observations suggest that our proposed skip-based
setting coupled with our multiplicative payment mechanisms have potential to work in practice; the
underlying fundamental theory ensures that the system cannot be gamed in the long run.
5
Discussion and Conclusions
In an extended version of this paper [SZ14], we generalize the ?skip-based? setting considered here
to one where we also elicit the workers? confidence about their answers. Moreover, in a companion
paper [SZP15], we construct mechanisms to elicit the support of worker?s beliefs.
Our mechanism offers some additional benefits. The pattern of skips of the workers provide a reasonable estimate of the difficulty of each question. In practice, the questions that are estimated to
be more difficult may now be delegated to an expert or to additional non-expert workers. Secondly,
the theoretical guarantees of our mechanism may allow for better post-processing of the data, incorporating the confidence information and improving the overall accuracy. Developing statistical
aggregation algorithms or augmenting existing ones (e.g., [RYZ+ 10, KOS11, LPI12, ZLP+ 15]) for
this purpose is a useful direction of research. Thirdly, the simplicity of our mechanisms may facilitate an easier adoption among the workers. In conclusion, given the uniqueness and optimality
in theory, simplicity, and good performance observed in practice, we envisage our multiplicative
payment mechanisms to be of interest to practitioners as well as researchers who employ crowdsourcing.
3
See the extended version of this paper [SZ14] for additional experiments involving free-form responses,
such as text transcription.
8
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5,167 | 5,678 | Local Expectation Gradients for Black Box
Variational Inference
Michalis K. Titsias
Athens University of Economics and Business
[email protected]
Miguel L?azaro-Gredilla
Vicarious
[email protected]
Abstract
We introduce local expectation gradients which is a general purpose stochastic
variational inference algorithm for constructing stochastic gradients by sampling
from the variational distribution. This algorithm divides the problem of estimating
the stochastic gradients over multiple variational parameters into smaller sub-tasks
so that each sub-task explores intelligently the most relevant part of the variational
distribution. This is achieved by performing an exact expectation over the single
random variable that most correlates with the variational parameter of interest
resulting in a Rao-Blackwellized estimate that has low variance. Our method
works efficiently for both continuous and discrete random variables. Furthermore,
the proposed algorithm has interesting similarities with Gibbs sampling but at the
same time, unlike Gibbs sampling, can be trivially parallelized.
1
Introduction
Stochastic variational inference has emerged as a promising and flexible framework for performing large scale approximate inference in complex probabilistic models. It significantly extends the
traditional variational inference framework [7, 1] by incorporating stochastic approximation [16]
into the optimization of the variational lower bound. Currently, there exist two major research directions in stochastic variational inference. The first one (data stochasticity) attempts to deal with
massive datasets by constructing stochastic gradients by using mini-batches of training examples
[5, 6]. The second direction (expectation stochasticity) aims at dealing with the intractable expectations under the variational distribution that are encountered in non-conjugate probabilistic models
[12, 14, 10, 18, 8, 15, 20]. Unifying these two ideas, it is possible to use stochastic gradients to address both massive datasets and intractable expectations. This results in a doubly stochastic estimation approach, where the mini-batch source of stochasticity can be combined with the stochasticity
associated with sampling from the variational distribution.
In this paper, we are interested to further investigate the expectation stochasticity that in practice is
dealt with by drawing samples from the variational distribution. A challenging issue here is concerned with the variance reduction of the stochastic gradients. Specifically, while the method based
on the log derivative trick is currently the most general one, it has been observed to severely suffer
from high variance problems [12, 14, 10] and thus it is only applicable together with sophisticated
variance reduction techniques based on control variates. However, the construction of efficient control variates can be strongly dependent on the form of the probabilistic model. Therefore, it would be
highly desirable to introduce more black box procedures, where simple stochastic gradients can work
well for any model, thus allowing the end-user not to worry about having to design model-dependent
variance reduction techniques. Notice, that for continuous random variables and differentiable functions the reparametrization approach [8, 15, 20] offers a simple black box procedure [20, 9] which
does not require further model-dependent variance reduction. However, reparametrization is neither
applicable for discrete spaces nor for non-differentiable models and this greatly limits its scope of
applicability.
1
In this paper, we introduce a simple black box algorithm for stochastic optimization in variational
inference which provides stochastic gradients having low variance and without needing any extra
variance reduction. This method is based on a new trick referred to as local expectation or integration. The key idea here is that stochastic gradient estimation over multiple variational parameters
can be divided into smaller sub-tasks where each sub-task requires different amounts of information
about different parts of the variational distribution. More precisely, each sub-task aims at exploiting
the conditional independence structure of the variational distribution. Based on this intuitive idea
we introduce the local expectation gradients algorithm that provides a stochastic gradient over a
variational parameter vi by performing an exact expectation over the associated latent variable xi
while using a single sample from the remaining latent variables. Essentially, this consists of a RaoBlackwellized estimate that allows to dramatically reduce the variance of the stochastic gradient
so that, for instance, for continuous spaces the new stochastic gradient is guaranteed to have lower
variance than the stochastic gradient corresponding to the reparametrization method where the latter
utilizes a single sample. Furthermore, the local expectation algorithm has interesting similarities
with Gibbs sampling with the important difference, that unlike Gibbs sampling, it can be trivially
parallelized.
2
Stochastic variational inference
Here, we discuss the main ideas behind current algorithms on stochastic variational inference and
particularly methods that sample from the variational distribution in order to approximate intractable
expectations using Monte Carlo. Given a joint probability distribution p(y, x) where y are observations and x are latent variables (possibly including model parameters that consist of random
variables) and a variational distribution qv (x), the objective is to maximize the lower bound
F(v) = Eqv (x) [log p(y, x) ? log qv (x)] ,
= Eqv (x) [log p(y, x)] ? Eqv (x) [log qv (x)] ,
(1)
(2)
with respect to the variational parameters v. Ideally, in order to tune v we would like to have a
closed-form expression for the lower bound so that we could subsequently maximize it by using
standard optimization routines such as gradient-based algorithms. However, for many probabilistic models and forms of the variational distribution at least one of the two expectations in (2) is
intractable. Therefore, in general we are faced with the following intractable expectation
e
F(v)
= Eqv (x) [f (x)] ,
(3)
where f (x) can be either log p(y, x), ? log qv (x) or log p(y, x) ? log qv (x), from which we would
like to efficiently estimate the gradient over v in order to apply gradient-based optimization.
e
The most general method for estimating the gradient ?v F(v)
is based on the log derivative trick,
also called likelihood ratio or REINFORCE, that has been invented in control theory and reinforcement learning [3, 21, 13] and used recently for variational inference [12, 14, 10]. Specifically, this
makes use of the property ?v qv (x) = qv (x)?v log qv (x), which allows to write the gradient as
e
?v F(v)
= Eqv (x) [f (x)?v log qv (x)]
(4)
and then obtain an unbiased estimate according to
S
1X
f (x(s) )?v log qv (x(s) ),
S s=1
(5)
where each x(s) is an independent draw from qv (x). While this estimate is unbiased, it has been
observed to severely suffer from high variance so that in practice it is necessary to consider variance
reduction techniques such as those based on control variates [12, 14, 10].
The second approach is suitable for continuous spaces where f (x) is a differentiable function of
x [8, 15, 20]. It is based on a simple transformation of (3) which allows to move the variational
parameters v inside f (x) so that eventually the expectation is taken over a base distribution that does
not depend on the variational parameters any more. For example, if the variational distribution is the
e
Gaussian N (x|?, LL> ) where v = (?, L), the expectation in (3) can be re-written as F(?,
L) =
2
R
N (z|0, I)f (? + Lz)dz and subsequently the gradient over (?, L) can be approximated by the
following unbiased Monte Carlo estimate
S
1X
?(?,L) f (? + Lz(s) ),
S s=1
(6)
where each z(s) is an independent sample from N (z|0, I). This estimate makes efficient use of the
slope of f (x) which allows to perform informative moves in the space of (?, L). Furthermore, it has
been shown experimentally in several studies [8, 15, 20, 9] that the estimate in (6) has relatively low
variance and can lead to efficient optimization even when a single sample is used at each iteration.
Nevertheless, a limitation of the approach is that it is only applicable to models where x is continuous
and f (x) is differentiable. Even within this subset of models we are also additionally restricted to
using certain classes of variational distributions for which reparametrization is possible.
Next we introduce an approach that is applicable to a broad class of models (both discrete and
continuous), has favourable scaling properties and provides low-variance stochastic gradients.
3
Local expectation gradients
Suppose that the n-dimensional latent vector x in the probabilistic model takes values in some space
S1 ? . . . Sn where each set Si can be continuous or discrete. We consider a variational distribution
over x that is represented as a directed graphical model having the following joint density
qv (x) =
n
Y
qvi (xi |pai ),
(7)
i=1
where qvi (xi |pai ) is the conditional factor over xi given the set of the parents denoted by pai .
We assume that each conditional factor has its own separate set of variational parameters vi and
v = (vi , . . . , vn ). The objective is then to obtain a stochastic approximation for the gradient of the
lower bound over each variational parameter vi .
Our method is motivated by the observation that each vi is influenced mostly by its corresponding
latent variable xi since vi determines the factor qvi (xi |pai ). Therefore, to get information about the
gradient of vi we should be exploring multiple possible values of xi and a rather smaller set of values
from the remaining latent variables x\i . Next we take this idea into the extreme where we will be
using infinite draws from xi (i.e. essentially an exact expectation) together with just a single sample
of x\i . More precisely, we factorize the variational distribution as qv (x) = q(xi |mbi )q(x\i ), where
mbi denotes the Markov blanket of xi . The gradient over vi can be written as
e
?vi F(v)
= Eq(x) [f (x)?vi log qvi (xi |pai )] = Eq(x\i ) Eq(xi |mbi ) [f (x)?vi log qvi (xi |pai )] ,
(8)
where in the second expression we used the law of iterated expectations. Then, an unbiased stochastic gradient, say at the t-th iteration of an optimization algorithm, can be obtained by drawing a
(t)
single sample x\i from q(x\i ) so that
h
i X
(t)
(t)
(t)
(t)
(t)
qe(xi |mbi )f (x\i , xi )?vi qvi (xi |pai ), (9)
Eq(xi |mb(t) ) f (x\i , xi )?vi log qvi (xi |pai ) =
i
xi
(t)
(t)
e(xi |mbi ) is the same as q(xi |mbi ) but with
xi denotes summation or integration and q
(t)
qvi (xi |pai ) removed from the numerator.1 The above is the expression for the proposed stochastic
gradient for the parameter vi . Notice that this estimate does not rely on the log derivative trick
(t)
since we never draw samples from q(xi |mbi ). Instead the trick here is to perform local expectation
(t)
(integration or summation). To get an independent sample x\i from q(x\i ) we can simply simulate a
full latent vector x(t) from qv (x) by applying the standard ancestral sampling procedure for directed
(t)
graphical models [1]. Then, the sub-vector x\i is by construction an independent draw from the
where
1
P
(t)
(t)
(t)
Notice that q(xi |mbi ) ? h(xi , mbi )qvi (xi |pai ) for some non-negative function h(?).
3
Algorithm 1 Stochastic variational inference using local expectation gradients
Input: f (x), qv (x).
Initialize v(0) , t = 0.
repeat
Set t = t + 1.
Draw pivot sample x(t) ? qv (x).
for i = 1 to n do
h
i
(t)
(t)
dvi = Eq(xi |mb(t) ) f (x\i , xi )?vi log qvi (xi |pai ) .
i
vi = vi + ?t dvi .
end for
until convergence criterion is met.
marginal q(x\i ). Furthermore, the sample x(t) can be thought of as a global or pivot sample that is
needed to be drawn once and then it can be re-used multiple times in order to compute all stochastic
gradients for all variational parameters (v1 , . . . , vn ) according to eq. (9).
When the variable xi takes discrete values, the expectation in eq. (9) reduces to a sum of terms
associated with all possible values of xi . On the other hand, when xi is a continuous variable
the expectation in (9) corresponds to an univariate integral that in general may not be analytically
tractable. In this case we shall use fast numerical integration methods.
We shall refer to the above algorithm for providing stochastic gradients over variational parameters
as local expectation gradients and pseudo-code of a stochastic variational inference scheme that
internally uses this algorithm is given in Algorithm 1. Notice that Algorithm 1 corresponds to the
case where f (x) = log p(y, x) ? log qv (x) while other cases can be expressed similarly.
In the next two sections we discuss some theoretical properties of local expectation gradients (Section 3.1) and draw interesting connections with Gibbs sampling (Section 3.2).
3.1
Properties of local expectation gradients
We first derive the variance of the stochastic estimates obtained by local expectation gradients. In
our analysis, we will focus on the case of fitting a fully factorized variational distribution (and leave
the more general case for future work) having the form
n
Y
qvi (xi ).
(10)
qv (x) =
i=1
For such case the local expectation gradient for each parameter vi from eq. (9) simplifies to
X
Eqvi (xi ) f (x\i , xi )?vi log qvi (xi ) =
?vi qvi (xi )f (x\i , xi ),
(11)
xi
(t)
where also for notational simplicity we write x\i as x\i . It would be useful to define the following
mean and covariance functions
m(xi ) = Eq(x\i ) [f (x\i , xi )],
(12)
Cov(xi , x0i ) = Eq(x\i ) [(f (x\i , xi ) ? m(xi ))(f (x\i , x0i ) ? m(x0i ))],
(13)
that characterize the variability of f (x\i , xi ) as x\i varies according to q(x\i ). Notice that based
on
P eq. (12) the exact gradient of the variational lower bound over vi can also be written as
xi ?vi qvi (xi )m(xi ), which has an analogous form to the local expectation gradient from (11)
with the difference that f (x\i , xi ) is now replaced by its mean value m(xi ).
We can now characterize the variance of the stochastic gradient and describe some additional properties. All proofs for the following statements are given in the Supplementary Material.
Proposition 1. The variance of the stochastic gradient in (11) can be written as
X
?vi qvi (xi )?vi qvi (x0i )Cov(xi , x0i ).
xi ,x0i
4
(14)
This gives us some intuition about when we expect the variance of the estimate to be small. For
instance, two simple cases are: i) when the covariance function Cov(xi , x0i ) takes small values,
which can occur when q(x\i ) has low entropy, or ii) when Cov(xi , x0i ) is approximately constant. In
fact, when Cov(xi , x0i ) is exactly constant, then the variance is zero (so that the stochastic gradient
is exact) as the following proposition states.
Proposition 2. If Cov(xi , x0i ) = c for all xi and x0i then the variance in (14) is equal to zero.
A case for which the condition Cov(xi , x0i ) = c holds exactly is when the function f (x) factorizes
as f (x\i , xi ) = fi (xi ) + f\i (x\i ) (see Supplementary Material for a proof). Such a factorization essentially implies that xi is independent from the remaining random variables, which results the local
expectation gradient to be exact. In contrast, in order to get exactness by using the standard Monte
Carlo stochastic gradient from eq. (5) (and any of its improvements that apply variance reduction)
we will typically need to draw infinite number of samples.
To further analyze local expectation gradients we can contrast them with stochastic gradients obtained by the reparametrization trick [8, 15, 20]. Suppose that we can reparametrize the random
variable xi ? qvi (xi ) according to xi = g(vi , zi ), where zi ? qi (zi ) and qi (zi ) is a suitable base
distribution. We further assume that the function f (x\i , xi ) is differentiable with respect to xi and
g(vi , zi ) is differentiable with respect to vi . Then, the exact gradient with respect to the variational
parameter vi can be reparametrized as
Z
Z
?vi
q(x\i )qvi (xi )f (x\i , xi )dx = q(x\i )qi (zi )?vi f (x\i , g(vi , zi ))dx\i dzi ,
(15)
while a single-sample stochastic estimate that follows from this expression is
?vi f (x\i , g(vi , zi )), x\i ? q(x\i ), zi ? qi (zi ).
(16)
The following statement gives us a clear understanding about how this estimate compares with the
corresponding local expectation gradient.
Proposition 3. Given that we can reparametrize xi as described above (and all differentiability
conditions mentioned above hold), the gradient from (11) can be equivalently written as
Z
qi (zi )?vi f (x\i , g(vi , zi ))dzi , x\i ? q(x\i ).
(17)
Clearly, the above expression is an expectation of the reparametrization gradient from eq. (16), and
therefore based on the standard Rao-Blackwellization argument the variance of the local expectation
gradient will always be lower or equal than the variance of a single-sample estimate based on the
reparametrization method. Notice that the reparametrization method is only applicable to continuous
random variables and differentiable functions f (x). However, for such cases, reparametrization
could be computationally more efficient than local expectation gradients since the latter approach
will require to apply 1-D numerical integration to estimate the integral in (11) or the integral in (17)2
which could be computationally more expensive.
3.2
Connection with Gibbs sampling
There are interesting similarities between local expectation gradients and Gibbs sampling. Firstly,
notice that carrying out Gibbs sampling in the variational distribution in eq. (7) requires iteratively
sampling from each conditional q(xi |mbi ), for i = 1, . . . , n, and clearly the same conditional appears also in local expectation gradients with the obvious difference that instead of sampling from
q(xi |mbi ) we now average under this distribution. Of course, in practice, we never perform Gibbs
sampling on a variational distribution but instead on the true posterior distribution which is proportional to ef (x) (where we assumed that ? log qv (x) is not part of f (x)). Specifically, at each Gibbs
step we simulate a new value for some xi from the posterior conditional distribution that is propor(t)
f (x
,x )
(t)
tional to e \i i and where x\i are the fixed values for the remaining random variables. We can
observe that an update in local expectation gradients is quite similar, because now we also condi(t)
tion on some fixed remaining values x\i in order to update the parameter vi towards the direction
2
The exact value of the two integrals is the same. However, approximation of these two integrals based on
numerical integration will typically not give the same value.
5
(t)
where q(xi |mbi ) gets closer to the corresponding true posterior conditional distribution. Despite
these similarities, there is a crucial computational difference between the two procedures. While
in local expectation gradients it is perfectly valid to perform all updates of the variational parameters in parallel, given the pivot sample x(t) , in Gibbs sampling all updates need to be executed in a
serial manner. This difference is essentially a consequence of the fundamental difference between
variational inference and Gibbs sampling where the former relies on optimization while the latter on
convergence of a Markov chain.
4
Experiments
In this section, we apply local expectation gradients (LeGrad) to two different types of stochastic
variational inference problems and we compare it against the standard stochastic gradient based
on the log derivative trick (LdGrad), that incorporates also variance reduction3 , as well as the
reparametrization-based gradient (ReGrad) given by eq. (6). In Section 4.1, we consider a two-class
classification problem using two digits from the MNIST database and we approximate a Bayesian
logistic regression model using stochastic variational inference. Then, in Section 4.2 we consider
sigmoid belief networks [11] and we fit them to the binarized version of the MNIST digits.
4.1
Bayesian logistic regression
In this section we compare the three approaches in a challenging binary classification problem using
n
Bayesian logistic regression. Specifically, given a dataset D ? {zj , yj }m
j=1 , where zj ? R is
the input and yj ? {?1, +1} the class label,
Q we model the joint
distribution over the observed
M
>
labels and the parameters w by p(y, w) =
m=1 ?(ym zm w) p(w), where ?(a) is the sigmoid
function and p(w) denotes a zero-mean Gaussian prior on the weights w. We wish to apply the
three algorithms in order to approximate the posterior overQthe regression parameters by a factorized
n
variational Gaussian distribution of the form qv (w) = i=1 N (wi |?i , `2i ). In the following we
consider a subset of the MNIST dataset that includes all 12660 training examples from the digit
classes 2 and 7, each with 784 pixels so that by including the bias the number of weights is n = 785.
To obtain the local expectation gradient for each (?i , `i ) we need to apply 1-D numerical integration.
We used the quadrature rule having K = 5 nodes4 so that LeGrad was using S = 785 ? 5 function
evaluations per gradient estimation. For LdGrad we also set the number of samples to S = 785 ? 5
so that LeGrad and LdGrad match exactly in the number of function evaluations and roughly in
computational cost. When using the ReGrad approach based on (6) we construct the stochastic
gradient using K = 5 target function gradient samples. This matches the computational cost, but
ReGrad still has the unmatched advantage of having access to the gradient of the target function.
The variance of the stochastic gradient for parameter ?1 is shown in Figure 1(a)-(b). It is much
smaller for LeGrad than for LdGrad, despite having almost similar computational cost and use the
same amount of information about the target function. The evolution of the bound in Figure 1(c)
clearly shows the advantage of using less noisy gradients. LdGrad will need a huge number of
iterations to find the global optimum, despite having optimized the step size of its stochastic updates.
4.2
Sigmoid belief networks
In the second example we consider sigmoid belief networks (SBNs) [11] and i) compare our approach with LdGrad in terms of variance and optimization efficiency and then ii) we perform density estimation experiments by training sigmoid belief nets with fully connected hidden units using
LeGrad. Note that ReGrad cannot be used on discrete models.
3
As discussed in [19], there are multiple unbiased sample-based estimators of (4), and using (5) directly
tends to have a large variance. We use instead the estimator given by eq. (8) in [19]. Though other estimators
with even lower variance exist, we restrict ourselves to those with the same scalability as the proposed LeGrad,
requiring at most O(S|v|) computation per gradient estimation.
4
Gaussian quadrature with K grid points integrates exactly polynomials up to 2K ? 1 degree.
6
2
100
?200
80
?400
1
Lower bound
Variance
Variance
1.5
60
40
?600
?800
0.5
20
?1000
0
0
5000
10000
Iterations
0
0
(a)
5000
10000
Iterations
0
5000
10000
Iterations
15000
(b)
Figure 1: (a) Variance of the gradient for the variational parameter ?1 for LeGrad (red line) and
ReGrad (blue line). (b) Variance of the gradient for the variational parameter ?1 for LdGrad (green
line). (c) Evolution of the stochastic value of the lower bound.
For the variance reduction comparison we consider a network with an unstructured hidden layer
where binary observed vectors yi ? {0, 1}D are generated independently according to
p(y|W ) =
D
XY
yd
1?yd
?(wd> x)
1 ? ?(wd> x)
p(x),
(18)
x d=1
where x ? {0, 1}K is a vector of hidden variables that follows a uniform distribution. The matrix W
(which includes bias terms) contains the parameters to be estimated by fitting the model to the data.
In theory we could use the EM algorithm to learn the parameters W , however, such an approach is
not feasible because at the E step we need to compute the posterior distribution p(xi |yi , W ) over
each hidden variable which clearly is intractable since each xi takes 2K values. Therefore, we need
to apply approximate inference and next we consider stochastic variational inference using the local
expectation gradients algorithm and compare this with the method in [19] eq. (8), which has the
same scalability properties and have been denoting as LdGrad.
More precisely, we shall consider a variational distribution that consists of a recognition model
[4, 2, 10, 8, 15] which is parametrized by a ?reverse? sigmoid network that predicts the latent vector
xik
1?xik
QK
xi from the associated observation yi : qV (xi ) = k=1 ?(v>
1 ? ?(v>
. The
k yi )
k yi )
variational parameters are contained in matrix V (also the bias terms). The application of stochastic
variational inference boils down to constructing a separate lower bound for each pair (yi , xi ) so
that the full bound is the sum of these individual terms (see Supplementary Material for explicit
expressions). Then, the maximization of the bound proceeds by performing stochastic gradient
updates for the model weights W and the variational parameters V . The update for W reduces to
a logistic regression type of update, based upon drawing a single sample from the full variational
distribution. On the other hand, obtaining effective and low variance stochastic gradients for the
variational parameters V is considered to be a very highly challenging task and current advanced
methods are based on covariates that employ neural networks as auxiliary models [10]. In contrast,
the local expectation gradient for each variational parameter vk only requires evaluating
?
?
(t)
>
n
n
D
?e
yid wd
(xi\k ,xik =0)
X
X
X
1
?
?
1
+
e
ik ?
+ log
yi ,
?v k F =
?v k F i =
?ik (1 ? ?ik ) ?
log
> (x(t) ,x =1)
?e
yid wd
?
ik
ik
i\k
1+e
i=1
i=1
d=1
(19)
?(v>
k yi )
where ?ik =
and yeid is the {?1, 1} encoding of yid . This expression is a weighted sum
across data terms where each term is a difference induced by the directions xik = 1 and xik = 0 for
all hidden units {xik }ni=1 associated with the variational factors that depend on vk .
Based on the above model, we compare the performance of LeGrad and LdGrad when simultaneously optimizing V and W for a small set of 100 random binarized MNIST digits [17]. The
evolution of the instantaneous bound for H = 40 hidden units can be seen in Figure 2(a), where
once again LeGrad shows superior performance and increased stability.
In the second series of experiments we consider a more complex sigmoid belief network where
the prior p(x) over the hidden units becomes a fully connected distribution parametrized by an
7
Table 1: NLL scores in the test data for the binarized MNIST dataset. The left part of the table
shows results based on sigmoid belief nets (SBN) constructed and trained based on the approach
from [10], denoted as NVIL, or by using the LeGrad algorithm. The right part of the table gives the
performance of alternative state of the art models (reported in Table 1 in [10]).
SBN
NVIL
NVIL
NVIL
LeGrad
LeGrad
LeGrad
Dim
200-200
200-200-200
200-200-500
200
300
500
Test NLL
99.8
96.7
97.0
96.0
95.1
94.9
Model
FDARN
NADE
DARN
RBM(CD3)
RBM(CD25)
MOB
Dim
400
500
400
500
500
500
Test NLL
96.3
88.9
93.0
105.5
86.3
137.6
additional set of K(K + 1)/2 model weights (see Supplementary Material). Such a model can
better capture the dependence structure of the hidden units and provide a good density estimator for
high dimensional data. We trained this model using the 5 ? 104 training examples of the binarized
MNIST by using mini-batches of size 100 and assuming different numbers of hidden units: H =
200, 300, 500. Table 1 provides negative log likelihood (NLL) scores for LegGrad and several other
methods reported in [10]. Notice that for LeGrad the NLLs are essentially variational upper bounds
of the exact NLLs obtained by Monte Carlo approximation of the variational bound (an estimate
also considered in [10]). From Table 1 we can observe that LeGrad outperforms the advanced NVIL
technique proposed in [10]. Finally, Figure 2(b) and 2(c) displays model weights and few examples
of digits generated after having trained the model with H = 200 units, respectively.
?30
?40
Lower bound
?50
?60
?70
?80
?90
?100
?110
0
1000
2000
3000
Iterations
4000
5000
(a)
(b)
(c)
Figure 2: (a) LeGrad (red) and LdGrad (green) convergence for the SBN model on a single minibatch of 100 MNIST digits. (b) Weights W (filters) learned by LeGrad when training an SBN with
H = 200 units in the full MNIST training set. (c) New digits generated from the trained model.
5
Discussion
Local expectation gradients is a generic black box stochastic optimization algorithm that can be
used to maximize objective functions of the form Eqv (x) [f (x)], a problem that arises in variational
inference. The idea behind this algorithm is to exploit the conditional independence structure of
the variational distribution qv (x). Also this algorithm is mostly related to stochastic optimization
schemes that make use of the log derivative trick that has been invented in reinforcement learning
[3, 21, 13] and has been recently used for variational inference [12, 14, 10]. The approaches in
[12, 14, 10] can be thought of as following a global sampling strategy, where multiple samples are
drawn from qv (x) and then variance reduction is built a posteriori in a subsequent stage through the
use of control variates. In contrast, local expectation gradients reduce variance by directly changing
the sampling strategy, so that instead of working with a global set of samples drawn from qv (x),
the strategy now is to exactly marginalize out the random variable that has the largest influence on a
specific gradient of interest while using a single sample for the remaining random variables.
We believe that local expectation gradients can be applied to a great range of stochastic optimization
problems that arise in variational inference and in other domains. Here, we have demonstrated its
use for variational inference in logistic regression and sigmoid belief networks.
8
References
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[2] Jrg Bornschein and Yoshua Bengio. Reweighted wake-sleep. CoRR, pages ?1?1, 2014.
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84, October 1990.
[4] Geoffrey Hinton, Peter Dayan, Brendan J Frey, and Radford M Neal. The wake-sleep algorithm for
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[5] Matthew D. Hoffman, David M. Blei, and Francis R. Bach. Online learning for latent dirichlet allocation.
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[11] Radford M. Neal. Connectionist learning of belief networks. Artif. Intell., 56(1):71?113, July 1992.
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[13] J. Peters and S. Schaal. Policy gradient methods for robotics. In Proceedings of the IEEE International
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[14] Rajesh Ranganath, Sean Gerrish, and David Blei. Black box variational inference. In Proceedings of the
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2014.
[15] Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In The 31st International Conference on Machine Learning
(ICML 2014), 2014.
[16] Herbert Robbins and Sutton Monro. A Stochastic Approximation Method. The Annals of Mathematical
Statistics, 22(3):400?407, 1951.
[17] Ruslan Salakhutdinov and Iain Murray. On the quantitative analysis of Deep Belief Networks. In Andrew McCallum and Sam Roweis, editors, Proceedings of the 25th Annual International Conference on
Machine Learning (ICML 2008), pages 872?879. Omnipress, 2008.
[18] Tim Salimans and David A. Knowles. Fixed-form variational posterior approximation through stochastic
linear regression. Bayesian Anal., 8(4):837?882, 12 2013.
[19] Tim Salimans and David A. Knowles. On Using Control Variates with Stochastic Approximation for
Variational Bayes and its Connection to Stochastic Linear Regression, January 2014.
[20] Michalis K. Titsias and Miguel L?azaro-Gredilla. Doubly stochastic variational bayes for non-conjugate
inference. In The 31st International Conference on Machine Learning (ICML 2014), 2014.
[21] Ronald J. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement
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9
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5,168 | 5,679 | Learning with a Wasserstein Loss
Charlie Frogner? Chiyuan Zhang?
Center for Brains, Minds and Machines
Massachusetts Institute of Technology
[email protected], [email protected]
Mauricio Araya-Polo
Shell International E & P, Inc.
[email protected]
Hossein Mobahi
CSAIL
Massachusetts Institute of Technology
[email protected]
Tomaso Poggio
Center for Brains, Minds and Machines
Massachusetts Institute of Technology
[email protected]
Abstract
Learning to predict multi-label outputs is challenging, but in many problems there
is a natural metric on the outputs that can be used to improve predictions. In this
paper we develop a loss function for multi-label learning, based on the Wasserstein
distance. The Wasserstein distance provides a natural notion of dissimilarity for
probability measures. Although optimizing with respect to the exact Wasserstein
distance is costly, recent work has described a regularized approximation that is
efficiently computed. We describe an efficient learning algorithm based on this
regularization, as well as a novel extension of the Wasserstein distance from probability measures to unnormalized measures. We also describe a statistical learning
bound for the loss. The Wasserstein loss can encourage smoothness of the predictions with respect to a chosen metric on the output space. We demonstrate this
property on a real-data tag prediction problem, using the Yahoo Flickr Creative
Commons dataset, outperforming a baseline that doesn?t use the metric.
1
Introduction
We consider the problem of learning to predict a non-negative measure over a finite set. This problem includes many common machine learning scenarios. In multiclass classification, for example,
one often predicts a vector of scores or probabilities for the classes. And in semantic segmentation [1], one can model the segmentation as being the support of a measure defined over the pixel
locations. Many problems in which the output of the learning machine is both non-negative and
multi-dimensional might be cast as predicting a measure.
We specifically focus on problems in which the output space has a natural metric or similarity structure, which is known (or estimated) a priori. In practice, many learning problems have such structure. In the ImageNet Large Scale Visual Recognition Challenge [ILSVRC] [2], for example, the
output dimensions correspond to 1000 object categories that have inherent semantic relationships,
some of which are captured in the WordNet hierarchy that accompanies the categories. Similarly, in
the keyword spotting task from the IARPA Babel speech recognition project, the outputs correspond
to keywords that likewise have semantic relationships. In what follows, we will call the similarity
structure on the label space the ground metric or semantic similarity.
Using the ground metric, we can measure prediction performance in a way that is sensitive to relationships between the different output dimensions. For example, confusing dogs with cats might
?
1
Authors contributed equally.
Code and data are available at http://cbcl.mit.edu/wasserstein.
1
Divergence
3
4
Wasserstein
5
Distance
Distance
0.3
0.2
0.1
0.0
6
7
0.4
0.3
0.2
0.1
Divergence
0.1
Grid Size
0.2
0.3
Wasserstein
0.4
0.5
0.6
0.7
0.8
0.9
Noise
Figure 2: The Wasserstein loss encourages predictions that are similar to ground truth, robustly
to incorrect labeling of similar classes (see Appendix E.1). Shown is Euclidean distance between
prediction and ground truth vs. (left) number of classes, averaged over different noise levels and
(right) noise level, averaged over number of classes. Baseline is the multiclass logistic loss.
be more severe an error than confusing breeds of dogs. A loss function that incorporates this metric
might encourage the learning algorithm to favor predictions that are, if not completely accurate, at
least semantically similar to the ground truth.
In this paper, we develop a loss function for multi-label learning that measures the Wasserstein distance between a prediction
and the target label, with respect to a chosen metric on the output space. The Wasserstein distance is defined as the cost of
the optimal transport plan for moving the mass in the predicted
measure to match that in the target, and has been applied to a
wide range of problems, including barycenter estimation [3], label propagation [4], and clustering [5]. To our knowledge, this
paper represents the first use of the Wasserstein distance as a
loss for supervised learning.
Siberian husky
Eskimo dog
Figure 1: Semantically nearequivalent classes in ILSVRC
We briefly describe a case in which the Wasserstein loss improves learning performance. The setting
is a multiclass classification problem in which label noise arises from confusion of semantically
near-equivalent categories. Figure 1 shows such a case from the ILSVRC, in which the categories
Siberian husky and Eskimo dog are nearly indistinguishable. We synthesize a toy version of this
problem by identifying categories with points in the Euclidean plane and randomly switching the
training labels to nearby classes. The Wasserstein loss yields predictions that are closer to the ground
truth, robustly across all noise levels, as shown in Figure 2. The standard multiclass logistic loss is
the baseline for comparison. Section E.1 in the Appendix describes the experiment in more detail.
The main contributions of this paper are as follows. We formulate the problem of learning with prior
knowledge of the ground metric, and propose the Wasserstein loss as an alternative to traditional
information divergence-based loss functions. Specifically, we focus on empirical risk minimization
(ERM) with the Wasserstein loss, and describe an efficient learning algorithm based on entropic
regularization of the optimal transport problem. We also describe a novel extension to unnormalized
measures that is similarly efficient to compute. We then justify ERM with the Wasserstein loss
by showing a statistical learning bound. Finally, we evaluate the proposed loss on both synthetic
examples and a real-world image annotation problem, demonstrating benefits for incorporating an
output metric into the loss.
2
Related work
Decomposable loss functions like KL Divergence and `p distances are very popular for probabilistic [1] or vector-valued [6] predictions, as each component can be evaluated independently, often
leading to simple and efficient algorithms. The idea of exploiting smoothness in the label space
according to a prior metric has been explored in many different forms, including regularization [7]
and post-processing with graphical models [8]. Optimal transport provides a natural distance for
probability distributions over metric spaces. In [3, 9], the optimal transport is used to formulate
the Wasserstein barycenter as a probability distribution with minimum total Wasserstein distance
to a set of given points on the probability simplex. [4] propagates histogram values on a graph by
minimizing a Dirichlet energy induced by optimal transport. The Wasserstein distance is also used
to formulate a metric for comparing clusters in [5], and is applied to image retrieval [10], contour
2
matching [11], and many other problems [12, 13]. However, to our knowledge, this is the first time
it is used as a loss function in a discriminative learning framework. The closest work to this paper is a theoretical study [14] of an estimator that minimizes the optimal transport cost between the
empirical distribution and the estimated distribution in the setting of statistical parameter estimation.
3
3.1
Learning with a Wasserstein loss
Problem setup and notation
We consider the problem of learning a map from X ? RD into the space Y = RK
+ of measures over
a finite set K of size |K| = K. Assume K possesses a metric dK (?, ?), which is called the ground
metric. dK measures semantic similarity between dimensions of the output, which correspond to
the elements of K. We perform learning over a hypothesis space H of predictors h? : X ! Y,
parameterized by ? 2 ?. These might be linear logistic regression models, for example.
In the standard statistical learning setting, we get an i.i.d. sequence of training examples S =
((x1 , y1 ), . . . , (xN , yN )), sampled from an unknown joint distribution PX ?Y . Given a measure of
performance (a.k.a. risk) E(?, ?), the goal is to find the predictor h? 2 H that minimizes the expected
risk E[E(h? (x), y)]. Typically E(?, ?) is difficult to optimize directly and the joint distribution PX ?Y
is unknown, so learning is performed via empirical risk minimization. Specifically, we solve
(
)
N
1 X
?
min ES [`(h? (x), y) =
`(h? (xi ), yi )
(1)
h? 2H
N i=1
with a loss function `(?, ?) acting as a surrogate of E(?, ?).
3.2
Optimal transport and the exact Wasserstein loss
Information divergence-based loss functions are widely used in learning with probability-valued outputs. Along with other popular measures like Hellinger distance and 2 distance, these divergences
treat the output dimensions independently, ignoring any metric structure on K.
Given a cost function c : K ? K ! R, the optimal transport distance [15] measures the cheapest
way to transport the mass in probability measure ?1 to match that in ?2 :
Z
Wc (?1 , ?2 ) =
inf
c(?1 , ?2 ) (d?1 , d?2 )
(2)
2?(?1 ,?2 )
K?K
where ?(?1 , ?2 ) is the set of joint probability measures on K?K having ?1 and ?2 as marginals. An
important case is that in which the cost is given by a metric dK (?, ?) or its p-th power dpK (?, ?) with p
1. In this case, (2) is called a Wasserstein distance [16], also known as the earth mover?s distance
[10]. In this paper, we only work with discrete measures. In the case of probability measures, these
are histograms in the simplex K . When the ground truth y and the output of h both lie in the
simplex K , we can define a Wasserstein loss.
Definition 3.1 (Exact Wasserstein Loss). For any h? 2 H, h? : X ! K , let h? (?|x) = h? (x)? be
the predicted value at element ? 2 K, given input x 2 X . Let y(?) be the ground truth value for ?
given by the corresponding label y. Then we define the exact Wasserstein loss as
Wpp (h(?|x), y(?)) =
inf
T 2?(h(x),y)
hT, M i
(3)
where M 2 RK?K
is the distance matrix M?,?0 = dpK (?, ?0 ), and the set of valid transport plans is
+
?(h(x), y) = {T 2 RK?K
: T 1 = h(x), T > 1 = y}
+
(4)
where 1 is the all-one vector.
Wpp is the cost of the optimal plan for transporting the predicted mass distribution h(x) to match
the target distribution y. The penalty increases as more mass is transported over longer distances,
according to the ground metric M .
3
Algorithm 1 Gradient of the Wasserstein loss
Given h(x), y, , K. ( a , b if h(x), y unnormalized.)
u
1
while u has
8 not converged do
>
>
< h(x) ? K y ? K u
?
? a +1
a
b
u
a
>
+1 ?
b +1
>
a
h(x)
K
y
?
K
u
:
if h(x), y normalized
if h(x), y unnormalized
end while
b
b
If h(x), y unnormalized: v
y b +1 ? K> u b +1
?
log u
log u> 1
if h(x), y normalized
K 1
@Wpp /@h(x)
(diag(u)Kv) ? h(x)) if h(x), y unnormalized
a (1
4
Efficient optimization via entropic regularization
To do learning, we optimize the empirical risk minimization functional (1) by gradient descent.
Doing so requires evaluating a descent direction for the loss, with respect to the predictions h(x).
Unfortunately, computing a subgradient of the exact Wasserstein loss (3), is quite costly, as follows.
The exact Wasserstein loss (3) is a linear program and a subgradient of its solution can be computed
using Lagrange duality. The dual LP of (3) is
d
Wpp (h(x), y) =
sup ?> h(x) +
>
?, 2CM
y,
CM = {(?, ) 2 RK?K : ?? +
?0
? M?,?0 }. (5)
As (3) is a linear program, at an optimum the values of the dual and the primal are equal (see, e.g.
[17]), hence the dual optimal ? is a subgradient of the loss with respect to its first argument.
Computing ? is costly, as it entails solving a linear program with O(K 2 ) contraints, with K being
the dimension of the output space. This cost can be prohibitive when optimizing by gradient descent.
4.1
Entropic regularization of optimal transport
Cuturi [18] proposes a smoothed transport objective that enables efficient approximation of both the
transport matrix in (3) and the subgradient of the loss. [18] introduces an entropic regularization
term that results in a strictly convex problem:
X
1
Wpp (h(?|x), y(?)) =
inf
hT, M i
H(T ), H(T ) =
T?,?0 log T?,?0 . (6)
T 2?(h(x),y)
?,?0
Importantly, the transport matrix that solves (6) is a diagonal scaling of a matrix K = e
?
T = diag(u)Kdiag(v)
for u = e
?
and v = e
, where ? and
M
1
:
(7)
are the Lagrange dual variables for (6).
Identifying such a matrix subject to equality constraints on the row and column sums is exactly a
matrix balancing problem, which is well-studied in numerical linear algebra and for which efficient
iterative algorithms exist [19]. [18] and [3] use the well-known Sinkhorn-Knopp algorithm.
4.2
Extending smoothed transport to the learning setting
When the output vectors h(x) and y lie in the simplex, (6) can be used directly in place of (3), as
(6) can approximate the exact Wasserstein distance closely for large enough [18]. In this case, the
>
gradient ? of the objective can be obtained from the optimal scaling vector u as ? = log u log uK 1 1.
1
A Sinkhorn iteration for the gradient is given in Algorithm 1.
1
Note that ? is only defined up to a constant shift: any upscaling of the vector u can be paired with a
corresponding downscaling of the vector v (and vice versa) without altering the matrix T ? . The choice ? =
log u
log u> 1
1 ensures that ? is tangent to the simplex.
K
4
(a) Convergence to smoothed trans- (b) Approximation
port.
Wasserstein.
of
exact (c) Convergence of alternating projections ( = 50).
Figure 3: The relaxed transport problem (8) for unnormalized measures.
For many learning problems, however, a normalized output assumption is unnatural. In image segmentation, for example, the target shape is not naturally represented as a histogram. And even when
the prediction and the ground truth are constrained to the simplex, the observed label can be subject
to noise that violates the constraint.
There is more than one way to generalize optimal transport to unnormalized measures, and this is a
subject of active study [20]. We will develop here a novel objective that deals effectively with the
difference in total mass between h(x) and y while still being efficient to optimize.
4.3
Relaxed transport
We propose a novel relaxation that extends smoothed transport to unnormalized measures. By replacing the equality constraints on the transport marginals in (6) with soft penalties with respect to
KL divergence, we get an unconstrained approximate transport problem. The resulting objective is:
,
a, b
WKL (h(?|x), y(?)) =
min hT, M i
1
T 2RK?K
+
H(T ) +
f
f T > 1ky (8)
a KL (T 1kh(x)) + b KL
f (wkz) = w> log(w ? z) 1> w + 1> z is the generalized KL divergence between
where KL
w, z 2 RK
+ . Here ? represents element-wise division. As with the previous formulation, the optimal
transport matrix with respect to (8) is a diagonal scaling of the matrix K.
Proposition 4.1. The transport matrix T ? optimizing (8) satisfies T ? = diag(u)Kdiag(v), where
u = (h(x) ? T ? 1) a , v = y ? (T ? )> 1 b , and K = e M 1 .
And the optimal transport matrix is a fixed point for a Sinkhorn-like iteration.
Proposition 4.2. T ? = diag(u)Kdiag(v) optimizing (8) satisfies: i) u = h(x)
and ii) v = y
b
b +1
K> u
b
b +1
, where
2
a
a +1
(Kv)
a
a +1
,
represents element-wise multiplication.
Unlike the previous formulation, (8) is unconstrained with respect to h(x). The gradient is given by
rh(x) WKL (h(?|x), y(?)) = a (1 T ? 1 ? h(x)). The iteration is given in Algorithm 1.
When restricted to normalized measures, the relaxed problem (8) approximates smoothed transport
(6). Figure 3a shows, for normalized h(x) and y, the relative distance between the values of (8) and
(6) 3 . For large enough, (8) converges to (6) as a and b increase.
(8) also retains two properties of smoothed transport (6). Figure 3b shows that, for normalized
outputs, the relaxed loss converges to the unregularized Wasserstein distance as , a and b increase
4
. And Figure 3c shows that convergence of the iterations in (4.2) is nearly independent of the
dimension K of the output space.
2
Note that, although the iteration suggested by Proposition 4.2 is observed empirically to converge (see
Figure 3c, for example), we have not proven a guarantee that it will do so.
3
In figures 3a-c, h(x), y and M are generated as described in [18] section 5. In 3a-b, h(x) and y have
dimension 256. In 3c, convergence is defined as in [18]. Shaded regions are 95% intervals.
4
The unregularized Wasserstein distance was computed using FastEMD [21].
5
0
1
0
1
2
3
Posterior Probability
Posterior Probability
0.20
0.18
0.16
0.14
0.12
0.10
0.08
2
3
0.20
0.18
0.16
0.14
0.12
0.10
0.08
4
2
3
4
0
1
p-th norm
2
3
5
6
4
p-th norm
(a) Posterior predictions for images of digit 0.
(b) Posterior predictions for images of digit 4.
Figure 4: MNIST example. Each curve shows the predicted probability for one digit, for models
trained with different p values for the ground metric.
5
Statistical Properties of the Wasserstein loss
Let S = ((x1 , y1 ), . . . , (xN , yN )) be i.i.d. samples and h?? be the empirical risk minimizer
(
)
N
X
? p
?
1
? S W (h? (?|x), y) =
h?? = argmin E
W p (hx ?(?|xi ), yi ) .
p
N i=1 p
h? 2H
Further assume H = s Ho is the composition of a softmax s and a base hypothesis space Ho of
functions mapping into RK . The softmax layer outputs a prediction that lies in the simplex K .
Theorem 5.1. For p = 1, and any
> 0, with probability at least 1
, it holds that
r
? 1
?
? 1
?
log(1/ )
o
E W1 (h??(?|x), y) ? inf E W1 (h? (?|x), y) + 32KCM RN (H ) + 2CM
h? 2H
2N
(9)
with the constant CM = max?,?0 M?,?0 . RN (Ho ) is the Rademacher complexity [22] measuring
the complexity of the hypothesis space Ho .
The Rademacher complexity RN (Ho ) for commonly used models like neural networks and kernel
machines [22] decays with the training set size. This theorem guarantees that the expected Wasserstein loss of the empirical risk minimizer approaches the best achievable loss for H.
As an important special case, minimizing the empirical risk with Wasserstein loss is also good for
multiclass classification. Let y = ? be the ?one-hot? encoded label vector for the groundtruth class.
Proposition 5.2. In the multiclass classification setting, for p = 1 and any
at least 1
, it holds that
?
?
Ex,? dK (???(x), ?) ? inf
h? 2H
> 0, with probability
KE[W11 (h? (x), y)] + 32K 2 CM RN (Ho ) + 2CM K
r
log(1/ )
(10)
2N
where the predictor is ???(x) = argmax? h??(?|x), with h?? being the empirical risk minimizer.
Note that instead of the classification error Ex,? [ {???(x) 6= ?}], we actually get a bound on the
expected semantic distance between the prediction and the groundtruth.
6
6.1
Empirical study
Impact of the ground metric
In this section, we show that the Wasserstein loss encourages smoothness with respect to an artificial
metric on the MNIST handwritten digit dataset. This is a multi-class classification problem with
output dimensions corresponding to the 10 digits, and we apply a ground metric dp (?, ?0 ) = |?
?0 |p , where ?, ?0 2 {0, . . . , 9} and p 2 [0, 1). This metric encourages the recognized digit to be
numerically close to the true one. We train a model independently for each value of p and plot the
average predicted probabilities of the different digits on the test set in Figure 4.
6
1.00
0.95
0.95
0.90
0.90
0.85
top-K Cost
top-K Cost
1.00
Loss Function
0.80
Divergence
Wasserstein (?=0.5)
0.75
0.70
5
10
0.85
Loss Function
0.80
Divergence
Wasserstein (?=0.5)
Wasserstein (?=0.3)
0.75
Wasserstein (?=0.3)
Wasserstein (?=0.1)
0.70
Wasserstein (?=0.1)
15
20
5
K (# of proposed tags)
10
15
20
K (# of proposed tags)
(a) Original Flickr tags dataset.
(b) Reduced-redundancy Flickr tags dataset.
Figure 5: Top-K cost comparison of the proposed loss (Wasserstein) and the baseline (Divergence).
Note that as p ! 0, the metric approaches the 0 1 metric d0 (?, ?0 ) = ?6=?0 , which treats all
incorrect digits as being equally unfavorable. In this case, as can be seen in the figure, the predicted
probability of the true digit goes to 1 while the probability for all other digits goes to 0. As p
increases, the predictions become more evenly distributed over the neighboring digits, converging
to a uniform distribution as p ! 1 5 .
6.2
Flickr tag prediction
We apply the Wasserstein loss to a real world multi-label learning problem, using the recently released Yahoo/Flickr Creative Commons 100M dataset [23]. 6 Our goal is tag prediction: we select
1000 descriptive tags along with two random sets of 10,000 images each, associated with these tags,
for training and testing. We derive a distance metric between tags by using word2vec [24] to
embed the tags as unit vectors, then taking their Euclidean distances. To extract image features we
use MatConvNet [25]. Note that the set of tags is highly redundant and often many semantically
equivalent or similar tags can apply to an image. The images are also partially tagged, as different
users may prefer different tags. We therefore measure the prediction performance by the top-K cost,
PK
defined as CK = 1/K k=1 minj dK (?
?k , ?j ), where {?j } is the set of groundtruth tags, and {?
?k }
are the tags with highest predicted probability. The standard AUC measure is also reported.
We find that a linear combination of the Wasserstein loss Wpp and the standard multiclass logistic loss
KL yields the best prediction results. Specifically, we train a linear model by minimizing Wpp + ?KL
on the training set, where ? controls the relative weight of KL. Note that KL taken alone is our
baseline in these experiments. Figure 5a shows the top-K cost on the test set for the combined loss
and the baseline KL loss. We additionally create a second dataset by removing redundant labels
from the original dataset: this simulates the potentially more difficult case in which a single user
tags each image, by selecting one tag to apply from amongst each cluster of applicable, semantically
similar tags. Figure 3b shows that performance for both algorithms decreases on the harder dataset,
while the combined Wasserstein loss continues to outperform the baseline.
In Figure 6, we show the effect on performance of varying the weight ? on the KL loss. We observe
that the optimum of the top-K cost is achieved when the Wasserstein loss is weighted more heavily
than at the optimum of the AUC. This is consistent with a semantic smoothing effect of Wasserstein,
which during training will favor mispredictions that are semantically similar to the ground truth,
sometimes at the cost of lower AUC 7 . We finally show two selected images from the test set in
Figure 7. These illustrate cases in which both algorithms make predictions that are semantically
relevant, despite overlapping very little with the ground truth. The image on the left shows errors
made by both algorithms. More examples can be found in the appendix.
5
To avoid numerical issues, we scale down the ground metric such that all of the distance values are in the
interval [0, 1).
6
The dataset used here is available at http://cbcl.mit.edu/wasserstein.
7
The Wasserstein loss can achieve a similar trade-off by choosing the metric parameter p, as discussed in
Section 6.1. However, the relationship between p and the smoothing behavior is complex and it can be simpler
to implement the trade-off by combining with the KL loss.
7
0.0
0.5
K=2
K=3
1.0
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.64 0.0
0.62
0.60
0.58
0.56
0.54
0.5
0.0
0.5
K=4
1.5
2.0
Wasserstein AUC
Divergence AUC
1.0
Top-K cost
0.5
AUC
0.64 0.0
0.62
0.60
0.58
0.56
0.54
K=1
AUC
Top-K cost
0.95
0.90
0.85
0.80
0.75
0.70
0.65
1.5
2.0
?
K=1
K=2
K=3
1.0
K=4
1.5
2.0
Wasserstein AUC
Divergence AUC
1.0
1.5
2.0
?
(a) Original Flickr tags dataset.
(b) Reduced-redundancy Flickr tags dataset.
Figure 6: Trade-off between semantic smoothness and maximum likelihood.
(a) Flickr user tags: street, parade, dragon; our
proposals: people, protest, parade; baseline proposals: music, car, band.
(b) Flickr user tags: water, boat, reflection, sunshine; our proposals: water, river, lake, summer;
baseline proposals: river, water, club, nature.
Figure 7: Examples of images in the Flickr dataset. We show the groundtruth tags and as well as
tags proposed by our algorithm and the baseline.
7
Conclusions and future work
In this paper we have described a loss function for learning to predict a non-negative measure over a
finite set, based on the Wasserstein distance. Although optimizing with respect to the exact Wasserstein loss is computationally costly, an approximation based on entropic regularization is efficiently
computed. We described a learning algorithm based on this regularization and we proposed a novel
extension of the regularized loss to unnormalized measures that preserves its efficiency. We also
described a statistical learning bound for the loss. The Wasserstein loss can encourage smoothness
of the predictions with respect to a chosen metric on the output space, and we demonstrated this
property on a real-data tag prediction problem, showing improved performance over a baseline that
doesn?t incorporate the metric.
An interesting direction for future work may be to explore the connection between the Wasserstein
loss and Markov random fields, as the latter are often used to encourage smoothness of predictions,
via inference at prediction time.
8
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9
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5,169 | 568 | Networks with Learned Unit Response Functions
John Moody and Norman Yarvin
Yale Computer Science, 51 Prospect St.
P.O. Box 2158 Yale Station, New Haven, CT 06520-2158
Abstract
Feedforward networks composed of units which compute a sigmoidal function of a weighted sum of their inputs have been much investigated. We
tested the approximation and estimation capabilities of networks using
functions more complex than sigmoids. Three classes of functions were
tested: polynomials, rational functions, and flexible Fourier series. Unlike sigmoids, these classes can fit non-monotonic functions. They were
compared on three problems: prediction of Boston housing prices, the
sunspot count, and robot arm inverse dynamics. The complex units attained clearly superior performance on the robot arm problem, which is
a highly non-monotonic, pure approximation problem. On the noisy and
only mildly nonlinear Boston housing and sunspot problems, differences
among the complex units were revealed; polynomials did poorly, whereas
rationals and flexible Fourier series were comparable to sigmoids.
1
Introduction
A commonly studied neural architecture is the feedforward network in which each
unit of the network computes a nonlinear function g( x) of a weighted sum of its
inputs x
wtu. Generally this function is a sigmoid, such as g( x)
tanh x or
g(x) = 1/(1 + e(x-9?). To these we compared units of a substantially different
type: they also compute a nonlinear function of a weighted sum of their inputs,
but the unit response function is able to fit a much higher degree of nonlinearity
than can a sigmoid. The nonlinearities we considered were polynomials, rational
functions (ratios of polynomials), and flexible Fourier series (sums of cosines.) Our
comparisons were done in the context of two-layer networks consisting of one hidden
layer of complex units and an output layer of a single linear unit.
=
1048
=
Networks with Learned Unit Response Functions
This network architecture is similar to that built by projection pursuit regression
(PPR) [1, 2], another technique for function approximation. The one difference is
that in PPR the nonlinear function of the units of the hidden layer is a nonparametric smooth. This nonparametric smooth has two disadvantages for neural modeling:
it has many parameters, and, as a smooth, it is easily trained only if desired output
values are available for that particular unit. The latter property makes the use of
smooths in multilayer networks inconvenient. If a parametrized function of a type
suitable for one-dimensional function approximation is used instead of the nonparametric smooth, then these disadvantages do not apply. The functions we used are
all suitable for one-dimensional function approximation.
2
Representation
A few details of the representation of the unit response functions are worth noting.
Polynomials: Each polynomial unit computed the function
g(x) = alX + a2x2 + ... + anx n
=
with x wT u being the weighted sum of the input. A zero'th order term was not
included in the above formula, since it would have been redundant among all the
units. The zero'th order term was dealt with separately and only stored in one
location.
Rationals: A rational function representation was adopted which could not have
zeros in the denominator. This representation used a sum of squares of polynomials,
as follows:
ao + alx + ... + anx n
9 (x ) - 1 + (b o + b1x)2 + (b 2x + b3 x2)2 + (b 4x + b5x 2 + b6X3 + b7x4)2 + .,.
This representation has the qualities that the denominator is never less than 1,
and that n parameters are used to produce a denominator of degree n. If the above
formula were continued the next terms in the denominator would be of degrees eight,
sixteen, and thirty-two. This powers-of-two sequence was used for the following
reason: of the 2( n - m) terms in the square of a polynomial p = am xm + '" + anx n ,
it is possible by manipulating am ... a n to determine the n - m highest coefficients,
with the exception that the very highest coefficient must be non-negative. Thus
if we consider the coefficients of the polynomial that results from squaring and
adding together the terms of the denominator of the above formula, the highest
degree squared polynomial may be regarded as determining the highest half of the
coefficients, the second highest degree polynomial may be regarded as determining
the highest half of the rest of the coefficients, and so forth. This process cannot set
all the coefficients arbitrarily; some must be non-negative.
Flexible Fourier series: The flexible Fourier series units computed
n
g(x) =
L: ai COS(bi X + Ci)
i=O
where the amplitudes ai, frequencies bi and phases Ci were unconstrained and could
assume any value.
1049
1050
Moody and Yarvin
Sigmoids: We used the standard logistic function:
g(x) = 1/(1 + e(x-9))
3
Training Method
All the results presented here were trained with the Levenberg-Marquardt modification of the Gauss-Newton nonlinear least squares algorithm. Stochastic gradient
descent was also tried at first, but on the problems where the two were compared,
Levenberg- Marquardt was much superior both in convergence time and in quality of
result. Levenberg-Marquardt required substantially fewer iterations than stochastic gradient descent to converge. However, it needs O(p2) space and O(p 2n) time
per iteration in a network with p parameters and n input examples, as compared
to O(p) space and O(pn) time per epoch for stochastic gradient descent. Further
details of the training method will be discussed in a longer paper.
With some data sets, a weight decay term was added to the energy function to be
optimized. The added term was of the form A L~=l
When weight decay was
used, a range of values of A was tried for every network trained.
w;.
Before training, all the data was normalized: each input variable was scaled so that
its range was (-1,1), then scaled so that the maximum sum of squares of input
variables for any example was 1. The output variable was scaled to have mean zero
and mean absolute value 1. This helped the training algorithm, especially in the
case of stochastic gradient descent.
4
Results
We present results of training our networks on three data sets: robot arm inverse
dynamics, Boston housing data, and sunspot count prediction. The Boston and
sunspot data sets are noisy, but have only mild nonlinearity. The robot arm inverse
dynamics data has no noise, but a high degree of nonlinearity. Noise-free problems
have low estimation error. Models for linear or mildly nonlinear problems typically
have low approximation error. The robot arm inverse dynamics problem is thus a
pure approximation problem, while performance on the noisy Boston and sunspots
problems is limited more by estimation error than by approximation error.
Figure la is a graph, as those used in PPR, of the unit response function of a oneunit network trained on the Boston housing data. The x axis is a projection (a
weighted sum of inputs wT u) of the 13-dimensional input space onto 1 dimension,
using those weights chosen by the unit in training. The y axis is the fit to data. The
response function of the unit is a sum ofthree cosines. Figure Ib is the superposition
of five graphs of the five unit response functions used in a five-unit rational function
solution (RMS error less than 2%) of the robot arm inverse dynamics problem. The
domain for each curve lies along a different direction in the six-dimensional input
space. Four of the five fits along the projection directions are non-monotonic, and
thus can be fit only poorly by a sigmoid.
Two different error measures are used in the following. The first is the RMS error,
normalized so that error of 1 corresponds to no training. The second measure is the
Networks with Learned Unit Response Functions
Robot arm fit to data
40
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b
a
square of the normalized RMS error, otherwise known as the fraction of explained
varIance. We used whichever error measure was used in earlier work on that data
set.
4.1
Robot arm inverse dynamics
This problem is the determination of the torque necessary at the joints of a twojoint robot arm required to achieve a given acceleration of each segment of the
arm , given each segment's velocity and position. There are six input variables to
the network, and two output variables. This problem was treated as two separate
estimation problems, one for the shoulder torque and one for the elbow torque. The
shoulder torque was a slightly more difficult problem, for almost all networks. The
1000 points in the training set covered the input space relatively thoroughly. This,
together with the fact that the problem had no noise, meant that there was little
difference between training set error and test set error.
Polynomial networks of limited degree are not universal approximators, and that
is quite evident on this data set; polynomial networks of low degree reached their
minimum error after a few units. Figure 2a shows this. If polynomial, cosine, rational, and sigmoid networks are compared as in Figure 2b, leaving out low degree
polynomials , the sigmoids have relatively high approximation error even for networks with 20 units. As shown in the following table, the complex units have more
parameters each, but still get better performance with fewer parameters total.
Type
degree 7 polynomial
degree 6 rational
2 term cosine
sigmoid
sigmoid
Units
5
5
6
10
20
Parameters
65
95
73
81
161
Error
.024
.027
.020
.139
.119
Since the training set is noise-free, these errors represent pure approximation error .
1051
1052
Moody and Yarvin
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The superior performance of the complex units on this problem is probably due to
their ability to approximate non-monotonic functions.
4.2
Boston housing
The second data set is a benchmark for statistical algorithms: the prediction of
Boston housing prices from 13 factors [3]. This data set contains 506 exemplars and
is relatively simple; it can be approximated well with only a single unit. Networks
of between one and six units were trained on this problem. Figure 3a is a graph
of training set performance from networks trained on the entire data set; the error
measure used was the fraction of explained variance. From this graph it is apparent
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a
b
Networks with Learned Unit Response Functions
1053
that training set performance does not vary greatly between different types of units,
though networks with more units do better.
On the test set there is a large difference. This is shown in Figure 3b. Each point
on the graph is the average performance of ten networks of that type. Each network
was trained using a different permutation of the data into test and training sets, the
test set being 1/3 of the examples and the training set 2/3. It can be seen that the
cosine nets perform the best, the sigmoid nets a close second, the rationals third,
and the polynomials worst (with the error increasing quite a bit with increasing
polynomial degree.)
It should be noted that the distribution of errors is far from a normal distribution,
and that the training set error gives little clue as to the test set error. The following
table of errors, for nine networks of four units using a degree 5 polynomial, is
somewhat typical:
Set
training
test
Error
0.091
0.395
I
Our speculation on the cause of these extremely high errors is that polynomial approximations do not extrapolate well; if the prediction of some data point results in
a polynomial being evaluated slightly outside the region on which the polynomial
was trained, the error may be extremely high. Rational functions where the numerator and denominator have equal degree have less of a problem with this, since
asymptotically they are constant. However, over small intervals they can have the
extrapolation characteristics of polynomials. Cosines are bounded, and so, though
they may not extrapolate well if the function is not somewhat periodic, at least do
not reach large values like polynomials.
4.3
Sunspots
The third problem was the prediction of the average monthly sunspot count in a
given year from the values of the previous twelve years. We followed previous work
in using as our error measure the fraction of variance explained, and in using as
the training set the years 1700 through 1920 and as the test set the years 1921
through 1955. This was a relatively easy test set - every network of one unit which
we trained (whether sigmoid, polynomial, rational, or cosine) had, in each of ten
runs, a training set error between .147 and .153 and a test set error between .105
and .111. For comparison, the best test set error achieved by us or previous testers
was about .085. A similar set of runs was done as those for the Boston housing
data, but using at most four units; similar results were obtained. Figure 4a shows
training set error and Figure 4b shows test set error on this problem.
4.4
Weight Decay
The performance of almost all networks was improved by some amount of weight
decay. Figure 5 contains graphs of test set error for sigmoidal and polynomial units,
1054
Moody and Yarvin
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number of WIlle
Figure 4:
a
2
3
Dumb .... of unit.
b
using various values of the weight decay parameter A. For the sigmoids, very little
weight decay seems to be needed to give good results, and there is an order of
magnitude range (between .001 and .01) which produces close to optimal results.
For polynomials of degree 5, more weight decay seems to be necessary for good
results; in fact, the highest value of weight decay is the best. Since very high values
of weight decay are needed, and at those values there is little improvement over
using a single unit, it may be supposed that using those values of weight decay
restricts the multiple units to producing a very similar solution to the one-unit
solution. Figure 6 contains the corresponding graphs for sunspots. Weight decay
seems to help less here for the sigmoids, but for the polynomials, moderate amounts
of weight decay produce an improvement over the one-unit solution.
Acknowledgements
The authors would like to acknowledge support from ONR grant N00014-89-J1228, AFOSR grant 89-0478, and a fellowship from the John and Fannie Hertz
Foundation. The robot arm data set was provided by Chris Atkeson.
References
[1] J. H. Friedman, W. Stuetzle, "Projection Pursuit Regression", Journal of the
American Statistical Association, December 1981, Volume 76, Number 376,
817-823
[2] P. J. Huber, "Projection Pursuit", The Annals of Statistics, 1985 Vol. 13 No.
2,435-475
[3] L. Breiman et aI, Classification and Regression Trees, Wadsworth and Brooks,
1984, pp217-220
Networks with Learned Unit Response Functions
0.30
Boston housin
hi decay
r-T"=::...:..:;.:;:....:r:-=::;.5I~;=::::..:;=:-;;..:..:..::.....;;-=..:.!ar:......::=~...,
00
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00
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0.001
1.0
0.01
X.l
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0.25
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0.5
0.15
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moids wilh wei hl decay
0. 16
0. 111
1.8
0.14
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O.IB
00
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0 .001
0 .01
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0.08
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Figure 6: Sunspot test error with various amounts of weight decay
1055
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5,170 | 5,680 | Principal Geodesic Analysis for Probability Measures
under the Optimal Transport Metric
Vivien Seguy
Graduate School of Informatics
Kyoto University
[email protected]
Marco Cuturi
Graduate School of Informatics
Kyoto University
[email protected]
Abstract
Given a family of probability measures in P (X ), the space of probability measures on a Hilbert space X , our goal in this paper is to highlight one ore more
curves in P (X ) that summarize efficiently that family. We propose to study this
problem under the optimal transport (Wasserstein) geometry, using curves that
are restricted to be geodesic segments under that metric. We show that concepts
that play a key role in Euclidean PCA, such as data centering or orthogonality of
principal directions, find a natural equivalent in the optimal transport geometry,
using Wasserstein means and differential geometry. The implementation of these
ideas is, however, computationally challenging. To achieve scalable algorithms
that can handle thousands of measures, we propose to use a relaxed definition
for geodesics and regularized optimal transport distances. The interest of our approach is demonstrated on images seen either as shapes or color histograms.
1
Introduction
Optimal transport distances (Villani, 2008), a.k.a Wasserstein or earth mover?s distances, define
a powerful geometry to compare probability measures supported on a metric space X . The
Wasserstein space P (X )?the space of probability measures on X endowed with the Wasserstein
distance?is a metric space which has received ample interest from a theoretical perspective. Given
the prominent role played by probability measures and feature histograms in machine learning, the
properties of P (X ) can also have practical implications in data science. This was shown by Agueh
and Carlier (2011) who described first Wasserstein means of probability measures. Wasserstein
means have been recently used in Bayesian inference (Srivastava et al., 2015), clustering (Cuturi
and Doucet, 2014), graphics (Solomon et al., 2015) or brain imaging (Gramfort et al., 2015). When
X is not just metric but also a Hilbert space, P (X ) is an infinite-dimensional Riemannian manifold
(Ambrosio et al. 2006, Chap. 8; Villani 2008, Part II). Three recent contributions by Boissard et al.
(2015, ?5.2), Bigot et al. (2015) and Wang et al. (2013) exploit directly or indirectly this structure to
extend Principal Component Analysis (PCA) to P (X ). These important seminal papers are, however, limited in their applicability and/or the type of curves they output. Our goal in this paper is to
propose more general and scalable algorithms to carry out Wasserstein principal geodesic analysis
on probability measures, and not simply dimensionality reduction as explained below.
Principal Geodesics in P (X ) vs. Dimensionality Reduction on P (X ) We provide in Fig. 1 a
simple example that illustrates the motivation of this paper, and which also shows how our approach
differentiates itself from existing dimensionality reduction algorithms (linear and non-linear) that
draw inspiration from PCA. As shown in Fig. 1, linear PCA cannot produce components that remain in P (X ). Even more advanced tools, such as those proposed by Hastie and Stuetzle (1989),
fall slightly short of that goal. On the other hand, Wasserstein geodesic analysis yields geodesic
components in P (X ) that are easy to interpret and which can also be used to reduce dimensionality.
1
P (X )
Wasserstein Principal Geodesics
Euclidean Principal Components
Principal Curve
Figure 1: (top-left) Dataset: 60 ? 60 images of a single Chinese character randomly translated,
scaled and slightly rotated (36 images displayed out of 300 used). Each image is handled as a
normalized histogram of 3, 600 non-negative intensities. (middle-left) Dataset schematically drawn
on P (X ). The Wasserstein principal geodesics of this dataset are depicted in red, its Euclidean
components in blue, and its principal curve (Verbeek et al., 2002) in yellow. (right) Actual curves
(blue colors depict negative intensities, green intensities ? 1). Neither the Euclidean components
nor the principal curve belong to P (X ), nor can they be interpreted as meaningful axis of variation.
Foundations of PCA and Riemannian Extensions Carrying out PCA on a family (x1 , . . . , xn )
of points taken in a space X can be described in abstract terms as: (i) define a mean element x
?
for that dataset; (ii) define a family of components in X, typically geodesic curves, that contain x
?;
(iii) fit a component by making it follow the xi ?s as closely as possible, in the sense that the sum
of the distances of each point xi to that component is minimized; (iv) fit additional components
by iterating step (iii) several times, with the added constraint that each new component is different
(orthogonal) enough to the previous components. When X is Euclidean and the xi ?s are vectors in
Rd , the (n + 1)-th component vn+1 can be computed iteratively by solving:
vn+1 ? argmin
v?Vn? ,||v||2 =1
N
X
i=1
def.
def.
min kxi ? (?
x + tv)k22 , where V0 = ?, and Vn = span{v1 , ? ? ? , vn }. (1)
t?R
Since PCA is known to boil down to a simple eigen-decomposition when X is Euclidean or Hilbertian (Sch?olkopf et al., 1997), Eq. (1) looks artificially complicated. This formulation is, however,
extremely useful to generalize PCA to Riemannian manifolds (Fletcher et al., 2004). This generalization proceeds first by replacing vector means, lines and orthogonality conditions using respectively Fr?echet means (1948), geodesics, and orthogonality in tangent spaces. Riemannian PCA
builds then upon the knowledge of the exponential map at each point x of the manifold X. Each exponential map expx is locally bijective between the tangent space Tx of x and X. After computing
the Fr?echet mean x
? of the dataset, the logarithmic map logx? at x
? (the inverse of expx? ) is used to map
all data points xi onto Tx? . Because Tx? is a Euclidean space by definition of Riemannian manifolds,
the dataset (logx? xi )i can be studied using Euclidean PCA. Principal geodesics in X can then be
recovered by applying the exponential map to a principal component v ? , {expx? (tv ? ), |t| < ?}.
From Riemannian PCA to Wasserstein PCA: Related Work As remarked by Bigot et al.
(2015), Fletcher et al.?s approach cannot be used as it is to define Wasserstein geodesic PCA, because P (X ) is infinite dimensional and because there are no known ways to define exponential
maps which are locally bijective between Wasserstein tangent spaces and the manifold of probability measures. To circumvent this problem, Boissard et al. (2015), Bigot et al. (2015) have proposed
to formulate the geodesic PCA problem directly as an optimization problem over curves in P (X ).
2
Boissard et al. and Bigot et al. study the Wasserstein PCA problem in restricted scenarios: Bigot
et al. focus their attention on measures supported on X = R, which considerably simplifies their
analysis since it is known in that case that the Wasserstein space P (R) can be embedded isometrically in L1 (R); Boissard et al. assume that each input measure has been generated from a single
template density (the mean measure) which has been transformed according to one ?admissible deformation? taken in a parameterized family of deformation maps. Their approach to Wasserstein
PCA boils down to a functional PCA on such maps. Wang et al. proposed a more general approach:
given a family of input empirical measuresP(?1 , . . . , ?N ), they propose to compute first a ?template
measure? ?
? using k-means clustering on i ?i . They consider next all optimal transport plans ?i
between that template ?
? and each of the measures ?i , and propose to compute the barycentric projection (see Eq. 8) of each optimal transport plan ?i to recover Monge maps Ti , on which standard
PCA can be used. This approach is computationally attractive since it requires the computation of
only one optimal transport per input measure. Its weakness lies, however, in the fact that the curves
in P (X ) obtained by displacing ?
? along each of these PCA directions are not geodesics in general.
Contributions and Outline We propose a new algorithm to compute Wasserstein Principal
Geodesics (WPG) in P (X ) for arbitrary Hilbert spaces X . We use several approximations?both
of the optimal transport metric and of its geodesics?to obtain tractable algorithms that can scale
to thousands of measures. We provide first in ?2 a review of the key concepts used in this paper,
namely Wasserstein distances and means, geodesics and tangent spaces in the Wasserstein space.
We propose in ?3 to parameterize a Wasserstein principal component (PC) using two velocity fields
defined on the support of the Wasserstein mean of all measures, and formulate the WPG problem
as that of optimizing these velocity fields so that the average distance of all measures to that PC
is minimal. This problem is non-convex and non-smooth. We propose to optimize smooth upperbounds of that objective using entropy regularized optimal transport in ?4. The practical interest of
our approach is demonstrated in ?5 on toy samples, datasets of shapes and histograms of colors.
Notations We write hA, B i for the Frobenius dot-product of matrices A and B. D(u) is the
diagonal matrix of vector u. For a mapping f : Y ? Y, we say that f acts on a measure ? ? P (Y)
through the pushforward operator # to define a new measure f #? ? P (Y). This measure is
characterized by the identity (f #?)(B) = ?(f ?1 (B)) for any Borel set B ? Y. We write p1 and
p2 for the canonical projection operators X 2 ? X , defined as p1 (x1 , x2 ) = x1 and p2 (x1 , x2 ) = x2 .
2
Background on Optimal Transport
Wasserstein Distances We start this section with the main mathematical object of this paper:
Definition 1. (Villani, 2008, Def. 6.1) Let P (X ) the space of probability measures on a Hilbert
space X . Let ?(?, ?) be the set of probability measures on X 2 with marginals ? and ?, i.e. p1 #? =
? and p2 #? = ?. The squared 2-Wasserstein distance between ? and ? in P (X ) is defined as:
Z
kx ? yk2X d?(x, y).
(2)
W22 (?, ?) =
inf
X2
???(?,?)
Wasserstein Barycenters Given a family of N probability measures (?1 , ? ? ? , ?N ) in P (X ) and
weights ? ? RN
?, the Wasserstein barycenter of these measures:
+ , Agueh and Carlier (2011) define ?
?
? ? argmin
N
X
?i W22 (?i , ?).
??P (X ) i=1
Our paper relies on several algorithms which have been recently proposed (Benamou et al., 2015;
Bonneel et al., 2015; Carlier et al., 2015; Cuturi and Doucet, 2014) to compute such barycenters.
Wasserstein Geodesics Given two measures ? and ?, let ?? (?, ?) be the set of optimal couplings
for Eq. (2). Informally speaking, it is well known that if either ? or ? are absolutely continuous
measures, then any optimal coupling ? ? ? ?? (?, ?) is degenerated in the sense that, assuming for
instance that ? is absolutely continuous, for all x in the support of ? only one point y ? X is
such that d? ? (x, y) > 0. In that case, the optimal transport is said to have no mass splitting, and
3
there exists an optimal mapping T : X ? X such that ? ? can be written, using a pushforward, as
? ? = (id ? T )#?. When there is no mass splitting to transport ? to ?, McCann?s interpolant (1997):
gt = ((1 ? t)id + tT )#?, t ? [0, 1],
(3)
defines a geodesic curve in the Wasserstein space, i.e. (gt )t is locally the shortest path between
any two measures located on the geodesic, with respect to W2 . In the more general case, where no
optimal map T exists and mass splitting occurs (for some locations x one may have d? ? (x, y) > 0
for several y), then a geodesic can still be defined, but it relies on the optimal plan ? ? instead:
gt = ((1 ? t)p1 + tp2 )#? ? , t ? [0, 1], (Ambrosio et al., 2006, ?7.2). Both cases are shown in Fig. 2.
0.4
0.7
0.3
0.2
0.1
0
1.3
1.4
1.5
1.6
?
?
geodesic
g1/3
0.65
g2/3
0.55
1.7
?
?
geodesic
g1/3
0.6
1.8
0.5
g2/3
0.8
1
1.2
1.4
1.6
Figure 2: Both plots display geodesic curves between two empirical measures ? and ? on R2 . An
optimal map exists in the left plot (no mass splitting occurs), whereas some of the mass of ? needs
to be split to be transported onto ? on the right plot.
Tangent Space and Tangent Vectors We briefly describe in this section the tangent spaces of
P (X ), and refer to (Ambrosio et al., 2006, Chap. 8) for more details. Let ? : I ? R ? P (X )
be a curve in P (X ). For a given time t, the tangent space of P (X ) at ?t is a subset of L2 (?t , X ),
the space of square-integrable velocity fields supported on Supp(?t ). At any t, there exists tangent
vectors vt in L2 (?t , X ) such that limh?0 W2 (?t+h , (id + hvt )#?t )/|h| = 0. Given a geodesic
curve in P (X ) parameterized as Eq. (3), its corresponding tangent vector at time zero is v = T ? id.
3
Wasserstein Principal Geodesics
Geodesic Parameterization The goal of principal geodesic analysis is to define geodesic curves
in P (X ) that go through the mean ?
? and which pass close enough to all target measures ?i . To that
end, geodesic curves can be parameterized with two end points ? and ?. However, to avoid dealing
with the constraint that a principal geodesic needs to go through ?
?, one can start instead from ?
?, and
consider a velocity field v ? L2 (?
?, X ) which displaces all of the mass of ?
? in both directions:
def.
gt (v) = (id + tv) #?
?, t ? [?1, 1].
(4)
Lemma 7.2.1 of Ambrosio et al. (2006) implies that any geodesic going through ?
? can be written
as Eq. (4). Hence, we do not lose any generality using this parameterization. However, given an
arbitrary vector field v, the curve (gt (v))t is not necessarily a geodesic. Indeed, the maps id ? v are
def.
not necessarily in the set C?? = {r ? L2 (?
?, X )|(id ? r)#?
? ? ?? (?
?, r#?
?)} of maps that are optimal
when moving mass away from ?
?. Ensuring thus, at each step of our algorithm, that v is still such
that (gt (v))t is a geodesic curve is particularly challenging. To relax this strong assumption, we
propose to use a generalized formulation of geodesics, which builds upon not one but two velocity
fields, as introduced by Ambrosio et al. (2006, ?9.2):
Definition 2. (adapted from (Ambrosio et al., 2006, ?9.2)) Let ?, ?, ? ? P (X ), and assume there
is an optimal mapping T (?,?) from ? to ? and an optimal mapping T (?,?) from ? to ?. A generalized
geodesic, illustrated in Fig. 3 between ? and ? with base ? is defined by,
gt = (1 ? t)T (?,?) + tT (?,?) #?, t ? [0, 1].
Choosing ?
? as the base measure in Definition 2, and two fields v1 , v2 such that id ? v1 , id + v2 are
optimal mappings (in C?? ), we can define the following generalized geodesic gt (v1 , v2 ):
def.
gt (v1 , v2 ) = (id ? v1 + t(v1 + v2 )) #?
?, for t ? [0, 1].
4
(5)
Generalized geodesics become true geodesics when v1 and v2 are positively proportional. We can
thus consider a regularizer that controls the deviation from that property by defining ?(v1 , v2 ) =
(hv1 , v2 iL2 (??,X ) ? kv1 kL2 (??,X ) kv2 kL2 (??,X ) )2 , which is minimal when v1 and v2 are indeed positively proportional. We can now formulate the WPG problem as computing, for n ? 0, the (n + 1)th
principal (generalized) geodesic component of a family of measures (?i )i by solving, with ? > 0:
(
N
X
id ? v1 , id + v2 ? C?? ,
2
min
??(v1 , v2 ) +
min W2 (gt (v1 , v2 ), ?i ), s.t.
(i)
(i)
2
t?[0,1]
v1 +v2 ? span({v1 + v2 }i?n )? .
v1 ,v2 ?L (?
?,X )
i=1
(6)
This problem is not convex in v1 , v2 . We pro- 1.4
pose to find an approximation of that minimum
by a projected gradient descent, with a projec- 1.2
tion that is to be understood in terms of an al1
ternative metric on the space of vector fields
2
L (?
?, X ). To preserve the optimality of the 0.8
mappings id ? v1 and id + v2 between itera?
?
tions, we introduce in the next paragraph a suit- 0.6
?
able projection operator on L2 (?
?, X ).
g? ? ?
Remark 1. A trivial way to ensure that (gt (v))t 0.4
g? ? ?
is geodesic is to impose that the vector field v is
g
g 1/3
a translation, namely that v is uniformly equal 0.2
g 2/3
to a vector ? on all of Supp(?
?). One can show
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
in that case that the WPG problem described
in Eq. (6) outputs an optimal vector ? which is
the Euclidean principal component of the fam- Figure 3: Generalized geodesic interpolation between two empirical measures ? and ? using the
ily formed by the means of each measure ?i .
base measure ?, all defined on X = R2 .
Projection on the Optimal Mapping Set We use a projected gradient descent method to solve
Eq. (6) approximately. We will compute the gradient of a local upper-bound of the objective of
Eq. (6) and update v1 and v2 accordingly. We then need to ensure that v1 and v2 are such that id ? v1
and id + v2 belong to the set of optimal mappings C?? . To do so, we would ideally want to compute
the projection r2 of id + v2 in C??
r2 = argmin k(id + v2 ) ? rk2L2 (??,X ) ,
(7)
r?C?
?
to update v2 ? r2 ? id. Westdickenberg (2010) has shown that the set of optimal mappings C?? is a
convex closed cone in L2 (?
?, X ), leading to the existence and the unicity of the solution of Eq. (7).
However, there is to our knowledge no known method to compute the projection r2 of id + v2 .
There is nevertheless a well known and efficient approach to find a mapping r2 in C?? which is close
to id + v2 . That approach, known as the the barycentric projection, requires to compute first an
optimal coupling ? ? between ?
? and (id + v2 )#?
?, to define then a (conditional expectation) map
Z
def.
T?? (x) =
yd? ? (y|x).
(8)
X
Ambrosio et al. (2006, Theorem 12.4.4) or Reich (2013, Lemma 3.1) have shown that T?? is indeed
an optimal mapping between ?
? and T?? #?
?. We can thus set the velocity field as v2 ? T?? ? id to
carry out an approximate projection. We show in the supplementary material that this operator can
be in fact interpreted as a projection under a pseudo-metric GW?? on L2 (?
?, X ).
4
Computing Principal Generalized Geodesics in Practice
We show in this section that when X = Rd , the steps outlined above can be implemented efficiently.
Input Measures and Their Barycenter Each input measure in the family (?1 , ? ? ? , ?N ) is a finite
weighted sum of Diracs, described by ni points contained in a matrix Xi of size d ? ni , and a (nonnegative) weight vector aiP
of dimension ni summing to 1. The Wasserstein mean of these measures
p
is given and equal to ?
? = k=1 bk ?yk , where the nonnegative vector b = (b1 , ? ? ? , bp ) sums to one,
d?p
and Y = [y1 , ? ? ? , yp ] ? R
is the matrix containing locations of ?
?.
5
Generalized Geodesic Two velocity vectors for each of the p points in ?
? are needed to parameterize a generalized geodesic. These velocity fields will be represented by two matrices
V1 = [v11 , ? ? ? , vp1 ] and V2 = [v12 , ? ? ? , vp2 ] in Rd?p . Assuming that these velocity fields yield optimal
mappings, the points at time t of that generalized geodesic are the measures parameterized by t,
p
X
def.
gt (V1 , V2 ) =
bk ?zkt , with locations Zt = [z1t , . . . , zpt ] = Y ? V1 + t(V1 + V2 ).
k=1
The squared 2-Wasserstein distance between datum ?i and a point gt (V1 , V2 ) on the geodesic is:
W22 (gt (V1 , V2 ), ?i ) = min hP, MZt Xi i,
(9)
P ?U (b,ai )
p?ni
, P 1ni
? R+
= b, P T 1p = ai }, and MZt Xi
where U (b, ai ) is the transportation polytope {P
stands for the p ? ni matrix of squared-Euclidean distances between the p and ni column vectors of
Zt and Xi respectively. Writing zt = D(ZtT Zt ) and xi = D(XiT Xi ), we have that
MZt Xi = zt 1Tni + 1p xTi ? 2ZtT Xi ? Rp?ni ,
which, by taking into account the marginal conditions on P ? U (b, ai ), leads to,
hP, MZt Xi i = bT zt + aTi xi ? 2hP, ZtT Xi i.
(10)
1. Majorization of the Distance of each ?i to the Principal Geodesic Using Eq. (10), the distance between each ?i and the PC (gt (V1 , V2 ))t can be cast as a function fi of (V1 , V2 ):
def.
T
T
T
fi (V1 , V2 ) = min b zt + ai xi + min ?2hP, (Y ? V1 + t(V1 + V2 )) Xi i . (11)
t?[0,1]
P ?U (b,ai )
where we have replaced Zt above by its explicit form in t to highlight that the objective above
is quadratic convex plus piecewise linear concave as a function of t, and thus neither convex nor
concave. Assume that we are given P ] and t] that are approximate arg-minima for fi (V1 , V2 ). For
any A, B in Rd?p , we thus have that each distance fi (V1 , V2 ) appearing in Eq. (6), is such that
def.
fi (A, B) 6 miV1 V2 (A, B) = hP ] , MZt] Xi i.
(12)
We can thus use a majorization-minimization procedure (Hunter and Lange, 2000) to minimize the
sum of terms fi by iteratively creating majorization functions miV1 V2 at each iterate (V1 , V2 ). All
functions mVi 1 V2 are quadratic convex. Given that we need to ensure that these velocity fields yield
optimal mappings, and that they may also need to satisfy orthogonality constraints with respect to
lower-order principal components, we use gradient steps to update V1 , V2 , which can be recovered
using (Cuturi and Doucet, 2014, ?4.3) and the chain rule as:
?1 mVi 1 V2 = 2(t] ? 1)(Zt] ? Xi P ]T D(b?1 )), ?2 miV1 V2 = 2t] (Zt] ? Xi P ]T D(b?1 )). (13)
2. Efficient Approximation of P ] and t] As discussed above, gradients for majorization functions mVi 1 V2 can be obtained using approximate minima P ] and t] for each function fi . Because the
objective of Eq. (11) is not convex w.r.t. t, we propose to do an exhaustive 1-d grid search with K
values in [0, 1]. This approach would still require, in theory, to solve K optimal transport problems
to solve Eq. (11) for each of the N input measures. To carry out this step efficiently, we propose
to use entropy regularized transport (Cuturi, 2013), which allows for much faster computations and
efficient parallelizations to recover approximately optimal transports P ] .
3. Projected Gradient Update Velocity fields are updated with a gradient stepsize ? > 0,
!
!
N
N
X
X
V1 V2
V1 V2
V1 ? V1 ? ?
?1 m i
+ ??1 ? , V2 ? V2 ? ?
?2 m i
+ ??2 ? ,
i=1
i=1
(1)
(1)
span(V1 + V2 , ? ? ?
(n)
(n)
, V1 + V2 )?
followed by a projection step to enforce that V1 and V2 lie in
2
th
in the L (?
?, X ) sense when computing the (n + 1) PC. We finally apply the barycentric projection
operator defined in the end of ?3. We first need to compute two optimal transport plans,
P1? ? argmin hP, MY (Y ?V1 ) i, P2? ? argmin hP, MY (Y +V2 ) i,
(14)
P ?U (b,b)
P ?U (b,b)
to form the barycentric projections, which then yield updated velocity vectors:
V1 ? ? (Y ? V1 )P1?T D(b?1 ) ? Y , V2 ? (Y + V2 )P2?T D(b?1 ) ? Y.
(15)
We repeat steps 1,2,3 until convergence. Pseudo-code is given in the supplementary material.
6
5
Experiments
3
?
?
0.5
?
?
2
?1
?1
?2
0
?3
-0.5
-1
-1
1
?2
0
?3
?4
pc1
pc1 -1
pc2
-2
0
1
2
3
-3
-6
-4
-2
0
2
4
6
Figure 4: Wasserstein mean ?
? and first PC computed on a dataset of four (left) and three (right)
empirical measures. The second PC is also displayed in the right figure.
Toy samples: We first run our algorithm on two simple synthetic examples. We consider respectively 4 and 3 empirical measures supported on a small number of locations in X = R2 , so
that we can compute their exact Wasserstein means, using the multi-marginal linear programming
formulation given in (Agueh and Carlier, 2011, ?4). These measures and their mean (red squares)
are shown in Fig. 4. The first principal component on the left example is able to capture both the
variability of average measure locations, from left to right, and also the variability in the spread
of the measure locations. On the right example, the first principal component captures the overall
elliptic shape of the supports of all considered measures. The second principal component reflects
the variability in the parameters of each ellipse on which measures are located. The variability in
the weights of each location is also captured through the Wasserstein mean, since each single line
of a generalized geodesic has a corresponding location and weight in the Wasserstein mean.
MNIST: For each of the digits ranging from 0 to 9, we sample 1,000 images in the MNIST
database representing that digit. Each image, originally a 28x28 grayscale image, is converted into a
probability distribution on that grid by normalizing each intensity by the total intensity in the image.
We compute the Wasserstein mean for each digit using the approach of Benamou et al. (2015). We
then follow our approach to compute the first three principal geodesics for each digit. Geodesics
for four of these digits are displayed in Fig. 5 by showing intermediary (rasterized) measures on the
curves. While some deformations in these curves can be attributed to relatively simple rotations
around the digit center, more interesting deformations appear in some of the curves, such as the the
loop on the bottom left of digit 2. Our results are easy to interpret, unlike those obtained with Wang
et al.?s approach (2013) on these datasets, see supplementary material. Fig. 6 displays the first PC
obtained on a subset of MNIST composed of 2,000 images of 2 and 4 in equal proportions.
P C1
P C2
P C3
t=0
t=1
Figure 5: 1000 images for each of the digits 1,2,3,4 were sampled from the MNIST database. We
display above the first three PCs sampled at times tk = k/4, k = 0, . . . , 4 for each of these digits.
Color histograms: We consider a subset of the Caltech-256 Dataset composed of three image
categories: waterfalls, tomatoes and tennis balls, resulting in a set of 295 color images. The pixels
7
Figure 6: First PC on a subset of MNIST composed of one thousand 2s and one thousand 4s.
contained in each image can be seen as a point-cloud in the RGB color space [0, 1]3 . We use k-means
quantization to reduce the size of these uniform point-clouds into a set of k = 128 weighted points,
using cluster assignments to define the weights of each of the k cluster centroids. Each image can be
thus regarded as a discrete probability measure of 128 atoms in the tridimensional RGB space. We
then compute the Wasserstein barycenter of these measures supported on p = 256 locations using
(Cuturi and Doucet, 2014, Alg.2). Principal components are then computed as described in ?4. The
computation for a single PC is performed within 15 minutes on an iMac (3.4GHz Intel Core i7).
Fig. 7 displays color palettes sampled along each of the first three PCs. The first PC suggests that
the main source of color variability in the dataset is the illumination, each pixel going from dark to
light. Second and third PCs display the variation of colors induced by the typical images? dominant
colors (blue, red, yellow). Fig. 8 displays the second PC, along with three images projected on that
curve. The projection of a given image on a PC is obtained by finding first the optimal time t? such
that the distance of that image to the PC at t? is minimum, and then by computing an optimal color
transfer (Piti?e et al., 2007) between the original image and the histogram at time t? .
Figure 7: Each row represents a PC displayed at regular time intervals from t = 0 (left) to t = 1
(right), from the first PC (top) to the third PC (bottom).
Figure 8: Color palettes from the second PC (t = 0 on the left, t = 1 on the right) displayed at times
t = 0, 31 , 23 , 1. Images displayed in the top row are original; their projection on the PC is displayed
below, using a color transfer with the palette in the PC to which they are the closest.
Conclusion We have proposed an approximate projected gradient descent method to compute generalized geodesic principal components for probability measures. Our experiments suggest that these
principal geodesics may be useful to analyze shapes and distributions, and that they do not require
any parameterization of shapes or deformations to be used in practice.
Aknowledgements MC acknowledges the support of JSPS young researcher A grant 26700002.
8
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9
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absolutely:2 srivastava:2 |
5,171 | 5,681 | Fast and Accurate Inference of Plackett?Luce Models
Lucas Maystre
EPFL
[email protected]
Matthias Grossglauser
EPFL
[email protected]
Abstract
We show that the maximum-likelihood (ML) estimate of models derived from
Luce?s choice axiom (e.g., the Plackett?Luce model) can be expressed as the
stationary distribution of a Markov chain. This conveys insight into several recently
proposed spectral inference algorithms. We take advantage of this perspective and
formulate a new spectral algorithm that is significantly more accurate than previous
ones for the Plackett?Luce model. With a simple adaptation, this algorithm can
be used iteratively, producing a sequence of estimates that converges to the ML
estimate. The ML version runs faster than competing approaches on a benchmark
of five datasets. Our algorithms are easy to implement, making them relevant for
practitioners at large.
1
Introduction
Aggregating pairwise comparisons and partial rankings are important problems with applications in
econometrics [1], psychometrics [2, 3], sports ranking [4, 5] and multiclass classification [6]. One
possible approach to tackle these problems is to postulate a statistical model of discrete choice. In
this spirit, Luce [7] stated the axiom of choice in a foundational work published over fifty years ago.
Denote p(i | A) the probability of choosing item i when faced with alternatives in the set A. Given
two items i and j, and any two sets of alternatives A and B containing i and j, the axiom posits that
p(i | B)
p(i | A)
=
.
p(j | A)
p(j | B)
(1)
In other words, the odds of choosing item i over item j are independent of the rest of the alternatives.
This simple assumption directly leads to a unique parametric choice model, known as the Bradley?
Terry model in the case of pairwise comparisons, and the Plackett?Luce model in the generalized case
of k-way rankings. In this paper, we highlight a connection between the maximum-likelihood (ML)
estimate under these models and the stationary distribution of a Markov chain parameterized by the
observed choices. Markov chains were already used in recent work [8, 9, 10] to aggregate pairwise
comparisons and rankings. These approaches reduce the problem to that of finding a stationary
distribution. By formalizing the link between the likelihood of observations under the choice model
and a certain Markov chain, we unify these algorithms and explicate them from an ML inference
perspective. We will also take a detour, and use this link in the reverse direction to give an alternative
proof to a recent result on the error rate of the ML estimate [11], by using spectral analysis techniques.
Beyond this, we make two contributions to statistical inference for this model. First, we develop
a simple, consistent and computationally efficient spectral algorithm that is applicable to a wide
range of models derived from the choice axiom. The exact formulation of the Markov chain used
in the algorithm is distinct from related work [9, 10] and achieves a significantly better statistical
efficiency at no additional computational cost. Second, we observe that with a small adjustment, the
algorithm can be used iteratively, and it then converges to the ML estimate. An evaluation on five
real-world datasets reveals that it runs consistently faster than competing approaches and has a much
more predictable performance that does not depend on the structure of the data. The key step, finding
a stationary distribution, can be offloaded to commonly available linear-algebra primitives, which
1
makes our algorithms scale well. Our algorithms are intuitively pleasing, simple to understand and
implement, and they outperform the state of the art, hence we believe that they will be highly useful
to practitioners.
The rest of the paper is organized as follows. We begin by introducing some notations and presenting
a few useful facts about the choice model and about Markov chains. By necessity, our exposition is
succinct, and the reader is encouraged to consult Luce [7] and Levin et al. [12] for a more thorough
exposition. In Section 2, we discuss related work. In Section 3, we present our algorithms, and in
Section 4 we evaluate them on synthetic and real-world data. We conclude in Section 5.
Discrete choice model. Denote by n the number of items. Luce?s axiom of choice implies that
each item P
i ? {1, . . . , n} can be parameterized by a positive strength ?i ? R>0 such that p(i |
A) = ?i / j?A ?j for any A containing i. The strengths ? = [?i ] are defined up to a multiplicative
P
factor; for identifiability, we let i ?i = 1. An alternative parameterization of the model is given by
?i = log(?i ), in which case the model is sometimes referred to as conditional logit [1].
Markov chain theory. We represent a finite, stationary, continuous-time Markov chain by a directed
graph G = (V, E), where V is the set of states and E is the set of transitions with positive rate. If G is
strongly connected, the Markov chain is said to be ergodic and admits a unique stationary distribution
?. The global balance equations relate the transition rates ?ij to the stationary distribution as follows:
X
X
?i ?ij =
?j ?ji ?i.
(2)
j6=i
j6=i
The stationary distribution is therefore invariant to changes in the time scale, i.e., to a rescaling of the
transition rates. In the supplementary file, we briefly discuss how to find ? given [?ij ].
2
Related work
Spectral methods applied to ranking and scoring items from noisy choices have a long-standing
history. To the best of our knowledge, Saaty [13] is the first to suggest using the leading eigenvector
of a matrix of inconsistent pairwise judgments to score alternatives. Two decades later, Page et al.
[14] developed PageRank, an algorithm that ranks Web pages according to the stationary distribution
of a random walk on the hyperlink graph. In the same vein, Dwork et al. [8] proposed several variants
of Markov chains for aggregating heterogeneous rankings. The idea is to construct a random walk
that is biased towards high-ranked items, and use the ranking induced by the stationary distribution.
More recently, Negahban et al. [9] presented Rank Centrality, an algorithm for aggregating pairwise
comparisons close in spirit to that of [8]. When the data is generated under the Bradley?Terry model,
this algorithm asymptotically recovers model parameters with only ?(n log n) pairwise comparisons.
For the more general case of rankings under the Plackett?Luce model, Azari Soufiani et al. [10]
propose to break rankings into pairwise comparisons and to apply an algorithm similar to Rank
Centrality. They show that the resulting estimator is statistically consistent. Interestingly, many of
these spectral algorithms can be related to the method of moments, a broadly applicable alternative to
maximum-likelihood estimation.
The history of algorithms for maximum-likelihood inference under Luce?s model goes back even
further. In the special case of pairwise comparisons, the same iterative algorithm was independently
discovered by Zermelo [15], Ford [16] and Dykstra [17]. Much later, this algorithm was explained
by Hunter [18] as an instance of minorization-maximization (MM) algorithm, and extended to the
more general choice model. Today, Hunter?s MM algorithm is the de facto standard for ML inference
in Luce?s model. As the likelihood is concave, off-the-shelf optimization procedures such as the
Newton-Raphson method can also be used, although they have been been reported to be slower and
less practical [18]. Recently, Kumar et al. [19] looked at the problem of finding the transition matrix
of a Markov chain, given its stationary distribution. The problem of inferring Luce?s model parameters
from data can be reformulated in their framework, and the ML estimate is the solution to the inversion
of the stationary distribution. Their work stands out as the first to link ML inference to Markov chains,
albeit very differently from the way presented in our paper. Beyond algorithms, properties of the
maximum-likelihood estimator in this model were studied extensively. Hajek et al. [11] consider the
Plackett?Luce model for k-way rankings. They give an upper bound to the estimation error and show
that the ML estimator is minimax-optimal. In summary, they show that only ?(n/k log n) samples
are enough to drive the mean-square error down to zero, as n increases. Rajkumar and Agarwal [20]
2
consider the Bradley?Terry model for pairwise comparisons. They show that the ML estimator is able
to recover the correct ranking, even when the data is generated as per another model, e.g., Thurstone?s
[2], as long as a so-called low-noise condition is satisfied. We also mention that as an alternative to
likelihood maximization, Bayesian inference has also been proposed. Caron and Doucet [21] present
a Gibbs sampler, and Guiver and Snelson [22] propose an approximate inference algorithm based on
expectation propagation.
In this work, we provide a unifying perspective on recent advances in spectral algorithms [9, 10] from
a maximum-likelihood estimation perspective. It turns out that this perspective enables us to make
contributions on both sides: On the one hand, we develop an improved and more general spectral
ranking algorithm, and on the other hand, we propose a faster procedure for ML inference by using
this algorithm iteratively.
3
Algorithms
We begin by expressing the ML estimate under the choice model as the stationary distribution
of a Markov chain. We then take advantage of this formulation to propose novel algorithms for
model inference. Although our derivation is made in the general choice model, we will also discuss
implications for the special cases of pairwise data in Section 3.3 and k-way ranking data in Section 3.4.
Suppose that we collect d independent observations in the multiset D = {(c` , A` ) | ` = 1, . . . , d}.
Each observation consists of a choice c` among a set of alternatives A` ; we say that i wins over j
and denote by i j whenever i, j ? A and c` = i. We define the directed comparison graph as
GD = (V, E), with V = {1, . . . , n} and (j, i) ? E if i wins at least once over j in D. In order to
ensure that the ML estimate is well-defined, we make the standard assumption that GD is strongly
connected [16, 18]. In practice, if this assumption does not hold, we can consider each strongly
connected component separately.
3.1
ML estimate as a stationary distribution
For simplicity, we denote the model parameter associated with item c` by ?` . The log-likelihood of
parameters ? given observations D is
?
?
d
X
X
?log ?` ? log
log L(? | D) =
?j ? .
(3)
j?A`
`=1
.
.
For each item, we define two sets of indices. Let Wi = {` | i ? A` , c` = i} and Li = {` |
i ? A` , c` 6= i} be the indices of the observations where item i wins over and loses against the
alternatives, respectively. The log-likelihood function is strictly concave and the model admits a
? The optimality condition ??? log L = 0 implies
unique ML estimate ?.
!
X
X
? log L
1
1
1
P
=
?P
?
= 0 ?i
(4)
??
?i
?
?i
?j
?j
j?A` ?
j?A` ?
`?Wi
`?Li
?
?
X
X
X
?
?
?
?
?
? = 0 ?i.
P j
P i
??
?
(5)
?
?
?t
t?A` t
t?A` ?
j6=i
`?Wi ?Lj
`?Wj ?Li
In order to go from (4) to (5), we multiply by ?
?i and rearrange the terms. To simplify the notation, let
us further introduce the function
1
. X
P
f (S, ?) =
,
(6)
i?A ?i
A?S
which takes observations S ? D and an instance of model parameters ?, and returns a non-negative
.
real number. Let Di j = {(c` , A` ) ? D | ` ? Wi ? Lj }, i.e., the set of observations where i wins
over j. Then (5) can be rewritten as
X
X
? =
? ?i.
?
?i ? f (Dj i , ?)
?
?j ? f (Di j , ?)
(7)
j6=i
j6=i
3
Algorithm 1 Luce Spectral Ranking
Require: observations D
1: ? ? 0n?n
2: for (i, A) ? D do
3:
for j ? A \ {i} do
4:
?ji ? ?ji + n/|A|
5:
end for
6: end for
? ? stat. dist. of Markov chain ?
7: ?
?
8: return ?
Algorithm 2 Iterative Luce Spectral Ranking
Require: observations D
1: ? ? [1/n, . . . , 1/n]|
2: repeat
3:
? ? 0n?n
4:
for (i, A) ? D do
5:
for j ? A \ {i} doP
6:
?ji ? ?ji + 1/ t?A ?t
7:
end for
8:
end for
9:
? ? stat. dist. of Markov chain ?
10: until convergence
This formulation conveys a new viewpoint on the ML estimate. It is easy to recognize the global
balance equations (2) of a Markov chain on n states (representing the items), with transition rates
? and stationary distribution ?.
? These transition rates have an interesting inter?ji = f (Di j , ?)
pretation: f (Di j , ?) is the count of how many times i wins over j, weighted by the strength of
the alternatives. At this point, it is useful to observe that for any parameters ?, f (Di j , ?) ? 1
if (j, i) ? E, and 0 otherwise. Combined with the assumption that GD is strongly connected, it
follows that any ? parameterizes the transition rates of an ergodic (homogeneous) Markov chain. The
ergodicity of the inhomogeneous Markov chain, where the transition rates are constantly updated to
reflect the current distribution over states, is shown by the following theorem.
Theorem 1. The Markov chain with inhomogeneous transition rates ?ji = f (Di j , ?) converges to
? for any initial distribution in the open probability simplex.
the maximum-likelihood estimate ?,
? is the unique invariant distribution of the Markov chain. In the supplemenProof (sketch). By (7), ?
tary file, we look at an equivalent uniformized discrete-time chain. Using the contraction mapping
principle, one can show that this chain converges to the invariant distribution.
3.2
Approximate and exact ML inference
We approximate the Markov chain described in (7) by considering a priori that all alternatives have
.
equal strength. That is, we set the transition rates ?ji = f (Di j , ?) by fixing ? to [1/n, . . . , 1/n]| .
For i 6= j, the contribution of i winning over j to the rate of transition ?ji is n/|A|. In other words,
for each observation, the winning item is rewarded by a fixed amount of incoming rate that is evenly
split across the alternatives (the chunk allocated to itself is discarded.) We interpret the stationary
? as an estimate of model parameters. Algorithm 1 summarizes this procedure, called
distribution ?
Luce Spectral Ranking (LSR.) If we consider a growing number of observations, LSR converges to
the true model parameters ? ? , even in the restrictive case where the sets of alternatives are fixed.
Theorem 2. Let A = {A` } be a collection of sets of alternatives such that for any partition of A into
two non-empty sets S and T , (?A?S A) ? (?A?T A) 6= ?1 . Let d` be the number of choices observed
? ? ? ? as d` ? ? ?`.
over alternatives A` . Then ?
Proof (sketch). The condition on A ensures that asymptotically GD is strongly connected. Let
d ? ? be a shorthand for d` ? ? ?`. We can show that if items i and j are compared in at least one
set of alternatives, the ratio of transition rates satisfies limd?? ?ij /?ji = ?j? /?i? . It follows that in
the limit of d ? ?, the stationary distribution is ? ? . A rigorous proof is given in the supplementary
file.
Starting from the LSR estimate, we can iteratively refine the transition rates of the Markov chain and
? We
obtain a sequence of estimates. By (7), the only fixed point of this iteration is the ML estimate ?.
call this procedure I-LSR and describe it in Algorithm 2.
LSR (or one iteration of I-LSR) entails (a) P
filling a matrix of (weighted) pairwise counts and
.
(b) finding a stationary distribution. Let D = ` |A` |, and let S be the running time of finding the
stationary distribution. Then LSR has running time O(D + S). As a comparison, one iteration of
1
This is equivalent to stating that the hypergraph H = (V, A) is connected.
4
the MM algorithm [18] is O(D). Finding the stationary distribution can be implemented in different
ways. For example, in a sparse regime where D n2 , the stationary distribution can be found with
the power method in a few O(D) sparse matrix multiplications. In the supplementary file, we give
more details about possible implementations. In practice, whether D or S turns out to be dominant in
the running time is not a foregone conclusion.
3.3
Aggregating pairwise comparisons
A widely-used special case of Luce?s choice model occurs when all sets of alternatives contain exactly
two items, i.e., when the data consists of pairwise comparisons. This model was proposed by Zermelo
[15], and later by Bradley and Terry [3]. As the stationary distribution is invariant to changes in the
.
time-scale, we can rescale the transition rates and set ?ji = |Di j | when using LSR on pairwise
data. Let S be the set containing the pairs of items that have been compared at least once. In the
case where each pair (i, j) ? S has been compared exactly p times, LSR is strictly equivalent to a
continuous-time Markov-chain formulation of Rank Centrality [9]. In fact, our derivation justifies
Rank Centrality as an approximate ML inference algorithm for the Bradley?Terry model. Furthermore,
we provide a principled extension of Rank Centrality to the case where the number of comparisons
observed is unbalanced. Rank Centrality considers transition rates proportional to the ratio of wins,
whereas (7) justifies making transition rates proportional to the count of wins.
Negahban et al. [9] also provide an upper bound on the error rate of Rank Centrality, which essentially
shows that it is minimax-optimal. Because the two estimators are equivalent in the setting of balanced
pairwise comparisons, the bound also applies to LSR. More interestingly, the expression of the ML
estimate as a stationary distribution enables us to reuse the same analytical techniques to bound the
error of the ML estimate. In the supplementary file, we therefore provide an alternative proof of the
recent result of Hajek et al. [11] on the minimax-optimality of the ML estimate.
3.4
Aggregating partial rankings
Another case of interest is when observations do not consist of only a single choice, but of a ranking
over the alternatives. We now suppose m observations consisting of k-way rankings, 2 ? k ? n.
For conciseness, we suppose that k is the same for all observations. Let one such observation
be ?(1) . . . ?(k), where ?(p) is the item with p-th rank. Luce [7] and later Plackett [4]
independently proposed a model of rankings where
P (?(1) . . . ?(k)) =
k
Y
??(r)
.
Pk
r=1
p=r ??(p)
(8)
In this model, a ranking can be interpreted as a sequence of k ? 1 independent choices: Choose the
first item, then choose the second among the remaining alternatives, etc. With this point of view in
mind, LSR and I-LSR can easily accommodate data consisting of k-way rankings, by decomposing
the m observations into d = m(k ? 1) choices.
Azari Soufiani et al. [10] provide a class of consistent estimators for the Plackett?Luce model, using
the idea of breaking rankings into pairwise comparisons. Although they explain their algorithms from
a generalized-method-of-moments perspective, it is straightforward to reinterpret their estimators as
stationary distributions of particular Markov chains. In fact, for k = 2, their algorithm GMM-F is
identical to LSR. When k > 2 however, breaking a ranking into k2 pairwise comparisons implicitly
makes the (incorrect) assumption that these comparisons are statistically independent. The Markov
chain that LSR builds breaks rankings into pairwise rate contributions, but weights the contributions
differently depending on the rank of the winning item. In Section 4, we show that this weighting
turns out to be crucial. Our approach yields a significant improvement in statistical efficiency, yet
keeps the same attractive computational cost and ease of use.
3.5
Applicability to other models
Several other variants and extensions of Luce?s choice model have been proposed. For example, Rao
and Kupper [23] extend the Bradley?Terry model to the case where a comparison between two items
can result in a tie. In the supplementary file, we show that the ML estimate in the Rao?Kupper model
can also be formulated as a stationary distribution, and we provide corresponding adaptations of LSR
5
and I-LSR. We believe that our algorithms can be generalized to further models that are based on the
choice axiom. However, this axiom is key, and other choice models (such as Thurstone?s [2]) do not
admit the stationary-distribution interpretation we derive here.
4
Experimental evaluation
In this section, we compare LSR and I-LSR to other inference algorithms in terms of (a) statistical
efficiency, and (b) empirical performance. In order to understand the efficiency of the estimators,
we generate synthetic data from a known ground truth. Then, we look at five real-world datasets
and investigate the practical performance of the algorithms in terms of accuracy, running time and
convergence rate.
Error metric. As the probability of i winning over j depends on the ratio of strengths ?i /?i , the
strengths are typically logarithmically spaced. In order to evaluate the accuracy of an estimate ? to
ground truth parameters ? ? , we therefore use a log transformation, reminiscent of the random-utility.
theoretic
formulation of the choice model [1, 11]. Define ? = [log ?i ? t], with t chosen such that
P
i ?i = 0. We will consider the root-mean-squared error (RMSE)
?
(9)
ERMS = k? ? ? ? k2 / n.
4.1
Statistical efficiency
To assess the statistical efficiency of LSR and other algorithms, we follow the experimental procedure
of Hajek et al. [11]. We consider n = 1024 items, and draw ? ? uniformly at random in [?2, 2]n .
We generate d = 64 full rankings over the n items from a Plackett-Luce model parameterized with
? ? [e?i ]. For a given k ? {21 , . . . , 210 }, we break down each of the full rankings as follows. First,
we partition the items into n/k subsets of size k uniformly at random. Then, we store the k-way
rankings induced by the full ranking on each of those subsets. As a result, we obtain m = dn/k
statistically independent k-way partial rankings. For a given estimator, this data produces an estimate
?, for which we record the root-mean-square error to ? ? . We consider four estimators. The first two
(LSR and ML) work on the ranking data directly. The remaining two follow Azari Soufiani et al. [10],
who suggest breaking down k-way rankings into k2 pairwise comparisons. These comparisons are
then used by LSR, resulting in Azari Soufiani et al.?s GMM-F estimator, and by an ML estimator
(ML-F.) In short, the four estimators vary according to (a) whether they use as-is rankings or derived
comparisons, and (b) whether the model is fitted using an approximate spectral algorithm or using
exact maximum likelihood. Figure 1 plots ERMS for increasing sizes of partial rankings, as well as
a lower bound to the error of any estimator for the Plackett-Luce model (see Hajek et al. [11] for
details.) We observe that breaking the rankings into pairwise comparisons (*-F estimators) incurs a
significant efficiency loss over using the k-way rankings directly (LSR and ML.) We conclude that by
correctly weighting pairwise rates in the Markov chain, LSR distinctly outperforms the rank-breaking
approach as k increases. We also observe that the ML estimate is always more efficient. Spectral
estimators such as LSR provide a quick, asymptotically consistent estimate of parameters, but this
observation justifies calling them approximate inference algorithms.
4.2
Empirical performance
We investigate the performance of various inference algorithms on five real-world datasets. The
NASCAR [18] and sushi [24] datasets contain multiway partial rankings. The YouTube, GIFGIF
and chess datasets2 contain pairwise comparisons. Among those, the chess dataset is particular in
that it features 45% of ties; in this case we use the extension of the Bradley?Terry model proposed
by Rao and Kupper [23]. We preprocess each dataset by discarding items that are not part of the
largest strongly connected component in the comparison graph. The number of items n, the number
of rankings m, as well as the size of a partial rankings k for each dataset are given in Table 1.
Additional details on the experimental setup are given in the supplementary material. We first
compare the estimates produced by three approximate ML inference algorithms, LSR, GMM-F and
Rank Centrality (RC.) Note that RC applies only to pairwise comparisons, and that LSR is the only
2
See https://archive.ics.uci.edu/ml/machine-learning-databases/00223/,
http://www.gif.gf/ and https://www.kaggle.com/c/chess.
6
RMSE
0.4
lower bound
ML-F
GMM-F
ML
LSR
0.2
0.1
21
22
23
24
25
26
k
27
28
29
210
Figure 1: Statistical efficiency of different estimators for increasing sizes of partial rankings. As k
grows, breaking rankings into pairwise comparisons becomes increasingly inefficient. LSR remains
efficient at no additional computational cost.
algorithm able to infer the parameters in the Rao-Kupper model. Also note that in the case of pairwise
comparisons, GMM-F and LSR are strictly equivalent. In Table 1, we report the root-mean-square
deviation to the ML estimate ?? and the running time T of the algorithm.
Table 1: Performance of approximate ML inference algorithms
LSR
Dataset
NASCAR
Sushi
YouTube
GIFGIF
Chess
GMM-F
RC
n
m
k
ERMS
T [s]
ERMS
T [s]
ERMS
T [s]
83
100
36
5 000
43
10
0.194
0.034
0.03
0.22
0.751
0.130
0.06
0.19
?
?
?
?
16 187
5 503
1 128 704
95 281
2
2
0.417
1.286
34.18
1.90
0.417
1.286
34.18
1.90
0.432
1.295
41.91
2.84
6 174
63 421
2
0.420
2.90
?
?
?
?
The smallest value of ERMS is highlighted in bold for each dataset. We observe that in the case of
multiway partial rankings, LSR is almost four times more accurate than GMM-F on the datasets
considered. In the case of pairwise comparisons, RC is slightly worse than LSR and GMM-F, because
the number of comparisons per pair is not homogeneous (see Section 3.3.) The running time of the
three algorithms is comparable.
Next, we turn our attention to ML inference and consider three iterative algorithms: I-LSR, MM and
Newton-Raphson. For Newton-Raphson, we use an off-the-shelf solver. Each algorithm is initialized
with ? (0) = [1/n, . . . , 1/n]| , and convergence is declared when ERMS < 0.01. In Table 2, we report
the number of iterations I needed to reach convergence, as well as the total running time T of the
algorithm.
Table 2: Performance of iterative ML inference algorithms.
I-LSR
Dataset
MM
Newton
?D
I
T [s]
I
T [s]
I
T [s]
NASCAR
Sushi
0.832
0.890
3
2
0.08
0.42
4
4
0.10
1.09
?
3
?
10.45
YouTube
GIFGIF
0.002
0.408
12
10
414.44
22.31
8 680
119
22 443.88
109.62
?
5
?
72.38
Chess
0.007
15
43.69
181
55.61
3
49.37
The smallest total running time T is highlighted in bold for each dataset. We observe that NewtonRaphson does not always converge, despite the log-likelihood being strictly concave3 . I-LSR consis3
On the NASCAR dataset, this has also been noted by Hunter [18]. Computing the Newton step appears to
be severely ill-conditioned for many real-world datasets. We believe that it can be addressed by a careful choice
7
tently outperforms MM and Newton-Raphson in running time. Even if the average running time per
iteration is in general larger than that of MM, it needs considerably fewer iterations: For the YouTube
dataset, I-LSR yields an increase in speed of over 50 times.
The slow convergence of minorization-maximization algorithms is known [18], yet the scale of the
issue and its apparent unpredictability is surprising. In Hunter?s MM algorithm, updating a given ?i
involves only parameters of items to which i has been compared. Therefore, we speculate that the
convergence rate of MM is dependent on the expansion properties of the comparison graph GD . As
an illustration, we consider the sushi dataset. To quantify the expansion properties, we look at the
spectral gap ?D of a simple random walk on GD ; intuitively, the larger the spectral gap is, the better
the expansion properties are [12]. The original comparison graph is almost complete, and ?D = 0.890.
By breaking each 10-way ranking into 5 independent pairwise comparisons, we effectively sparsify
the comparison graph. As a result, the spectral gap decreases to 0.784. In Figure 2, we show the
convergence rate of MM and I-LSR for the original (k = 10) and modified (k = 2) datasets. We
observe that both algorithms display linear convergence, however the rate at which MM converges
appears to be sensitive to the structure of the comparison graph. In contrast, I-LSR is robust to
changes in the structure. The spectral gap of each dataset is listed in Table 2.
100
10?2
RMSE
10?4
10?6
MM, k = 10
MM, k = 2
I-LSR, k = 10
I-LSR, k = 2
10?8
10?10
10?12
1
2
3
4
5
6
iteration
7
8
9
10
Figure 2: Convergence rate of I-LSR and MM on the sushi dataset. When partial rankings (k = 10)
are broken down into independent comparisons (k = 2), the comparison graph becomes sparser.
I-LSR is robust to this change, whereas the convergence rate of MM significantly decreases.
5
Conclusion
In this paper, we develop a stationary-distribution perspective on the maximum-likelihood estimate
of Luce?s choice model. This perspective explains and unifies several recent spectral algorithms from
an ML inference point of view. We present our own spectral algorithm that works on a wider range of
data, and show that the resulting estimate significantly outperforms previous approaches in terms of
accuracy. We also show that this simple algorithm, with a straighforward adaptation, can produce a
sequence of estimates that converge to the ML estimate. On real-world datasets, our ML algorithm is
always faster than the state of the art, at times by up to two orders of magnitude.
Beyond statistical and computational performance, we believe that a key strength of our algorithms
is that they are simple to implement. As an example, our implementation of LSR fits in ten lines
of Python code. The most complex operation?finding a stationary distribution?can be readily
offloaded to commonly available and highly optimized linear-algebra primitives. As such, we believe
that our work is very useful for practitioners.
Acknowledgments
We thank Holly Cogliati-Bauereis, Ksenia Konyushkova and Brunella Spinelli for careful proofreading
and comments on the text.
of starting point, step size, or by monitoring the numerical stability; however, these modifications are non-trivial
and impose an additional burden on the practitioner.
8
References
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9
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5,172 | 5,682 | BACK S HIFT :
Learning causal cyclic graphs from
unknown shift interventions
Dominik Rothenh?ausler?
Seminar f?ur Statistik
ETH Z?urich, Switzerland
[email protected]
Jonas Peters
Max Planck Institute for Intelligent Systems
T?ubingen, Germany
[email protected]
Christina Heinze?
Seminar f?ur Statistik
ETH Z?urich, Switzerland
[email protected]
Nicolai Meinshausen
Seminar f?ur Statistik
ETH Z?urich, Switzerland
[email protected]
Abstract
We propose a simple method to learn linear causal cyclic models in the presence
of latent variables. The method relies on equilibrium data of the model recorded
under a specific kind of interventions (?shift interventions?). The location and
strength of these interventions do not have to be known and can be estimated from
the data. Our method, called BACK S HIFT, only uses second moments of the data
and performs simple joint matrix diagonalization, applied to differences between
covariance matrices. We give a sufficient and necessary condition for identifiability of the system, which is fulfilled almost surely under some quite general
assumptions if and only if there are at least three distinct experimental settings,
one of which can be pure observational data. We demonstrate the performance on
some simulated data and applications in flow cytometry and financial time series.
1
Introduction
Discovering causal effects is a fundamentally important yet very challenging task in various disciplines, from public health research and sociological studies, economics to many applications in the
life sciences. There has been much progress on learning acyclic graphs in the context of structural
equation models [1], including methods that learn from observational data alone under a faithfulness assumption [2, 3, 4, 5], exploiting non-Gaussianity of the data [6, 7] or non-linearities [8].
Feedbacks are prevalent in most applications, and we are interested in the setting of [9], where we
observe the equilibrium data of a model that is characterized by a set of linear relations
x = Bx + e,
(1)
where x 2 R is a random vector and B 2 R
is the connectivity matrix with zeros on the diagonal (no self-loops). Allowing for self-loops would lead to an identifiability problem, independent
of the method. See Section B in the Appendix for more details on this setting. The graph corresponding to B has p nodes and an edge from node j to node i if and only if Bi,j 6= 0. The error
terms e are p-dimensional random variables with mean 0 and positive semi-definite covariance matrix ?e = E(eeT ). We do not assume that ?e is a diagonal matrix which allows the existence of
latent variables.
p
p?p
The solutions to (1) can be thought of as the deterministic equilibrium solutions (conditional on the
noise term) of a dynamic model governed by first-order difference equations with matrix B in the
?
Authors contributed equally.
1
sense of [10]. For well-defined equilibrium solutions of (1), we need that I B is invertible. Usually
we also want (1) to converge to an equilibrium when iterating as x(new)
Bx(old) + e or in other
words limm!1 Bm ? 0. This condition is equivalent to the spectral radius of B being strictly
smaller than one [11]. We will make an assumption on cyclic graphs that restricts the strength of the
feedback. Specifically, let a cycle of length ? be given by (m1 , . . . , m?+1 = m1 ) 2 {1, . . . , p}1+?
and mk 6= m` for 1 ? k < ` ? ?. We define the cycle-product CP (B) of a matrix B to be the
maximum over cycles of all lengths 1 < ? ? p of the path-products
Y
CP (B) :=
max
Bmk+1 ,mk .
(2)
(m1 ,...,m? ,m?+1 ) cycle
1?k??
1<??p
The cycle-product CP (B) is clearly zero for acyclic graphs. We will assume the cycle-product to be
strictly smaller than one for identifiability results, see Assumption (A) below. The most interesting
graphs are those for which CP (B) < 1 and for which the spectral radius of B is strictly smaller than
one. Note that these two conditions are identical as long as the cycles in the graph do not intersect,
i.e., there is no node that is part of two cycles (for example if there is at most one cycle in the graph).
If cycles do intersect, we can have models for which either (i) CP (B) < 1 but the spectral radius
is larger than one or (ii) CP (B) > 1 but the spectral radius is strictly smaller than one. Models in
situation (ii) are not stable in the sense that the iterations will not converge under interventions. We
can for example block all but one cycle. If this one single unblocked cycle has a cycle-product larger
than 1 (and there is such a cycle in the graph if CP (B) > 1), then the solutions of the iteration are
not stable1 . Models in situation (i) are not stable either, even in the absence of interventions. We
can still in theory obtain the now instable equilibrium solutions to (1) as (I B) 1 e and the theory
below applies to these instable equilibrium solutions. However, such instable equilibrium solutions
are arguably of little practical interest. In summary: all interesting feedback models that are stable
under interventions satisfy both CP (B) < 1 and have a spectral radius strictly smaller than one. We
will just assume CP (B) < 1 for the following results.
It is impossible to learn the structure B of this model from observational data alone without making
further assumptions. The L INGAM approach has been extended in [11] to cyclic models, exploiting
a possible non-Gaussianity of the data. Using both experimental and interventional data, [12, 9]
could show identifiability of the connectivity matrix B under a learning mechanism that relies on
data under so-called ?surgical? or ?perfect? interventions. In their framework, a variable becomes
independent of all its parents if it is being intervened on and all incoming contributions are thus effectively removed under the intervention (also called do-interventions in the classical sense of [13]).
The learning mechanism makes then use of the knowledge where these ?surgical? interventions occurred. [14] also allow for ?changing? the incoming arrows for variables that are intervened on;
but again, [14] requires the location of the interventions while we do not assume such knowledge.
[15] consider a target variable and allow for arbitrary interventions on all other nodes. They neither
permit hidden variables nor cycles.
Here, we are interested in a setting where we have either no or just very limited knowledge about
the exact location and strength of the interventions, as is often the case for data observed under
different environments (see the example on financial time series further below) or for biological data
[16, 17]. These interventions have been called ?fat-hand? or ?uncertain? interventions [18]. While
[18] assume acyclicity and model the structure explicitly in a Bayesian setting, we assume that the
data in environment j are equilibrium observations of the model
xj = Bxj + cj + ej ,
(3)
where the random intervention shift cj has a mean and covariance ?c,j . The location of these
interventions (or simply the intervened variables) are those components of cj that are not zero
with probability one. Given these locations, the interventions simply shift the variables by a value
determined by cj ; they are therefore not ?surgical? but can be seen as a special case of what is
called an ?imperfect?, ?parametric? [19] or ?dependent? intervention [20] or ?mechanism change?
[21]. The matrix B and the error distribution of ej are assumed to be identical in all environments.
In contrast to the covariance matrix for the noise term ej , we do assume that ?c,j is a diagonal
1
The blocking of all but one cycle can be achieved by do-interventions on appropriate variables under the
following condition: for every pair of cycles in the graph, the variables in one cycle cannot be a subset of the
variables in the other cycle. Otherwise the blocking could be achieved by deletion of appropriate edges.
2
matrix, which is equivalent to demanding that interventions at different variables are uncorrelated.
This is a key assumption necessary to identify the model using experimental data. Furthermore, we
will discuss in Section 4.2 how a violation of the model assumption (3) can be detected and used to
estimate the location of the interventions.
In Section 2 we show how to leverage observations under different environments with different interventional distributions to learn the structure of the connectivity matrix B in model (3). The method
rests on a simple joint matrix diagonalization. We will prove necessary and sufficient conditions for
identifiability in Section 3. Numerical results for simulated data and applications in flow cytometry
and financial data are shown in Section 4.
2
2.1
Method
Grouping of data
Let J be the set of experimental conditions under which we observe equilibrium data from
model (3). These different experimental conditions can arise in two ways: (a) a controlled experiment was conducted where the external input or the external imperfect interventions have been
deliberately changed from one member of J to the next. An example are the flow cytometry data
[22] discussed in Section 4.2. (b) The data are recorded over time. It is assumed that the external input is changing over time but not in an explicitly controlled way. The data are grouped into
consecutive blocks j 2 J of observations, see Section 4.3 for an example.
2.2
Notation
Assume we have nj observations in each setting j 2 J . Let Xj be the (nj ? p)-matrix of observations from model (3). For general random variables aj 2 Rp , the population covariance matrix
in setting j 2 J is called ?a,j = Cov(aj ), where the covariance is under the setting j 2 J .
Furthermore, the covariance on all settings except setting j 2 J P
is defined as an average over all
environments except for the j-th environment, (|J | 1)?c, j := j 0 2J \{j} ?c,j 0 . The population
? a,j be the empirical
Gram matrix is defined as Ga,j = E(aj aj T ). Let the (p ? p)-dimensional ?
covariance matrix of the observations Aj 2 Rnj ?p of variable aj in setting j 2 J . More precisely,
? j be the column-wise mean-centered version of Aj . Then ?
? a,j := (nj 1) 1 A
?TA
? j . The
let A
j
1
T
? a,j := n A Aj .
empirical Gram matrix is denoted by G
j
j
2.3
Assumptions
The main assumptions have been stated already but we give a summary below.
(A) The data are observations of the equilibrium observations of model (3). The matrix I B is
invertible and the solutions to (3) are thus well defined. The cycle-product (2) CP (B) is strictly
smaller than one. The diagonal entries of B are zero.
(B) The distribution of the noise ej (which includes the influence of latent variables) and the connectivity matrix B are identical across all settings j 2 J . In each setting j 2 J , the intervention shift cj and the noise ej are uncorrelated.
(C) Interventions at different variables in the same setting are uncorrelated, that is ?c,j is an (unknown) diagonal matrix for all j 2 J .
We will discuss a stricter version of (C) in Section D in the Appendix that allows the use of Gram
matrices instead of covariance matrices. The conditions above imply that the environments are
characterized by different interventions strength, as measured by the variance of the shift c in each
setting. We aim to reconstruct both the connectivity matrix B from observations in different environments and also aim to reconstruct the a-priori unknown intervention strength and location in
each environment. Additionally, we will show examples where we can detect violations of the model
assumptions and use these to reconstruct the location of interventions.
2.4
Population method
The main idea is very simple. Looking at the model (3), we can rewrite
(I B)xj = cj + ej .
3
(4)
The population covariance of the transformed observations are then for all settings j 2 J given by
(I B)?x,j (I B)T = ?c,j + ?e .
(5)
The last term ?e is constant across all settings j 2 J (but not necessarily diagonal as we allow
hidden variables). Any change of the matrix on the left-hand side thus stems from a shift in the
covariance matrix ?c,j of the interventions. Let us define the difference between the covariance
of c and x in setting j as
?c,j := ?c,j ?c, j , and
?x,j := ?x,j ?x, j .
(6)
Assumption (B) together with (5) implies that
(I B) ?x,j (I B)T = ?c,j
8j 2 J .
(7)
Using assumption (C), the random intervention shifts at different variables are uncorrelated and the
right-hand side in (7) is thus a diagonal matrix for all j 2 J . Let D ? Rp?p be the set of all
invertible matrices. We also define a more restricted space Dcp which only includes those members
of D that have entries all equal to one on the diagonal and have a cycle-product less than one,
n
o
D := D 2 Rp?p : D invertible
(8)
n
o
Dcp := D 2 Rp?p : D 2 D and diag(D) ? 1 and CP (I D) < 1 .
(9)
Under Assumption (A), I
B 2 Dcp . Motivated by (7), we now consider the minimizer
X
X
D = argminD0 2Dcp
L(D0 ?x,j D0T ), where L(A) :=
A2k,l
j2J
(10)
k6=l
is the loss L for any matrix A and defined as the sum of the squared off-diagonal elements. In
Section 3, we present necessary and sufficient conditions on the interventions under which D =
I B is the unique minimizer of (10). In this case, exact joint diagonalization is possible so that
L(D ?x,j DT ) = 0 for all environments j 2 J . We discuss an alternative that replaces covariance
with Gram matrices throughout in Section D in the Appendix. We now give a finite-sample version.
2.5
Finite-sample estimate of the connectivity matrix
In practice, we estimate B by minimizing Algorithm 1 BACK S HIFT
the empirical counterpart of (10) in two
Input: Xj 8j 2 J
steps. First, the solution of the optimiza?x,j 8j 2 J
tion is only constrained to matrices in D. 1: Compute d
? = FFD IAG( d
Subsequently, we enforce the constraint on 2: D
?x,j )
the solution to be a member of Dcp . The 3: D
? = PermuteAndScale(D)
?
BACK S HIFT algorithm is presented in Al?
?
gorithm 1 and we describe the important 4: B = I D
?
Output: B
steps in more detail below.
Steps 1 & 2. First, we minimize the following empirical, less constrained variant of (10)
X
? := argmin 0
D
L(D0 ( d
?x,j )D0T ),
D 2D
(11)
j2J
where the population differences between covariance matrices are replaced with their empirical
? 2 D. For the
counterparts and the only constraint on the solution is that it is invertible, i.e. D
optimization we use the joint approximate matrix diagonalization algorithm FFD IAG [23].
Step 3. The constraint on the cycle product and the diagonal elements of D is enforced by (a)
? Part (b) simply scales the rows so that the diagonal
permuting and (b) scaling the rows of D.
?
elements of the resulting matrix D are all equal to one. The more challenging first step (a) consists
of finding a permutation such that under this permutation the scaled matrix from part (b) will have
a cycle product as small as possible (as follows from Theorem 3, at most one permutation can lead
to a cycle product less than one). This optimization problem seems computationally challenging at
first, but we show that it can be solved by a variant of the linear assignment problem (LAP) (see e.g.
[24]), as proven in Theorem 3 in the Appendix. As a last step, we check whether the cycle product
? is less than one, in which case we have found the solution. Otherwise, no solution satisfying
of D
the model assumptions exists and we return a warning that the model assumptions are not met. See
Appendix B for more details.
4
Computational cost. The computational complexity of BACK S HIFT is O(|J |?n?p2 ) as computing
the covariance matrices costs O(|J |?n?p2 ), FFD IAG has a computational cost of O(|J |?p2 ) and both
the linear assignment problem and computing the cycle product can be solved in O(p3 ) time. For
instance, this complexity is achieved when using the Hungarian algorithm for the linear assignment
problem (see e.g. [24]) and the cycle product can be computed with a simple dynamic programming
approach.
2.6
Estimating the intervention variances
One additional benefit of BACK S HIFT is that the location and strength of the interventions can be
estimated from the data. The empirical, plug-in version of Eq. (7) is given by
(I
? d
B)
?x,j (I
? T = d
b c,j
B)
?c,j = ?
b c,
?
j
8j 2 J .
(12)
So the element ( d
?c,j )kk is an estimate for the difference between the variance of the intervention at variable k in environment j, namely (?c,j )kk , and the average in all other environments,
(?c, j )kk . From these differences we can compute the intervention variance for all environments
up to an offset. By convention, we set the minimal intervention variance across all environments
equal to zero. Alternatively, one can let observational data, if available, serve as a baseline against
which the intervention variances are measured.
3
Identifiability
Let for simplicity of notation,
? j,k := ( ?c,j )kk
be the variance of the random intervention shifts cj at node k in environment j 2 J as per the
definition of ?c,j in (6). We then have the following identifiability result (the proof is provided
in Appendix A).
Theorem 1. Under assumptions (A), (B) and (C), the solution to (10) is unique if and only if for all
k, l 2 {1, . . . , p} there exist j, j 0 2 J such that
? j,k ? j 0 ,l 6= ? j,l ? j 0 ,k .
(13)
If none of the intervention variances ? j,k vanishes, the uniqueness condition is equivalent to demanding that the ratio between the intervention variances for two variables k, l must not stay identical across all environments, that is there exist j, j 0 2 J such that
? j,k
? j 0 ,k
6=
,
(14)
? j,l
? j 0 ,l
which requires that the ratio of the variance of the intervention shifts at two nodes k, l is not identical
across all settings. This leads to the following corollary.
Corollary 2. (i) The identifiability condition (13) cannot be satisfied if |J | = 2 since then ? j,k =
? j 0 ,k for all k and j 6= j 0 . We need at least three different environments for identifiability.
(ii) The identifiability condition (13) is satisfied for all |J | 3 almost surely if the variances of the
intervention cj are chosen independently (over all variables and environments j 2 J ) from a
distribution that is absolutely continuous with respect to Lebesgue measure.
Condition (ii) can be relaxed but shows that we can already achieve full identifiability with a very
generic setting for three (or more) different environments.
4
Numerical results
In this section, we present empirical results for both synthetic and real data sets. In addition to
estimating the connectivity matrix B, we demonstrate various ways to estimate properties of the
interventions. Besides computing the point estimate for BACK S HIFT, we use stability selection [25]
to assess the stability of retrieved edges. We attach R-code with which all simulations and analyses
can be reproduced2 .
2
An R-package called ?backShift? is available from CRAN.
5
5
0.76
3
?0.65
6
3
10
E1
9
2.1
8
?0.69
1
1
0.52
W
1
1
0.34
7
0.75
X3
E2
2
X1
X2
2
I1
0.50
Interv. strength
?0
? 0.5
?1
0.25
Sample size
? 1000
? 10000
0.00
Hidden vars.
? FALSE
TRUE
I2
(a)
Method
? BACKSHIFT
LING
?
?
1.00
I3
0.67
?0.73
0.54
E3
3 0.46
0.72
2
?
PRECISION
4
0.00
0.25
0.50
RECALL
(b)
0.75
1.00
(c)
Figure 1: Simulated data. (a) True network. (b) Scheme for data generation. (c) Performance metrics for the
settings considered in Section 4.1. For BACK S HIFT, precision and recall values for Settings 1 and 2 coincide.
BACK S HIFT
Setting 1
n = 1000
no hidden vars.
mI = 1
4
Setting 2
n = 10000
no hidden vars.
mI = 1
4
3
5
10
6
7
7
L ING
4
10
6
7
9
8
7
9
8
1
SHD = 14, |t| = 0.68
4
10
6
7
9
8
1
SHD = 16, |t| = 0.98
7
4
3
5
10
6
9
8
1
SHD = 5, |t| = 0.25
2
7
9
8
3
5
2
10
1
SHD = 12
2
3
6
9
8
3
4
5
10
7
SHD = 2, |t| = 0.25
2
4
5
2
10
1
Setting 5
n = 10000
no hidden vars.
mI = 0.5
3
6
9
8
6
1
SHD = 17, |t| = 0.91
7
3
5
2
10
1
SHD = 0, |t| = 0.25
3
5
4
5
2
6
9
8
1
SHD = 0, |t| = 0.25
4
10
Setting 4
n = 10000
no hidden vars.
mI = 0
3
4
5
2
6
9
8
3
5
2
Setting 3
n = 10000
hidden vars.
mI = 1
1
SHD = 8, |t| = 0.25
2
10
6
7
9
8
1
SHD = 7, |t| = 0.29
Figure 2: Point estimates of
BACK S HIFT and L ING for synthetic data. We threshold the point estimate of
at t = ?0.25 to exclude those entries which are close to zero. We then threshold the estimate of
L ING so that the two estimates have the same number of edges. In Setting 4, we threshold L ING at t = ?0.25
as BACK S HIFT returns the empty graph. In Setting 3, it is not possible to achieve the same number of edges as
all remaining coefficients in the point estimate of L ING are equal to one in absolute value. The transparency
of the edges illustrates the relative magnitude of the estimated coefficients. We report the structural Hamming
distance (SHD) for each graph. Precision and recall values are shown in Figure 1(c).
BACK S HIFT
4.1
Synthetic data
We compare the point estimate of BACK S HIFT against L ING [11], a generalization of L INGAM to
the cyclic case for purely observational data. We consider the cyclic graph shown in Figure 1(a) and
generate data under different scenarios. The data generating mechanism is sketched in Figure 1(b).
Specifically, we generate ten distinct environments with non-Gaussian noise. In each environment,
the random intervention variable is generated as (cj )k = kj Ikj , where 1j , . . . , pj are drawn i.i.d.
from Exp(mI ) and I1j , . . . , Ipj are independent standard normal random variables. The intervention
shift thus acts on all observed random variables. The parameter mI regulates the strength of the
intervention. If hidden variables exist, the noise term (ej )k of variable k in environment j is equal to
j
k W , where the weights 1 , . . . , p are sampled once from a N (0, 1)-distribution and the random
variable W j has a Laplace(0, 1) distribution. If no hidden variables are present, then (ej )k , k =
1, . . . , p is sampled i.i.d. Laplace(0, 1). In this set of experiments, we consider five different settings
(described below) in which the sample size n, the intervention strength mI as well as the existence
of hidden variables varies.
We allow for hidden variables in only one out of five settings as L ING assumes causal sufficiency
and can thus in theory not cope with hidden variables. If no hidden variables are present, the pooled
data can be interpreted as coming from a model whose error variables follow a mixture distribution.
But if one of the error variables comes from the second mixture component, for example, the other
6
PIP2
PIP2
PLCg
PIP3
PIP2
PLCg
PIP3
Erk
Mek
Erk
Raf
Raf
Akt
JNK
p38
JNK
(a)
PKA
PKC
Raf
Akt
JNK
PKA
PKC
Mek
Erk
Raf
Akt
PKA
PLCg
PIP3
Mek
Erk
Akt
PIP2
PLCg
PIP3
Mek
p38
(b)
JNK
PKA
PKC
(c)
p38
PKC
p38
(d)
Figure 3: Flow cytometry data. (a) Union of the consensus network (according to [22]), the reconstruction by
[22] and the best acyclic reconstruction by [26]. The edge thickness and intensity reflect in how many of these
three sources that particular edge is present. (b) One of the cyclic reconstructions by [26]. The edge thickness
and intensity reflect the probability of selecting that particular edge in the stability selection procedure. For
more details see [26]. (c) BACK S HIFT point estimate, thresholded at ?0.35. The edge intensity reflects the
relative magnitude of the coefficients and the coloring is a comparison to the union of the graphs shown in
panels (a) and (b). Blue edges were also found in [26] and [22], purple edges are reversed and green edges were
not previously found in (a) or (b). (d) BACK S HIFT stability selection result with parameters E(V ) = 2 and
?thr = 0.75. The edge thickness illustrates how often an edge was selected in the stability selection procedure.
error variables come from the second mixture component, too. In this sense, the data points are not
independent anymore. This poses a challenge for L ING which assumes an i.i.d. sample. We also
cover a case (for mI = 0) in which all assumptions of L ING are satisfied (Scenario 4).
Figure 2 shows the estimated connectivity matrices for five different settings and Figure 1(c) shows
the obtained precision and recall values. In Setting 1, n = 1000, mI = 1 and there are no hidden
variables. In Setting 2, n is increased to 10000 while the other parameters do not change. We observe
that BACK S HIFT retrieves the correct adjacency matrix in both cases while L ING?s estimate is not
very accurate. It improves slightly when increasing the sample size. In Setting 3, we do include
hidden variables which violates the causal sufficiency assumption required for L ING. Indeed, the
estimate is worse than in Setting 2 but somewhat better than in Setting 1. BACK S HIFT retrieves
two false positives in this case. Setting 4 is not feasible for BACK S HIFT as the distribution of the
variables is identical in all environments (since mI = 0). In Step 2 of the algorithm, FFD IAG does
not converge and therefore the empty graph is returned. So the recall value is zero while precision
is not defined. For L ING all assumptions are satisfied and the estimate is more accurate than in
the Settings 1?3. Lastly, Setting 5 shows that when increasing the intervention strength to 0.5,
BACK S HIFT returns a few false positives. Its performance is then similar to L ING which returns its
most accurate estimate in this scenario. The stability selection results for BACK S HIFT are provided
in Figure 5 in Appendix E.
In short, these results suggest that the BACK S HIFT point estimates are close to the true graph if the
interventions are sufficiently strong. Hidden variables make the estimation problem more difficult
but the true graph is recovered if the strength of the intervention is increased (when increasing mI
to 1.5 in Setting 3, BACK S HIFT obtains a SHD of zero). In contrast, L ING is unable to cope with
hidden variables but also has worse accuracy in the absence of hidden variables under these shift
interventions.
4.2
Flow cytometry data
The data published in [22] is an instance of a data set where the external interventions differ between the environments in J and might act on several compounds simultaneously [18]. There are
nine different experimental conditions with each containing roughly 800 observations which correspond to measurements of the concentration of biochemical agents in single cells. The first setting
corresponds to purely observational data.
In addition to the original work by [22], the data set has been described and analyzed in [18] and
[26]. We compare against the results of [26], [22] and the ?well-established consensus?, according
to [22], shown in Figures 3(a) and 3(b). Figure 3(c) shows the (thresholded) BACK S HIFT point
estimate. Most of the retrieved edges were also found in at least one of the previous studies. Five
edges are reversed in our estimate and three edges were not discovered previously. Figure 3(d) shows
the corresponding stability selection result with the expected number of falsely selected variables
7
7.0
S&P 500
NASDAQ
S&P 500
2001
2003
2005
2007
2009
2011
2001
TIME
(a) Prices (logarithmic)
2003
2005
2007
2009
2011
EST. INTERVENTION VARIANCE
LOG?RETURNS
8.0
NASDAQ
7.5
EST. INTERVENTION VARIANCE
9.0
DAX
6.5
LOG(PRICE)
8.5
DAX
DAX
S&P 500
NASDAQ
2001
2003
TIME
2005
2007
2009
2011
TIME
(b) Daily log-returns
(c)
BACK S HIFT
DAX
S&P 500
NASDAQ
2001
2003
2005
2007
2009
2011
TIME
(d) L ING
Figure 4: Financial time series with three stock indices: NASDAQ (blue; technology index), S&P 500 (green;
American equities) and DAX (red; German equities). (a) Prices of the three indices between May 2000 and end
of 2011 on a logarithmic scale. (b) The scaled log-returns (daily change in log-price) of the three instruments
are shown. Three periods of increased volatility are visible starting with the dot-com bust on the left to the
financial crisis in 2008 and the August 2011 downturn. (c) The scaled estimated intervention variance with
the estimated BACK S HIFT network. The three down-turns are clearly separated as originating in technology,
American and European equities. (d) In contrast, the analogous L ING estimated intervention variances have a
peak in American equities intervention variance during the European debt crisis in 2011.
E(V ) = 2. This estimate is sparser in comparison to the other ones as it bounds the number of false
discoveries. Notably, the feedback loops between PIP2 $ PLCg and PKC $ JNK were also found
in [26].
It is also noteworthy that we can check the model assumptions of shift interventions, which is important for these data as they can be thought of as changing the mechanism or activity of a biochemical
agent rather than regulate the biomarker directly [26]. If the shift interventions are not appropriate, we are in general not able to diagonalize the differences in the covariance matrices. Large
off-diagonal elements in the estimate of the r.h.s in (7) indicate a mechanism change that is not just
explained by a shift intervention as in (1). In four of the seven interventions environments with
known intervention targets the largest mechanism violation happens directly at the presumed intervention target, see Appendix C for details. It is worth noting again that the presumed intervention
target had not been used in reconstructing the network and mechanism violations.
4.3
Financial time series
Finally, we present an application in financial time series where the environment is clearly changing
over time. We consider daily data from three stock indices NASDAQ, S&P 500 and DAX for a
period between 2000-2012 and group the data into 74 overlapping blocks of 61 consecutive days
each. We take log-returns, as shown in panel (b) of Figure 4 and estimate the connectivity matrix,
which is fully connected in this case and perhaps of not so much interest in itself. It allows us,
however, to estimate the intervention strength at each of the indices according to (12), shown in
panel (c). The intervention variances separate very well the origins of the three major down-turns of
the markets on the period. Technology is correctly estimated by BACK S HIFT to be at the epicenter
of the dot-com crash in 2001 (NASDAQ as proxy), American equities during the financial crisis in
2008 (proxy is S&P 500) and European instruments (DAX as best proxy) during the August 2011
downturn.
5
Conclusion
We have shown that cyclic causal networks can be estimated if we obtain covariance matrices of
the variables under unknown shift interventions in different environments. BACK S HIFT leverages
solutions to the linear assignment problem and joint matrix diagonalization and the part of the computational cost that depends on the number of variables is at worst cubic. We have shown sufficient
and necessary conditions under which the network is fully identifiable, which require observations
from at least three different environments. The strength and location of interventions can also be
reconstructed.
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9
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5,173 | 5,683 | Learning with Relaxed Supervision
Percy Liang
Stanford University
[email protected]
Jacob Steinhardt
Stanford University
[email protected]
Abstract
For weakly-supervised problems with deterministic constraints between the latent
variables and observed output, learning necessitates performing inference over latent variables conditioned on the output, which can be intractable no matter how
simple the model family is. Even finding a single latent variable setting that satisfies the constraints could be difficult; for instance, the observed output may be
the result of a latent database query or graphics program which must be inferred.
Here, the difficulty lies in not the model but the supervision, and poor approximations at this stage could lead to following the wrong learning signal entirely.
In this paper, we develop a rigorous approach to relaxing the supervision, which
yields asymptotically consistent parameter estimates despite altering the supervision. Our approach parameterizes a family of increasingly accurate relaxations,
and jointly optimizes both the model and relaxation parameters, while formulating constraints between these parameters to ensure efficient inference. These
efficiency constraints allow us to learn in otherwise intractable settings, while
asymptotic consistency ensures that we always follow a valid learning signal.
1
Introduction
We are interested in the problem of learning from intractable supervision. For example, for a
question answering application, we might want to learn a semantic parser that maps a question
x (e.g., ?Which president is from Arkansas??) to a logical form z (e.g., USPresident(e) ?
PlaceOfBirth(e, Arkansas)) that executes to the answer y (e.g., BillClinton). If we are only
given (x, y) pairs as training data [1, 2, 3], then even if the model p? (z | x) is tractable, it is still
intractable to incorporate the hard supervision constraint [S(z, y) = 1] since z and y live in a large
space and S(z, y) can be complex (e.g., S(z, y) = 1 iff z executes to y on a database). In addition to
semantic parsing, intractable supervision also shows up in inverse graphics [4, 5, 6], relation extraction [7, 8], program induction [9], and planning tasks with complex, long-term goals [10]. As we
scale to weaker supervision and richer output spaces, such intractabilities will become the norm.
One can handle the intractable constraints in various ways: by relaxing them [11], by applying them
in expectation [12], or by using approximate inference [8]. However, as these constraints are part of
the supervision rather than the model, altering them can fundamentally change the learning process;
this raises the question of when such approximations are faithful enough to learn a good model.
In this paper, we propose a framework that addresses these questions formally, by constructing a
relaxed supervision function with well-characterized statistical and computational properties. Our
approach is sketched in Figure 1: we start with an intractable supervision function q? (y | z) (given
by the constraint S), together with a model family p? (z | x). We then replace q? by a family of
functions q? (y | z) which contains q? , giving rise to a joint model p?,? (y, z | x). We ensure
tractability of inference by constraining p? (z | x) and p?,? (z | x, y) to stay close together, so that
the supervision y is never too surprising to the model. Finally, we optimize ? and ? subject to this
tractability constraint; when q? (y | z) is properly normalized, there is always pressure to use the true
1
more exact
intractable region
?
less exact
y
tor
jec
a
g tr
nin
r
lea
tractable region
?
less accurate
Figure 1: Sketch of our approach; we define
a family of relaxations q? of the supervision,
and then jointly optimize both ? and ?. If the
supervision q? is too harsh relative to the accuracy of the current model p? , inference becomes intractable. In Section 4, we formulate
constraints to avoid this intractable region and
learn within the tractable region.
more accurate
supervision q? , and we can prove that the global optimum of p?,? is an asymptotically consistent
estimate of the true model.
Section 2 introduces the relaxed supervision model q? (y | z) ? exp(? > ?(z, y)), where ?(z, y) = 0
iff the constraint S(z, y) is satisfied (the original supervision is then obtained when ? = ?). Section 3 studies the statistical properties of this relaxation, establishing asymptotic consistency as well
as characterizing the properties for any fixed ?: we show roughly that both the loss and statistical
?1
efficiency degrade by a factor of ?min
, the inverse of the smallest coordinate of ?. In Section 4, we
introduce novel tractability constraints, show that inference is efficient if the constraints are satisfied, and present an EM-like algorithm for constrained optimization of the likelihood. Finally, in
Section 5, we explore the empirical properties of this algorithm on two illustrative examples.
2
Framework
We assume that we are given a partially supervised problem x ? z ? y where (x, y) ? X ? Y
are observed and z ? Z is unobserved. We model z given x as an exponential family p? (z | x) =
exp(?> ?(x, z)?A(?; x)), and assume that y = f (z) is a known deterministic function of z. Hence:
X
p? (y | x) =
S(z, y) exp(?> ?(x, z) ? A(?; x)),
(1)
z
where S(z, y) ? {0, 1} encodes the constraint [f (z) = y]. In general, f could have complicated
structure, rendering inference (i.e., computing p? (z | x, y), which is needed for learning) intractable.
To alleviate this, we consider projections ?j mapping Y to some smaller set Yj ; we then obtain the
def
(hopefully simpler) constraint that f (z) and y match under ?j : Sj (z, y) = [?j (f (z)) = ?j (y)]. We
Vk
assume ?1 ? ? ? ? ? ?k is injective, which implies that S(z, y) equals the conjunction j=1 Sj (z, y).
We also assume that some part of S (call it T(z, y)) can be imposed tractably. We can always take
T ? 1, but it is better to include as much of S as possible because T will be handled exactly while S
will be approximated. We record our assumptions below:
Definition 2.1. Let S(z, y) encode the constraint f (z) = y. We say that (T, ?1 , . . . , ?k ) logically
decomposes S if (1) S implies T and (2) ?1 ? ? ? ? ? ?k is injective.
Before continuing, we give three examples to illustrate the definitions above.
Example 2.2 (Translation from unordered supervision). Suppose that given an input sentence x,
each word is passed through the same unknown 1-to-1 substitution cipher to obtain an enciphered
sentence z, and then ordering is removed to obtain an output y = multiset(z). For example, we
might have x = abaa, z = dcdd, and y = {c : 1, d : 3}. Suppose the vocabulary is {1, . . . , V }. Our
constraint is S(z, y) = [y = multiset(z)], which logically decomposes as
?j (y)
f (z)
V
z
z }| {
}|
{
^
[y = multiset(z)] ?? [zi ? y for all i] ?
[count(z, j) = count(y, j)],
{z
}
|
{z
}
{z
}
|
|
j=1
S(z,y)
T(z,y)
(2)
Sj (z,y)
where count(?, j) counts the number of occurrences of the word j. The constraint T is useful because
it lets us restrict attention to words in y (rather than all of {1, . . . , V }), which dramatically reduces
the search space. If each sentence has length L, then Yj = ?j (Y) = {0, . . . , L}.
Example 2.3 (Conjunctive semantic parsing). Suppose again that x is an input sentence, and that
each input word xi ? {1, . . . , V } maps to a predicate (set) zi ? {Q1 , . . . , Qm }, and the meaning y
2
of the sentence is the intersection of the predicates. For instance, if the sentence x is ?brown dog?,
and Q6 is the set of all brown objects and Q11 is the set of all dogs, then z1 = Q6 , z2 = Q11 , and
def
y = Q6 ? Q11 is the set of all brown dogs. In general, we define y = JzK = z1 ? ? ? ? ? zl . This is a
simplified form of learning semantic parsers from denotations [2].
We let Y be every set that is obtainable as an intersection of predicates Q, and define ?j (y) = [y ?
Qj ] for j = 1, . . . , m (so Yj = {0, 1}). Note that for all y ? Y, we have y = ?j:?j (y)=1 Qj , so
?1 ? ? ? ? ? ?m is injective. We then have the following logical decomposition:
?j (y)
m
^
z }| {
y = JzK ?? [zi ? y for all i] ?
[JzK ? Qj ] = [y ? Qj ] .
| {z }
|
{z
}
|
{z
}
j=1
S(z,y)
(3)
Sj (z,y)
T(z,y)
The first constraint T factors across i, so it can be handled tractably.
Example 2.4 (Predicate abstraction). Next, we consider a program induction task; here
the input x might be ?smallest square divisible by six larger than 1000?, z would be
argmin{i1 | mod(i1,6) = 0 and i1 = i2*i2 and i1 > 1000}, and y would be
1296; hence S(z, y) = 1 if z evaluates to y. Suppose that we have a collection of predicates ?j ,
such as ?1 (y) = mod(y, 6), ?2 (y) = isPrime(y), etc. These predicates are useful for giving partial
credit; for instance, it is easier to satisfy mod(y, 6) = 0 than y = 1296, but many programs that
satisfy the former will have pieces that are also in the correct z. Using the ?j to decompose S will
therefore provide a more tractable learning signal that still yields useful information.
Relaxing the supervision. Returning to the general framework, let us now use Sj and T to relax
S, and thus also p? (y | x). First, define
penalty features ?j (z, y) = Sj (z, y) ? 1, and also define
q? (y | z) ? T(z, y) exp ? > ?(z, y) for any vector ? ? 0. Then, ? log q? (y | z) measures how far
S(z, y) is from being satisfied: for each violated Sj , we incur a penalty ?j (or infinite penalty if T is
violated). Note that the original q? (y | z) = S(z, y) corresponds to ?1 = ? ? ? = ?k = +?.
Normalization
constant.
The log-normalization constant A(?; z) for q? is equal to
P
log( y?Y T(z, y) exp(? > ?(z, y))); this is in general difficult to compute, since ? could have arbitrary structure. Fortunately, we can uniformly upper-bound A(?; z) by a tractable quantity A(?):
Proposition 2.5. For any z, we have the following bound:
A(?; z) ?
k
X
def
log (1 + (|Yj |?1) exp(??j )) = A(?).
(4)
j=1
See the supplement for proof; the intuition is that, by injectivity of ?1 ? ? ? ? ? ?k , we can bound Y
Qk
by the product set j=1 Yj . We now define our joint model, which is a relaxation of (1):
q? (y | z) = T(z, y) exp ? > ?(z, y) ? A(?) ,
(5)
X
p?,? (y | x) =
T(z, y) exp(?> ?(x, z) + ? > ?(z, y) ? A(?; x) ? A(?)),
(6)
z
L(?, ?) = Ex,y?p? [? log p?,? (y | x)], where p? is the true distribution.
(7)
The relaxation parameter ? provides a trade-off between faithfulness to the original objective (large
?) and tractability (small ?). Importantly, p?,? (y | x) produces valid probabilities which can be
meaningfully compared
across different ?; this will be important later in allowing us to optimize ?.
P
(Note that while y p?,? (y | x) < 1 if the bound (4) is not tight, this gap vanishes as ? ? ?.)
3
Analysis
We now analyze the effects of relaxing supervision (i.e., taking ? < ?); proofs may be found in the
supplement. We will analyze the following properties:
1. Effect on loss: How does the value of the relaxation parameter ? affect the (unrelaxed) loss of
the learned parameters ? (assuming we had infinite data and perfect optimization)?
3
2. Amount of data needed to learn: How does ? affect the amount of data needed in order to
identify the optimal parameters?
3. Optimizing ? and consistency: What happens if we optimize ? jointly with ?? Is there natural
pressure to increase ? and do we eventually recover the unrelaxed solution?
Notation. Let Ep? denote the expectation under x, y ? p? , and let L(?, ?) denote the unrelaxed
loss (see (5)?(7)). Let L? = inf ? L(?, ?) be the optimal unrelaxed loss and ?? be the minimizing
argument. Finally, let E? and Cov? denote the expectation and covariance, respectively, under
p? (z | x). To simplify expressions, we will often omit the arguments from ?(x, z) and ?(z, y), and
use S and ?S for the events [S(z, y) = 1] and [S(z, y) = 0]. For simplicity, assume that T(z, y) ? 1.
Effect on loss. Suppose we set ? to some fixed value (?1 , . . . , ?k ) and let ??? be the minimizer of
L(?, ?). Since ??? is optimized for L(?, ?) rather than L(?, ?), it is possible that L(??? , ?) is very
large; indeed, if p??? (y | x) is zero for even a single outlier (x, y), then L(??? , ?) will be infinite.
However, we can bound ??? under an alternative loss that is less sensitive to outliers:
Proposition 3.1. Let ?min = minkj=1 ?j . Then, Ep? [1 ? p??? (y | x)] ?
L?
1?exp(??min ) .
The key idea in the proof is that replacing S with exp(? > ?) in p?,? does not change the loss too
much, in the sense that S ? exp(? > ?) ? exp(??min ) + (1 ? exp(??min ))S.
?
?
?1
L
When ?min 1, 1?exp(??
. If
? ?Lmin . Hence, the error increases roughly linearly with ?min
min )
?
?min is large and the original loss L is small, then L(?, ?) is a good surrogate. Of particular interest
is the case L? = 0 (perfect predictions); in this case, the relaxed loss L(?, ?) also yields a perfect
predictor for any ? > 0. Note conversely that Proposition 3.1 is vacuous when L? ? 1.
We show in the supplement that Proposition 3.1 is essentially tight:
Lemma 3.2. For any 0 < ?min < L? , there exists a model with loss L? and a relaxation parameter
? = (?min , ?, . . . , ?), such that Ep? [p??? (y | x)] = 0.
Amount of data needed to learn. To estimate how much data is needed to learn, we compute the
def
Fisher information I? = ?2? L(??? , ?), which measures the statistical efficiency of the maximum
likelihood estimator [13]. All of the equations below follow from standard properties of exponential
families [14], with calculations in the supplement. For the unrelaxed loss, the Fisher information is:
I? = Ep? [P?? [?S] (E?? [? ? ? | ?S] ? E?? [? ? ? | S])] .
(8)
Hence ?? is easy to estimate if the features have high variance when S = 0 and low variance when
S = 1. This should be true if all z with S(z, y) = 1 have similar feature values while the z with
S(z, y) = 0 have varying feature values.
In the relaxed case, the Fisher information can be written to first order as
h
i
I? = Ep? Cov??? ?(x, z) ? ?(x, z), ?? > ?(z, y) + O ? 2 .
(9)
In other words, I? , to first order, is the covariance of the penalty ?? > ? with the second-order
statistics of ?. To interpret this, we will make the simplifying assumptions that (1) ?j = ?min for all
j, and (2) the events ?Sj are all disjoint. In this case, ?? > ? = ?min ?S, and the covariance in (9)
simplifies to
Cov??? ? ? ?, ?? > ? = ?min P??? [S]P??? [?S] E??? [? ? ? | ?S] ? E??? [? ? ? | S] .
(10)
Relative to (8), we pick up a ?P??? [S] factor. If we further assume that P??? [S] ? 1, we see that the
?1
amount of data required to learn under the relaxation increases by a factor of roughly ?min
.
Optimizing ?. We now study the effects of optimizing both ? and ? jointly. Importantly, joint
optimization recovers the true distribution p?? in the infinite data limit:
Proposition 3.3. Suppose the model is well-specified: p? (y | x) = p?? (y | x) for all x, y. Then, all
global optima of L(?, ?) satisfy p?,? (y | x) = p? (y | x); one such optimum is ? = ?? , ? = ?.
4
There is thus always pressure to send ? to ? and ? to ?? . The key fact in the proof is that the log-loss
L(?, ?) is never smaller than the conditional entropy Hp? (y | x), with equality iff p?,? = p? .
Summary. Based on our analyses above, we can conclude that relaxation has the following impact:
?1
? Loss: The loss increases by a factor of ?min
in the worst case.
?1
? Amount of data: In at least one regime, the amount of data needed to learn is ?min
times larger.
The general theme is that the larger ? is, the better the statistical properties of the maximumlikelihood estimator. However, larger ? also makes the distribution p?,? less tractable, as q? (y | z)
becomes concentrated on a smaller set of y?s. This creates a trade-off between computational efficiency (small ?) and statistical accuracy (large ?). We explore this trade-off in more detail in the
next section, and show that in some cases we can get the best of both worlds.
4
Constraints for Efficient Inference
In light of the previous section, we would like to make ? as large as possible; on the other hand,
if ? is too large, we are back to imposing S exactly and inference becomes intractable. We would
therefore like to optimize ? subject to a tractability constraint ensuring that we can still perform
efficient inference, as sketched earlier in Figure 1. We will use rejection sampling as the inference
procedure, with the acceptance rate as a measure of tractability.
To formalize our approach, we assume that the model p? (z | x) and the constraint T(z, y) are jointly
tractable, so that we can efficiently draw exact samples from
def
p?,T (z | x, y) = T(z, y) exp ?> ?(x, z) ? AT (?; x, y) ,
(11)
P
where AT (?; x, y) = log( z T(z, y) exp(?> ?(x, z))). Most learning algorithms require the conditional expectations of ? and ? given x and y; we therefore need to sample the distribution
p?,? (z | x, y) = T(z, y) exp ?> ?(x, z) + ? > ?(z, y) ? A(?, ?; x, y) , where
(12)
!
X
def
A(?, ?; x, y) = log
T(z, y) exp(?> ?(x, z) + ? > ?(z, y)) .
(13)
z
Since ? > ? ? 0, we can draw exact samples from p?,? using rejection sampling: (1) sample z from
p?,T (? | x, y), and (2) accept with probability exp(? > ?(z, y)). If the acceptance rate is high, this
algorithm lets us tractably sample from (12). Intuitively, when ? is far from the optimum, the model
p? and constraints Sj will clash, necessitating a small value of ? to stay tractable. As ? improves,
more of the constraints Sj will be satisfied automatically under p? , allowing us to increase ?.
Formally, the expected number of samples is the inverse of the acceptance probability and can be
expressed as (see the supplement for details)
X
?1
p?,T (z | x, y) exp(? > ?(z, y))
= exp (AT (?; x, y) ? A(?, ?; x, y)) .
(14)
z
We can then minimize the loss L(?, ?) = A(?; x) + A(?) ? A(?, ?; x, y) (see (6)?(7) and (13))
subject to the tractability constraint Ex,y [exp (AT (?; x, y) ? A(?, ?; x, y))] ? ? , where ? is our
computational budget. While one might have initially worried that rejection sampling will perform
poorly, this constraint guarantees that it will perform well by bounding the number of rejections.
Implementation details. To minimize L subject to a constraint on (14), we will develop an EM-like
algorithm; the algorithm maintains an inner approximation to the constraint set as well as an upper
bound on the loss, both of which will be updated with each iteration of the algorithm. These bounds
? ?)
? we have by convexity:
are obtained by linearizing A(?, ?; x, y); more precisely, for any (?,
? ?;
? x, y) + (? ? ?)
? > ?? + (? ? ?)
? > ?,
?
? ?; x, y) def
A(?, ?; x, y) ? A(?,
= A(?,
X
X
def
def
where ?? =
p?,
?? =
p?,
? ?? (z | x, y)?(x, z),
? ?? (z | x, y)?(z, y).
z
z
5
(15)
? on the loss L, as well as a tractability constraint C1 , which are both convex:
We thus obtain a bound L
h
i
? ?; x, y)
?
minimize Ep? A(?; x) + A(?) ? A(?,
(L)
h
i
? ?; x, y) ? ?.
subject to Ep? exp AT (?; x, y) ? A(?,
(C1 )
? and C1 using the minimizing
We will iteratively solve the above minimization, and then update L
(?, ?) from the previous step. Note that the minimization itself can be done without inference; we
? Since inference is tractable at (?,
? ?)
? by design,
only need to do inference when updating ?? and ?.
?
?
we can obtain unbiased estimates of ? and ? using the rejection sampler described earlier. We can
? ?;
? x, y) at the same time by using samples from p ? and the relation (14).
also estimate A(?,
?,T
? ?).
? It is
A practical issue is that C1 becomes overly stringent when (?, ?) is far away from (?,
therefore difficult to make large moves in parameter space, which is especially bad for getting started
initially. We can solve this using the trivial constraint
k
X
exp
?j ? ?,
(C0 )
j=1
which will also ensure tractability. We use (C0 ) for several initial iterations, then optimize the rest
of the way using (C1 ). To avoid degeneracies at ? = 0, we also constrain ? ? in all iterations. We
will typically take = 1/k, which is feasible for (C0 ) assuming ? ? exp(1).1
To summarize, we have obtained an iterative algorithm for jointly minimizing L(?, ?), such that
p?,? (z | x, y) always admits efficient rejection sampling. Pseudocode is provided in Algorithm 1;
note that all population expectations Ep? should now be replaced with sample averages.
Algorithm 1 Minimizing L(?, ?) while guaranteeing tractable inference.
Input training data (x(i) , y (i) )ni=1 .
Initialize ?? = 0, ??j = for j = 1, . . . , k.
while not converged do
? ?;
? x(i) , y (i) ) for i = 1, . . . , n by sampling p ? ? (z | x(i) , y (i) ).
Estimate ??(i) , ??(i) , and A(?,
?,?
? ?; x(i) , y (i) ) using the output from the preceding step.
Estimate the functions A(?,
? ?)
? be the solution to
Let (?,
n
1 X
? ?; x(i) , y (i) )
minimize
A(?; x(i) ) + A(?) ? A(?,
?,?
n i=1
subject to (C0 ), ?j ? for j = 1, . . . , k
? ?)
? ? (?,
? ?).
?
Update (?,
end while
Repeat the same loop as above, with the constraint (C0 ) replaced by (C1 ).
? ?).
?
Output (?,
5
Experiments
We now empirically explore our method?s behavior. All of our code, data, and experiments may be
found on the CodaLab worksheet for this paper at https://www.codalab.org/worksheets/
0xc9db508bb80446d2b66cbc8e2c74c052/, which also contains more detailed plots beyond
those shown here. We would like to answer the following questions:
? Fixed ?: For a fixed ?, how does the relaxation parameter ? affect the learned parameters?
What is the trade-off between accuracy and computation as we vary ??
1
If only some of the constraints Sj are active for each y (e.g. for translation we only have to worry about
the words that actually appear in the output sentence), then we need only include those ?j in the sum for (C0 ).
This can lead to substantial gains, since now k is effectively the sentence length rather than the vocabulary size.
6
1.0
1.0
AdaptFull(50)
AdaptTied(50)
Fixed(0.8)
Fixed(0.5)
Fixed(0.2)
Fixed(0.1)
0.6
0.8
accuracy
accuracy
0.8
0.4
0.2
0.0
AdaptFull(200)
AdaptTied(200)
AdaptFull(100)
AdaptFull(50)
Fixed(0.5)
Fixed(0.3)
Fixed(0.2)
0.6
0.4
0.2
104
105
106
number of samples
107
0.0
108
(a)
104
105
106
107
number of samples
108
109
(b)
Figure 2: (a) Accuracy versus computation (measured by number of samples drawn by the rejection
sampler) for the unordered translation task. (b) Corresponding plot for the conjunctive semantic
parsing task. For both tasks, the F IXED method needs an order of magnitude more samples to
achieve comparable accuracy to either adaptive method.
? Adapting ?: Does optimizing ? affect performance? Is the per-coordinate adaptivity of our
relaxation advantageous, or can we set all coordinates of ? to be equal? How does the computational budget ? (from C0 and C1 ) impact the optimization?
To answer these questions, we considered using a fixed ? (F IXED(?)), optimizing ? with a computational constraint ? (A DAPT F ULL(? )), and performing the same optimization with all coordinates of
? constrained to be equal (A DAPT T IED(? )). For optimization, we used Algorithm 1, using S = 50
samples to approximate each ??(i) and ??(i) , and using the solver SNOPT [15] for the inner optimization. We ran Algorithm 1 for 50 iterations; when ? is not fixed, we apply the constraint (C0 ) for
the first 10 iterations and (C1 ) for the remaining 40 iterations; when it is fixed, we do not apply any
constraint.
Unordered translation. We first consider the translation task from Example 2.2. Recall that we
def
are given a vocabulary [V ] = {1, . . . , V }, and wish to recover an unknown 1-1 substitution cipher
c : [V ] ? [V ]. Given an input sentence x1:L , the latent z is the result of applying c, where zi is
c(xi ) with probability 1 ? ? and uniform over [V ] with probability ?. To model this, we define a
feature ?u,v (x, z) that counts the number of times that xi = u and zi = v; hence, p? (z | x) ?
PL
exp( i=1 ?xi ,zi ). Recall also that the output y = multiset(z).
In our experiments, we generated n = 100 sentences of length L = 20 with vocabulary size V =
102. For each pair of adjacent words (x2i?1 , x2i ), we set x2i?1 = 3j + 1 with j drawn from a power
law distribution on {0, . . . , V /3 ? 1} with exponent r ? 0; we then set x2i to 3j + 2 or 3j + 3 with
equal probability. This ensures that there are pairs of words that co-occur often (without which the
constraint T would already solve the problem).
We set r = 1.2 and ? = 0.1, which produces a moderate range of word frequencies as well as
a moderate noise level (we also considered setting either r or ? to 0, but omitted these results because essentially all methods achieved ceiling accuracy; the interested reader may find them in our
CodaLab worksheet). We set the computational budget ? = 50 for the constraints C0 and C1 , and
= L1 as the lower bound on ?. To measure accuracy, we look at the fraction of words whose modal
prediction under the model corresponds to the correct mapping.
We plot accuracy versus computation (i.e., cumulative number of samples drawn by the rejection
sampler up through the current iteration) in Figure 2a; note that the number of samples is plotted on a
log-scale. For the F IXED methods, there is a clear trade-off between computation and accuracy, with
multiplicative increases in computation needed to obtain additive increases in accuracy. The adaptive
methods completely surpass this trade-off curve, achieving higher accuracy than F IXED(0.8) while
using an order of magnitude less computation. The A DAPT F ULL and A DAPT T IED methods achieve
similar results to each other; in both cases, all coordinates of ? eventually obtained their maximum
value of 5.0, which we set as a cap for numerical reasons, and which corresponds closely to imposing
the exact supervision signal.
7
Conjunctive semantic parsing. We also ran experiments on the semantic parsing task from Example 2.3. We used vocabulary size V = 150, and represented each predicate Q as a subset of [U ],
where U = 300. The five most common words in [V ] mapped to the empty predicate Q = [U ], and
the remaining words mapped to a random subset of 85% of [U ]. We used n = 100 and sentence
length L = 25. Each word in the input was drawn independently from a power law with r = 0.8. A
word was mapped to its correct predicate with probability 1 ? ? and to a uniformly random predicate
with probability ?, with ? = 0.1. We constrained the denotation y = JzK to have non-zero size by
re-generating each examples until this constraint held. We used the same model p? (z | x) as before,
and again measured accuracy based on the fraction of the vocabulary for which the modal prediction
was correct. We set ? = 50, 100, 200 to compare the effect of different computational budgets.
Results are shown in Figure 2b. Once again, the adaptive methods substantially outperform the
F IXED methods. We also see that the accuracy of the algorithm is relatively invariant to the computational budget ? ? indeed, for all of the adaptive methods, all coordinates of ? eventually obtained
their maximum value, meaning that we were always using the exact supervision signal by the end
of the optimization. These results are broadly similar to the translation task, suggesting that our
method generalizes across tasks.
6
Related Work and Discussion
For a fixed relaxation ?, our loss L(?, ?) is similar to the Jensen risk bound defined by Gimpel and
Smith [16]. For varying ?, our framework is similar in spirit to annealing, where the entire objective
is relaxed by exponentiation, and the relaxation is reduced over time. An advantage of our method
is that we do not have to pick a fixed annealing schedule; it falls out of learning, and moreover, each
constraint can be annealed at its own pace.
Under model well-specification, optimizing the relaxed likelihood recovers the same distribution as
optimizing the original likelihood. In this sense, our approach is similar in spirit to approaches such
as pseudolikelihood [17, 18] and, more distantly, reward shaping in reinforcement learning [19].
There has in the past been considerable interest in specifying and learning under constraints on
model predictions, leading to a family of ideas including constraint-driven learning [11], generalized
expectation criteria [20, 21], Bayesian measurements [22], and posterior regularization [23]. These
ideas are nicely summarized in Section 4 of [23], and involve relaxing the constraint either by using
a variational approximation or by applying the constraint in expectation rather than pointwise (e.g.,
replacing the constraint h(x, z, y) ? 1 with E[h(x, z, y)] ? 1). This leads to tractable inference
when the function h can be tractably incorporated as a factor in the model, which is the case for many
problems of interest (including the translation task in this paper). In general, however, inference will
be intractable even under the relaxation, or the relaxation could lead to different learned parameters;
this motivates our framework, which handles a more general class of problems and has asymptotic
consistency of the learned parameters.
The idea of learning with explicit constraints on computation appears in the context of prioritized
search [24], MCMC [25, 26], and dynamic feature selection [27, 28, 29]. These methods focus on
keeping the model tractable; in contrast, we assume a tractable model and focus on the supervision.
While the parameters of the model can be informed by the supervision, relaxing the supervision as
we do could fundamentally alter the learning process, and requires careful analysis to ensure that
we stay grounded to the data. As an analogy, consider driving a car with a damaged steering wheel
(approximate model) versus not being able to see the road (approximate supervision); intuitively,
the latter appears to pose a more fundamental challenge.
Intractable supervision is a key bottleneck in many applications, and will only become more so as
we incorporate more sophisticated logical constraints into our statistical models. While we have
laid down a framework that grapples with this issue, there is much to be explored?e.g., deriving
stochastic updates for optimization, as well as tractability constraints for more sophisticated inference methods.
Acknowledgments. The first author was supported by a Fannie & John Hertz Fellowship and an
NSF Graduate Research Fellowship. The second author was supported by a Microsoft Research
Faculty Fellowship. We are also grateful to the referees for their valuable comments.
8
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9
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5,174 | 5,684 | M -Statistic for Kernel Change-Point Detection
Shuang Li, Yao Xie
H. Milton Stewart School of
Industrial and Systems Engineering
Georgian Institute of Technology
[email protected]
[email protected]
Hanjun Dai, Le Song
Computational Science and Engineering
College of Computing
Georgia Institute of Technology
[email protected]
[email protected]
Abstract
Detecting the emergence of an abrupt change-point is a classic problem in statistics and machine learning. Kernel-based nonparametric statistics have been proposed for this task which make fewer assumptions on the distributions than traditional parametric approach. However, none of the existing kernel statistics has
provided a computationally efficient way to characterize the extremal behavior of
the statistic. Such characterization is crucial for setting the detection threshold, to
control the significance level in the offline case as well as the average run length
in the online case. In this paper we propose two related computationally efficient
M -statistics for kernel-based change-point detection when the amount of background data is large. A novel theoretical result of the paper is the characterization
of the tail probability of these statistics using a new technique based on change-ofmeasure. Such characterization provides us accurate detection thresholds for both
offline and online cases in computationally efficient manner, without the need to
resort to the more expensive simulations such as bootstrapping. We show that our
methods perform well in both synthetic and real world data.
1 Introduction
Detecting the emergence of abrupt change-points is a classic problem in statistics and machine
learning. Given a sequence of samples, x1 , x2 , . . . , xt , from a domain X , we are interested in
detecting a possible change-point ? , such that before ? , the samples xi ? P i.i.d. for i ? ? , where
P is the so-called background distribution, and after the change-point, the samples xi ? Q i.i.d. for
i ? +1, where Q is a post-change distribution. Here the time horizon t can be either a fixed number
t = T0 (called an offline or fixed-sample problem), or t is not fixed and we keep getting new samples
(called a sequential or online problem). Our goal is to detect the existence of the change-point in
the offline setting, or detect the emergence of a change-point as soon as possible after it occurs in
the online setting. We will restrict our attention to detecting one change-point, which arises often in
monitoring problems. One such example is the seismic event detection [9], where we would like to
detect the onset of the event precisely in retrospect to better understand earthquakes or as quickly as
possible from the streaming data. Ideally, the detection algorithm can also be robust to distributional
assumptions as we wish to detect all kinds of seismic events that are different from the background.
Typically we have a large amount of background data (since seismic events are rare), and we want
the algorithm to exploit these data while being computationally efficient.
Classical approaches for change-point detection are usually parametric, meaning that they rely on
strong assumptions on the distribution. Nonparametric and kernel approaches are distribution free
and more robust as they provide consistent results over larger classes of data distributions (they
can possibly be less powerful in settings where a clear distributional assumption can be made).
In particular, many kernel based statistics have been proposed in the machine learning literature [5, 2, 18, 6, 7, 1] which typically work better in real data with few assumptions. However,
none of these existing kernel statistics has provided a computationally efficient way to characterize
1
the tail probability of the extremal value of these statistics. Characterization such tail probability is
crucial for setting the correct detection thresholds for both the offline and online cases. Furthermore,
efficiency is also an important consideration since typically the amount of background data is very
large. In this case, one has the freedom to restructure and sample the background data during the
statistical design to gain computational efficiency. On the other hand, change-point detection problems are related to the statistical two-sample test problems; however, they are usually more difficult
in that for change-point detection, we need to search for the unknown change-point location ? . For
instance, in the offline case, this corresponds to taking a maximum of a series of statistics each corresponding to one putative change-point location (a similar idea was used in [5] for the offline case),
and in the online case, we have to characterize the average run length of the test statistic hitting the
threshold, which necessarily results in taking a maximum of the statistics over time. Moreover, the
statistics being maxed over are usually highly correlated. Hence, analyzing the tail probabilities of
the test statistic for change-point detection typically requires more sophisticated probabilistic tools.
In this paper, we design two related M -statistics for change-point detection based on kernel maximum mean discrepancy (MMD) for two-sample test [3, 4]. Although MMD has a nice unbiased
and minimum variance U -statistic estimator (MMDu ), it can not be directly applied since MMDu
costs O(n2 ) to compute based on a sample of n data points. In the change-point detection case, this
translates to a complexity quadratically grows with the number of background observations and the
detection time horizon t. Therefore, we adopt a strategy inspired by the recently developed B-test
statistic [17] and design a O(n) statistic for change-point detection. At a high level, our methods
sample N blocks of background data of size B, compute quadratic-time MMDu of each reference
block with the post-change block, and then average the results. However, different from the simple
two-sample test case, in order to provide an accurate change-point detection threshold, the background block needs to be designed in a novel structured way in the offline setting and updated
recursively in the online setting.
Besides presenting the new M -statistics, our contributions also include: (1) deriving accurate approximations to the significance level in the offline case, and average run length in the online case,
for our M -statistics, which enable us to determine thresholds efficiently without recurring to the
onerous simulations (e.g. repeated bootstrapping); (2) obtaining a closed-form variance estimator
which allows us to form the M -statistic easily; (3) developing novel structured ways to design background blocks in the offline setting and rules for update in the online setting, which also leads to
desired correlation structures of our statistics that enable accurate approximations for tail probability.
To approximate the asymptotic tail probabilities, we adopt a highly sophisticated technique based
on change-of-measure, recently developed in a series of paper by Yakir and Siegmund et al. [16].
The numerical accuracy of our approximations are validated by numerical examples. We demonstrate the good performance of our method using real speech and human activity data. We also find
that, in the two-sample testing scenario, it is always beneficial to increase the block size B as the
distribution for the statistic under the null and the alternative will be better separated; however, this
is no longer the case in online change-point detection, because a larger block size inevitably causes
a larger detection delay. Finally, we point to future directions to relax our Gaussian approximation
and correct for the skewness of the kernel-based statistics.
2 Background and Related Work
We briefly review kernel-based methods and the maximum mean discrepancy. A reproducing kernel
Hilbert space (RKHS) F on X with a kernel k(x, x0 ) is a Hilbert space of functions f (?) : X 7! R
with inner product h?, ?iF . Its element k(x, ?) satisfies the reproducing property: hf (?), k(x, ?)iF =
f (x), and consequently, hk(x, ?), k(x0 , ?)iF = k(x, x0 ), meaning that we can view the evaluation of
a function f at any point x 2 X as an inner product.
Assume there are two sets with n observations from a domain X , where X = {x1 , x2 , . . . , xn }
are drawn i.i.d. from distribution P , and Y = {y1 , y2 , . . . , yn } are drawn i.i.d. from distribution Q. The maximum mean discrepancy (MMD) is defined as [3] MMD0 [F, P, Q] :=
supf 2F {Ex [f (x)] Ey [f (y)]} . An unbiased estimate of MMD20 can be obtained using U -statistic
MMD2u [F, X, Y ] =
1
n(n
1)
2
n
X
i,j=1,i6=j
h(xi , xj , yi , yj ),
(1)
where h(?) is the kernel of the U -statistic defined as h(xi , xj , yi , yj ) = k(xi , xj ) + k(yi , yj )
k(xi , yj ) k(xj , yi ). Intuitively, the empirical test statistic MMD2u is expected to be small (close
to zero) if P = Q, and large if P and Q are far apart. The complexity for evaluating (1) is O(n2 )
since we have to form the so-called Gram matrix for the data. Under H0 (P = Q), the U -statistic is
degenerate and distributed the same as an infinite sum of Chi-square variables.
To improve the computational efficiency and obtain an easy-to-compute threshold for hypothesis
testing, recently, [17] proposed an alternative statistic for MMD20 called B-test. The key idea of the
approach is to partition the n samples from P and Q into N non-overlapping blocks, X1 , . . . , XN
and Y1 , . . . , YN , each of constant size B. Then MMD2u [F, Xi , Yi ] is computed for each pair of
PN
blocks and averaged over the N blocks to result in MMD2B [F, X, Y ] = N1 i=1 MMD2u [F, Xi , Yi ].
Since B is constant, N ? O(n), and the computational complexity of MMD2B [F, X, Y ] is O(B 2 n),
a significant reduction compared to MMD2u [F, X, Y ]. Furthermore, by averaging MMD2u [F, Xi , Yi ]
over independent blocks, the B-statistic is asymptotically normal leveraging over the central limit
theorem. This latter property also allows a simple threshold to be derived for the two-sample test
rather than resorting to more expensive bootstrapping approach. Our later statistics are inspired
by B-statistic. However, the change-point detection setting requires significant new derivations
to obtain the test threshold since one cares about the maximum of MMD2B [F, X, Y ] computed
at different point in time. Moreover, the change-point detection case consists of a sum of highly
correlated MMD statistics, because these MMD2B are formed with a common test block of data.
This is inevitable in our change-point detection problems because test data is much less than the
reference data. Hence, we cannot use the central limit theorem (even a martingale version), but have
to adopt the aforementioned change-of-measure approach.
Related work. Other nonparametric change-point detection approach has been proposed in the
literature. In the offline setting, [5] designs a kernel-based test statistic, based on a so-called running
maximum partition strategy to test for the presence of a change-point; [18] studies a related problem
in which there are s anomalous sequences out of n sequences to be detected and they construct a
test statistic using MMD. In the online setting, [6] presents a meta-algorithm that compares data in
some ?reference window? to the data in the current window, using some empirical distance measures
(not kernel-based); [1] detects abrupt changes by comparing two sets of descriptors extracted online
from the signal at each time instant: the immediate past set and the immediate future set; based on
soft margin single-class support vector machine (SVM), they build a dissimilarity measure (which is
asymptotically equivalent to the Fisher ratio in the Gaussian case) in the feature space between those
sets without estimating densities as an intermediate step; [7] uses a density-ratio estimation to detect
change-point, and models the density-ratio using a non-parametric Gaussian kernel model, whose
parameters are updated online through stochastic gradient decent. The above work lack theoretical
analysis for the extremal behavior of the statistics or average run length.
3
M -statistic for offline and online change-point detection
Give a sequence of observations {. . . , x 2 , x 1 , x0 , x1 , . . . , xt }, xi 2 X , with {. . . , x 2 , x 1 , x0 }
denoting the sequence of background (or reference) data. Assume a large amount of reference data is
available. Our goal is to detect the existence of a change-point ? , such that before the change-point,
samples are i.i.d. with a distribution P , and after the change-point, samples are i.i.d. with a different
distribution Q. The location ? where the change-point occurs is unknown. We may formulate this
problem as a hypothesis test, where the null hypothesis states that there is no change-point, and
the alternative hypothesis is that there exists a change-point at some time ? . We will construct our
kernel-based M -statistic using the maximum mean discrepancy (MMD) to measure the difference
between distributions of the reference and the test data.
We denote by Y the block of data which potentially contains a change-point (also referred to as the
post-change block or test block). In the offline setting, we assume the size of Y can be up to Bmax ,
and we want to search for a location of the change-point ? within Y such that observations after ?
are from a different distribution. Inspired by the idea of B-test [17], we sample N reference blocks
of size Bmax independently from the reference pool, and index them as XiBmax , i = 1, . . . , N . Since
we search for a location B (2 ? B ? Bmax ) within Y for a change-point, we construct sub-block
from Y by taking B contiguous data points, and denote them as Y B . To form the statistic, we
correspondingly construct sub-blocks from each reference block by taking B contiguous data points
(B)
out of that block, and index these sub-blocks as Xi (illustrated in Fig. 1(a)). We then compute
3
Block containing
potential change
point
Pool of reference data
B
MMD2u XN max , Y (Bmax
??
B
XN max
??
Bmax
Bmax
X3
Bmax
Bmax
Bmax
Y
X1
X2
Bmax
Bmax
MMD2u ??
??
Bmax
Bmax
B
B
B
Block containing
potential change
point
, ? (?0,?
Y (B0,t
Pool of reference data
?
sample
? time
??
B
B0 ,t
Y (B0,t+1
Pool of reference data
?+1
sample
2
2
2
2
time
?0
Xi
B
?0 ,?
Xi
2
(a): offline
B0 ,t+1
(b): sequential
Figure 1: Illustration of (a) offline case: data are split into blocks of size Bmax , indexed backwards from time
t, and we consider blocks of size B, B = 2, . . . , Bmax ; (b) online case. Assuming we have large amount of
reference or background data that follows the null distribution.
(B)
MMD2u between (Xi
ZB :=
, Y (B) ), and average over blocks
N
1
1 X
(B)
MMD2u (Xi , Y (B) ) =
N i=1
N B(B
1)
(B)
N
X
B
X
(B)
(B)
(B)
h(Xi,j , Xi,l , Yj
(B)
, Yl
), (2)
i=1 j,l=1,j6=l
(B)
B
where Xi,j
denotes the jth sample in Xi , and Yj
denotes the j th sample in Y B . Due to the
property of MMD2u , under the null hypothesis, E[ZB ] = 0. Let Var[ZB ] denote the variance of ZB
under the null. The expression of ZB is given by (6) in the following section. We see the variance
depends on the block size B and the number of blocks N . As B increases Var[ZB ] decreases (also
illustrated in Figure 5 in the appendix). Considering this, we standardize the statistic, maximize over
all values of B to define the offline M -statistic, and detect a change-point whenever the M -statistic
exceeds the threshold b > 0:
p
M :=
max
ZB / Var[ZB ] > b, {offline change-point detection}
(3)
{z
}
B2{2,3,...,Bmax } |
0
ZB
where varying the block-size from 2 to Bmax corresponds to searching for the unknown change-point
location. In the online setting, suppose the post-change block Y has size B0 and we construct it
using a sliding window. In this case, the potential change-point is declared as the end of each block
Y . To form the statistic, we take N B0 samples without replacement (since we assume the reference
data are i.i.d.with distribution P ) from the reference pool to form N reference blocks, compute
the quadratic MMD2u statistics between each reference block and the post-change block, and then
average them. When there is a new sample (time moves from t to t + 1), we append the new sample
in the reference block, remove the oldest sample from the post-change block, and move it to the
reference pool. The reference blocks are also updated accordingly: the end point of each reference
block is moved to the reference pool, and a new point is sampled and appended to the front of each
reference block, as shown in Fig. 1(b). Using the sliding window scheme described above, similarly,
we may define an online M -statistic by forming a standardized average of the MMD2u between the
post-change block in a sliding window and the reference block:
ZB0 ,t :=
N
1 X
(B ,t)
MMD2u (Xi 0 , Y (B0 ,t) ),
N i=1
(4)
(B ,t)
where B0 is the fixed block-size, Xi 0 is the ith reference block of size B0 at time t, and Y (B0 ,t) is
the the post-change block of size B0 at time t. In the online case, we have to characterize the average
run length of the test statistic hitting the threshold, which necessarily results in taking a maximum
of the statistics over time. The online change-point detection procedure is a stopping time, where
we detect a change-point whenever the normalized ZB0 ,t exceeds a pre-determined threshold b > 0:
p
T = inf{t : ZB0 ,t / Var[ZB0 ] > b}. {online change-point detection}
(5)
|
{z
}
Mt
Note in the online case, we actually take a maximum of the standardized statistics over time. There is
a recursive way to calculate the online M -statistic efficiently, explained in Section A in the appendix.
At the stopping time T , we claim that there exists a change-point. There is a tradeoff in choosing
the block size B0 in online setting: a small block size will incur a smaller computational cost, which
may be important for the online case, and it also enables smaller detection delay for strong change4
point magnitude; however, the disadvantage of a small B0 is a lower power, which corresponds to
a longer detection delay when the change-point magnitude is weak (for example, the amplitude of
the mean shift is small). Examples of offline and online M -statistics are demonstrated in Fig. 2
based on synthetic data and a segment of the real seismic signal. We see that the proposed offline
M -statistic powerfully detects the existence of a change-point and accurately pinpoints where the
change occurs; the online M -statistic quickly hits the threshold as soon as the change happens.
0
100
Normal (0,1)
Laplace (0,1)
Seismic Signal
0
Normal (0,1)
Signal
5
Laplace (0,1)
0
?5
?10
0
100
200
300
Time
400
500
10
?10
0
100
200
300
Time
400
500
300
B
200
100
(a): Offline, null
0
5 b=3.34
500
100
200
300
Time
400
500
10
Peak
0
400
?10
0
Statistic
Statistic
0
500
0
?50
10
5 b=3.34
50
400
300
B
200
100
0
(b): Offline, ? = 250
?100
200
400
600
Time
100
Statistic
?5
Statistic
10
10
Normal (0,1)
5
Signal
Signal
10
5 b=3.55
800
1000
Bandw=Med
Bandw=100Med
Bandw=0.1Med
50
b=3.55
0
0
0
100
200
300
Time
400
500
(c): Online, ? = 250
200
400
600
Time
800
1000
(d) seismic signal
Figure 2: Examples of offline and online M -statistic with N = 5: (a) and (b), offline case without and with a
change-point (Bmax = 500 and the maximum is obtained when B = 263); (c) online case with a change-point
at ? = 250, stopping-time T = 268 (detection delay is 18), and we use B0 = 50; (d) a real seismic signal and
M -statistic with different kernel bandwidth. All thresholds are theoretical values and are marked in red.
4 Theoretical Performance Analysis
We obtain an analytical expression for the variance Var[ZB ] in (3) and (5), by leveraging the correspondence between the MMD2u statistics and U -statistic [11] (since ZB is some form of U -statistic),
and exploiting the known properties of U -statistic. We also derive the covariance structure for the
online and offline standardized ZB statistics, which is crucial for proving theorems 3 and 4.
Lemma 1 (Variance of ZB under the null.) Given any fixed block size B and number of blocks
N , under the null hypothesis,
? ? 1?
B
1
N 1
Var[ZB ] =
E[h2 (x, x0 , y, y 0 )] +
Cov [h(x, x0 , y, y 0 ), h(x00 , x000 , y, y 0 )] , (6)
2
N
N
where x, x0 , x00 , x000 , y, and y 0 are i.i.d. with the null distribution P .
Lemma 1 suggests an easy way to estimate the variance Var[ZB ] from the reference data. To estimate (6), we need to first estimate E[h2 (x, x0 , y, y 0 )], by each time drawing four samples without replacement from the reference data, use them for x, x0 , y, y 0 , evaluate the sampled function value, and
then form a Monte Carlo average. Similarly, we may estimate Cov [h(x, x0 , y, y 0 ), h(x00 , x000 , y, y 0 )].
Lemma 2 (Covariance structure of the standardized ZB statistics.) Under the null hypothesis,
given u and v in [2, Bmax ], for the offline case
s? ?? ? ?
?
u v
u_v
0
0
,
(7)
ru,v := Cov (Zu , Zv ) =
2
2
2
where u _ v = max{u, v}, and for the online case,
0
ru,v
:= Cov(Mu , Mu+s ) = (1
s
)(1
B0
s
B0
1
), for s
0.
In the offline setting, the choice of the threshold b involves a tradeoff between two standard performance metrics: (i) the significant level (SL), which is the probability that the M -statistic exceeds
the threshold b under the null hypothesis (i.e., when there is no change-point); and (ii) power, which
is the probability of the statistic exceeds the threshold under the alternative hypothesis. In the online
setting, there are two analogous performance metrics commonly used for analyzing change-point
detection procedures [15]: (i) the expected value of the stopping time when there is no change, the
average run length (ARL); (ii) the expected detection delay (EDD), defined to be the expected stopping time in the extreme case where a change occurs immediately at ? = 0. We focus on analyzing
SL and ARL of our methods, since they play key roles in setting thresholds. We derive accurate
approximations to these quantities as functions of the threshold b, so that given a prescribed SL or
5
ARL, we can solve for the corresponding b analytically. Let P1 and E1 denote, respectively, the
probability measure and expectation under the null.
p
Theorem 3 (SL in offline case.) When b ! 1 and b/ Bmax ! c for some constant c, the significant level of the offline M -statistic defined in (3) is given by
s
(
)
!
BX
max
1 2
Z
(2B
1)
2B
1
B
p
p
P1
max
> b = b2 e 2 b ?
? b
+o(1),
B(B 1)
B2{2,3,...,Bmax }
2 2?B(B 1)
Var[ZB ]
B=2
(8)
(2/u)( (u/2) 0.5)
where the special function ?(u) ? (u/2) (u/2)+ (u/2) , is the probability density function and
(x) is the cumulative distribution function of the standard normal distribution, respectively.
The proof of theorem 3 uses a change-of-measure argument, which is based on the likelihood ratio
identity (see, e.g., [12, 16]). The likelihood ratio identity relates computing of the tail probability
under the null to computing a sum of expectations each under an alternative distribution indexed by
a particular parameter value. To illustrate, assume the probability density function (pdf) under the
null is f (u). Given a function g! (x), with ! in some index set ?,, we may introduce
R a family of alternative distributions with pdf f! (u) = e?g! (u) ! (?) f (u), where ! (?) := log e?g! (u) f (u)du
is the log moment generating function, and ? is the parameter that we may assign an arbitrary value.
It can be easily verified that f! (u) is a pdf. Using this family of alternative, we may calculate the
probability of an event A under the original distribution f , by calculating a sum of expectations:
?P
X
e `!
P{A} = E P!2? `s ; A =
E! [e`! ; A],
s2? e
!2?
where E[U ; A] := E[U I{A}], the indicator function I{A} is one when event A is true and zero
otherwise, E! is the expectation using pdf f! (u), `! = log[f (u)/f! (u)] = ?g! (u)
! (?), is the
log-likelyhood ratio, and we have the freedom to choose a different ? value for each f! .
0
The basic idea of change-of-measure in our setting is to treat ZB
:= ZB /Var[ZB ], as a random field
indexed by B. Then to characterize SL, we need to study the tail probability of the maximum of this
0
random field. Relate this to the setting above, ZB
corresponds to g! (u), B corresponds to !, and
A corresponds to the threshold crossing event. To compute the expectations under the alternative
measures, we will take a few steps. First, we choose a parameter value ?B for each pdf associated
with a parameter value B, such that ? B (?B ) = b. This is equivalent to setting the mean under each
0
alternative probability to the threshold b: EB [ZB
] = b and it allows us to use the local central limit
theorem since under the alternative measure boundary cross has much larger probability. Second,
we will express the random quantities involved in the expectations, as a functions of the so-called
local field terms: {`B `s : s = B, B ? 1, . . .}, as well as the re-centered log-likelihood ratios:
`?Bp= `B b. We show that they are asymptotically independent as b ! 1 and b grows on the order
of B, and this further simplifies our calculation. The last step is to analyze the covariance structure
of the random field (Lemma 2 in the following), and approximate it using a Gaussian random field.
Note that the terms Zu0 and Zv0 have non-negligible correlation due to our construction: they share
the same post-change block Y (B) . We then apply the localization theorem (Theorem 5.2 in [16]) to
obtain the final result.
p
Theorem 4 (ARL in online case.) When b ! 1 and b/ B0 ! c0 for some constant c0 , the average run length (ARL) of the stopping time T defined in (5) is given by
s
(
!) 1
2
eb /2
(2B0 1)
2(2B0 1)
1
E [T ] = 2 ? p
?? b
+ o(1).
(9)
b
B0 (B0 1)
2?B0 (B0 1)
Proof for Theorem 4 is similar to that for Theorem 3, due to the fact that for a given m > 0,
?
P1 {T ? m} = P1 max Mt > b .
1?t?m
(10)
Hence,pwe also need to study the tail probability of the maximum of a random field Mt =
ZB0 ,t / ZB0 ,t for a fixed block size B0 . A similar change-of-measure approach can be used, except that the covariance structure of Mt in the online case is slightly different from the offline case.
This tail probability turns out to be in a form of P1 {T ? m} = m + o(1). Using similar argu6
ments as those in [13, 14], we may see that T is asymptotically exponentially distributed. Hence,
P1 {T ? m} [1 exp( m)] ! 0. Consequently E1 {T } ? 1 , which leads to (9).
p
2
Theorem 4 shows that ARL ? O(eb ) and, hence, b ? O( log ARL). On the other hand, the
EDD is typically on the order of b/ using the Wald?s identity [12] (although a more careful analysis should be carried out in the future work), where
is the Kullback-Leibler (KL) divergence
between the null and alternative distributions (on a order of a constant). Hence, given a desired ARL
(typically on the order of 5000 or 10000), the error made in the estimated threshold will only be
translated linearly to EDD. This is a blessing to us and it means typically a reasonably accurate b
2
will cause little performance loss in EDD. Similarly, Theorem 3 shows that SL ? O(e b ) and a
similar argument can be made for the offline case.
5 Numerical examples
We test the performance of the M -statistic using simulation and real world data. Here we only
highlight the main results. More details can be found in Appendix C. In the following examples, we
use a Gaussian kernel: k(Y, Y 0 ) = exp kY Y 0 k2 /2 2 , where > 0 is the kernel bandwidth
and we use the ?median trick? [10, 8] to get the bandwidth which is estimated using the background
data.
Accuracy of Lemma 1 for estimating Var[ZB ]. Fig. 5 in the appendix shows the empirical distributions of ZB when B = 2 and B = 200, when N = 5. In both cases, we generate 10000
random instances, which are computed from data following N (0, I), I 2 R20?20 to represent the
null distribution. Moreover, we also plot the Gaussian pdf with sample mean and sample variance,
which matches well with the empirical distribution. Note the approximation works better when the
block size decreases. (The skewness of the statistic can be corrected; see discussions in Section 7).
Accuracy of theoretical results for estimating threshold. For the offline case, we compare the
thresholds obtained from numerical simulations, bootstrapping, and using our approximation in
Theorem 3, for various SL values ?. We choose the maximum block size to be Bmax = 20. In
the appendix, Fig. 6(a) demonstrates how a threshold is obtained by simulation, for ? = 0.05, the
threshold b = 2.88 corresponds to the 95% quantile of the empirical distribution of the offline M statistic. For a range of b values, Fig. 6(b) compares the empirical SL value ? from simulation with
that predicted by Theorem 3, and shows that theory is quite accurate for small ?, which is desirable
as we usually care of small ??s to obtain thresholds. Table 1 shows that our approximation works
quite well to determine thresholds given ??s: thresholds obtained by our theory matches quite well
with that obtained from Monte Carlo simulation (the null distribution is N (0, I), I 2 R20?20 ), and
even from bootstrapping for a real data scenario. Here, the ?bootstrap? thresholds are for a speech
signal from the CENSREC-1-C dataset. In this case, the null distribution P is unknown, and we only
have 3000 samples speech signals. Thus we generate bootstrap samples to estimate the threshold,
as shown in Fig. 7 in the appendix. These b?s obtained from theoretical approximations have little
performance degradation, and we will discuss how to improve in Section 7.
Table 1: Comparison of thresholds for offline case, determined by simulation, bootstrapping and theory respectively, for various SL value ?.
?
0.20
0.15
0.10
b (sim)
1.78
2.02
2.29
Bmax = 10
b (boot)
1.77
2.05
2.45
b (the)
2.00
2.18
2.40
b (sim)
1.97
2.18
2.47
Bmax = 20
b (boot)
2.29
2.63
3.09
b (the)
2.25
2.41
2.60
b (sim)
2.21
2.44
2.70
Bmax = 50
b (boot)
2.47
2.78
3.25
b (the)
2.48
2.62
2.80
For the online case, we also compare the thresholds obtained from simulation (using 5000 instances)
for various ARL and from Theorem 4, respectively. As predicated by theory, the threshold is consistently accurate for various null distributions (shown in Fig. 3). Also note from Fig. 3 that the
precision improves as B0 increases. The null distributions we consider include N (0, 1), exponential distribution with mean 1, a Erdos-Renyi random graph with 10 nodes and probability of 0.2 of
forming random edges, and Laplace distribution.
Expected detection delays (EDD). In the online setting, we compare EDD (with the assumption
? = 0) of detecting a change-point when the signal is 20 dimensional and the transition happens
7
7
b
5
Gaussian(0,I)
Exp(1)
Random Graph (Node=10, p=0.2)
Laplace(0,1)
Theory
6
5
b
6
4
7
Gaussian(0,I)
Exp(1)
Random Graph (Node=10, p=0.2)
Laplace(0,1)
Theory
6
5
b
7
4
4
3
3
3
2
2
2
1
0
0.2
0.4
0.6
ARL(104)
0.8
1
1
0
0.2
0.4
0.6
ARL(104)
0.8
1
Gaussian(0,I)
Exp(1)
Random Graph (Node=10, p=0.2)
Laplace(0,1)
Theory
1
0
0.2
0.4
0.6
ARL(104)
0.8
1
(a): B0 = 10
(b): B0 = 50
(c): B0 = 200
Figure 3: In online case, for a range of ARL values, comparison b obtained from simulation and from Theorem
4 under various null distributions.
from a zero-mean Gaussian N (0, I20 ) to a non-zero mean Gaussian N (?, I20 ), where the postchange mean vector ? is element-wise equal to a constant mean shift. In this setting, Fig. 10(a)
demonstrates the tradeoff in choosing a block size: when block size is too small the statistical power
of the M -statistic is weak and hence EDD is large; on the other hand, when block size is too large,
although statistical power is good, EDD is also large because the way we update the test block.
Therefore, there is an optimal block size for each case. Fig. 10(b) shows the optimal block size
decreases as the mean shift increases, as expected.
6 Real-data
We test the performance of our M -statistics using real data. Our datasets include: (1) CENSREC1-C: a real-world speech dataset in the Speech Resource Consortium (SRC) corpora provided by
National Institute of Informatics (NII)1 ; (2) Human Activity Sensing Consortium (HASC) challenge
2011 data2 . We compare our M -statistic with a state-of-the-art algorithm, the relative densityratio (RDR) estimate [7] (one limitation of the RDR algorithm, however, is that it is not suitable
for high-dimensional data because estimating density ratio in the high-dimensional setting is illposed). To achieve reasonable performance for the RDR algorithm, we adjust the bandwidth and
the regularization parameter at each time step and, hence, the RDR algorithm is computationally
more expensive than the M -statistics method. We use the Area Under Curve (AUC) [7] (the larger
the better) as a performance metric. Our M -statistics have competitive performance compared with
the baseline RDR algorithm in the real data testing. Here we report the main results and the details
can be found in Appendix D. For the speech data, our goal is to detect the onset of speech signal
emergent from the background noise (the background noises are taken from real acoustic signals,
such as background noise in highway, airport and subway stations). The overall AUC for the M statistic is .8014 and for the baseline algorithm is .7578. For human activity detection data, we aim at
detection the onset of transitioning from one activity to another. Each data consists of human activity
information collected by portable three-axis accelerometers. The overall AUC for the M -statistic is
.8871 and for the baseline algorithm is .7161.
7 Discussions
We may be able to improve the precision of the tail probability approximation in theorems 3 and 4 to
0
account for skewness of ZB
. In the change-of-measurement argument, we need to choose parameter
0
values ?B such that ? B (?B ) = b. Currently, we use a Gaussian assumption ZB
? N (0, 1) and,
2
hence, B (?) = ? /2, and ?B = b. We may improve the precision if we are able to estimate
0
0
skewness ?(ZB
) for ZB
. In particular, we can include the skewness in the log moment generating
0
function approximation B (?) ? ?2 /2+?(ZB
)?3 /6 when we estimate the change-of-measurement
0
parameter: setting the derivative of this to b and solving a quadratic equation ?(ZB
)?2 /2 + ? = b
0
0
0
0
b2 /2
(?
)
for ?B . This will change the leading exponent term in (8) from e
to be e B B ?B b . A similar
improvement can be done for the ARL approximation in Theorem 4.
Acknowledgments
This research was supported in part by CMMI-1538746 and CCF-1442635 to Y.X.; NSF/NIH BIGDATA
1R01GM108341, ONR N00014-15-1-2340, NSF IIS-1218749, NSF CAREER IIS-1350983 to L.S..
1
2
Available from http://research.nii.ac.jp/src/en/CENSREC-1-C.html
Available from http://hasc.jp/hc2011
8
References
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9
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5,175 | 5,685 | Fast Two-Sample Testing with Analytic
Representations of Probability Measures
Kacper Chwialkowski
Gatsby Computational Neuroscience Unit, UCL
[email protected]
Dino Sejdinovic
Dept of Statistics, University of Oxford
[email protected]
Aaditya Ramdas
Dept. of EECS and Statistics, UC Berkeley
[email protected]
Arthur Gretton
Gatsby Computational Neuroscience Unit, UCL
[email protected]
Abstract
We propose a class of nonparametric two-sample tests with a cost linear in the
sample size. Two tests are given, both based on an ensemble of distances between analytic functions representing each of the distributions. The first test uses
smoothed empirical characteristic functions to represent the distributions, the second uses distribution embeddings in a reproducing kernel Hilbert space. Analyticity implies that differences in the distributions may be detected almost surely at a
finite number of randomly chosen locations/frequencies. The new tests are consistent against a larger class of alternatives than the previous linear-time tests based
on the (non-smoothed) empirical characteristic functions, while being much faster
than the current state-of-the-art quadratic-time kernel-based or energy distancebased tests. Experiments on artificial benchmarks and on challenging real-world
testing problems demonstrate that our tests give a better power/time tradeoff than
competing approaches, and in some cases, better outright power than even the
most expensive quadratic-time tests. This performance advantage is retained even
in high dimensions, and in cases where the difference in distributions is not observable with low order statistics.
1 Introduction
Testing whether two random variables are identically distributed without imposing any parametric
assumptions on their distributions is important in a variety of scientific applications. These include
data integration in bioinformatics [6], benchmarking for steganography [20] and automated model
checking [19]. Such problems are addressed in the statistics literature via two-sample tests (also
known as homogeneity tests).
Traditional approaches to two-sample testing are based on distances between representations of the
distributions, such as density functions, cumulative distribution functions, characteristic functions or
mean embeddings in a reproducing kernel Hilbert space (RKHS) [27, 26]. These representations are
infinite dimensional objects, which poses challenges when defining a distance between distributions.
Examples of such distances include the classical Kolmogorov-Smirnov distance (sup-norm between
cumulative distribution functions); the Maximum Mean Discrepancy (MMD) [9], an RKHS norm
of the difference between mean embeddings, and the N-distance (also known as energy distance)
[34, 31, 4], which is an MMD-based test for a particular family of kernels [25] . Tests may also be
based on quantities other than distances, an example being the Kernel Fisher Discriminant (KFD)
[12], the estimation of which still requires calculating the RKHS norm of a difference of mean
embeddings, with normalization by an inverse covariance operator.
1
In contrast to consistent two-sample tests, heuristics based on pseudo-distances, such as the difference between characteristic functions evaluated at a single frequency, have been studied in the
context of goodness-of-fit tests [13, 14]. It was shown that the power of such tests can be maximized
against fully specified alternative hypotheses, where test power is the probability of correctly rejecting the null hypothesis that the distributions are the same. In other words, if the class of distributions
being distinguished is known in advance, then the tests can focus only at those particular frequencies where the characteristic functions differ most. This approach was generalized to evaluating the
empirical characteristic functions at multiple distinct frequencies by [8], thus improving on tests
that need to know the single ?best? frequency in advance (the cost remains linear in the sample size,
albeit with a larger constant). This approach still fails to solve the consistency problem, however:
two distinct characteristic functions can agree on an interval, and if the tested frequencies fall in that
interval, the distributions will be indistinguishable.
In Section 2 of the present work, we introduce two novel distances between distributions, which
both use a parsimonious representation of the probability measures. The first distance builds on
the notion of differences in characteristic functions with the introduction of smooth characteristic
functions, which can be though of as the analytic analogues of the characteristics functions. A
distance between smooth characteristic functions evaluated at a single random frequency is almost
surely a distance (Definition 1 formalizes this concept) between these two distributions. In other
words, there is no need to calculate the whole infinite dimensional representation - it is almost
surely sufficient to evaluate it at a single random frequency (although checking more frequencies will
generally result in more powerful tests). The second distance is based on analytic mean embeddings
of two distributions in a characteristic RKHS; again, it is sufficient to evaluate the distance between
mean embeddings at a single randomly chosen point to obtain almost surely a distance. To our
knowledge, this representation is the first mapping of the space of probability measures into a finite
dimensional Euclidean space (in the simplest case, the real line) that is almost surely an injection,
and as a result almost surely a metrization. This metrization is very appealing from a computational
viewpoint, since the statistics based on it have linear time complexity (in the number of samples)
and constant memory requirements.
We construct statistical tests in Section 3, based on empirical estimates of differences in the analytic
representations of the two distributions. Our tests have a number of theoretical and computational
advantages over previous approaches. The test based on differences between analytic mean embeddings is a.s. consistent for all distributions, and the test based on differences between smoothed
characteristic functions is a.s. consistent for all distributions with integrable characteristic functions (contrast with [8], which is only consistent under much more onerous conditions, as discussed
above). This same weakness was used by [1] in justifying a test that integrates over the entire frequency domain (albeit at cost quadratic in the sample size), for which the quadratic-time MMD is
a generalization [9]. Compared with such quadratic time tests, our tests can be conducted in linear
time ? hence, we expect their power/computation tradeoff to be superior.
We provide several experimental benchmarks (Section 4) for our tests. First, we compare test power
as a function of computation time for two real-life testing settings: amplitude modulated audio
samples, and the Higgs dataset, which are both challenging multivariate testing problems. Our
tests give a better power/computation tradeoff than the characteristic function-based tests of [8],
the previous sub-quadratic-time MMD tests [11, 32], and the quadratic-time MMD test. In terms
of power when unlimited computation time is available, we might expect worse performance for
the new tests, in line with findings for linear- and sub-quadratic-time MMD-based tests [15, 9, 11,
32]. Remarkably, such a loss of power is not the rule: for instance, when distinguishing signatures
of the Higgs boson from background noise [3] (?Higgs dataset?), we observe that a test based on
differences in smoothed empirical characteristic functions outperforms the quadratic-time MMD.
This is in contrast to linear- and sub-quadratic-time MMD-based tests, which by construction are less
powerful than the quadratic-time MMD. Next, for challenging artificial data (both high-dimensional
distributions, and distributions for which the difference is very subtle), our tests again give a better
power/computation tradeoff than competing methods.
2 Analytic embeddings and distances
In this section we consider mappings from the space of probability measures into a sub-space of
real valued analytic functions. We will show that evaluating these maps at J randomly selected
2
points is almost surely injective for any J > 0. Using this result, we obtain a simple (randomized)
metrization of the space of probability measures. This metrization is used in the next section to
construct linear-time nonparametric two-sample tests.
To motivate our approach, we begin by recalling an integral family of distances between distributions, denoted Maximum Mean Discrepancies (MMD) [9]. The MMD is defined as
?Z
Z
MMD(P, Q) = sup
f dP
f dQ ,
(1)
f 2Bk
E
E
where P and Q are probability measures on E, and Bk is the unit ball in the RKHS Hk associated
with a positive definite kernel k : E ? E ! R. A popular choice of k is the Gaussian kernel
k(x, y) = exp( kx yk2 / 2 ) with bandwidth parameter . It can be shown that the MMD is equal
to the RKHS distance between so called mean embeddings,
MMD(P, Q) = k?P
?Q kHk ,
where ?P is an embedding of the probability measure P to Hk ,
Z
?P (t) =
k(x, t)dP (x),
(2)
(3)
E
and k ? kHk denotes the norm in the RKHS Hk . When k is translation invariant, i.e., k (x, y) =
?(x y), the squared MMD can be written [27, Corollary 4]
Z
2
2
MMD (P, Q) =
|'P (t) 'Q (t)| F 1 ?(t)dt,
(4)
Rd
where F denotes the Fourier transform, F 1 is the inverse Fourier transform, and 'P , 'Q are
the characteristic functions of P , Q, respectively. From [27, Theorem 9], a kernel k is called
characteristic when the MMD for Hk satisfies
MMD(P, Q) = 0 iff P = Q.
(5)
Any bounded, continuous, translation-invariant kernel whose inverse Fourier transform is almost
everywhere non-zero is characteristic [27]. By representation (2), it is clear that the MMD with a
characteristic kernel is a metric.
Pseudometrics based on characteristic functions. A practical limitation when using the MMD
in testing is that an empirical estimate is expensive to compute, this being the sum of two U-statistics
and an empirical average, with cost quadratic in the sample size [9, Lemma 6]. We might instead
consider a finite dimensional approximation to the MMD, achieved by estimating the integral (4),
with the random variable
d2',J (P, Q) =
J
J
1X
|'P (Tj )
J j=1
'Q (Tj )|2 ,
(6)
where {Tj }j=1 are sampled independently from the distribution with a density function F 1 ?. This
type of approximation is applied to various kernel algorithms under the name of random Fourier
features [21, 17]. In the statistical testing literature, the quantity d',J (P, Q) predates the MMD by
a considerable time, and was studied in [13, 14, 8], and more recently revisited in [33]. Our first
proposition is that d2',J (P, Q) can be a poor choice of distance between probability measures, as it
fails to distinguish a large class of measures. The following result is proved in the Appendix.
J
Proposition 1. Let J 2 N and let {Tj }j=1 be a sequence of real valued i.i.d. random variables
with a distribution which is absolutely continuous with respect to the Lebesgue measure. For any
0 < ? < 1, there exists an uncountable set A of mutually distinct probability measures (on the real
line) such that for any P, Q 2 A, P d2',J (P, Q) = 0
1 ?.
We are therefore motivated to find distances of the form (6) that can distinguish larger classes of
distributions, yet remain efficient to compute. These distances are characterized as follows:
Definition 1 (Random Metric). A random process d with values in R, indexed with pairs from the
set of probability measures M, i.e., d = {d(P, Q) : P, Q 2 M}, is said to be a random metric
if it satisfies all the conditions for a metric with qualification ?almost surely?. Formally, for all
P, Q, R 2 M, random variables d(P, Q), d(P, R), d(R, Q) must satisfy
3
1. d(P, Q)
0 a.s.
2. if P = Q, then d(P, Q) = 0 a.s, if P 6= Q then d(P, Q) 6= 0 a.s.
3. d(P, Q) = d(Q, P ) a.s.
4. d(P, Q) ? d(P, R) + d(R, Q) a.s.
1
From the statistical testing point of view, the coincidence axiom of a metric d, d(P, Q) = 0 if and
only if P = Q, is key, as it ensures consistency against all alternatives. The quantity d',J (P, Q) in
(6) violates the coincidence axiom, so it is only a random pseudometric (other axioms are trivially
satisfied). We remedy this problem by replacing the characteristic functions by smooth characteristic
functions:
Definition 2. A smooth characteristic function P (t) of a measure P is a characteristic function of
P convolved with an analytic smoothing kernel l, i.e.
Z
'P (w)l(t w)dw,
t 2 Rd .
(7)
P (t) =
Rd
Proposition 3 shows that smooth characteristic function can be estimated in a linear time. The
analogue of d',J (P, Q) for smooth characteristic functions is simply
d2 ,J (P, Q) =
J
1X
|
J j=1
P (Tj )
Q (Tj )|
2
,
(8)
J
where {Tj }j=1 are sampled independently from the absolutely continuous distribution (returning
to our earlier example, this might be F 1 ?(t) if we believe this to be an informative choice). The
following theorem, proved in the Appendix, demonstrates that the smoothing greatly increases the
class of distributions we can distinguish.
Theorem 1. Let l be an analytic, integrable kernel with an inverse Fourier transform that is nonzero almost everywhere. Then, for any J > 0, d ,J is a random metric on the space of probability
measures with integrable characteristic functions, and P is an analytic function.
This result is primarily a consequence of analyticity of smooth characteristic functions and the fact
that analytic functions are ?well behaved?. There is an additional, practical advantage to smoothing:
when the variability in the difference of the characteristic functions is high, and these differences are
local, smoothing distributes the difference in CFs more broadly in the frequency domain (a simple
illustration is in Fig. A.1, Appendix), making it easier to find by measurement at a small number of
randomly chosen points. This accounts for the observed improvements in test power in Section 4,
over differences in unsmoothed CFs.
Metrics based on mean embeddings. The key step which leads us to the construction of a random
metric d ,J is the convolution of the original characteristic functions with an analytic smoothing kernel. This idea need not be restricted to the representations of probability measures in the frequency
domain. We may instead directly convolve the probability measure with a positive definite kernel k
(that need not be translation invariant), yielding its mean embedding into the associated RKHS,
Z
?P (t) =
k(x, t)dP (x).
(9)
E
We say that a positive definite kernel k : RD ?RD ! R is analytic on its domain if for all x 2 RD ,
the feature map k(x, ?) is an analytic function on RD . By using embeddings with characteristic and
analytic kernels, we obtain particularly useful representations of distributions. As for the smoothed
CF case, we define
J
1X
d2?,J (P, Q) =
(?P (Tj ) ?Q (Tj ))2 .
(10)
J j=1
The following theorem ensures that d?,J (P, Q) is also a random metric.
1
Note that this does not imply that realizations of d are distances on M, but it does imply that they are
almost surely distances for all arbitrary finite subsets of M.
4
Theorem 2. Let k be an analytic, integrable and characteristic kernel. Then for any J > 0, d?,J is
a random metric on the space of probability measures (and ?P is an analytic function).
Note that this result is stronger than the one presented in Theorem 1, since it is not restricted to the
class of probability measures with integrable characteristic functions. Indeed, the assumption that
the characteristic function is integrable implies the existence and boundedness of a density. Recalling the representation of MMD in (2), we have proved that it is almost always sufficient to measure
difference between ?P and ?Q at a finite number of points, provided our kernel is characteristic
and analytic. In the next section, we will see that metrization of the space of probability measures
using random metrics d?,J , d ,J is very appealing from the computational point of view. It turns
out that the statistical tests that arise from these metrics have linear time complexity (in the number
of samples) and constant memory requirements.
3 Hypothesis Tests Based on Distances Between Analytic Functions
In this section, we provide two linear-time two-sample tests: first, a test based on analytic mean
embeddings, and next a test based on smooth characteristic functions. We further describe the
relation with competing alternatives. Proofs of all propositions are in Appendix B.
Difference in analytic functions In the previous section we described the random metric based
PJ
on a difference in analytic mean embeddings, d2?,J (P, Q) = J1 j=1 (?P (Tj ) ?Q (Tj ))2 . If we
P
n
replace ?P with the empirical mean embedding ?
?P = n1 i=1 k(Xi , ?) it can be shown that for any
J
sequence of unique {tj }j=1 , under the null hypothesis, as n ! 1,
J
p X
n
(?
?P (tj )
?
?Q (tj ))2
(11)
j=1
converges in distribution to a sum of correlated chi-squared variables. Even for fixed {tj }Jj=1 , it is
very computationally costly to obtain quantiles of this distribution, since this requires a bootstrap
or permutation procedure. We will follow a different approach based on Hotelling?s T 2 -statistic
[16]. The Hotelling?s T 2 -squared statistic of a normally distributed, zero mean, Gaussian vector
W = (W1 , ? ? ? , WJ ), with a covariance matrix ?, is T 2 = W ? 1 W . The compelling property of
the statistic is that it is distributed as a 2 -random variable with J degrees of freedom. To see a link
PJ
between T 2 and equation (11), consider a random variable i=j Wj2 : this is also distributed as a
sum of correlated chi-squared variables. In our case W is replaced with a difference of normalized
empirical mean embeddings, and ? is replaced with the empirical covariance of the difference of
mean embeddings. Formally, let Zi denote the vector of differences between kernels at tests points
Tj ,
Zi = (k(Xi , T1 ) k(Yi , T1 ), ? ? ? , k(Xi , TJ ) k(Yi , TJ )) 2 RJ .
(12)
Pn
1
We define the vector of mean empirical differences Wn = n i=1 Zi , and its covariance matrix
P
?n = n1 i (Zi Wn )(Zi Wn )T . The test statistic is
Sn = nWn ?n 1 Wn .
(13)
The computation of Sn requires inversion of a J ? J matrix ?n , but this is fast and numerically
stable: J will typically be small, and is less than 10 in our experiments. The next proposition
demonstrates the use of Sn as a two-sample test statistic.
Proposition 2 (Asymptotic behavior of Sn ). Let d2?,J (P, Q) = 0 a.s. and let {Xi }ni=1 and {Yi }ni=1
be i.i.d. samples from P and Q respectively. If ?n 1 exists for n large enough, then the statistic Sn
is a.s. asymptotically distributed as a 2 -random variable with J degrees of freedom (as n ! 1
with d fixed). If d2?,J (P, Q) > 0 a.s., then a.s. for any fixed r, P(Sn > r) ! 1 as n ! 1 .
We now apply the above proposition to obtain a statistical test.
Test 1 (Analytic mean embedding ). Calculate Sn . Choose a threshold r? corresponding to the 1 ?
quantile of a 2 distribution with J degrees of freedom, and reject the null hypothesis whenever Sn
is larger than r? .
5
J
There are a number of valid sampling schemes for the test points {Tj }j=1 to evaluate the differences
in mean embeddings: see Section 4 for a discussion.
Difference in smooth characteristic functions From the convolution definition of a smooth characteristic function (7) it is not immediately obvious how to calculate its estimator in linear time. In
the next proposition, however, we show that a smooth characteristic function is an expected value of
some function (with respect to the given measure), which can be estimated in a linear time.
Proposition 3. Let k be an integrable translation-invariant kernel and f its inverse Fourier transR
>
form. Then the smooth characteristic function of P can be written as P (t) = Rd eit x f (x)dP (x).
It is now clear that a test based on the smooth characteristic functions is similar to the test based on
mean embeddings. The main difference is in the definition of the vector of differences Zi :
Zi = (f (Xi ) sin(Xi T1 ) f (Yi ) sin(Yi T1 ), f (Xi ) cos(Xi T1 ) f (Yi ) cos(Yi T1 ), ? ? ? ) 2 R2J (14)
p
p
>
>
The imaginary and real part of the e 1Tj Xi f (Xi ) e 1Tj
to ensure that Wn , ?n and Sn as all real-valued quantities.
Yi
f (Yi ) are stacked together, in order
Proposition 4. Let d2 ,J (P, Q) = 0 and let {Xi }ni=1 and {Yi }ni=1 be i.i.d. samples from P and
Q respectively. Then the statistic Sn is almost surely asymptotically distributed as a 2 -random
variable with 2J degrees of freedom (as n ! 1 with J fixed). If d2 ,J (P, Q) > 0 , then almost
surely for any fixed r, P (Sn > r) ! 1 as n ! 1.
Other tests. The test [8] based on empirical characteristic functions was constructed originally
for one test point and then generalized to many points - it is quite similar to our second test, but
does not perform smoothing (it is also based on a T 2 -Hotelling statistic). The block MMD [32] is
a sub-quadratic test, which can be trivially linearized by fixing the block size, as presented in the
Appendix. Finally, another alternative is the MMD, an inherently quadratic
time test. We scale
p
MMD to linear time by sub-sampling our data set, and choosing only n points, so that the MMD
complexity becomes O(n). Note, however, that the true complexity of MMD involves a permutation
calculation of the null distribution at cost O(bn n), where the number of permutations bn grows with
n. See Appendix C for a detailed description of alternative tests.
4 Experiments
In this section we compare two-sample tests on both artificial benchmark data and on real-world
data. We denote the smooth characteristic function test as ?Smooth CF?, and the test based on the
analytic mean embeddings as ?Mean Embedding?. We compare against several alternative testing approaches: block MMD
p (?Block MMD?), a characteristic functions based test (?CF?), a sub-sampling
MMD test (?MMD( n)?), and the quadratic-time MMD test (?MMD(n)?).
Experimental setup. For all the experiments, D is the dimensionality of samples in a dataset, n
is a number of samples in the dataset (sample size) and J is number of test frequencies. Parameter
selection is required for all the tests. The table summarizes the main choices of the parameters made
for the experiments. The first parameter is the test function, used to calculate the particular statistic.
The scalar represents the length-scale of the observed data. Notice that for the kernel tests we
2
y 2
recover the standard parameterization exp( k x
k ) = exp( kx 2yk ). The original CF test
was proposed without any parameters, hence we added to ensure a fair comparison - for this test
varying is equivalent to adjusting the variance of the distribution of frequencies Tj . For all tests,
the value of the scaling parameter was chosen so as to minimize a p-value estimate on a held-out
training set: details are described in Appendix D. We chose not to optimize the sampling scheme
for the Mean Embeddingp
and Smooth CF tests, since this would give them an unfair advantage over
the Block MMD, MMD( n) and CF tests. The block size in the Block MMD test and the number
of test frequencies in the Mean Embedding, Smooth CF, and CF tests, were always set to the same
value (not greater than 10) to maintain exactly the same time complexity. Note that we did not use
the popular median heuristic for kernel bandwidth choice (MMD and B-test), since it gives poor
results for the Blobs and AM Audio datasets [11]. We do not run MMD(n) test for ?Simulation 1?
or ?Amplitude Modulated Music?, since the sample size is 10000, and too large for a quadratic-time
test with permutation sampling for the test critical value.
6
Figure 1: Higgs dataset. Left: Test power vs. sample size. Right: Test power vs. execution time.
It is important to verify that Type I error is indeed at the design level, set at ? = 0.05 in this paper.
This is verified in the Appendix, Figure A.2. Also shown in the plots is the 95% percent confidence
intervals for the results, as averaged over 4000 runs.
Test
Mean Embedding
Smooth CF
p
MMD(n),MMD( n)
Block MMD
CF
Test Function
exp( k 1 (x t)k2 )
exp(it> 1 x k 1 x tk2 )
exp( k 1 (x t)k2 )
exp( k 1 (x t)k2 )
exp(it> 1 x)
Sampling scheme
Tj ? N (0D , ID )
Tj ? N (0D , ID )
not applicable
not applicable
Tj ? N (0D , ID )
Other parameters
J - no. of test frequencies
J - no. of test frequencies
b -bootstraps
B-block size
J - no. of test frequencies
Real Data 1: Higgs dataset, D = 4, n varies, J = 10. The first experiment we consider is on
the UCI Higgs dataset [18] described in [3] - the task is to distinguish signatures of processes that
produce Higgs bosons from background processes that do not. We consider a two-sample test on
certain extremely low-level features in the dataset - kinematic properties measured by the particle
detectors, i.e., the joint distributions of the azimuthal angular momenta ' for four particle jets. We
denote by P the jet '-momenta distribution of the background process (no Higgs bosons), and by
Q the corresponding distribution for the process that produces Higgs bosons (both are distributions
on R4 ). As discussed in [3, Fig. 2], '-momenta, unlike transverse momenta pT , carry very little
discriminating information for recognizing whether Higgs bosons were produced. Therefore, we
would like to test the null hypothesis that the distributions of angular momenta P (no Higgs boson
observed) and Q (Higgs boson observed) might yet be rejected. The results for different algorithms
are presented in the Figure 1. We observe that the joint distribution of the angular momenta is in
fact discriminative. Sample size varies from 1000 to 12000. The Smooth CF test has significantly
higher power than the other tests, including the quadratic-time MMD, which we could only run
on up to 5100 samples due to computational limitations. The leading performance of the Smooth
CF test is especially remarkable given it is several orders of magnitude faster than the quadratictime MMD(n), even though we used the fastest quadratic-time MMD implementation, where the
asymptotic distribution is approximated by a Gamma density .
Real Data 2: Amplitude Modulated Music, D = 1000, n = 10000, J = 10. Amplitude modulation is the earliest technique used to transmit voice over the radio. In the following experiment
observations were one thousand dimensional samples of carrier signals that were modulated with
two different input audio signals from the same album, song P and song Q (further details of these
data are described in [11, Section 5]). To increase the difficulty of the testing problem, independent
Gaussian noise of increasing variance (in the range 1 to 4.0) was added to the signals. The results
are presented in the Figure 2. Compared to the other tests, the Mean Embedding and Smooth CF
tests are more robust to the moderate noise contamination.
Simulation 1: High Dimensions, D varies, n = 10000, J = 3. It has recently been shown, in
theory and in practice, that the two-sample problem gets more difficult for an increasing number of
dimensions increases on which the distributions do not differ [22, 23]. In the following experiment,
we study the power of the two-sample tests as a function of dimension of the samples. We run twosample tests on two datasets of Gaussian random vectors which differ only in the first dimension,
Dataset I:
Dataset II:
P = N (0D , ID )
P = N (0D , ID )
Q = N ((1, 0, ? ? ? , 0), ID )
Q = N (0D , diag((2, 1, ? ? ? , 1))) ,
vs.
vs.
7
Figure 2: Music Dataset.Left: Test power vs. added noise. Right: four samples from P and Q.
Figure 3: Power vs. redundant dimensions comparison for tests on high dimensional data.
where 0d is a D-dimensional vector of zeros, ID is a D-dimensional identity matrix, and diag(v)
is a diagonal matrix with v on the diagonal. The number of dimensions (D) varies from 50 to
2500 (Dataset I) and from 50 to 1200 (Dataset II). The power of the different two-sample tests is
presented in Figure 3. The Mean Embedding test yields best performance for both datasets, where
the advantage is especially large for differences in variance.
Simulation 2: Blobs, D = 2, n varies, J = 5. The Blobs dataset is a grid of two dimensional
Gaussian distributions (see Figure 4), which is known to be a challenging two-sample testing task.
The difficulty arises from the fact that the difference in distributions is encoded at a much smaller
lengthscale than the overall data. In this experiment both P and Q are four by four grids of Gaussians, where P has unit covariance matrix in each mixture component, while each component of Q
has direction of the largest variance rotated by ?/4 and amplified to 4. It was demonstrated by [11]
that a good choice of kernel is crucial for this task. Figure 4 presents the results of two-sample tests
on the Blobs dataset. The number of samples varies from 50 to 14000 ( MMD(n) reached test power
one with n = 1400). We found that the MMD(n) test has the best power as function of the sample
size, but the worst power/computation tradeoff. By contrast, random distance based tests have the
best power/computation tradeoff.
Acknowledgment. We would like thank Bharath Sriperumbudur and Wittawat Jitkrittum for insightful comments.
Figure 4: Blobs Dataset. Left: test power vs. sample size. Center: test power vs. execution time.
Right: illustration of the blob dataset.
8
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9
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5,176 | 5,686 | Adversarial Prediction Games for Multivariate Losses
Hong Wang
Wei Xing
Kaiser Asif
Brian D. Ziebart
Department of Computer Science
University of Illinois at Chicago
Chicago, IL 60607
{hwang27, wxing3, kasif2, bziebart}@uic.edu
Abstract
Multivariate loss functions are used to assess performance in many modern prediction tasks, including information retrieval and ranking applications. Convex
approximations are typically optimized in their place to avoid NP-hard empirical risk minimization problems. We propose to approximate the training data
instead of the loss function by posing multivariate prediction as an adversarial
game between a loss-minimizing prediction player and a loss-maximizing evaluation player constrained to match specified properties of training data. This avoids
the non-convexity of empirical risk minimization, but game sizes are exponential
in the number of predicted variables. We overcome this intractability using the
double oracle constraint generation method. We demonstrate the efficiency and
predictive performance of our approach on tasks evaluated using the precision at
k, the F-score and the discounted cumulative gain.
1
Introduction
For many problems in information retrieval and learning to rank, the performance of a predictor is
evaluated based on the combination of predictions it makes for multiple variables. Examples include the precision when limited to k positive predictions (P@k), the harmonic mean of precision
and recall (F-score), and the discounted cumulative gain (DCG) for assessing ranking quality. These
stand in contrast to measures like the accuracy and (log) likelihood, which are additive over independently predicted variables. Many multivariate performance measures are not concave functions of
predictor parameters, so maximizing them over empirical training data (or, equivalently, empirical
risk minimization over a corresponding non-convex multivariate loss function) is computationally
intractable [11] and can only be accomplished approximately using local optimization methods [10].
Instead, convex surrogates for the empirical risk are optimized using either an additive [21, 12, 22]
or a multivariate approximation [14, 24] of the loss function. For both types of approximations,
the gap between the application performance measure and the surrogate loss measure can lead to
substantial sub-optimality of the resulting predictions [4].
Rather than optimizing an approximation of the multivariate loss for available training data, we take
an alternate approach [26, 9, 1] that robustly minimizes the exact multivariate loss function using
approximations of the training data. We formalize this using a zero-sum game between a predictor
player and an adversarial evaluator player. Learned weights parameterize this game?s payoffs and
enable generalization from training data to new predictive settings. The key computational challenge
this approach poses is that the size of multivariate prediction games grows exponentially in the
number of variables. We leverage constraint generation methods developed for solving large zerosum games [20] and efficient methods for computing best responses [6] to tame this complexity.
In many cases, the structure of the multivariate loss function enables the zero-sum game?s Nash
equilibrium to be efficiently computed. We formulate parameter estimation as a convex optimization
problem and solve it using standard convex optimization methods. We demonstrate the benefits of
this approach on prediction tasks with P@k, F-score and DCG multivariate evaluation measures.
1
2
Background and Related Work
2.1
Notation and multivariate performance functions
We consider the general task of making a multivariate prediction for variables y =
{y1 , y2 , . . . , yn } ? Y n (with random variables denoted as Y = {Y1 , Y2 , . . . , Yn }) given some
contextual information x = {x1 , x2 , . . . , xn } ? X = {X1 , X2 , . . . , Xn } (with random variable,
X). Each xi is the information relevant to predicted variable yi . We denote the estimator?s predicted
? = {?
? when the
values as y
y1 , y?2 , . . . , y?n }. The multivariate performance measure when predicting y
true multivariate value is actually y is represented as a scoring function: score(?
y, y). Equivalently,
a complementary loss function for any score function based on the maximal score can be defined as:
loss(?
y, y) = maxy0 ,y00 score(y0 , y00 ) ? score(?
y, y).
For information retrieval, a vector of retrieved items from the pool of n items can be represented
? ? {0, 1}n and a vector of relevant items as y ? {0, 1}n with x = {x1 , x2 , . . . , xn } denotas y
ing side contextual information (e.g., search terms and document contents). Precision and recall
are important measures for information retrieval systems. However, maximizing either leads to
degenerate solutions (predict all to maximize recall or predict none to maximize precision). The
y||1 = k,
precision when limited to exactly k positive predictions, P@k(?
y, y) = y?k?y where ||?
is one popular multivariate performance measure that avoids these extremes. Another is the Fscore, which is the harmonic mean of the precision and recall often used in information retrieval tasks. Using this notation, the F-score for a set of items can be simply represented as:
y?y
F1 (?
y, y) = ||?y ||2?
and F1 (0, 0) = 1.
1 +||y||1
In other information retrieval tasks, a ranked list of retrieved items is desired. This can be represented as a permutation, ?, where ?(i) denotes the ith -ranked item (and ? ?1 (j) denotes the
rank of the j th item). Evaluation measures that emphasize the top-ranked items are used, e.g.,
to produce search engine results attuned to actual usage. The discounted cumulative gain (DCG)
measures the performance of item rankings with k relevancy scores, yi ? {0, . . . , k ? 1} as:
Pn y?? (i)
Pn 2y?? (i) ?1
0
? , y) = y?? (1) + i=2 log
DCG(?
? , y) = i=1 log
(i+1) or DCG (?
i.
2
2.2
2
Multivariate empirical risk minimization
Empirical risk minimization [28] is a common supervised learning approach that seeks a predictor
P? (?
y|x) (from, e.g., a set of predictors ?) that minimizes the loss under the empirical distribution of
? Y)]. Multivariate losses are oftraining data, denoted P? (y, x): minP? (?y|x)?? EP? (y,x)P? (?y|x) [loss(Y,
ten not convex and finding the optimal solution is computationally intractable for expressive classes
of predictors ? typically specified by some set of parameters ? (e.g., linear discriminant functions:
P? (?
y |x) = 1 if ? ? ?(x, y?) > ? ? ?(x, y 0 ) ?y 0 6= y?).
Given these difficulties, convex surrogates to P
the multivariate loss are instead employed that are
additive over y?i and yi (i.e., loss(?
y, y) =
yi , yi )). Employing the logarithmic loss,
i loss(?
?
?
loss(?
yi , yi ) = ? log P (Yi = yi ) yields the logistic regression model [9]. Using the hinge loss
yields support vector machines [5]. Structured support vector machines [27] employ a convex approximation of the multivariate loss over a training dataset D using the hinge loss function:
X
min ||?||2 + ?
?i such that ?i, y0 ? Y, ? ? [?(x(i) , y(i) ) ? ?(x(i) , y0 )] ? ?(y0 , y(i) ) ? ?i .
?,?i ?0
i
In other words, linear parameters ? for feature functions ?(?, ?) are desired that make the example
label y(i) have a potential value ? ? ?(x(i) , y(i) ) that is better than all alternative labels y0 by at
least the multivariate loss between y0 and y(i) , denoted ?(y0 , y(i) ). When this is not possible for
a particular example, a hinge loss penalty ?i is incurred that grows linearly with the difference in
potentials. Parameter ? controls a trade-off between obtaining a predictor with lower hinge loss or
better discrimination between training examples (the margin). The size of set Y is often too large
for explicit construction of the constraint set to be computationally tractable. Instead, constraint
generation methods are employed to find a smaller set of active constraints. This can be viewed as
either finding the most-violated constraint [27] or as a loss-augmented inference problem [25]. Our
2
approach employs similar constraint generation techniques?in the inference procedure rather than
the parameter learning procedure?to improve its efficiency.
3
Multivariate Prediction Games
We formulate a minimax game for multivariate loss optimization, describe our approach for limiting the computational complexity of solving this game, and describe algorithms for estimating
parameters of the game and making predictions using this framework.
3.1
Game formulation
Following a recent adversarial formulation for classification [1], we view multivariate prediction as
? making predictions and player Y
? determining the evaluation
a two-player game between player Y
?
distribution. Player Y first stochastically chooses a predictive distribution of variable assignments,
? stochastically chooses
P? (?
y|x), to maximize a multivariate performance measure, then player Y
?
?
an evaluation distribution, P (?
y|x), that minimizes the performance measure. Further, player Y
must choose the relevant items in a way that (approximately) matches in expectation with a set of
statistics, ?(x, y), measured from labeled data. We denote this set as ?.
Definition 1. The multivariate prediction game (MPG) for n predicted variables is:
h
i
? Y)
? ,
max min EP? (x)P? (?y|x)P? (?y|x) score(Y,
y|x)??
P? (?
y|x) P? (?
(1)
where P? (?
y|x) and P? (?
y|x) are distributions over combinations of labelsfor the n predicted vari? = E?
ables and the set ? corresponds to the constraint: EP? (x)P (?y|x) ?(X, Y)
P (y,x) [?(X, Y)] .
Since the set ? constrains the adversary?s multivariate label distribution over the entire distribution
of inputs P? (x), solving this game directly is impractical when the number of training examples is
large. Instead, we employ the method of Lagrange multipliers in Theorem 1, which allows the set
of games to be independently solved given Lagrange multipliers ?.
Theorem 1. The multivariate prediction game?s value (Definition 1) can be equivalently obtained
by solving a set of unconstrained maximin games parameterized by Lagrange multipliers ?:
max
min
? (?
? (?
y|x)??
P
y|x) P
h
i
? Y)
? (a)
EP? (x)P? (?y|x)P? (?y|x) score(Y,
=
?
(b)
?
= max ?
?EP? (y,x) [? ? ?(X, Y)] +
?
min
h
i
? Y)
?
max EP? (x)P? (?y|x)P? (?y|x) score(Y,
? (?
? (?
P
y|x)?? P
y|x)
??
?
X
??
?
?
? ) ? ? ? ?(x, y
? )?
P? (x) min max ?
y, y
?score(?
?? ,
? (?
? (?
|
{z
}
P
y|x) P
y|x)
x?X
(2)
C 0y
? ,?
y
where: ?(x, y) is a vector of features characterizing the set of prediction variables {yi } and provided contextual variables {xi } each related to predicted variable yi .
Proof (sketch). Equality (a) is a consequence of duality in zero-sum games [29]. Equality (b) is
obtained by writing the Lagrangian and taking the dual. Strong Lagrangian duality is guaranteed
when a feasible solution exists on the relative interior of the convex constraint set ? [2]. (A small
amount of slack corresponds to regularization of the ? parameter in the dual and guarantees the
strong duality feasibility requirement is satisfied in practice.)
The resulting game?s payoff matrix can be expressed as the original game scores of Eq. (1) augmented with Lagrangian potentials. The combination defines a new payoff matrix with entries
? ) ? ? ? ?(x, y
? ), as shown in Eq. (2).
C 0y? ,?y = score(?
y, y
3.2
Example multivariate prediction games and small-scale solutions
Examples of the Lagrangian payoff matrices for the P@2, F-score, and DCG P
games are shown in Ta? ) = ni=1 ?(xi ) I(?
ble 1 for three variables. We employ additive feature functions, ?(x, y
yi = 1),
3
Table 1: The payoff matrices for the zero-sum games between player Y? choosing columns and
player Y? choosing rows with three variables for: precision at k (top); F-score (middle) and DCG
with binary relevance values, y?i ? {0, 1}, and we let lg 3 , log2 3 (bottom).
P@2 000 001
1
011 0
2 ??3
1
101 0
2 ??3
110 0 0??3
F1 000 001
000 1 0??3
001 0 1??3
010 0 0??3
2
011 0
3 ??3
100 0 0??3
2
101 0
3 ??3
110 0 0??3
1
111 0
2 ??3
DCG 000 001
1
123 0
2 ??3
132 0 lg13 ??3
1
213 0
2 ??3
231 0 lg13 ??3
312 0 1??3
321 0 1??3
010
0??2
1
2 ??2
011
1??2 ??3
1
2 ??2 ??3
1
2 ??2 ??3
100
0??1
1
2 ??1
1
2 ??1
1??1 ??3
1
2 ??1 ??3
1??1 ??2
111
1??1 ??2 ??3
1??1 ??2 ??3
1??1 ??2 ??3
010
0??2
0??2
1??2
2
3 ??2
0??2
0??2
2
3 ??2
1
2 ??2
011
0??2 ? ?3
2
3 ??2 ? ?3
2
3 ??2 ? ?3
1??2 ? ?3
0??2 ? ?3
1
2 ??2 ? ?3
1
2 ??2 ? ?3
4
5 ??2 ? ?3
100
0??1
0??1
0??1
0??1
1??1
2
3 ??1
2
3 ??1
1
2 ??1
101
0??1 ??3
2
3 ??1 ??3
0??1 ??3
1
2 ??1 ??3
2
3 ??1 ??3
1??1 ??3
1
2 ??1 ??3
4
5 ??1 ??3
110
0??1 ??2
0??1 ??2
2
3 ??1 ??2
1
2 ??1 ??2
2
3 ??1 ??2
1
2 ??1 ??2
1??1 ??2
4
5 ??1 ??2
111
0??1 ??2 ? ?3
1
2 ??1 ??2 ? ?3
1
2 ??1 ??2 ? ?3
4
5 ??1 ??2 ? ?3
1
2 ??1 ??2 ? ?3
4
5 ??1 ??2 ? ?3
4
5 ??1 ??2 ? ?3
1??1 ??2 ? ?3
010
011
100
101
110
1+ lg13 ??1 ??2
3
2 ??1 ??2
1+ lg13 ??1 ??2
3
2 ??1 ??2
1
1
+
2 lg 3 ??1 ??2
1
1
2 + lg 3 ??1 ??2
1
3
2 + lg 3 ??1 ??2 ??3
3
1
2 + lg 3 ??1 ??2 ??3
3
1
2 + lg 3 ??1 ??2 ??3
3
1
2 + lg 3 ??1 ??2 ??3
1
3
2 + lg 3 ??1 ??2 ??3
1
3
2 + lg 3 ??1 ??2 ??3
1
2 ??2
1
1
2 + lg 3 ??2 ??3 1??1
1
1
2 + lg 3 ??2 ??3 1??1
3
1
1??2
2 ??2 ??3
lg 3 ??1
1
1??2 1+ lg 3 ??2 ??3 12 ??1
1
3
1
2 ? ?2
2 ??2 ??3
lg 3 ??1
1
1
1
lg 3 ??2 1+ lg 3 ??2 ??3
2 ??1
1
lg 3 ??2
1
2 ? ?2
101
110
1
2 ??1 ??3
1
2 ??1 ??2
1
2 ??1 ??2
3
2 ??1 ??3
1+ lg13 ??1 ??3
1
1
2 + lg 3 ??1 ??3
1
1
2 + lg 3 ??1 ??3
1
1+ lg 3 ??1 ??3
3
2 ??1 ??3
111
in these examples (with indicator function I(?)). We compactly represent the Lagrangian potential
terms for each game with potential variables, ?i , ? ? ?(Xi = xi ) when Y?i = 1 (and 0 otherwise).
Zero-sum games such as these can be solved using a pair of linear programs that have a constraint
for each pure action (set of variable assignments) in the game [29]:
X
X
C 0y? ,?y ??
P? (?
y|x) = 1;
(3)
P? (?
y|x)C
y ? Y and
max v such that v ?
v,P? (?
y|x)?0
min
v such that v ?
v,P? (?
y|x)?0
? ?Y
y
? ?Y
y
X
C 0y? ,?y
P? (?
y|x)C
? ?Y
y
??
y ? Y and
X
P? (?
y|x) = 1,
(4)
? ?Y
y
where C 0 is the Lagrangian-augmented payoff and v is the value of the game. The second player to
act in a zero-sum game can maximize/minimize using a pure strategy (i.e., a single value assignment
to all variables). Thus, these LPs consider only the set of pure strategies of the opponent to find the
first player?s mixed equilibrium strategy. The equilibrium strategy for the predictor is a distribution
over rows and the equilibrium strategy for the adversary is a distribution over columns.
The size of each game?s payoff matrix grows exponentially with the number of variables, n: (2n ) nk
for the precision at k game; (2n )2 for the F-score game; and (n! k n ) for the DCG game with k
possible relevance levels. These sizes make explicit construction of the game matrix impractical for
all but the smallest of problems.
3.3
Large-scale strategy inference
More efficient methods for obtaining Nash equilibria are needed to scale our MPG approach to
large prediction tasks with exponentially-sized payoff matrices. Though much attention has focused
on efficiently computing -Nash equilibria (e.g., in O(1/) time or O(ln(1/)) time [8]), which
guarantee each player a payoff within of optimal, we employ an approach for finding an exact
equilibrium that works well in practice despite not having as strong theoretical guarantees [20].
4
Consider the reduced game matrices of Table 2. The Nash equilibrium for the precision at k game with
potentials
1 1Lagrangian
1
?
?
?
=
?
=
?
=
0.4
is:
P
(?
y
|x)
=
and
P
(?
y|x) =
3
3
3
11 1 2 1 3
2
3
3
3 ; with a game value of ? 15 . The Nash equilibrium for
the reduced F-score
?1 = ?2= ?3 =
game with no learning(i.e.,
0) is: P? (?
y|x) = 13 32 and P? (?
y|x) = 13 29 29 29 ; with a
game value of 23 . The reduced game equilibrium is also an equilibrium of the original game. Though the exact size of the subgame
and its specific actions depends on the values of ?, often a compact
sub-game with identical equilibrium or close approximation exists
[18]. Motivated by the compactness of the reduced game, we employ a constraint generation approach known as the double oracle
algorithm [20] to iteratively construct an appropriate reduced game
that provides the correct equilibrium but avoids the computational
complexity of the original exponentially sized game.
Table 2: The reduced precision at k game with ?1 =
?2 = ?3 = 0.4 (top) and Fscore game with ?1 = ?2 =
?3 = 0 (bottom).
011
101
110
0.2 -0.3 -0.3
101 -0.3 0.2 -0.3
111 -0.3 -0.3 0.2
011
000 001 010 100
000
111
0
1
1
1
1
1
2
1
2
1
2
Algorithm 1 Constraint generation game solver
?
?
Input: Lagrange potentials for
= {?1 , ?2 , . . . , ?n }; initial action sets S0 and S0
each variable, ?
y|x), P? (?
y|x)
Output: Nash equilibrium, P? (?
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
Initialize Player Y? ?s action set S? ? S?0 and Player Y? ?s action set S? ? S?0
? S,
? ?)
C 0 ? buildPayoffMatrix(S,
. Using Eq. (2) for the sub-game matrix of S? ? S?
repeat
C 0)
. Using the LP of Eq. (3)
[P? (?
y|x), vNash1 ] ? solveZeroSumGameY? (C
[?
a, v?BR ] ? findBestResponseAction(P (?
y|x), ?)
.a
? denotes the best response action
if (vNash1 6= v?BR ) then
. Check if best response provides improvement
?
?
S ?S?a
?
? S,
? ?)
C 0 ? buildPayoffMatrix(S,
. Add new row to game matrix
end if
C 0)
[P? (?
y|x), vNash2 ] ? solveZeroSumGameY? (C
. Using the LP of Eq. (4)
[?
a, v?BR ] ? findBestResponseAction(P (?
y|x), ?)
if (vNash2 6= v?BR ) then
S? ? S? ? a
?
? S,
? ?)
C 0 ? buildPayoffMatrix(S,
. Add new column to game matrix
end if
until (vNash1 = vNash2 = v?BR = v?BR )
. Stop if neither best response provides improvement
return [P? (?
y|x), P? (?
y|x)]
Neither player can improve upon their strategy with additional pure
strategies when Algorithm 1 terminates, thus the mixed strategies it returns are a Nash equilibrium
pair [20]. Additionally, the algorithm is efficient in practice so long as each player?s strategy is
compact (i.e., the number of actions with non-zero probability is a polynomial subset of the label combinations) and best responses to opponents? strategies can be obtained efficiently (i.e., in
polynomial time) for each player. Additionally, this algorithm can be modified to find approximate
?
equilibria by limiting the number of actions for each player?s set S? and S.
3.4
Efficiently computing best responses
The tractability of our approach largely rests on our ability to efficiently find best responses to oppoC 0y?,Y? ] and argminy??Y? EP? (?y|x) [C
C 0Y? ,?y ]. For some combinations
nent strategies: argmaxy??Y? EP? (?y|x) [C
of loss functions and features, finding the best response is trivial using, e.g., a greedy selection algorithm. Other loss function/feature combinations require specialized algorithms or are NP-hard. We
illustrate each situation.
Precision at k best response Many best responses can be obtained using greedy algorithms that
are based on marginal probabilities of the opponent?s strategy. For example, the expected payoff in
5
the precision at k game for the estimator player setting y?i = 1 is P? (?
yi = 1|x). Thus, the set of top k
variables with the largest marginal label probability provides the best response. For the adversary?s
best response, the Lagrangian terms must also be included. Since k is a known variable, as long as
the value of each included term, P? (?
yi = 1, ||?
y ||1 = k|x) ? k?i , is negative, the sum is the smallest,
and the corresponding response is the best for the adversary.
F-score game best response We leverage a recently developed method for efficiently maximizing
the F-score when a distribution over relevant documents is given [6]. The key insight is that the
problem can be separated into an inner greedy maximization over item sets of a certain size k and an
outer maximization to select the best set size k from {0, . . . , n}. This method can be directly applied
to find the best response of the estimator player, Y? , since the Lagrangian terms of the cost matrix
are invariant to the choice of y?. Algorithm 2 obtains the best response for the adversary player, Y? ,
using slight modifications to incorporate the Lagrangian potentials into the objective function.
Algorithm 2 Lagrangian-augmented F-measure Maximizer for adversary player Y?
Input: vector P? of estimator probabilities and Lagrange potentials ? (?1 , ?2 , ..., ?n )
1
1: define matrix W with element W s,k = s+k
,
s, k ? {1, ..., n}
1 T
n
?
F
W
2: construct matrix = P ?
? 2? ? 1
. 1n is the all ones 1 ? n vector
3: for k = 1 to n do
4:
solve the inner optimization
Pn problem:
Pn
?
5:
a(k) = argmina?Ak 2 i=1 ai fik
. Ak = {a ? {0, 1}n | i=1 ai = k}
(k)
6:
by setting ai = 1 for the k-th column of F ?s smallest k elements, and ai = 0 for the rest;
?
P
?
F (y, a(k) )] = 2 ni=1 a(k)
7:
store a value of Ey?p(Y|x)
[F
fik
?
i
8: end for
?
? = 0n |x)
F (y, 0n )] = p(Y
9: for k = 0 take a(k) = 0n , and Ey?P (Y|x)
[F
?
10: solve the outer optimization problem:
F (y, a)]
11: a? = argmina?{a(0)? ,...,a(n)? } Ey?p(Y|x)
[F
?
?
?
F
12: return a and Ey?p(Y|x)
[F (y, a )]
?
Order inversion best response Another common loss measure when comparing two rankings is
the number of pairs of items with inverted order across rankings (i.e., one variable may occur before
another in one ranking, but not in the other ranking). Only the marginal probabilities of pairwise
P
orderings, P? (?
? ?1 (i) > ?
? ?1 (j)) , ?? P? (?
? ) I(? ?1 (i) > ? ?1 (j)), are needed to construct the
portion of the payoff received for ?
? ranking item i over item j, P? (?
? ?1 (i) > ?
? ?1 (j))(1 + ?i>j ),
where ?i>j is a Lagrangian potential based on pair-wise features for ranking item i over item j.
One could construct a fully connected directed graph with edges weighted by these portions of the
payoff for ranking pairs of items. The best response for ?
? corresponds to a set of acyclic edges
with the smallest sum of edge weights. Unfortunately, this problem is NP-hard in general because
the NP-complete minimum feedback arc set problem [15], which seeks to form an acyclic graph by
removing the set of edges with the minimal sum of edge weights, can be reduced to it.
DCG best response Although we cannot find an efficient algorithm to get the best response using
order inversion, solving best response of DCG has a known efficient algorithm. In this problem the
maximizer is a permutation of the documents while the minimizer is the relevance score of each
document pair. The estimator?s best response ?
? maximizes:
!
!
n
n
y??
X
X
X
X
?
(i)
1
2
?1
P (?
y |x)
? ? ? ?(x, y?) =
P (?
y |x)2y??? (i) ? 1 ? c,
log
(i
+
1)
log
(i
+
1)
2
2
y?
y?
i=1
i=1
where c is a constant that has
with ?
? . Since 1/log2 (i + 1) is monotonically decreasPno relationship
ing, computing and sorting y? P (?
y |x)2y?i ? 1 with descending order and greedily assign the order
to ?
? is optimal. The adversary?s best response using additive features minimizes:
X
?
?
!
n
n
n
X
X
X
X
2y?i ? 1
2y??? (i) ? 1
P (?
? |x)
?
?i ? ?i (xi , y?i ) =
P (?
? |x)
? ?i ? ?i (xi , y?i ) .
log2 (i + 1) i=1
log2 (? ?1 (i) + 1)
i=1
i=1
?
?
6
Thus, by using the expectation of a function of each variable?s rank, 1/(log2 (? ?1 (i) + 1), which is
easily computed from P? (?), each variable?s relevancy score y?i can be independently chosen.
3.5
Parameter estimation
Predictive model parameters, ?, must be chosen to ensure that the adversarial distribution is similar
to training data. Though adversarial prediction can be posed as a convex optimization problem
[1], the objective function is not smooth. General subgradient methods require O(1/2 ) iterations
to provide an approximation to the optima. We instead employ L-BFGS [19], which has been
empirically shown to converge at a faster rate in many cases despite lacking theoretical guarantees
for non-smooth objectives [16]. We also employ L2 regularization to avoid overfitting to the training
data sample. The addition of the smooth regularizer often helps to improve the rate of convergence.
The gradient in these optimizations with L2 regularization, ? ?2 ||?||2 , for training dataset D =
{(x(i) , y(i))} is the difference between feature moments
with additional regularization term:
P|D|
P
1
(i)
(i)
(i)
(i)
? y|x )?(x , y
? ) ? ??. The adversarial strategies P? (?|x(i) )
? ?Y P (?
j=1 ?(x , y ) ?
y
|D|
needed for calculating this gradient are computed via Alg. 1.
4
Experiments
We evaluate our approach, Multivariate Prediction Games (MPG), on the three performance measures of interest in this work: precision at k, F-score, and DCG. Our primary point of comparison
is with structured support vector machines (SSVM)[27] to better understand the trade-offs between
convexly approximating the loss function with the hinge loss versus adversarially approximating the
training data using our approach. We employ an optical recognition of handwritten digits (OPTDIGITS) dataset [17] (10 classes, 64 features, 3,823 training examples, 1,797 test examples), an income
prediction dataset (?a4a? ADULT1 [17] (two classes, 123 features, 3,185 training examples, 29,376
test examples), and query-document pairs from the million query TREC 2007 (MQ2007) dataset
of LETOR4.0 [23] (1700 queries, 41.15 documents on average per query, 46 features per document). Following the same evaluation method used in [27] for OPTDIGITS, the multi-class dataset
is converted into multiple binary datasets and we report the macro-average of the performance of all
classes on test data. For OPTDIGITS/ADULT, we use a random 31 of the training data as a holdout
validation data to select the L2 regularization parameter trade-off C ? {2?6 , 2?5 , ..., 26 }.
We evaluate the performance of our approach and comparison methods (SSVM variants2 and logistic regression Table 3: Precision at k (top) and F-score
(LR)) using precision at k, where k is half the number of performance (bottom).
positive examples (i.e. k = 12 P OS), and F-score. For
precision at k, we restrict the pure strategies of the adverPrecision@k OPTDIGITS ADULT
MPG
0.990
0.805
sary to select k positive labels. This prevents adversary
SSVM
0.956
0.638
strategies with no positive labels. From the results in TaSSVM?
0.989
0.805
ble 3, we see that our approach, MPG, works better than
F-score
OPTDIGITS
ADULT
SSVM on the OPTDGITS datasets: slightly better on preMPG
0.920
0.697
cision at k and more significantly better on F-measure.
SSVM
0.915
0.673
For the ADULT dataset, MPG provides equivalent perLR
0.914
0.639
formance for precision at k and better performance on Fmeasure. The nature of the running time required for validation and testing is very different for
SSVM, which must find the maximizing set of variable assignments, and MPG, which must interactively construct a game and its equilibrium. Model validation and testing require ? 30 seconds for
SSVM on the OPTDIGITS dataset and ? 3 seconds on the ADULT dataset, while requiring ? 9
seconds and ? 25 seconds for MPG precision at k and ? 1397 seconds and ? 252 seconds for
MPG F-measure optimization, respectively. For precision at k, MPG is within an order of magni1
http://www.csie.ntu.edu.tw/?cjlin/libsvmtools/datasets/binary.html)
For precision at k, the original SSVM?s implementation uses the restriction k during training, but not
during testing. We modified the code by ordering SSVM?s prediction value for each test example, and select
the top k predictions as positives, the rest are considered as negatives. We denote the original implementation
as SSVM, and the modified version as SSVM?.
2
7
tude (better for OPTDIGITS, worse for ADULT). For the more difficult problem of maximizing the
F-score of ADULT over 29, 376 test examples, the MPG game becomes quite large and requires
significantly more computational time. Though our MPG method is not as finely optimized as existing SSVM implementations, this difference in run times will remain as the game formulation is
inherently more computationally demanding for difficult prediction tasks.
We compare the performance of our approach and comparison methods using five-fold cross validation on the
MQ2007 dataset. We measure performance using Normalized DCG (NDCG), which divides the realized DCG
by the maximum possible DCG for the dataset, based on a
slightly different variant of DCG employed by LETOR4.0:
y?
Pn
? (i)
DCG00 (?
? , y) = 2y?? (1) ? 1 + i=2 2 log ?1
. The compari2i
son methods are: RankSVM-Struct [13], part of SVMstruct
which uses structured SVM to predict the rank; ListNet
[3], a list-wise ranking algorithm employing cross entropy loss; AdaRank-NDCG [30], a boosting method using ?weak rankers? and data reweighing to achieve good
NDCG performance; AdaRank-MAP uses Mean Average Figure 1: NDCG@K as K increases.
Precision (MAP) rather than NDCG; and RankBoost [7],
which reduces ranking to binary classification problems on instance pairs.
Table 4: MQ2007 NDCG Results.
Method
MPG
RankSVM
ListNet
AdaRank-NDCG
AdaRank-MAP
RankBoost
5
Mean NDCG
0.5220
0.4966
0.4988
0.4914
0.4891
0.5003
Table 4 reports the NDCG@K averaged over all values of K
(between 1 and, on average 41) while Figure 1 reports the results for each value of K between 1 and 10. From this, we can
see that our MPG approach provides better rankings on average than the baseline methods except when K is very small
(K = 1, 2). In other words, the adversary focuses most of its
effort in reducing the score received from the first item in the
ranking, but at the expense of providing a better overall NDCG
score for the ranking as a whole.
Discussion
We have extended adversarial prediction games [1] to settings with multivariate performance measures in this paper. We believe that this is an important step in demonstrating the benefits of this
approach in settings where structured support vector machines [14] are widely employed. Our future work will investigate improving the computational efficiency of adversarial methods and also
incorporating structured statistical relationships amongst variables in the constraint set in addition
to multivariate performance measures.
Acknowledgments
This material is based upon work supported by the National Science Foundation under Grant No.
#1526379, Robust Optimization of Loss Functions with Application to Active Learning.
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5,177 | 5,687 | Regressive Virtual Metric Learning
Micha?el Perrot, and Amaury Habrard
Universit?e de Lyon, Universit?e Jean Monnet de Saint-Etienne,
Laboratoire Hubert Curien, CNRS, UMR5516, F-42000, Saint-Etienne, France.
{michael.perrot,amaury.habrard}@univ-st-etienne.fr
Abstract
We are interested in supervised metric learning of Mahalanobis like distances.
Existing approaches mainly focus on learning a new distance using similarity
and dissimilarity constraints between examples. In this paper, instead of bringing closer examples of the same class and pushing far away examples of different
classes we propose to move the examples with respect to virtual points. Hence,
each example is brought closer to a a priori defined virtual point reducing the
number of constraints to satisfy. We show that our approach admits a closed form
solution which can be kernelized. We provide a theoretical analysis showing the
consistency of the approach and establishing some links with other classical metric learning methods. Furthermore we propose an efficient solution to the difficult
problem of selecting virtual points based in part on recent works in optimal transport. Lastly, we evaluate our approach on several state of the art datasets.
1
Introduction
The goal of a metric learning algorithm is to capture the idiosyncrasies in the data mainly by defining
a new space of representation where some semantic constraints between examples are fulfilled. In
the previous years the main focus of metric
learning algorithms has been to learn Mahalanobis like
p
distances of the form dM (x, x0 ) = (x ? x0 )T M(x ? x0 ) where M is a positive semi-definite
matrix (PSD) defining a set of parameters1 . Using a Cholesky decomposition M = LLT , one can
see that this is equivalent to learn a linear transformation from the input space.
Most of the existing approaches in metric learning use constraints of type must-link and cannot-link
between learning examples [1, 2]. For example, in a supervised classification task, the goal is to
bring closer examples of the same class and to push far away examples of different classes. The idea
is that the learned metric should affect a high value to dissimilar examples and a low value to similar
examples. Then, this new distance can be used in a classification algorithm like a nearest neighbor
classifier. Note that in this case the set of constraints is quadratic in the number of examples which
can be prohibitive when the number of examples increases. One heuristic is then to select only
a subset of the constraints but selecting such a subset is not trivial. In this paper, we propose to
consider a new kind of constraints where each example is associated with an a priori defined virtual
point. It allows us to consider the metric learning problem as a simple regression where we try
to minimize the differences between learning examples and virtual points. Fig. 1 illustrates the
differences between our approach and a classical metric learning approach. It can be noticed that
our algorithm only uses a linear number of constraints. However defining these constraints by hand
can be tedious and difficult. To overcome this problem, we present two approaches to automatically
define them. The first one is based on some recent advances in the field of Optimal Transport while
the second one uses a class-based representation space.
1
When M = I, the identity matrix, it corresponds to the Euclidean distance.
1
(a) Classical must-link cannot-link approach.
(b) Our virtual point-based regression formulation.
Figure 1: Arrows denote the constraints used by each approach for one particular example in a
binary classification task. The classical metric learning approach in Fig. 1(a) uses O(n2 ) constraints
bringing closer examples of the same class and pushing far away examples of different classes. On
the contrary, our approach presented in Fig. 1(b) moves the examples to the neighborhood of their
corresponding virtual point, in black, using only O(n) constraints. ( Best viewed in color )
Moreover, thanks to its regression-based formulation, our approach can be easily kernelized allowing
us to deal efficiently with non linear transformations which is a nice advantage in comparison to
some metric learning methods. We also provide a theoretical analysis showing the consistency of
our approach and establishing some relationships with a classical metric learning formulation.
This paper is organized as follows. In Section 2 we identify several related works. Then in Section 3
we present our approach, provide some theoretical results and give two solutions to generate the
virtual points. Section 4 is dedicated to an empirical evaluation of our method on several widely
used datasets. Finally, we conclude in Section 5.
2
Related work
For up-to-date surveys on metric learning see [3] and [4]. In this section we focus on algorithms
which are more closely related to our approach. First of all, one of the most famous approach
in metric learning is LMNN [5] where the authors propose to learn a PSD matrix to improve the
k-nearest-neighbours algorithm. In their work, instead of considering pairs of examples, they use
triplets (xi , xj , xk ) where xj and xk are in the neighborhood of xi and such that xi and xj are of
the same class and xk is of a different class. The idea is then to bring closer xi and xj while pushing
xk far away. Hence, if the number of constraints seems to be cubic, the authors propose to only
consider triplets of examples which are already close to each other. In contrast, the idea presented in
[6] is to collapse all the examples of the same class in a single point and to push infinitely far away
examples of different classes. The authors define a measure to estimate the probability of having an
example xj given an example xi with respect to a learned PSD matrix M. Then, they minimize,
w.r.t. M, the KL divergence between this measure and the best case where the probability is 1 if the
two examples are of the same class and 0 otherwise. It can be seen as collapsing all the examples
of the same class on an implicit virtual point. In this paper we use several explicit virtual points and
we collapse the examples on these points with respect to their classes and their distances to them.
A recurring issue in Mahalanobis like metric learning is to fulfill the PSD constraint on the learned
metric. Indeed, projecting a matrix on the PSD cone is not trivial and generally requires a costly
eigenvalues decomposition. To address this problem, in ITML [1] the authors propose to use a
LogDet divergence as the regularization term. The idea is to learn a matrix which is close to an a
priori defined PSD matrix. The authors then show that if the divergence is finite, then the learned
matrix is guaranteed to be PSD. Another approach, as proposed in [2], is to learn a matrix L such
that M = LLT , i.e. instead of learning the metric the authors propose to learn the projection. The
main drawback is the fact that most of the time the resulting optimization problem is not convex
[3, 4, 7] and is thus harder to optimize. In this paper, we are also interested in learning L directly.
However, because we are using constraints between examples and virtual points, we obtain a convex
problem with a closed form solution allowing us to learn the metric in an efficient way.
The problem of learning a metric such that the induced space is not linearly dependent of the input
space has been addressed in several works before. First, it is possible to directly learn an intrinsically
non linear metric as in ?2 -LMNN [8] where the authors propose to learn a ?2 distance rather than a
Mahalanobis distance. This distance is particularly relevant for histograms comparisons. Note that
this kind of approaches is close to the kernel learning problem which is beyond the scope of this
work. Second, another solution used by local metric learning methods is to split the input space
2
in several regions and to learn a metric in each region to introduce some non linearity, as in MMLMNN [7]. Similarly, in GB-LMNN [8] the authors propose to locally refine the metric learned
by LMNN by successively splitting the input space. A third kind of approach tries to project the
learning examples in a new space which is non linearly dependent of the input space. It can be done
in two ways, either by projecting a priori the learning examples in a new space with a KPCA [9]
or by rewriting the optimization problem in a kernelized form [1]. The first approach allows one to
include non linearity in most of the metric learning algorithms but imposes to select the interesting
features beforehand. The second method can be difficult to use as rewriting the optimization problem
is most of the times non trivial [4]. Indeed, if one wants to use the kernel trick it implies that the
access to the learning examples should only be done through dot products which is difficult when
working with pairs of examples as it is the case in metric learning. In this paper we show that using
virtual points chosen in a given target space allows us to kernelize our approach easily and thus to
work in a very high dimensional space without using an explicit projection thanks to the kernel trick.
Our method is based on a regression and can thus be linked, in its kernelized form, to several approaches in kernelized regression for structured output [10, 11, 12]. The idea behind these approaches is to minimize the difference between input examples and output examples using kernels,
i.e. working in a high dimensional space. In our case, the learning examples can be seen as input
examples and the virtual points as output examples. However, we only project the learning examples
in a high dimensional space, the virtual points already belong to the output space. Hence, we do not
have the pre-image problem [12]. Furthermore, our goal is not to predict a virtual point but to learn
a metric between examples and thus, after the learning step, the virtual points are discarded.
3
Contributions
The main idea behind our algorithm is to bring closer the learning examples to a set of virtual points.
We present this idea in three subsections. First we assume that we have access to a set of n learning
pairs (x,v) where x is a learning example and v is a virtual point associated to x and we present
both the linear and kernelized formulations of our approach called RVML. It boils down to solve
a regression in closed form, the main originality being the introduction of virtual points. In the
second subsection, we show that it is possible to theoretically link our approach to a classical metric
learning one based on [13]. In the last subsection, we propose two automatic methods to generate
the virtual points and to associate them with the learning examples.
3.1
Regressive Virtual Metric Learning (RVML)
Given a probability distribution D defined over X ? Y where X ? Rd and Y is a finite label
set, let S = {(xi , yi )}ni=1 be a set of examples drawn i.i.d. from D. Let fv : X ? Y ? V
0
where V ? Rd be the function which associates each example to a virtual point. We consider
the learning set Sv = {(xi , vi )}ni=1 where vi = fv (xi , yi ). For the sake of simplicity denote by
T
T
X = (x1 , . . . , xn ) and V = (v1 , . . . , vn ) the matrices containing respectively one example and
the associated virtual point on each line. In this section, we consider that the function fv is known.
We come back to its definition in Section 3.3. Let k ? kF be the Frobenius norm and k ? k2 be the l2
vector norm. Our goal is to learn a matrix L such that M = LLT and for this purpose we consider
the following optimisation problem:
min f (L, X, V) = min
L
L
1
kXL ? Vk2F + ?kLk2F .
n
(1)
The idea is to learn a new space of representation where each example is close to its associated
virtual point. Note that L is a d ? d0 matrix and if d0 < d we also perform dimensionality reduction.
Theorem 1. The optimal solution of Problem 1 can be found in closed form. Furthermore, we can
derive two equivalent solutions:
?1 T
L = XT X + ?nI
X V
(2)
?1
L = XT XXT + ?nI
V.
(3)
Proof. The proof of this theorem can be found in the supplementary material.
3
From Eq. 2 we deduce the matrix M:
M = LLT = XT X + ?nI
?1
XT VVT X XT X + ?nI
?1
.
(4)
Note that M is PSD by construction: xT Mx = xT LLT x = kLT xk22 ? 0.
So far, we have focused on the linear setting. We now present a kernelized version, showing that it
is possible to learn a metric in a very high dimensional space without an explicit projection.
Let ?(x) be a projection function and K(x, x0 ) = ?(x)T ?(x0 ) be its associated kernel. For the sake
T
of readability, let KX = ?(X)?(X)T where ?(X) = (?(x1 ), . . . , ?(xn )) . Given the solution
?1
?1
matrix L presented in Eq. 3, we have M = XT XXT + ?nI
VVT XXT + ?nI
X. Then,
MK the kernelized version of the matrix M is defined such that:
?1
?1
MK = ?(X)T (KX + ?nI)
VVT (KX + ?nI)
?(X).
T
The squared Mahalanobis distance can be written as d2M (x, x0 ) = xT Mx + x0 Mx0 ? 2xT Mx0 .
T
Thus we can obtain d2MK (?(x), ?(x0 )) = ?(x)T MK ?(x) + ?(x0 ) MK ?(x0 ) ? 2?(x)T MK ?(x0 )
the kernelized version by considering that:
?(x)T MK ?(x) = ?(x)T ?(X)T (KX + ?nI)?1 VVT (KX + ?nI)?1 ?(X)?(x)
= KX (x)T (KX + ?nI)?1 VVT (KX + ?nI)?1 KX (x)
T
where KX (x) = (K(x, x1 ), . . . , K(x, xn )) is the similarity vector to the examples w.r.t. K.
?1
Note that it is also possible to obtain a kernelized version of L: LK = ?(X)T (KX + ?nI)
V.
This result is close to a previous one already derived in [11] in a structured output setting. The main
difference is the fact that we do not use a kernel on the output (the virtual points here). Hence, it is
possible to compute the projection of an example x of dimension d in a new space of dimension d0 :
?1
?(x)LK = ?(x)T ?(X)T (KX + ?nI)
V = KX (x)T (KX + ?nI)
?1
V.
Recall that in this work we are interested in learning a distance between examples and not in the
prediction of the virtual points which only serve as a way to bring closer similar examples and push
far away dissimilar examples.
From a complexity standpoint, we can see that, assuming the kernel function as easy to calculate,
the main bottleneck when computing the solution in closed form is the inversion of a n ? n matrix.
3.2
Theoretical Analysis
In this section, we propose to theoretically show the interest of our approach by proving (i) that it is
consistent and (ii) that it is possible to link it to a more classical metric learning formulation.
3.2.1
Consistency
Let l(L, (x, v)) = kxT L ? vT k22 be our loss and let Dv be the probability distribution over X ? V
such that pDv (x, v) = pD (x, y|v = fv (x, y)). Showing the consistency boils down to bound with
?
high probability the true risk, denoted by R(L), by the empirical risk, denoted by R(L)
such that:
X
1
1
?
l(L, (x, v)) = kXL ? Vk2F .
R(L) = E(x,v)?Dv l(L, (x, v)) and R(L)
=
n
n
(x,v)?Sv
The empirical risk corresponds to the error of the learned matrix L on the learning set Sv . The true
risk is the error of L on the unknown distribution Dv . The consistency property ensures that with a
sufficient number of examples a low empirical risk implies a low true risk with high probability. To
show that our approach is consistent, we use the uniform stability framework [14].
Theorem 2. Let kvk2 ? Cv for any v ? V and kxk2 ? Cx for any x ? X . With probability 1 ? ?,
for any matrix L optimal solution of Problem 1, we have:
s
2
2 !
2 2
2
ln 1?
8C
C
C
C
16C
x
x
v x
x
?
R(L) ? R(L)
+
1+ ?
+
+ 1 Cv2 1 + ?
.
?n
?
2n
?
?
4
Proof. The proof of this theorem can be found in the supplementary material.
We obtain a rate of convergence in O
3.2.2
?1
n
which is standard with this kind of bounds.
Link with a Classical Metric Learning Formulation
In this section we show that it is possible to bound the true risk of a classical metric learning approach
with the empirical risk of our formulation. Most of the classical metric learning approaches make
use of a notion of margin between similar and dissimilar examples. Hence, similar examples have
to be close to each other, i.e. at a distance smaller than a margin ?1 , and dissimilar examples have to
be far from each other, i.e. at a distance greater than a margin ??1 . Let (xi , yi ) and (xj , yj ) be two
examples from X ? Y, using this notion of margin, we consider the following loss [13]:
h
i
l(L, (xi , yi ), (xj , yj )) = yij (d2 (LT xi , LT xj ) ? ?yij )
(5)
+
where yij = 1 if yi = yj and ?1 otherwise, [z]+ = max(0, z) is the hinge loss and ?yij is the
desired margin between examples. As introduced before, we consider that ?yij takes a big value
when the examples are dissimilar, i.e. when yij = ?1, and a small value when the examples are
similar, i.e. when yij = 1. In the following we show that, relating the notion of margin to the
distances between virtual points, it is possible to bound the true risk associated with this loss by the
empirical risk of our approach with respect to a constant.
0
Theorem 3. Let D be a distribution over X ? Y. Let V ? Rd be a finite set of virtual points and
fv is defined as fv (xi , yi ) = vi , vi ? V. Let kvk2 ? Cv for any v ? V and kxk2 ? Cx for any
x ? X . Let ?1 = 2 maxxk ,xl ,ykl =1 d2 (vk , vl ) and ??1 = 21 minxk ,xl ,ykl =?1 d2 (vk , vl ), we have:
E(xi ,yi )?D,(xj ,yj )?D yij (d2 (LT xi , LT xj ) ? ?yij ) +
?
?
s
2
2 !
1
2 2
2
ln
8Cv Cx
16Cx
Cx
Cx
? ?
?
? 8 ?R(L)
+
+ 1 Cv2 1 + ?
) .
1+ ?
+
?n
?
2n
?
?
Proof. The proof of this theorem can be found in the supplementary material.
In Theorem 3, we can notice that the margins are related to the distances between virtual points and
correspond to the ideal margins, i.e. the margins that we would like to achieve after the learning
step. Aside this remark, we can define ??1 and ???1 the observed margins obtained after the learning
step: All the similar examples
are in a
sphere centered in their corresponding virtual point and of
diameter ??1 = 2 max(x,v)
xT L ? vT
2 . Similarly, the distance between hyperspheres of dissimilar examples is ???1 = minv,v0 ,v6=v0 kv ? v0 k2 ? ??1 . As a consequence, even if we do not use
cannot-link constraints our algorithm is able to push reasonably far away dissimilar examples.
In the next subsection we present two different methods to select the virtual points.
3.3
Virtual Points Selection
Previously, we assumed to have access to the function fv : X ? Y ? V. In this subsection, we
present two methods for generating automatically the set of virtual points and the mapping fv .
3.3.1
Using Optimal Transport on the Learning Set
In this first approach, we propose to generate the virtual points by using a recent variation of the
Optimal Transport (OT) problem [15] allowing one to transport some examples to new points corresponding to a linear combination of a set of known instances. These new points will actually
correspond to our virtual points. Our approach works as follows. We begin by extracting a set of
landmarks S 0 from the training set S. For this purpose, we use an adaptation of the landmark selection method proposed in [16] allowing us to take into account some diversity among the landmarks.
To avoid to fix the number of landmarks in advance, we have just replaced it by a simple heuristic
saying that the number of landmarks must be greater than the number of classes and that the maximum distance between an example and a landmark must be lower than the mean of all pairwise
5
Algorithm 1: Selecting S 0 from a set of examples S.
input : S = {(xi , yi )}n
i=1 a set of examples; Y the label set.
output: S 0 a subset of S
begin
? = mean of distances between all the examples of S
xmax = arg maxkx ? 0k2
x?S
S 0 = {xmax }; S = S \ S 0
? = maxx?S minx0 ?S 0 kx ? x0 k2
while |S 0 | < |Y| or ? >X
? do
xmax = arg max
kx ? x0 k2
x?S
0
x0 ?S 0
0
S = S ? {xmax }; S = S \ S 0
? = maxx?S minx0 ?S 0 kx ? x0 k2
distances from the training set -allowing us to have a fully automatic procedure. It is summarized in
Algorithm 1.
Then we compute an optimal transport from the training set S to the landmark set S 0 . For this
purpose, we create a real matrix C of size |S| ? |S 0 | giving the cost to transport one training instance
to a landmark such that C(i, j) = kxi ? x0j k2 with xi ? S and x0j ? S 0 . The optimal transport is
0
found by learning a matrix ? ? R|S|?|S | able to minimize the cost of moving training examples to
0
the landmark points. Let S be the matrix of landmark points (one per line), the transport w.r.t. ? of
any training instance (xi , yi ) gives a new virtual point such that fv (xi , yi ) = ?(i)S0 , ?(i) designing
the ith line of ?. Note that this new virtual point is a linear combination of the landmark instances to
which the example is transported. The set of virtual points is then defined by V = ?S0 . The virtual
points are thus not defined a priori but are automatically learned by solving a problem of optimal
transport. Note that this transportation mode is potentially non linear since there is no guarantee that
there exists a matrix T such that V = XT. Our metric learning approach can, in this case, be seen
as a an approximation of the result given by the optimal transport.
To learn ?, we use the following optimization problem proposed in [17]:
arg min h?, CiF ?
?
XX
1
h(?) + ?
k?(yi = c, j)kpq
?
c
j
P
where h(?) = ? i,j ?(i, j) log(?(i, j)) is the entropy of ? that allows to solve the transportation
problem efficiently with the Sinkhorn-Knopp algorithm [18]. The second regularization term, where
?(yi = c, j) corresponds to the lines of the j th column of ? where the class of the input is c, has
been introduced in [17]. The goal of this term is to prevent input examples of different classes to
move toward the same output examples by promoting group sparsity in the matrix ? thanks to k ? kpq
corresponding to a lq -norm to the power of p used here with q = 1 and p = 21 .
3.3.2
Using a Class-based Representation Space
For this second approach, we propose to define virtual points as the unit vectors of a space of
dimension |Y|. Let ej ? R|Y| be such a unit vector (1 ? j ? |Y|) -i.e. a vector where all the
attributes are 0 except for one attribute j which is set to 1- to which we associate a class label from
Y. Then, for any learning example (xi , yi ), we define fv (xi , yi ) = e#yi where #yi = j if ej is
mapped with the class yi . Thus, we have exactly |Y| virtual points, each one corresponding to a unit
vector and a class label. We call this approach the class-based representation space method. If the
number of classes is smaller than the number of dimensions used to represent the learning examples,
then our method will perform dimensionality reduction for free. Furthermore, our approach will try
to project all the examples of one class on the same axis while examples of other classes will tend
to be projected on different axes. The underlying intuition behind the new space defined by L is to
make each attribute discriminant for one class.
6
Table 1: Comparison of our approach with several baselines in the linear setting.
4
Base
Amazon
Breast
Caltech
DSLR
Ionosphere
Isolet
Letters
Pima
Scale
Splice
Svmguide1
Wine
Webcam
1NN
41.51 ? 3.24
95.49 ? 0.79
18.04 ? 2.20
29.61 ? 4.38
86.23 ? 1.95
88.97
94.74 ? 0.27
69.91 ? 1.69
78.68 ? 2.66
71.17
95.12
96.18 ? 1.59
42.90 ? 4.19
Baselines
LMNN
65.50 ? 2.28
95.49 ? 0.89
49.68 ? 2.76
76.08 ? 4.79
88.02 ? 3.02
95.83
96.43 ? 0.28*
70.04 ? 2.20
78.20 ? 1.91
82.02
95.03
98.36 ? 1.03
85.81 ? 3.75
SCML
71.68 ? 1.86
96.50 ? 0.64*
52.84 ? 1.61
65.10 ? 9.00
90.38 ? 2.55*
89.61
96.13 ? 0.20
69.22 ? 2.60
93.39 ? 1.70*
85.43
87.38
96.91 ? 1.93
90.43 ? 2.70
mean
69.89
82.81
83.46
Our approach
RVML-Lin-OT
RVML-Lin-Class
71.62 ? 1.34
73.09 ? 2.49
95.24 ? 1.21
95.34 ? 0.95
52.51 ? 2.41
55.41 ? 2.55*
74.71 ? 5.27
75.29 ? 5.08
87.36 ? 3.12
82.74 ? 2.81
91.40
94.61
90.25 ? 0.60
95.51 ? 0.26
70.48 ? 3.19
69.57 ? 2.85
90.05 ? 2.13
87.94 ? 1.99
84.64
78.44
94.83
85.25
98.55 ? 1.67
98.18 ? 1.48
88.60 ? 3.63
88.60 ? 2.69
83.86
83.07
Experimental results
In this section, we evaluate our approach on 13 different datasets coming from either the UCI [19]
repository or used in recent works in metric learning [8, 20, 21]. For isolet, splice and svmguide1
we have access to a standard training/test partition, for the other datasets we use a 70% training/30%
test partition, we perform the experiments on 10 different splits and we average the result. We
normalize the examples with respect to the training set by subtracting for each attribute its mean
and dividing by 3 times its standard deviation. We set our regularization parameter ? with a 5-fold
cross validation. After the metric learning step, we use a 1-nearest neighbor classifier to assess the
performance of the metric and report the accuracy obtained.
We perform two series of experiments. First, we consider our linear formulation used with the
two virtual points selection methods presented in this paper: RVML-Lin-OT based on Optimal
Transport (Section 3.3.1) and RVML-Lin-Class using the class-based representation space method
(Section 3.3.2). We compare them to a 1-nearest neighbor classifier without metric learning (1NN),
and with two state of the art linear metric learning methods: LMNN [5] and SCML [20].
In a second series, we consider the kernelized versions of RVML, namely RVML-RBF-OT and
RVML-RBF-Class, based respectively on Optimal Transport and class-based representation space
methods, with a RBF kernel with the parameter ? fixed as the mean of all pairwise training set
Euclidean distances [16]. We compare them to non linear methods using a KPCA with a RBF kernel2
as a pre-process: 1NN-KPCA a 1-nearest neighbor classifier without metric learning and LMNNKPCA corresponding to LMNN in the KPCA-space. The number of dimensions is fixed as the one
of the original space for high dimensional datasets (more than 100 attributes), to 3 times the original
dimension when the dimension is smaller (between 5 and 100 attributes) and to 4 times the original
dimension for the lowest dimensional datasets (less than 5 attributes). We also consider some local
metric learning methods: GBLMNN [8] a non linear version of LMNN and SCMLLOCAL [20] the
local version of SCML. For all these methods, we use the implementations available online letting
them handle hyper-parameters tuning.
The results for linear methods are presented in Table 1 while Table 2 gives the results obtained
with the non linear approaches. In each table, the best result on each line is highlighted with a
bold font while the second best result is underlined. A star indicates either that the best baseline
is significantly better than our best result or that our best result is significantly better than the best
baseline according to classical significance tests (the p-value being fixed at 0.05).
We can make the following remarks. In the linear setting, our approaches are very competitive to the
state of the art and RVML-Lin-OT tends to be the best on average even though it must be noticed that
SCML is very competitive on some datasets (the average difference is not significant). RVML-LinClass performs slightly less on average. Considering now the non linear methods, our approaches
improve their performance and are significantly better than the others on average, RVML-RBF-Class
has the best average behavior in this setting. These experiments show that our regressive formulation
2
With the ? parameter fixed as previously to the mean of all pairwise training set Euclidean distances.
7
Table 2: Comparison of our approach with several baselines in the non-linear case.
Base
Amazon
Breast
Caltech
DSLR
Ionosphere
Isolet
Letter
Pima
Scale
Splice
Svmguide1
Wine
Webcam
1NN-KPCA
20.27 ? 2.42
92.43 ? 2.19
20.82 ? 8.29
64.90 ? 5.81
75.57 ? 2.79
68.70
95.39 ? 0.27
69.57 ? 2.64
78.36 ? 0.88
66.99
95.72
92.18 ? 1.23
73.55 ? 4.57
mean
70.34
Baselines
LMNN-KPCA
GBLMNN
53.16 ? 3.73
65.53 ? 2.32
95.39 ? 1.32
95.58 ? 0.87
29.88 ? 10.89
49.91 ? 2.80
73.92 ? 7.57
76.08 ? 4.79
85.66 ? 2.55
87.36 ? 3.02
96.28
96.02
97.17* ? 0.18
96.51 ? 0.25
69.48 ? 2.04
69.52 ? 2.27
88.10 ? 2.26
77.88 ? 2.43
88.97
82.21
95.60
95.00
95.82 ? 2.98
98.00 ? 1.34
84.52 ? 3.83
85.81 ? 3.75
81.07
82.72
SCMLLOCAL
69.14 ? 1.74
96.31 ? 0.66
50.56 ? 1.62
62.55 ? 6.94
90.94 ? 3.02
91.40
96.63 ? 0.26
68.40 ? 2.75
93.86 ? 1.78
87.13
87.40
96.55 ? 2.00
88.71 ? 2.83
83.04
Our approach
RVML-RBF-OT
RVML-RBF-Class
73.51 ? 0.83
76.22 ? 2.09*
95.73 ? 0.97
95.78 ? 0.92
54.39 ? 1.89
57.98 ? 2.22*
70.39 ? 4.48
76.67 ? 4.57
90.66 ? 3.10
93.11 ? 3.30*
95.96
96.73
91.26 ? 0.50
96.09 ? 0.21
69.35 ? 2.95
70.74 ? 2.36
95.19 ? 1.46*
94.07 ? 2.02
88.51
88.32
95.67
95.05
98.91 ? 1.53
98.00 ? 1.81
88.71 ? 4.28
88.92 ? 2.91
85.25
86.74
is very competitive and is even able to improve state of the art performances in a non linear setting
and consequently that our virtual point selection methods automatically select correct instances.
Considering the virtual point selection, we can observe that the OT formulation performs better than
the class-based representation space one in the linear case, while it is the opposite in the non-linear
case. We think that this can be explained by the fact that the OT approach generates more virtual
points in a potentially non linear way which brings more expressiveness for the linear case. On the
other hand, in the non linear one, the relative small number of virtual points used by the class-based
method seems to induce a better regularization. In Section 4 of the supplementary material, we
provide additional experiments showing the interest of using explicit virtual points and the need of
a careful association between examples and virtual points. We also provide some graphics showing
2D projections of the space learned by RVML-Lin-Class and RVML-RBF-Class on the Isolet dataset
illustrating the capability of these approaches to learn discriminative attributes.
In terms of computational cost, our approach -implemented in closed form [22]- is competitive
with classical methods but does not yield to significant improvements. Indeed, in practice, classical
approaches only consider a small number of constraints e.g. c times the number of examples, where
c is a small constant, in the case of SCML. Thus, the practical computational complexity of both our
approach and classical methods is linearly dependant on the number of examples.
5
Conclusion
We present a new metric learning approach based on a regression and aiming at bringing closer
the learning examples to some a priori defined virtual points. The number of constraints has the
advantage to grow linearly with the size of the learning set in opposition to the quadratic grow of
standard must-link cannot-link approaches. Moreover, our method can be solved in closed form and
can be easily kernelized allowing us to deal with non linear problems. Additionally, we propose
two methods to define the virtual points: One making use of recent advances in the field of optimal
transport and one based on unit vectors of a class-based representation space allowing one to perform
directly some dimensionality reduction. Theoretically, we show that our approach is consistent and
we are able to link our empirical risk to the true risk of a classical metric learning formulation.
Finally, we empirically show that our approach is competitive with the state of the art in the linear
case and outperforms some classical approaches in the non-linear one.
We think that this work opens the door to design new metric learning formulations, in particular the
definition of the virtual points can bring a way to control some particular properties of the metric
(rank, locality, discriminative power, . . . ). As a consequence, this aspect opens new issues which are
in part related to landmark selection problems but also to the ability to embed expressive semantic
constraints to satisfy by means of the virtual points. Other perspectives include the development of
a specific solver, of online versions, the use of low rank-inducing norms or the conception of new
local metric learning methods. Another direction would be to study similarity learning extensions
to perform linear classification such as in [21, 23].
8
References
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9
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5,178 | 5,688 | Halting in Random Walk Kernels
Karsten M. Borgwardt
D-BSSE, ETH Z?urich
Basel, Switzerland
[email protected]
Mahito Sugiyama
ISIR, Osaka University, Japan
JST, PRESTO
[email protected]
Abstract
Random walk kernels measure graph similarity by counting matching walks in
two graphs. In their most popular form of geometric random walk kernels, longer
walks of length k are downweighted by a factor of ?k (? < 1) to ensure convergence of the corresponding geometric series. We know from the field of link
prediction that this downweighting often leads to a phenomenon referred to as
halting: Longer walks are downweighted so much that the similarity score is
completely dominated by the comparison of walks of length 1. This is a na??ve
kernel between edges and vertices. We theoretically show that halting may occur
in geometric random walk kernels. We also empirically quantify its impact in simulated datasets and popular graph classification benchmark datasets. Our findings
promise to be instrumental in future graph kernel development and applications of
random walk kernels.
1
Introduction
Over the last decade, graph kernels have become a popular approach to graph comparison [4, 5, 7, 9,
12, 13, 14], which is at the heart of many machine learning applications in bioinformatics, imaging,
and social-network analysis. The first and best-studied instance of this family of kernels are random
walk kernels, which count matching walks in two graphs [5, 7] to quantify their similarity. In particular, the geometric random walk kernel [5] is often used in applications as a baseline comparison
method on graph benchmark datasets when developing new graph kernels. These geometric random
walk kernels assign a weight ?k to walks of length k, where ? < 1 is set to be small enough to
ensure convergence of the corresponding geometric series.
Related similarity measures have also been employed in link prediction [6, 10] as a similarity score
between vertices [8]. However, there is one caveat regarding these approaches. Walk-based similarity scores with exponentially decaying weights tend to suffer from a problem referred to as halting [1]. They may downweight walks of lengths 2 and more, so much so that the similarity score is
ultimately completely dominated by walks of length 1. In other words, they are almost identical to
a simple comparison of edges and vertices, which ignores any topological information in the graph
beyond single edges. Such a simple similarity measure could be computed more efficiently outside
the random walk framework. Therefore, halting may affect both the expressivity and efficiency of
these similarity scores.
Halting has been conjectured to occur in random walk kernels [1], but its existence in graph kernels
has never been theoretically proven or empirically demonstrated. Our goal in this study is to answer
the open question if and when halting occurs in random walk graph kernels.
We theoretically show that halting may occur in graph kernels and that its extent depends on properties of the graphs being compared (Section 2). We empirically demonstrate in which simulated
datasets and popular graph classification benchmark datasets halting is a concern (Section 3). We
conclude by summarizing when halting occurs in practice and how it can be avoided (Section 4).
1
We believe that our findings will be instrumental in future applications of random walk kernels and
the development of novel graph kernels.
2
Theoretical Analysis of Halting
We theoretically analyze the phenomenon of halting in random walk graph kernels. First, we review
the definition of graph kernels in Section 2.1. We then present our key theoretical result regarding
halting in Section 2.2 and clarify the connection to linear kernels on vertex and edge label histograms
in Section 2.3.
2.1
Random Walk Kernels
Let G = (V, E, ?) be a labeled graph, where V is the vertex set, E is the edge set, and ? is a
mapping ? : V ? E ? ? with the range ? of vertex and edge labels. For an edge (u, v) ? E, we
identify (u, v) and (v, u) if G is undirected. The degree of a vertex v ? V is denoted by d(v).
The direct (tensor) product G? = (V? , E? , ?? ) of two graphs G = (V, E, ?) and G? =
(V ? , E ? , ?? ) is defined as follows [1, 5, 14]:
V? = { (v, v ? ) ? V ? V ? | ?(v) = ?? (v ? ) },
E? = { ((u, u? ), (v, v ? )) ? V? ? V? | (u, v) ? E, (u? , v ? ) ? E ? , and ?(u, v) = ?? (u? , v ? ) },
and all labels are inherited, or ?? ((v, v ? )) = ?(v) = ?? (v ? ) and ?? ((u, u? ), (v, v ? )) = ?(u, v) =
?? (u? , v ? ). We denote by A? the adjacency matrix of G? and denote by ?? and ?? the minimum
and maximum degrees of G? , respectively.
To measure the similarity between graphs G and G? , random walk kernels count all pairs of matching
walks on G and G? [2, 5, 7, 11]. If we assume a uniform distribution for the starting and stopping
probabilities over the vertices of G and G? , the number of matching walks is obtained through the
adjacency matrix A? of the product graph G? [14]. For each k ? N, the k-step random walk kernel
between two graphs G and G? is defined as:
]
[ k
|V? |
? ?
k
?
l
K? (G, G ) =
?l A?
i,j=1
l=0
ij
with a sequence of positive, real-valued weights ?0 , ?1 , ?2 , . . . , ?k assuming that A0? = I, the
?
identity matrix. Its limit K?
(G, G? ) is simply called the random walk kernel.
?
can be directly computed if weights are the geometric series, or ?l = ?l , resulting
Interestingly, K?
in the geometric random walk kernel:
]
[?
|V? |
|V? |
? ?
?[
]
l l
?
=
(I ? ?A? )?1 ij .
KGR (G, G ) =
? A?
i,j=1
l=0
ij
i,j=1
In the above equation, let (I ? ?A? )x = 0 for some value of x. Then, ?A? x = x and (?A? )l x =
x for any l ? N. If (?A? )l ?
converges to 0 as l ? ?, (I ? ?A? ) is invertible since x becomes 0.
?
Therefore, (I ? ?A? )?1 = l=0 ?l Al? from the equation (I ? ?A? )(I + ?A? + ?2 A2? + . . . ) =
I [5]. It is well-known that the geometric series of matrices, often called the Neumann series,
I + ?A? + (?A? )2 + ? ? ? converges only if the maximum eigenvalue of A? , denoted by ??,max , is
strictly smaller than 1/?. Therefore, the geometric random walk kernel KGR is well-defined only if
? < 1/??,max .
There is a relationship for the minimum and maximum degrees ?? and ?? of G? [3]: ?? ?
d? ? ??
?,max ? ?? , where d? is the average of the vertex degrees of G? , or d? =
(1/|V? |) v?V? d(v). In practice, it is sufficient to set the parameter ? < 1/?? .
In the inductive learning setting, since we do not know a priori target graphs that a learner will
receive in the future, ? should be small enough so ? < 1/??,max for any pair of unseen graphs.
Otherwise, we need to re-compute the full kernel matrix and re-train the learner. In the transductive
2
setting, we are given a collection G of graphs beforehand. We can explicitly compute the upper
bound of ?, which is (maxG,G? ?G ??,max )?1 with the maximum of the maximum eigenvalues over
all pairs of graphs G, G? ? G.
2.2
Halting
The geometric random walk kernel KGR is one of the most popular graph kernels, as it can take
walks of any length into account [5, 14]. However, the fact that it weights walks of length k by the
kth power of ?, together with the condition that ? < (??,max )?1 < 1, immediately tells us that the
contribution of longer walks is significantly lowered in KGR . If the contribution of walks of length
2 and more to the kernel value is even completely dominated by the contribution of walks of length
1, we would speak of halting. It is as if the random walks halt after one step.
Here, we analyze under which conditions this halting phenomenon may occur in geometric random
walk kernels. We obtain the following key theoretical statement by comparing KGR to the one-step
1
random walk kernel K?
.
Theorem 1 Let ?0 = 1 and ?1 = ? in the random walk kernel. For a pair of graphs G and G? ,
1
1
K?
(G, G? ) ? KGR (G, G? ) ? K?
(G, G? ) + ?,
where
(??? )2
,
? = |V? |
1 ? ???
and ? monotonically converges to 0 as ? ? 0.
Proof. Let d(v) be the degree of a vertex v in G? and N (v) be the set of neighboring vertices of v,
that is, N (v) = {u ? V? | (u, v) ? E? }. Since A? is the adjacency matrix of G? , the following
relationships hold:
|V? |
?
[A? ]ij =
i,j=1
|V? |
?
d(v) ? |V? |?? ,
[A3? ]ij =
i,j=1
?
[A2? ]ij =
i,j=1
v?V?
|V? |
?
?
?
?
?
?
v?V?
v ? ?N (v)
d(v ?? ) ? |V? |?3? , . . . ,
v?V? v ? ?N (v) v ?? ?N (v ? )
d(v ? ) ? |V? |?2? ,
|V? |
?
[An? ]ij ? |V? |?n? .
i,j=1
From the assumption that ??? < 1, we have
KGR (G, G? ) =
|V? |
?
1
[I + ?A? + ?2 A2? + . . . ]ij = K?
(G, G? ) +
i,j=1
|V? |
?
[?2 A2? + ?3 A3? + . . . ]ij
i,j=1
1
K?
(G, G? )
?
+ |V? |?
+ ??? + ?
It is clear that ? monotonically goes to 0 when ? ? 0.
2
?2? (1
2
?2?
1
+ . . . ) = K?
(G, G? ) + ?.
?
1
(G, G? ).
Moreover, we can normalize ? by dividing KGR (G, G? ) by K?
Corollary 1 Let ?0 = 1 and ?1 = ? in the random walk kernel. For a pair of graphs G and G? ,
KGR (G, G? )
?
1?
1 (G, G? ) ? 1 + ? ,
K?
where
(??? )2
?? =
(1 ? ??? )(1 + ?d? )
and d? is the average of vertex degrees of G? .
Proof. Since we have
1
K?
(G, G? ) = |V? | + ?
?
d(v) = |V? |(1 + ?d? ),
v?V?
1
it follows that ?/K?
(G, G? ) = ?? .
?
k
Theorem 1 can be easily generalized to any k-step random walk kernel K?
.
3
Corollary 2 Let ?(k) = |V? |(??? )k /(1 ? ??? ). For a pair of graphs G and G? , we have
k
k
K?
(G, G? ) ? KGR (G, G? ) ? K?
(G, G? ) + ?(k + 1).
Our results imply that, in the geometric random walk kernel KGR , the contribution of walks of
length longer than 2 diminishes for very small choices of ?. This can easily happen in real-world
graph data, as ? is upper-bounded by the inverse of the maximum degree of the product graph.
2.3
Relationships to Linear Kernels on Label Histograms
Next, we clarify the relationship between KGR and basic linear kernels on vertex and edge label
histograms. We show that halting KGR leads to the convergence of it to such linear kernels.
Given a pair of graphs G and G? , let us introduce two linear kernels on vertex and edge histograms.
Assume that the range of labels ? = {1, 2, . . . , s} without loss of generality. The vertex label
histogram of a graph G = (V, E, ?) is a vector f = (f1 , f2 , . . . , fs ), such that fi = |{v ? V |
?(v) = i}| for each i ? ?. Let f and f ? be the vertex label histograms of graphs G and G? ,
respectively. The vertex label histogram kernel KVH (G, G? ) is then defined as the linear kernel
between f and f ? :
?s
KVH (G, G? ) = ?f , f ? ? = i=1 fi fi? .
Similarly, the edge label histogram is a vector g = (g1 , g2 , . . . , gs ), such that gi = |{(u, v) ? E |
?(u, v) = i}| for each i ? ?. The edge label histogram kernel KEH (G, G? ) is defined as the linear
kernel between g and g ? , for respective histograms:
?s
KEH (G, G? ) = ?g, g ? ? = i=1 gi gi? .
Finally, we introduce the vertex-edge label histogram. Let h = (h111 , h211 , . . . , hsss ) be a histogram vector, such that hijk = |{(u, v) ? E | ?(u, v) = i, ?(u) = j, ?(v) = k}| for each
i, j, k ? ?. The vertex-edge label histogram kernel KVEH (G, G? ) is defined as the linear kernel
between h and h? for the respective histograms of G and G? :
?s
KVEH (G, G? ) = ?h, h? ? = i,j,k=1 hijk h?ijk .
Notice that KVEH (G, G? ) = KEH (G, G? ) if vertices are not labeled.
From the definition of the direct product of graphs, we can confirm the following relationships
between histogram kernels and the random walk kernel.
Lemma 1 For a pair of graphs G, G? and their direct product G? , we have
1 0
KVH (G, G? ) =
K (G, G? ) = |V? |.
?0 ?
|V? |
?
1 1
?0 0
?
?
KVEH (G, G ) =
K? (G, G ) ? K? (G, G ) =
[A? ]ij .
?1
?1
i,j=1
?
Proof. The first equation KVH (G, G? ) = |V? | can be proven from the following:
?
KVH (G, G? ) =
|{ v ? ? V ? | ?(v) = ?? (v ? ) }| = |{ (v, v ? ) ? V ? V ? | ?(v) = ?? (v ? ) }|
v?V
1 0
K (G, G? ).
?0 ?
We can prove the second equation in a similar fashion:
?
KVEH (G, G? ) = 2
|{ (u? , v ? ) ? E ? | ?(u, v) = ?? (u? , v ? ), ?(u) = ?? (u? ), ?(v) = ?? (v ? ) }|
= |V? | =
(u,v)?E
{
(
)
= 2 (u, v), (u? , v ? ) ? E ? E ?
|V? |
= 2|E? | =
?
i,j=1
[A? ]ij =
}
?(u, v) = ?? (u? , v ? ),
?(u) = ?? (u? ), ?(v) = ?? (v ? )
?0 0
1 1
K (G, G? ) ? K?
(G, G? ).
?1 ?
?1
4
?
Finally, let us define a new kernel
KH (G, G? ) := KVH (G, G? ) + ?KVEH (G, G? )
(1)
1
with a parameter ?. From Lemma 1, since KH (G, G? ) = K?
(G, G? ) holds if ?0 = 1 and ?1 = ?
1
in the one-step random walk kernel K? , we have the following relationship from Theorem 1.
Corollary 3 For a pair of graphs G and G? , we have
KH (G, G? ) ? KGR (G, G? ) ? KH (G, G? ) + ?,
where ? is given in Theorem 1.
To summarize, our results show that if the parameter ? of the geometric random walk kernel KGR is
small enough, random walks halt, and KGR reduces to KH , which finally converges to KVH . This
is based on vertex histograms only and completely ignores the topological structure of the graphs.
3
Experiments
We empirically examine the halting phenomenon of the geometric random walk kernel on popular
real-world graph benchmark datasets and semi-simulated graph data.
3.1
Experimental Setup
Environment. We used Amazon Linux AMI release 2015.03 and ran all experiments on a single
core of 2.5 GHz Intel Xeon CPU E5-2670 and 244 GB of memory. All kernels were implemented
in C++ with Eigen library and compiled with gcc 4.8.2.
Datasets. We collected five real-world graph classification benchmark datasets:1 ENZYMES, NCI1,
NCI109, MUTAG, and D&D, which are popular in the graph-classification literature [13, 14].
ENZYMES and D&D are proteins, and NCI1, NCI109, and MUTAG are chemical compounds.
Statistics of these datasets are summarized in Table 1, in which we also show the maximum of
maximum degrees of product graphs maxG,G? ?G ?? for each dataset G. We consistently used
?max = (maxG,G? ?G ?? )?1 as the upper bound of ? in geometric random walk kernels, in which
the gap was less than one order as the lower bound of ?. The average degree of the product graph,
the lower bound of ?, were 18.17, 7.93, 5.60, 6.21, and 13.31 for ENZYMES, NCI1, NCI109,
MUTAG, and DD, respectively.
Kernels. We employed the following graph kernels in our experiments: We used linear kernels on
vertex label histograms KVH , edge label histograms KEH , vertex-edge label histograms KVEH , and
the combination KH introduced in Equation (1). We also included a Gaussian RBF kernel between
vertex-edge label histograms, denoted as KVEH,G . From the family of random walk kernels, we
k
used the geometric random walk kernel KGR and the k-step random walk kernel K?
. Only the
k
and ?k was fixed to 1 for all k. We used
number k of steps were treated as a parameter in K?
fix-point iterations [14, Section 4.3] for efficient computation of KGR . Moreover, we employed the
Weisfeiler-Lehman subtree kernel [13], denoted as KWL , as the state-of-the-art graph kernel, which
has a parameter h of the number of iterations.
3.2
Results on Real-World Datasets
We first compared the geometric random walk kernel KGR to other kernels in graph classification.
The classification accuracy of each graph kernel was examined by 10-fold cross validation with
multiclass C-support vector classification (libsvm2 was used), in which the parameter C for CSVC and a parameter (if one exists) of each kernel were chosen by internal 10-fold cross validation
(CV) on only the training dataset. We repeated the whole experiment 10 times and reported average
1
The code and all datasets are available at:
http://www.bsse.ethz.ch/mlcb/research/machine-learning/graph-kernels.html
2
http://www.csie.ntu.edu.tw/?cjlin/libsvm/
5
Table 1: Statistics of graph datasets, |?V | and |?E | denote the number of vertex and edge labels.
Dataset
Size
#classes
avg.|V |
avg.|E|
max|V |
max|E|
|?V |
|?E |
max??
ENZYMES
NCI1
NCI109
MUTAG
D&D
600
4110
4127
188
1178
6
2
2
2
2
32.63
29.87
29.68
17.93
284.32
62.14
32.3
32.13
19.79
715.66
126
111
111
28
5748
149
119
119
33
14267
3
37
38
7
82
1
3
3
11
1
65
16
17
10
50
(i) Comparison of various graph kernels (ii)
30
20
KVH KEH KVEH KH KVEH,G KGR Kxk
40
30
20
KWL
k-step Kxk
50
Accuracy
40
(iii)
KGR
KH
50
Accuracy
Accuracy
50
Comparison of KGR with KH
40
30
20
10?5
Label histogram Random walk
10?4
10?3
Parameter ?
10?2
1
3
5
7
9
Number of steps k
(a) ENZYMES
Comparison of KGR with KH
85
80
80
75
70
(iii)
KGR
KH
75
70
65
65
KVH KEH KVEH KH KVEH,G KGR Kxk
KWL
k-step Kxk
85
Accuracy
85
Accuracy
Accuracy
(i) Comparison of various graph kernels (ii)
80
75
70
65
10?5
Label histogram Random walk
10?4 10?3 10?2 0.0625
Parameter ?
1
3
5
7
9
Number of steps k
(b) NCI1
Comparison of KGR with KH
85
80
80
75
70
KGR
KH
75
70
65
65
KVH KEH KVEH KH KVEH,G KGR Kxk
KWL
Label histogram Random walk
(iii)
k-step Kxk
85
Accuracy
85
Accuracy
Accuracy
(i) Comparison of various graph kernels (ii)
80
75
70
65
10?5
10?4 10?3 10?2 0.0588
Parameter ?
1
3
5
7
9
Number of steps k
(c) NCI109
Figure 1: Classification accuracy on real-world datasets (Means ? SD).
classification accuracies with their standard errors. The list of parameters optimized by the internal
CV is as follows: C ? {2?7 , 2?5 , . . . , 25 , 27 } for C-SVC, the width ? ? {10?2 , . . . , 102 } in
k
the RBF kernel KVEH,G , the number of steps k ? {1, . . . , 10} in K?
, the number of iterations
?5
?2
h ? {1, . . . , 10} in KWL , and ? ? {10 , . . . , 10 , ?max } in KH and KGR , where ?max =
(maxG,G? ?G ?? )?1 .
Results are summarized in the left column of Figure 1 for ENZYMES, NCI1, and NCI109. We
present results on MUTAG and D&D in the Supplementary Notes, as different graph kernels do
not give significantly different results (e.g., [13]). Overall, we could observe two trends. First,
the Weisfeiler-Lehman subtree kernel KWL was the most accurate, which confirms results in [13],
6
(b)
ENZYMES
(c)
NCI1
NCI109
50
50
40
40
40
30
20
10
0
?4 ?3 ?2 ?1 0
log10 ??
1
Percentage
50
Percentage
Percentage
(a)
30
20
10
0
2
?4 ?3 ?2 ?1 0
log10 ??
1
30
20
10
0
2
?4 ?3 ?2 ?1 0
log10 ??
1
2
Figure 2: Distribution of log10 ?? , where ?? is defined in Corollary 1, in real-world datasets.
Accuracy
45
40
(b)
Sim-ENZYMES
KGR
KH
KVH
35
30
25
(c)
Sim-NCI1
KGR
KH
KVH
80
75
70
KGR
KH
KVH
75
70
65
65
0 10 20
50
100
Number of added edges
Sim-NCI109
80
Accuracy
50
Accuracy
(a)
0 10 20
50
100
Number of added edges
0 10 20
50
100
Number of added edges
Figure 3: Classification accuracy on semi-simulated datasets (Means ? SD).
k
Second, the two random walk kernels KGR and K?
show greater accuracy than na??ve linear kernels
on edge and vertex histograms, which indicates that halting is not occurring in these datasets. It is
also noteworthy that employing a Gaussian RBF kernel on vertex-edge histograms leads to a clear
improvement over linear kernels on all three datasets. On ENZYMES, the Gaussian kernel is even
on par with the random walks in terms of accuracy.
To investigate the effect of halting in more detail, we show the accuracy of KGR and KH in the
center column of Figure 1 for various choices of ?, from 10?5 to its upper bound. We can clearly
see that halting occurs for small ?, which greatly affects the performance of KGR . More specifically,
if it is chosen to be very small (smaller than 10?3 in our datasets), the accuracies are close to the
na??ve baseline KH that ignores the topological structure of graphs. However, accuracies are much
closer to that reached by the Weisfeiler-Lehman kernel if ? is close to its theoretical maximum. Of
course, the theoretical maximum of ? depends on unseen test data in reality. Therefore, we often
have to set ? conservatively so that we can apply the trained model to any unseen graph data.
Moreover, we also investigated the accuracy of the random walk kernel as a function of the number
k
of steps k of the random walk kernel K?
. Results are shown in the right column of Figure 1. In
all datasets, accuracy improves with each step, up to four to five steps. The optimal number of
k
steps in K?
and the maximum ? give similar accuracy levels. We also confirmed Theorem 1 that
conservative choices of ? (10?3 or less) give the same accuracy as a one-step random walk.
In addition, Figure 2 shows histograms of log10 ?? , where ?? is given in Corollary 1 for ? =
(max ?? )?1 for all pairs of graphs in the respective datasets. The value ?? can be viewed as the
deviation of KGR from KH in percentages. Although ?? is small on average (about 0.1 percent in
ENZYMES and NCI datasets), we confirmed the existence of relatively large ?? in the plot (more
than 1 percent), which might cause the difference between KGR and KH .
3.3
Results on Semi-Simulated Datasets
To empirically study halting, we generated semi-simulated graphs from our three benchmark
datasets (ENZYMES, NCI1, and NCI109) and compared the three kernels KGR , KH , and KVH .
In each dataset, we artificially generated denser graphs by randomly adding edges, in which
the number of new edges per graph was determined from a normal distribution with the mean
7
m ? {10, 20, 50, 100} and the distribution of edge labels was unchanged. Note that the accuracy of
the vertex histogram kernel KVH stays always the same, as we only added edges.
Results are plotted in Figure 3. There are two key observations. First, by adding new false edges
to the graphs, the accuracy levels drop for both the random walk kernel and the histogram kernel.
However, even after adding 100 new false edges per graph, they are both still better than a na??ve
classifier that assigns all graphs to the same class (accuracy of 16.6 percent on ENZYMES and
approximately 50 percent on NCI1 and NCI109). Second, the geometric random walk kernel quickly
approaches the accuracy level of KH when new edges are added. This is a strong indicator that
halting occurs. As graphs become denser, the upper bound for ? gets smaller, and the accuracy of
the geometric random walk kernel KGR rapidly drops and converges to KH . This result confirms
Corollary 3, which says that both KGR and KH converge to KVH as ? goes to 0.
4
Discussion
In this work, we show when and where the phenomenon of halting occurs in random walk kernels.
Halting refers to the fact that similarity measures based on counting walks (of potentially infinite
length) often downweight longer walks so much that the similarity score is completely dominated
by walks of length 1, degenerating the random walk kernel to a simple kernel between edges and
vertices. While it had been conjectured that this problem may arise in graph kernels [1], we provide
the first theoretical proof and empirical demonstration of the occurrence and extent of halting in
geometric random walk kernels.
We show that the difference between geometric random walk kernels and simple edge kernels depends on the maximum degree of the graphs being compared. With increasing maximum degree,
the difference converges to zero. We empirically demonstrate on simulated graphs that the comparison of graphs with high maximum degrees suffers from halting. On real graph data from popular
graph classification benchmark datasets, the maximum degree is so low that halting can be avoided
if the decaying weight ? is set close to its theoretical maximum. Still, if ? is set conservatively to a
low value to ensure convergence, halting can clearly be observed, even on unseen test graphs with
unknown maximum degrees.
There is an interesting connection between halting and tottering [1, Section 2.1.5], a weakness of
random walk kernels described more than a decade ago [11]. Tottering is the phenomenon that a
walk of infinite length may go back and forth along the same edge, thereby creating an artificially
inflated similarity score if two graphs share a common edge. Halting and tottering seem to be opposing effects. If halting occurs, the effect of tottering is reduced and vice versa. Halting downweights
these tottering walks and counteracts the inflation of the similarity scores. An interesting point is that
the strategies proposed to remove tottering from walk kernels did not lead to a clear improvement
in classification accuracy [11], while we observed a strong negative effect of halting on the classification accuracy in our experiments (Section 3). This finding stresses the importance of studying
halting.
Our theoretical and empirical results have important implications for future applications of random
walk kernels. First, if the geometric random walk kernel is used on a graph dataset with known
maximum degree, ? should be close to the theoretical maximum. Second, simple baseline kernels
based on vertex and edge label histograms should be employed to check empirically if the random
walk kernel gives better accuracy results than these baselines. Third, particularly in datasets with
high maximum degree, we advise using a fixed-length-k random walk kernel rather than a geometric random walk kernel. Optimizing the length k by cross validation on the training dataset led to
competitive or superior results compared to the geometric random walk kernel in all of our experiments. Based on these results and the fact that by definition the fixed-length kernel does not suffer
from halting, we recommend using the fixed-length random walk kernel as a comparison method in
future studies on novel graph kernels.
Acknowledgments. This work was supported by JSPS KAKENHI Grant Number 26880013 (MS),
the Alfried Krupp von Bohlen und Halbach-Stiftung (KB), the SNSF Starting Grant ?Significant
Pattern Mining? (KB), and the Marie Curie Initial Training Network MLPM2012, Grant No. 316861
(KB).
8
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Journal of Machine Learning Research, 11:1201?1242, 2010.
9
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5,179 | 5,689 | Rate-Agnostic (Causal) Structure Learning
David Danks
Carnegie-Mellon University
Pittsburgh, PA
[email protected]
Sergey Plis
The Mind Research Network,
Albuquerque, NM
[email protected]
Vince Calhoun
The Mind Research Network
ECE Dept., University of New Mexico
Albuquerque, NM
[email protected]
Cynthia Freeman
The Mind Research Network,
CS Dept., University of New Mexico
Albuquerque, NM
[email protected]
Abstract
Causal structure learning from time series data is a major scientific challenge. Extant algorithms assume that measurements occur sufficiently quickly; more precisely, they assume approximately equal system and measurement timescales. In
many domains, however, measurements occur at a significantly slower rate than
the underlying system changes, but the size of the timescale mismatch is often
unknown. This paper develops three causal structure learning algorithms, each
of which discovers all dynamic causal graphs that explain the observed measurement data, perhaps given undersampling. That is, these algorithms all learn causal
structure in a ?rate-agnostic? manner: they do not assume any particular relation
between the measurement and system timescales. We apply these algorithms to
data from simulations to gain insight into the challenge of undersampling.
1
Introduction
Dynamic causal systems are a major focus of scientific investigation in diverse domains, including neuroscience, economics, meteorology, and education. One significant limitation in all of these
sciences is the difficulty of measuring the relevant variables at an appropriate timescale for the particular scientific domain. This challenge is particularly salient in neuroimaging: standard fMRI
experiments sample the brain?s bloodflow approximately every one or two seconds, though the underlying neural activity (i.e., the major driver of bloodflow) occurs much more rapidly. Moreover,
the precise timescale of the underlying causal system is unknown; it is almost certainly faster than
the fMRI measurements, but it is unknown how much faster.
In this paper, we aim to learn the causal structure of a system that evolves at timescale ?S , given
measurements at timescale ?M . We focus on the case in which ?S is faster than ?M to an unknown
degree. We assume that the underlying causal structure can be modeled as a directed graphical
model G without simultaneous influence. There has been substantial work on modeling the statistics of time series, but relatively less on learning causal structure, and almost all of that assumes that
the measurement and causal timescales match [1?5]. The problem of causal learning from ?undersampled? time series data was explicitly addressed by [6, 7], but they assumed that the degree of
undersampling?i.e., the ratio of ?S to ?M ?was both known and small. In contrast, we focus on
the significantly harder challenge of causal learning when that ratio is unknown.
We provide a formal specification of the problem and representational framework in Section 2. We
then present three different Rate-Agnostic Structure Learning (RASL) algorithms in Section 3. We
finish in Section 4 by exploring their performance on synthetic data.
1
2
Representation and Formalism
A dynamic causal graphical model consists of a graph G over random variables V at the current time
t, as well as nodes for V at all previous (relative) timesteps that contain a direct cause of a variable
at the current timestep.1 The Markov order of the system is the largest k such that Vit?k ? Vjt ,
where superscripts denote timestep. We assume throughout that the ?true? underlying causal system
is Markov order 1, and that all causally relevant variables are measured.2 Finally, we assume that
there are no isochronal causal edges Vit ? Vjt ; causal influences inevitably take time to propagate,
and so any apparent isochronal edge will disappear when measured sufficiently finely. Since we do
not assume that the causal timescale ?S is known, this is a relatively innocuous assumption.
G is thus over 2V nodes, where the only edges are Vit?1 ? Vjt , where possibly i = j. There
is additionally a conditional probability distribution or density, P (Vt |Vt?1 ), which we assume
to be time-independent. We do not, however, assume stationarity of P (Vt ). Finally, we assume
appropriate versions of the Markov (?Variable V is independent of non-descendants given parents?)
and Faithfulness/Stability (?The only independencies are those implied by Markov?) assumptions,
such that the graph and probability distribution/density mutually constrain each other.
Let {t0 , t1 , . . . , tk , . . .} be the measurement timesteps. We undersample at rate u when we measure
only timesteps {t0 , tu , . . . , tuk , . . .}; the causal timescale is thus ?undersampled at rate 1.? We
denote the causal graph resulting from undersampling at rate u by Gu . To obtain Gu , we ?unroll?
G1 by introducing nodes for Vt?2 that bear the same graphical and parametric relations to Vt?1
as those variables bear to Vt , and iterate until we have included Vt?u . We then marginalize out all
variables except those in Vt and Vt?u .
Marginalization yields an Acyclic Directed Mixed Graph (ADMG) Gu containing both directed and
bidirected edges [8]. Vit?u ? Vjt in Gu iff there is a directed path from Vit?u to Vjt in the unrolled
graph. Define a trek to be a pair of directed paths (?1 , ?2 ) such that both have the same start variable.
Vit ? Vjt in Gu iff there is a trek between Vit and Vjt with length(?1 ) = length(?2 ) = k < u.
Clearly, if a bidirected edge occurs in Gm , then it occurs in Gu for all u ? m.
Unrolling-and-marginalizing can be computationally complex due to duplication of nodes, and so
we instead use compressed graphs that encode temporal relations in edges. For an arbitrary dynamic
causal graph H, H is its compressed graph representation: (i) H is over non-time-indexed nodes for
V; (ii) Vi ? Vj in H iff Vit?1 ? Vjt in H; and (iii) Vi ? Vj in H iff Vit ? Vjt in H. Compressed
graphs can be cyclic (Vi Vj for Vit?1 ? Vjt and Vjt?1 ? Vit ), including self-cycles. There is
clearly a 1-1 mapping between dynamic ADMGs and compressed graphs.
Computationally, the effects of undersampling at rate u can be computed in a compressed graph
simply by finding directed paths of length u in G 1 . More precisely, Vit?u ? Vjt in G u iff there is
a directed path of length u in G 1 . Similarly, Vit ? Vjt in G u iff there is a trek with length(?1 ) =
length(?2 ) = k < u in G 1 . We thus use compressed graphs going forward.
3
Algorithms
The core question of this paper is: given H = G u for unknown u, what can be inferred about G 1 ?
Let JHK = {G 1 : ?u G u = H} be the equivalence class of G 1 that could, for some undersample
rate, yield H. We are thus trying to learn JHK from H. An obvious brute-force algorithm is: for
each possible G 1 , compute the corresponding graphs for all u, and then output all G u = H. Equally
2
obviously, this algorithm will be computationally intractable for any reasonable n, as there are 2n
1
possible G and u can (in theory) be arbitrarily large. Instead, we pursue three different constructive
strategies that more efficiently ?build? the members of JHK (Sections 3.2, 3.3, and 3.4). Because
these algorithms make no assumptions about u, we refer to them each as RASL?Rate Agnostic
Structure Learner?and use subscripts to distinguish between different types. First, though, we provide some key theoretical results about forward inference that will be used by all three algorithms.
1
We use difference equations in our analyses. The results and algorithms will be applicable to systems of
differential equations to the extent that they can be approximated by a system of difference equations.
2
More precisely, we assume a dynamic variant of the Causal Sufficiency assumption, though it is more
complicated than just ?no unmeasured common causes.?
2
3.1
Nonparametric Forward Inference
For given G 1 and u, there is an efficient algorithm [9] for calculating G u , but it is only useful in
learning if we have stopping rules that constrain which G 1 and u should ever be considered. These
rules will depend on how G 1 changes as u ? ?. A key notion is a strongly connected component
(SCC) in G 1 : a maximal set of variables S ? V such that, for every X, Y ? S (possibly X = Y ),
there is a directed path from X to Y . Non-singleton SCCs are clearly cyclic and can provably
be decomposed into a set of (possibly overlapping) simple loops (i.e., those in which no node is
repeated): ?1 , . . . , ?s [10]. Let LS be the set of those simple loop lengths.
One stopping rule must specify, for given G 1 , which u to consider. For a single SCC, the greatest
common divisor of simple loop lengths (where gcd(LS ) = 1 for singleton S) is key: gcd(LS ) = 1
iff ?f s.t. ?u > f [G u = G f ]; that is, gcd() determines whether an SCC ?converges? to a fixedpoint graph as u ? ?. We can constrain u if there is such a fixed-point graph, and Theorem 3.1
generalizes [9, Theorem 5] to provide an upper bound on (interesting) u. (All proofs found in
supplement.)
Theorem 3.1. If gcd(LS ) = 1, then stabilization occurs at f ? nF + ? + d + 1.
where nF is the Frobenius number,3 d is the graph diameter, and ? is the transit number (see supplement). This is a theoretically useful bound, but is not practically helpful since neither ? nor nF
have a known analytic expression. Moreover, gcd(LS ) = 1 is a weak restriction, but a restriction
nonetheless. We instead use a functional stopping rule for u (Theorem 3.2) that holds for all G:
Theorem 3.2. If G u = G v for u > v, then ?w > u?kw < u[G w = G kw ].
That is, as u increases, if we find a graph that we previously encountered, then there cannot be
any new graphs as u ? ?. For a given G 1 , we can thus determine all possible corresponding
undersampled graphs by computing G 2 , G 3 , . . . until we encounter a previously-observed graph.
This stopping rule enables us to (correctly) constrain the u that are considered for each G 1 .
2
We also require a stopping rule for G 1 , as we cannot evaluate all 2n possible graphs for any reasonable n. The key theoretical result is:
Theorem 3.3. If G 1 ? J 1 , then ?u[G u ? J u ].
1
Let GE
be the graph resulting from adding the edges in E to G 1 . Since this is simply another graph,
1 u
1
it can be undersampled at rate u; denote the result (GE
) . Since GE
can always serve as J 1 in
Theorem 3.3, we immediately have the following two corollaries:
1 u
Corollary 3.4. If G u * H, then ?E[(GE
) * H]
1 u
Corollary 3.5. If ?u[G u * H], then ?E, u[(GE
) * H]
We thus have a stopping rule for some candidate G 1 : if G u is not an edge-subset of H for all u, then
do not consider any edge-superset of G 1 . This stopping rule fits very cleanly with ?constructive?
algorithms that iteratively add edge(s) to candidate G 1 . We now develop three such algorithms.
3.2
A recursive edgewise inverse algorithm
The two stopping rules naturally suggest a recursive structure learning algorithm with H as input
and JHK as output. Start with an empty graph. For each edge e (of n2 possible edges), construct G 1
containing only e. If G u * H for all u, then reject; else if G u = H for some u,4 then add G 1 to
JHK; else, recurse into non-conflicting graphs in order. Effectively, this is a depth first search (DFS)
algorithm on the solution tree; denote it as RASLre for ?recursive edgewise.? Figure 1a provides
pseudo-code, and Figure 1b shows how one DFS path in the search tree unfolds. We can prove:
Theorem 3.6. The RASLre algorithm is correct and complete.
One significant drawback of RASLre is that the same graph can be constructed in many different
ways, corresponding to different orders of edge addition; the search tree is actually a search latP
For set B of positive integers with gcd(B) = 1, nF is the max integer with nF 6= bi=1 ?i Bi for ?i ? 0.
4
This check requires at most min(eu , eH ) + 1 (fast) operations, where eu , eH are the number of edges in
G u , H, respectively. This equality check occurs relatively rarely, since G u and H must be non-conflicting.
3
3
1
Algorithm RecursiveEqClass
Input: H
Output: JHK
initialize empty graph G and set S
2
3
4
5
6
7
8
9
10
11
12
13
candidate edges
1 2 3 3 1 1 2 3 2
1 2 3 1 2 3 1 2 3
1
begin EdgeAdder G ? , H, L
if L has elements then
forall the edges in L do
if edge creates a conflict then
remove it from L
if L has elements then
forall the edges in L do
add the edge to G ?
if ?G ? {(G ? )u } s.t. G = H then
add G ? to S
EdgeAdder G ? , H, L \ the edge
remove the edge from G ?
3
constructed graph
pruned conflict-inducing
candidate edges
next edge to add
2
1 2 3 3 1 1 2 3 2
1 2 3 1 2 3 1 2 3
1
3
2
3 1 1 2 3 2
1 2 3 1 2 3
1
3
no graph constructed
along this branch
generates H
2
1 1 2 3 2
2 3 1 2 3
1
2
3
3 2
15
16
put all n2 edges into list L
EdgeAdder(G, H, L)
return S
1
3
a: RASLre algorithm
2
ground
truth
2 3
14
1
1
3
2
3
2
H
no more non-conflicting
candidates: backtrack
b: Branch of the search tree
Figure 1: RASLre algorithm 1a specification, and 1b search tree example
tice. The algorithm is thus unnecessarily inefficient, even when we use dynamic programming via
memoization of input graphs.
3.3
An iterative edgecentric inverse algorithm
To minimize multiple constructions of the same graph, we can use RASLie (?iterative edgewise?)
which generates, at stage i, all not-yet-eliminated G 1 with exactly i edges. More precisely, at stage
0, RASLie starts with the empty graph; if H is also empty, then it adds the empty graph to JHK.
Otherwise, it moves to stage 1. In general, at stage i + 1, RASLie (a) considers each graph G 1
resulting from a single edge addition to an acceptable graph at stage i; (b) rejects G 1 if it conflicts
(for all u) with H; (c) otherwise keeps G 1 as acceptable at i + 1; and (d) if ?u[G u = H], then adds
G 1 to JHK. RASLie continues until there are no more edges to add (or it reaches stage n2 + 1).
Figure 2 provides the main loop (Figure 2a) and core function of RASLie (Figure 2c), as well as an
example of the number of graphs potentially considered at each stage (Figure 2b). RASLie provides
significant speed-up and memory gains over RASLre (see Figure 3).
We optimize RASLie by tracking the single edges that could possibly still be added; for example,
if a single-edge graph is rejected in stage 1, then do not consider adding that edge at other stages.
Additional conflicts can be derived analytically, further reducing the graphs to consider. In general,
absence of an edge in H implies, for the corresponding (unknown) u, absence of length u paths in all
G 1 ? JHK. Since we do not know u, we cannot directly apply this constraint. However, lemmas 3.7
and 3.8 provide useful, special case constraints for u > 1 (implied by a single bidirected edge).
Lemma
3.7. If u > 1, then ?V 6?
W ? H, G 1 cannot contain
any of the following paths:
1. V ? W ; 2. V ? X ? W ; 3. V ? X ? W ; 4. V ? X ? W ; 5. V ? W .
Lemma 3.8. If u > 1, then ?V 6? W ? H @T [V ? T ? W ] ? G 1
3.4
An iterative loopcentric inverse algorithm
RASLie yields results in reasonable time for H with up to 8 nodes, though it is computationally
demanding. We can gain further computational advantages if we assume that H is an SCC. This
assumption is relatively innocuous, as it requires only that our time series be generated by a system
with (appropriate) feedback loops. As noted earlier, any SCC is composed of a set of simple loops,
and so we modify RASLie to iteratively add loops instead of edges; call the resulting algorithm
4
1
2
3
4
5
6
Algorithm IterativeEqClass
Input: H
Output: JHK
1
2
initialize empty sets S
init d as an empty graph and n2 edges
while d do
d, Si
N extIterationGraphs(d, H)
S = S ? Si
return S
3
4
5
=
6
7
8
9
10
a: RASLie main algorithm
11
RASL
input:
1
3
2
run 1
3
3
2
run 2
1 1
7 2
17 3
15 4
4 5
1 1
7 2
12 3
2 4
iteration
index
1
1
number of non-conflicting
graphs at the iteration
12
2
run 3
1 1
9 2
34 3
68 4
non-conflicting
74 5 graphs
46 6 histogram
15 7
2 8
13
14
15
16
Procedure NextIterationGraphs
Input: d graph:edges structure, and H
Output: dr and set S ? JHK
initialize empty structure dr and sets
S, Si
forall the graphs G in d do
forall the edges e in d(G) do
if e 6? G then
if e conflicts with G then
continue
add e to G
if G 6? Si then
add G to Si
if G conflicts with H
then
continue
if ?G? ? {G u } s.t. G? = H
then
add G to S
remove e from G
add non-conflicting graphs w/ edges
to dr
return dr , S
c: Core function of RASLie
b: Three runs of the algorithm
Figure 2: RASLie algorithm (a) main loop; (b) example of graphs considered; and (c) core function.
RASLil for ?iterative loopwise.? More precisely, RASLil uses the same algorithm as in Figure 2,
but successively attempts to add non-conflicting simple loops, rather than non-conflicting edges.
RASLil also incorporates the additional constraints due to lemmas 3.7 and 3.8.
RASLil is surprisingly
faster than RASLie even though, for
Pn much
n nodes, there are i=0 ni (i?1)! simple loops (compared to n2
edges). The key is that introducing a single simple loop induces
multiple constraints simultaneously, and so conflicting graphs are
discovered at a much earlier stage. As a result, RASLil checks
many fewer graphs in practice. For example, consider the G 1
in Figure 1, with corresponding H for u = 3. RASLre constructs (not counting pruned single edges) 28,661 graphs; RASLie
constructs only 249 graphs; and RASLil considers only 47. For
u = 2, these numbers are 413, 44, and 7 respectively. Unsurprisingly, these differences in numbers of examined graphs translate
directly into wall clock time differences (Figure 3).
Figure 3: Run-time comparison.
4
Results
All three RASL algorithms take a measurement timescale graph H as input. They are therefore compatible with any structure learning algorithm that outputs a measurement timescale graph, whether
Structural Vector Autoregression (SVAR) [11], direct Dynamic Bayes Net search [12], or modifications of standard causal structure learning algorithms such as PC [1, 13] and GES [14]. The problem
of learning a measurement timescale graph is a very hard one, but is also not our primary focus here.
Instead, we focus on the performance of the novel RASL algorithms.
First, we abstract away from learning measurement timescale structure and assume that the correct H
is provided as input. For these simulated graphs, we focus on SCCs, which are the most scientifically
interesting cases. For simplicity (and because within-SCC structure can be learned in parallel for a
complex H [9]), we employ single-SCC graphs. To generate random SCCs, we (i) build a single
simple loop over n nodes, and (ii) uniformly sample from the other n(n ? 1) possible edges until
5
we reach the specified density (i.e., proportion of the n2 total possible edges). We employ density
in order to measure graph complexity in an (approximately) n-independent way.
We can improve the runtime speed of RASLre using memoization, though it is then memory-constrained for n ? 6.
Figure 3 provides the wall-clock running times for all
three RASL algorithms applied to 100 random 5-node
graphs at each of three densities. This graph substantiates our earlier claims that RASLil is faster than RASLie ,
which is faster than RASLre . In fact, each is at least an
order of magnitude faster than the previous one.
RASLre would take over a year on the most difficult problems, so we focus exclusively on RASLil . Unsurprisingly, run-time complexity of all RASL algorithms depends on the density of H. For each of three density values (20%, 25%, 30%), we generated 100 random 6-node
SCCs, which were then undersampled at rates 2, 3, and 4
before being provided as input to RASLil . Figure 4 summarizes wall clock computation time as a function of H?s
density, with different plots based on density of G 1 and
undersampling rate. We also show three examples of H
with a range of computation runtime. Unsurprisingly, the
most difficult H is quite dense; H with densities below
50% typically require less than one minute.
Figure 4: Run-time behavior.
4.1
Equivalence classes
We first use RASLil to determine JHK size and composition for varying H; that is, we explore
the degree of underdetermination produced by undersampling. The worst-case underdetermination
occurs if H is a super-clique with every possible edge: ?X, Y [X ? Y & X ? Y ]. Any SCC
with gcd(LS ) = 1 becomes a super-clique as u ? ? [9], so JHK contains all such graphs for
super-clique H. We thus note when H is a super-clique, rather than computing the size of JHK.
6-node graphs
u=4
u=3
u=2
5-node graphs
25%
1
2
30%
4
>1000
35%
40%
45%
50%
density
superclique
20%
25%
30%
density
Figure 5: Size of equivalence classes for 100 random SCCs at each density and u ? {2, 3, 4}.
6
Figure 6: Size of equivalence classes for larger graphs n ? 7, 8, 10 for u ? 2, 3
Figures 5 and 6 plot equivalence class size as a function of both G 1 density and the true undersampling rate. For each n and density, we (i) generated 100 random G 1 ; (ii) undersampled each at
indicated u; (iii) passed G u = H to RASLil ; and (iv) computed the size of JHK. Interestingly, JHK
is typically quite small, sometimes even a singleton. For example, 5-node graphs at u = 2 typically
have singleton JHK up to 40% G 1 density. Even 10-node graphs often have a singleton JHK (though
with relatively sparse G 1 ). Increased undersampling and density both clearly worsen underdetermination, but often not intractably so, particularly since even nonsingleton JHK can be valuable if they
permit post hoc inspection or analysis.
...
equivalence class size
2
5
4
3
5-node 25% edge density graphs
S.C.
7
6
undersampling rate
8
9
10
11
Figure 7: Effect of the undersampling rate on equivalence class size.
To focus on the impact of undersampling, we generated 100 random 5-node SCCs with 25% density,
each of which was undersampled for u ? {2, . . . , 11}. Figure 7 plots the size of JHK as a function
of u for these graphs. For u ? 4, singleton JHK still dominate. Interestingly, even u = 11 still yields
some non-superclique H.
5-node graphs
6-node graphs
u=2
rate
u=3
u=4
...
25%
30%
35%
40%
45%
50%
20%
25%
30%
Figure 8: Distribution of u for G u = H for G 1 ? JHK for 5- and 6-node graphs
Finally, G 1 ? JHK iff ?u[G u = H], but the appropriate u need not be the same for all members
of JHK. Figure 8 plots the percentages of u-values appropriate for each G 1 ? JHK, for the H from
Figure 5. If actually utrue = 2, then almost all G 1 ? JHK are because of G 2 ; there are rarely
G 1 ? JHK due to u > 2. If actually utrue > 2, though, then many G 1 ? JHK are due to G u where
u 6= utrue . As density and utrue increase, there is increased underdetermination in both G 1 and u.
4.2
Synthetic data
7
In practice, we typically must learn H structure from finite sample data. As noted earlier, there are many algorithms for learning
H, as it is a measurement timescale structure (though small modifications are required
to learn bidirected edges). In pilot testing,
we found that structural vector autoregressive
(SVAR) model [11] optimization provided the
most accurate and stable solutions for H for our
simulation regime. We thus employ the SVAR
procedure here, though we note that other measurement timescale learning algorithms might Figure 9: The estimation and search errors on synwork better in different domains.
thetic data: 6-node graphs, u = 2, 20 per density.
To test the two-step procedure?SVAR learning
passed to RASLil ?we generated 20 random 6-node SCCs for each density in {25%, 30%, 35%}.
For each random graph, we generated a random transition matrix A by sampling weights for the
non-zero elements of the adjacency matrix, and controlling system stability (by keeping the maximal
eigenvalue at or below 1). We then generated time series data using a vector auto-regressive (VAR)
model [11] with A and random noise (? = 1). To simulate undersampling, datapoints were removed
to yield u = 2. SVAR optimization on the resulting time series yielded a candidate H that was passed
to RASLil to obtain JHK.
The space of possible H is a factor of n2 larger than the space of possible G 1 , and so SVAR
optimization can return an H such that JHK = ?. If RASLil returns ?, then we rerun it on all H? that
result from a single edge addition or deletion on H. If RASLil returns ? for all of those graphs, then
we consider the H? that result from two changes to H, then three changes. This search through the
3-step Hamming neighborhood of H essentially always finds an H? with JH? K 6= ?.
Figure 9 shows the results of the two-step process, where algorithm output is evaluated by two
error-types: omission error: the number of omitted edges normalized to the total number of edges
in the ground truth; comission error: number of edges not present in the ground truth normalized
to the total possible edges minus the number of those present in the ground truth. We also plot
the estimation errors of SVAR (on the undersampled data) to capture the dependence of RASLil
estimation errors on estimation errors for H. Interestingly, RASLil does not significantly increase
the error rates over those produced by the SVAR estimation. In fact, we find the contrary (similarly
to [6]): the requirement to use an H that could be generated by some undersampled G 1 functions as
a regularization constraint that corrects for some SVAR estimation errors.
5
Conclusion
Time series data are widespread in many scientific domains, but if the measurement and system
timescales differ, then we can make significant causal inference errors [9, 15]. Despite this potential
for numerous errors, there have been only limited attempts to address this problem [6, 7], and even
those methods required strong assumptions about the undersample rate.
We here provided the first causal inference algorithms that can reliably learn causal structure from
time series data when the system and measurement timescales diverge to an unknown degree. The
RASL algorithms are complex, but not restricted to toy problems. We also showed that underdetermination of G 1 is sometimes minimal, given the right methods. JHK was often small; substantial
system timescale causal structure could be learned from undersampled measurement timescale data.
Significant open problems remain, such as more efficient methods when H has JHK = ?. This paper
has, however, expanded our causal inference ?toolbox? to include cases of unknown undersampling.
Acknowledgments
SP & DD contributed equally. This work was supported by awards NIH R01EB005846 (SP); NSF
IIS-1318759 (SP); NSF IIS-1318815 (DD); & NIH U54HG008540 (DD) (from the National Human Genome Research Institute through funds provided by the trans-NIH Big Data to Knowledge
(BD2K) initiative). The content is solely the responsibility of the authors and does not necessarily
represent the official views of the National Institutes of Health.
8
References
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9
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5,180 | 569 | The Efficient Learning of Multiple Task
Sequences
Satinder P. Singh
Department of Computer Science
University of Massachusetts
Amherst, MA 01003
Abstract
I present a modular network architecture and a learning algorithm based
on incremental dynamic programming that allows a single learning agent
to learn to solve multiple Markovian decision tasks (MDTs) with significant transfer of learning across the tasks. I consider a class of MDTs,
called composite tasks, formed by temporally concatenating a number of
simpler, elemental MDTs. The architecture is trained on a set of composite and elemental MDTs. The temporal structure of a composite task is
assumed to be unknown and the architecture learns to produce a temporal decomposition. It is shown that under certain conditions the solution
of a composite MDT can be constructed by computationally inexpensive
modifications of the solutions of its constituent elemental MDTs.
1
INTRODUCTION
Most applications of domain independent learning algorithms have focussed on
learning single tasks. Building more sophisticated learning agents that operate in
complex environments will require handling multiple tasks/goals (Singh, 1992). Research effort on the scaling problem has concentrated on discovering faster learning
algorithms, and while that will certainly help, techniques that allow transfer of
learning across tasks will be indispensable for building autonomous learning agents
that have to learn to solve multiple tasks . In this paper I consider a learning agent
that interacts with an external, finite-state, discrete-time, stochastic dynamical environment and faces multiple sequences of Markovian decision tasks (MDTs).
251
252
Singh
Each MDT requires the agent to execute a sequence of actions to control the environment, either to bring it to a desired state or to traverse a desired state trajectory
over time. Let S be the finite set of states and A be the finite set of actions available
to the agent. l At each time step t, the agent observes the system's current state
Zt E S and executes action at E A. As a result, the agent receives a payoff with
expected value R(zt, at) E R and the system makes a transition to state Zt+l E S
with probability P:r:t:r:t+l (at). The agent's goal is to learn an optimal closed loop
control policy, i.e., a function assigning actions to states, that maximizes the agent's
objective. The objective used in this paper is J = E~o -yt R(zt, at), i.e., the sum
of the payoffs over an infinite horizon. The discount factor, 0 ~ "Y ~ I, allows
future payoff to be weighted less than more immediate payoff. Throughout this
paper, I will assume that the learning agent does not have access to a model of the
environment. Reinforcement learning algorithms such as Sutton's (1988) temporal
difference algorithm and Watkins's (1989) Q-Iearning algorithm can be used to learn
to solve single MDTs (also see Barto et al., 1991).
I consider compositionally-structured MDTs because they allow the possibility of
sharing knowledge across the many tasks that have common subtasks. In general,
there may be n elemental MDTs labeled T I , T2 , ??? , Tn. Elemental MDTs cannot be
decomposed into simpler subtasks. Compo8ite MDTs, labeled G I , G2 , ??? , Gm , are
produced by temporally concatenating a number of elemental MDTs. For example,
G; = [T(j, I)T(j, 2) ... T(j, k)] is composite task j made up of k elemental tasks that
have to be performed in the order listed. For 1 $ i $ k, T(j, i) E {TI' T2 , ??? , Tn} is
the itk elemental task in the list for task G;. The sequence of elemental tasks in a
composite task will be referred to as the decompo8ition of the composite task; the
decomposition is assumed to be unknown to the learning agent.
Compo8itional learning involves solving a composite task by learning to compose
the solutions of the elemental tasks in its decomposition. It is to be emphasized that
given the short-term, evaluative nature of the payoff from the environment (often
the agent gets informative payoff only at the completion of the composite task),
the task of discovering the decomposition of a composite task is formidable. In this
paper I propose a compositional learning scheme in which separate modules learn
to solve the elemental tasks, and a task-sensitive gating module solves composite
tasks by learning to compose the appropriate elemental modules over time.
2
ELEMENTAL AND COMPOSITE TASKS
All elemental tasks are MDTs that share the the same state set S, action set A, and
have the same environment dynamics. The payoff function for each elemental task
11, 1 ~ i ~ n, is ~(z, a) = EYES P:r:y(a)ri(Y) - c(z, a), where ri(Y) is a positive
reward associated with the state Y resulting from executing action a in state Z for
task 11, and c(z, a) is the positive cost of executing action a in state z. I assume
that ri(z) = 0 if Z is not the desired final state for 11. Thus, the elemental tasks
share the same cost function but have their own reward functions.
A composite task is not itself an MDT because the payoff is a function of both
lThe extension to the case where different sets of actions are available in different states
is straightforward.
The Efficient Learning of Multiple Task Sequences
the state and the current elemental task, instead of the state alone. Formally, the
new state set 2 for a composite task, S', is formed by augmenting the elements of
set S by n bits, one for each elemental task. For each z, E S', the projected 3tate
z E S is defined as the state obtained by removing the augmenting bits from z'.
The environment dynamics and cost function, c, for a composite task is defined by
assigning to each z, E S' and a E A the transition probabilities and cost assigned
to the projected state z E S and a E A. The reward function for composite task
C j , rj, is defined as follows. rj( z') ;::: 0 if the following are all true: i) the projected
state z is the final state for some elemental task in the decomposition of Cj, say
task Ii, ii) the augmenting bits of z' corresponding to elemental tasks appearing
before and including sub task Ti in the decomposition of C j are one, and iii) the rest
of the augmenting bits are zero; rj(z') 0 everywhere else.
=
3
COMPOSITIONAL Q-LEARNING
Following Watkins (1989), I define the Q-value, Q(z,a), for z E S and a E A, as the
expected return on taking action a in state z under the condition that an optimal
policy is followed thereafter. Given the Q-values, a greedy policy that in each state
selects an action with the highest associated Q-value, is optimal. Q-Iearning works
as follows. On executing action a in state z at time t, the resulting payoff and next
state are used to update the estimate of the Q-value at time t, Qt(z, a):
(1.0 - Qt)Qt(z, a)
+ ae[R(z, a) + l' max
Qt(Y, a')],
a'EA
(1)
where Y is the state at time t + 1, and at is the value of a positive learning rate
parameter at time t. Watkins and Dayan (1992) prove that under certain conditions
on the sequence {at}, if every state-action pair is updated infinitely often using
Equation 1, Qt converges to the true Q-values asymptotically.
Compositional Q-Iearning (CQ-Iearning) is a method for constructing the Q-values
of a composite task from the Q-values of the elemental tasks in its decomposition.
Let QT.(z,a) be the Q-value of (z,a), z E S and a E A, for elemental task Ii,
and let Q~:(z',a) be the Q-value of (z', a), for z' E S' and a E A, for task Ii
when performed as part of the composite task Cj
[T(j, 1) ... T(j, k)]. Assume
Ii = T(j, I) . Note that the superscript on Q refers to the task and the subscript
refers to the elemental task currently being performed. The absence of a superscript
implies that the task is elemental.
=
=
1) MDTs that have compositional structure
Consider a set of undiscounted (1'
and satisfy the following conditions:
(AI) Each elemental task has a single desired final state.
(A2) For all elemental and composite tasks, the expected value of undiscounted
return for an optimal policy is bounded both from above and below for all states.
(A3) The cost associated with each state-action pair is independent of the task
being accomplished.
2The theory developed in this paper does not depend on the particular extension of S
chosen, as long as the appropriate connection between the new states and the elements of
S can be made.
253
254
Singh
(A4) For each elemental task 71, the reward function ri is zero for all states except
the desired final state for that task. For each composite task C j , the reward function
rj is zero for all states except pouibly the final states of the elemental tasks in its
decomposition (Section 2).
Then, for any elemental task Ii and for all composite tasks C j containing elemental
task 71, the following holds:
Q~:(z',a)
QT.(Z, a)
+ K(Cj,T(j, I?,
(2)
for all z' E S' and a E A, where z E S is the projected state, and K (Cj, T(j, I? is a
function of the composite task Cj and subtask T(j, I), where Ti T(j, I). Note that
K( Cj , T(j, I? is independent of the state and the action. Thus, given solutions of
the elemental tasks, learning the solution of a composite task with n elemental tasks
requires learning only the values of the function K for the n different subtasks. A
proof of Equation 2 is given in Singh (1992).
=
a
WIll.
NoIN
N(O.G)
Q
Networtc
1
Q
Q
? ?? Networtt
n
Figure 1: The CQ-Learning Architecture (CQ-L). This figure is adapted from Jacobs
et al. (1991). See text for details.
Equation 2 is based on the assumption that the decomposition of the composite
tasks is known. In the next Section, I present a modular architecture and learning
algorithm that simultaneously discovers the decomposition of a composite task and
implements Equation 2.
4
CQ-L: CQ-LEARNING ARCHITECTURE
Jacobs (1991) developed a modular connectionist architecture that performs task
decomposition. Jacobs's gating architecture consists of several expert networks and
a gating network that has an output for each expert network. The architecture
has been used to learn multiple non-sequential tasks within the supervised learning
The Efficient Learning of Multiple Task Sequences
Table 1: Tasks. Tasks Tl, T2, and T3 are elemental tasks; tasks G l , G2 , and G3
are composite tasks. The last column describes the compositional structure of the
tasks.
Label
'11
T2
T3
01
C2
C3
Command
000001
000010
000100
001000
010000
100000
De.eription
A
B
C
VlS1t
VlS1t
V1S1t
VlSlt
VlS1t
V1S1t
Deeompo.ition
Tl
T2
T3
A and then C
1113
B and then C
A, then B and then C
T2 T 3
T1 T2T3
paradigm. I extend the modular network architecture to a CQ-Learning architecture (Figure I), called CQ-L, that can learn multiple compositionally-structured
sequential tasks even when training information required for supervised learning is
not available. CQ-L combines CQ-learning and the gating architecture to achieve
transfer of learning by "sharing" the solutions of elemental tasks across multiple
composite tasks. Only a very brief description of the CQ-L is provided in this
paper; details are given in Singh (1992) .
In CQ-L the expert networks are Q-learning networks that learn to approximate
the Q-values for the elemental tasks. The Q-networks receive as input both the
current state and the current action. The gating and bias networks (Figure 1)
receive as input the augmenting bits and the task command used to encode the
current task being performed by the architecture. The stochastic switch in Figure 1
selects one Q-network at each time step. CQ-L's output, Q, is the output of the
selected Q-network added to the output of the bias network.
The learning rules used to train the network perform gradient descent in the log
likelihood, L(t), of generating the estimate of the desired Q-value at time t, denoted
D(t), and are given below:
8 log L(t)
qj(t) + oQ 8qj(t) ,
Si(t)
+ Og
8 log L(t)
8Si(t)
,and
b(t) + ob(D(t) - Q(t)),
where qj is the output of the jt" Q-network, Si is the it" output of the gating
network, b is the output of the bias network, and 0Q, Ob and Og are learning rate
parameters. The backpropagation algorithm ( e.g., Rumelhart et al., 1986) was
used to update the weights in the networks. See Singh (1992) for details.
5
NAVIGATION TASK
To illustrate the utility of CQ-L, I use a navigational test bed similar to the one used
by Bachrach (1991) that simulates a planar robot that can translate simultaneously
255
256
Singh
c
G
Figure 2: Navigation Testbed. See text for details.
and independently in both ~ and y directions. It can move one radius in any
direction on each time step. The robot has 8 distance sensors and 8 gray-scale
sensors evenly placed around its perimeter. These 16 values constitute the state
vector. Figure 2 shows a display created by the navigation simulator. The bottom
portion of the figure shows the robot's environment as seen from above. The upper
panel shows the robot's state vector. Three different goal locations, A, B, and C,
are marked on the test bed. The set of tasks on which the robot is trained are shown
in Table 1. The elemental tasks require the robot to go to the given goal location
from a random starting location in minimum time. The composite tasks require the
robot to go to a goal location via a designated sequence of subgoallocations.
Task commands were represented by standard unit basis vectors (Table 1), and thus
the architecture could not "parse" the task command to determine the decomposition of a composite task. Each Q-network was a feedforward connectionist network
with a single hidden layer containing 128 radial basis units. The bias and gating
networks were also feedforward nets with a single hidden layer containing sigmoid
units. For all ~ E S U Sf and a E A, c(~, a)
-0.05. ri(~)
1.0 only if ~ is the
desired final state of elemental task Ii, or if ~ E Sf is the final state of composite
task Cii ri(~) = 0.0 in all other states. Thus, for composite tasks no intermediate
payoff for successful completion of subtasks was provided.
=
6
=
SIMULATION RESULTS
In the simulation described below, the performance of CQ-L is compared to the
performance of a "one-for-one" architecture that implements the "learn-each-taskseparately" strategy. The one-for-one architecture has a pre-assigned distinct net-
The Efficient Learning of Multiple Task Sequences
work for each task, which prevents transfer of learning. Each network of the onefor-one architecture was provided with the augmented state.
-
,oo
..,.
?
1I ..
8.
I
t
t??
.-..
- --
'I
COA.
ON.?FOA-ONE
"
'
1I
,
0
0
f .... ,
'I
'~ ,~' I',
1oo
'; V\
'I
?
.'t .
, , ,1,1
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...
'
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,
,- ,
Trial Nurrber (for T..k [AB))
- - ---
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o',
.. I,' I
TrW NIJ1rioer (for T..k A)
...
...
COA.
ONE-FOA.oNE
,
?
1
0
-- -
1-
..
,
C<>L
Til.. Number (fer TMk [ABC))
Figure 3: Learning Curves for Multiple tasks.
Both CQ-L and the one-for-one architecture were separately trained on the six
tasks T 1 , T2, T3 , C lI C 2 , and C 3 until they could perform the six tasks optimally.
CQ-L contained three Q-networks, and the one-for-one architecture contained six
Q-networks. For each trial, the starting state of the robot and the task identity
were chosen randomly. A trial ended when the robot reached the desired final state
or when there was a time-out. The time-out period was 100 for the elemental tasks,
200 for C 1 and C 2 , and 500 for task C 3 ? The graphs in Figure 3 show the number
of actions executed per trial. Separate statistics were accumulated for each task.
The rightmost graph shows the performance of the two architectures on elemental
task TI. Not surprisingly, the one-for-one architecture performs better because
it does not have the overhead of figuring out which Q-network to train for task
T 1 . The middle graph shows the performance on task C I and shows that the CQL architecture is able to perform better than the one-for-one architecture for a
composite task containing just two elemental tasks. The leftmost graph shows the
results for composite task C 3 and illustrates the main point of this paper. The onefor-one architecture is unable to learn the task, in fact it is unable to perform the
task more than a couple of times due to the low probability of randomly performing
the correct task sequence.
This simulation shows that CQ-L is able to learn the decomposition of a composite
task and that compositional learning, due to transfer of training across tasks, can
be faster than learning each composite task separately. More importantly, CQ-L
is able to learn to solve composite tasks that cannot be solved using traditional
schemes.
7
DISCUSSION
Learning to solve MDTs with large state sets is difficult due to the sparseness of the
evaluative information and the low probability that a randomly selected sequence
of actions will be optimal. Learning the long sequences of actions required to solve
such tasks can be accelerated considerably if the agent has prior knowledge of useful
subsequences. Such subsequences can be learned through experience in learning to
257
258
Singh
solve other tasks. In this paper, I define a class of MOTs, called composite MOTs,
that are structured as the temporal concatenation of simpler MOTs, called elemental MOTs. I present CQ-L, an architecture that combines the Q-Iearning algorithm
of Watkins (1989) and the modular architecture of Jacobs et al. (1991) to achieve
transfer of learning by sharing the solutions of elemental tasks across multiple composite tasks. Given a set of composite and elemental MOTs, the sequence in which
the learning agent receives training experiences on the different tasks determines the
relative advantage of CQ-L over other architectures that learn the tasks separately.
The simulation reported in Section 6 demonstrates that it is possible to train CQ-L
on intermixed trials of elemental and composite tasks. Nevertheless, the ability of
CQ-L to scale well to complex sets of tasks will depend on the choice of the training
sequence.
Acknowledgements
This work was supported by the Air Force Office of Scientific Research, Bolling
AFB, under Grant AFOSR-89-0526 and by the National Science Foundation under
Grant ECS-8912623. I am very grateful to Andrew Barto for his extensive help in
formulating these ideas and preparing this paper.
References
J . R. Bachrach. (1991) A connectionist learning control architecture for navigation. In R. P. Lippmann, J. E. Moody, and D. S. Touretzky, editors, Adv4nce6 in
Neural Information Proceuing Sy6tem6 3, pages 457-463, San Mateo, CA. Morgan
Kaufmann.
A. G. Barto, S. J. Bradtke, and S. P. Singh. (1991) Real-time learning and control
using asynchronous dynamic programming. Technical Report 91-57, University of
Massachusetts, Amherst, MA. Submitted to AI Journal.
R. A. Jacobs. (1990) T46lc decomp06ition through competition in a modular connectioni6t architecture. PhD thesis, COINS dept, Univ. of Massachusetts, Amherst,
Mass. U.S.A.
R. A. Jacobs, M. I. Jordan, S. J. Nowlan, and G. E. Hinton. (1991) Adaptive
mixtures of local experts. Neural Computation, 3( 1).
D. E. Rumelhart, G. E. Hinton, and R. J. Williams. (1986) Learning internal representations by error propagation. In D. E. Rumelhart and J. L. McClelland, editors,
Parallel Distributed Proceuing: E:cploration6 in the Micr06tructure of Cognition,
vol.1: Found4tion6. Bradford Books/MIT Press, Cambridge, MA.
S. P. Singh. (1992) Transfer of learning by composing solutions for elemental sequential tasks. Machine Learning.
R. S. Sutton. (1988) Learning to predict by the methods of temporal differences.
Machine Learning, 3:9-44.
C. J . C. H. Watkins. (1989) Learning from Delayed Rewards. PhD thesis, Cambridge Univ., Cambridge, England.
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5,181 | 5,690 | Online Prediction at the Limit of Zero Temperature
Mark Herbster
Stephen Pasteris
Department of Computer Science
University College London
London WC1E 6BT, England, UK
{m.herbster,s.pasteris}@cs.ucl.ac.uk
Shaona Ghosh
ECS
University of Southampton
Southampton, UK SO17 1BJ
[email protected]
Abstract
We design an online algorithm to classify the vertices of a graph. Underpinning
the algorithm is the probability distribution of an Ising model isomorphic to the
graph. Each classification is based on predicting the label with maximum marginal
probability in the limit of zero-temperature with respect to the labels and vertices
seen so far. Computing these classifications is unfortunately based on a #P complete problem. This motivates us to develop an algorithm for which we give
a sequential guarantee in the online mistake bound framework. Our algorithm is
optimal when the graph is a tree matching the prior results in [1]. For a general
graph, the algorithm exploits the additional connectivity over a tree to provide a
per-cluster bound. The algorithm is efficient, as the cumulative time to sequentially predict all of the vertices of the graph is quadratic in the size of the graph.
1
Introduction
Semi-supervised learning is now a standard methodology in machine learning. A common approach
in semi-supervised learning is to build a graph [2] from a given set of labeled and unlabeled data
with each datum represented as a vertex. The hope is that the constructed graph will capture either
the cluster [3] or manifold [4] structure of the data. Typically, an edge in this graph indicates the
expectation that the joined data points are more likely to have the same label. One method to
exploit this representation is to use the semi-norm induced by the Laplacian of the graph [5, 4, 6, 7].
A shared idea of the Laplacian semi-norm based approaches is that the smoothness of a boolean
labeling of the graph is measured via the ?cut?, which is just the number of edges that connect
disagreeing labels. In practice the semi-norm is then used as a regularizer in which the optimization
problem is relaxed from boolean to real values. Our approach also uses the ?cut?, but unrelaxed, to
define an Ising distribution over the vertices of the graph.
Predicting with the vertex marginals of an Ising distribution in the limit of zero temperature was
shown to be optimal in the mistake bound model [1, Section 4.1] when the graph is a tree. The exact
computation of marginal probabilities in the Ising model is intractable on non-trees [8]. However, in
the limit of zero temperature, a rich combinatorial structure called the Picard-Queyranne graph [9]
emerges. We exploit this structure to give an algorithm which 1) is optimal on trees, 2) has a
quadratic cumulative computational complexity, and 3) has a mistake bound on generic graphs that
is stronger than previous bounds in many natural cases.
The paper is organized as follows. In the remainder of this section, we introduce the Ising model
and lightly review previous work in the online mistake bound model for predicting the labeling of a
graph. In Section 2 we review our key technical tool the Picard-Queyranne graph [9] and explain the
required notation. In the body of Section 3 we provide a mistake bound analysis of our algorithm
as well as the intractable 0-Ising algorithm and then conclude with a detailed comparison to the
state of the art. In the appendices we provide proofs as well as preliminary experimental results.
Ising model in the limit zero temperature. In our setting, the parameters of the Ising model
are an n-vertex graph G = (V (G), E(G)) and a temperature parameter ? > 0, where V (G) =
1
{1, . . . , n} denotes the vertex set and E(G) denotes the edge set. Each vertex of this graph may
be labeled with one of two states {0, 1} and thus a labeling of a graph may be denoted by a vector
n
u ? {0, 1}P
where ui denotes the label of vertex i. The cutsize of a labeling u is defined as
?G (u) :=
Ising probability distribution over labelings of G is then
(i,j)?E(G) |ui ? uj |. The
defined as pG? (u) ? exp ? ?1 ?G (u) where ? > 0 is the temperature parameter. In our online
setting at the beginning of trial t + 1 we will have already received an example sequence, St , of t
vertex-label pairs (i1 , y1 ), . . . , (it , yt ) where pair (i, y) ? V (G) ? {0, 1}. We use pG? (uv = y|St ) :=
pG? (uv = y|ui1 = y1 , . . . , uit = yt ) to denote the marginal probability that vertex v has label y
given the previously labeled vertices of St . For convenience we also define the marginalized cutsize
?G (u|St ) to be equal to ?G (u) if ui1 = y1 , . . . , uit = yt and equal to undefined otherwise. Our
prediction y?t+1 of vertex it+1 is then the label with maximal marginal probability in the limit of
zero temperature, thus
0I
y?t+1
(it+1 |St ) := argmax lim pG? (uit+1 = y|ui1 = y1 , . . . , uit = yt ) .
y?{0,1} ? ?0
[0-Ising]
(1)
Note the prediction is undefined if the labels are equally probable. In low temperatures the mass
of the marginal is dominated by the labelings consistent with St and the proposed label of vertex
it+1 of minimal cut; as we approach zero, y?t+1 is the label consistent with the maximum number of
labelings of minimal cut. Thus if k := min n?G (u|S) then we have that
u?{0,1}
?
0I
y? (v|S) =
0
1
|u ? {0, 1}n : ?G (u|(S, (v, 0))) = k| > |u ? {0, 1}n : ?G (u|(S, (v, 1))) = k|
.
|u ? {0, 1}n : ?G (u|(S, (v, 0))) = k| < |u ? {0, 1}n : ?G (u|(S, (v, 1))) = k|
The problem of counting minimum label-consistent cuts was shown to be #P-complete in [10] and
further computing y?0I (v|S) is also NP-hard (see Appendix G). In Section 2.1 we introduce the
Picard-Queyranne graph [9] which captures the combinatorial structure of the set of minimum-cuts.
We then use this simplifying structure as a basis to design a heuristic approximation to y?0I (v|S) with
a mistake bound guarantee.
Predicting the labelling of a graph in the mistake bound model. We prove performance guarantees for our method in the mistake bound model introduced by Littlestone [11]. On the graph this
model corresponds to the following game. Nature presents a graph G; Nature queries a vertex
i1 ? V (G) = INn ; the learner predicts the label of the vertex y?1 ? {0, 1}; nature presents a
label y1 ; nature queries a vertex i2 ; the learner predicts y?2 ; and so forth. The learner?s goal is
to minimize the total number of mistakes M = |{t : y?t 6= yt }|. If nature is adversarial, the learner
will always make a ?mistake?, but if nature is regular or simple, there is hope that a learner may
incur only a few mistakes. Thus, a central goal of online learning is to design algorithms whose
total mistakes can be bounded relative to the complexity of nature?s labeling. The graph labeling
problem has been studied extensively in the online literature. Here we provide a rough discussion of
the two main approaches for graph label prediction, and in Section 3.3 we provide a more detailed
comparison. The first approach is based on the graph Laplacian [12, 13, 14]; it provides bounds that
utilize the additional connectivity of non-tree graphs, which are particularly strong when the graph
contains uniformly-labeled clusters of small (resistance) diameter. The drawbacks of this approach
are that the bounds are weaker on graphs with large diameter and that the computation times are
slower. The second approach is to estimate the original graph with an appropriately selected tree or
?path? graph [15, 16, 1, 17]; this leads to faster computation times, and bounds that are better on
graphs with large diameters. The algorithm treeOpt [1] is optimal on trees. These algorithms may
be extended to non-tree graphs by first selecting a spanning tree uniformly at random [16] and then
applying the algorithm to the sampled tree. This randomized approach enables expected mistake
bounds which exploit the cluster structure in the graph.
The bounds we prove for the NP-hard 0-Ising prediction and our heuristic are most similar to the
?small p? bounds proven for the p-seminorm interpolation algorithm [14]. Although these bounds
are not strictly comparable, a key strength of our approach is that the new bounds often improve
when the graph contains uniformly-labeled clusters of varying diameters. Furthermore, when the
graph is a tree we match the optimal bounds of [1]. Finally, the cumulative time required to compute
the complete labeling of a graph is quadratic in the size of the graph for our algorithm, while [14] requires the minimization of a non-strongly convex function (on every trial) which is not differentiable
when p ? 1.
2
2
Preliminaries
An (undirected) graph G is a pair of sets (V, E) such that E is a set of unordered pairs of distinct
elements from V . We say that R is a subgraph R ? G iff V (R) ? V (G) and E(R) = {(i, j) :
i, j ? V (R), (i, j) ? E(G)}. Given any subgraph R ? G, we define its boundary (or inner border)
?0 (R), its neighbourhood (or exterior border) ?e (R) respectively as ?0 (R) := {j : i 6? V (R), j ?
V (R), (i, j) ? E(G)}, and ?e (R) := {i : i 6? V (R), j ? V (R), (i, j) ? E(G)}, and its exterior
edge border ?eE (R) := {(i, j) : i 6? V (R), j ? V (R), (i, j) ? E(G)}. The length of a subgraph
P is denoted by |P| := |E(P)| and we denote the diameter of a graph by D(G). A pair of vertices
v, w ? V (G) are ?-connected if there exist ? edge-disjoint paths connecting them. The connectivity
of a graph, ?(G), is the maximal value of ? such that every pair of points in G is ?-connected. The
atomic number N? (G) of a graph at connectivity level ? is the minimum cardinality c of a partition
of G into subgraphs {R1 , . . . , Rc } such that ?(Ri ) ? ? for all 1 ? i ? c.
Our results also require the use of directed-, multi-, and quotient- graphs. Every undirected graph
also defines a directed graph where each undirected edge (i, j) is represented by directed edges (i, j)
and (j, i). An orientation of an undirected graph is an assignment of a direction to each edge, turning
the initial graph into a directed graph. In a multi-graph the edge set is now a multi-set and thus there
may be multiple edges between two vertices. A quotient-graph G is defined from a graph G and
N
a partition of its vertex set {Vi }N
i=1 so that V (G) := {Vi }i=1 (we often call these vertices supervertices to emphasize that they are sets) and the multiset E(G) := {(I, J) : I, J ? V (G), I 6= J, i ?
I, j ? J, (i, j) ? E(G)}. We commonly construct a quotient-graph G by ?merging? a collection of
super-vertices, for example, in Figure 2 from 2a to 2b where 6 and 9 are merged to ?6/9? and also
the five merges that transforms 2c to 2d.
The set of all label-consistent minimum-cuts in a graph with respect to an example sequence S is
UG? (S) := argminu?{0,1}n ?G (u|S). The minimum is typically non-unique. For example in Figure 2a, the vertex sets {v1 , . . . , v4 }, {v5 , . . . , v12 } correspond to one label-consistent minimum-cut
and {v1 , . . . , v5 , v7 , v8 }, {v6 , v9 . . . , v12 } to another (the cutsize is 3). The (uncapacitated) maximum flow is the number of edge-disjoint paths between a source and target vertex. Thus in Figure 2b
between vertex ?1? and vertex ?6/9? there are at most 3 simultaneously edge-disjoint paths; these are
also not unique, as one path must pass through either vertices hv11 , v12 i or vertices hv11 , v10 , v12 i.
Figure 2c illustrates one such flow F (just the directed edges). For convenience it is natural to view
the maximum flow or the label-consistent minimum-cut as being with respect to only two vertices
as in Figure 2a transformed to Figure 2b so that H ? merge(G, {v6 , v9 }). The ?flow? and the
?cut? are related by Menger?s theorem which states that the minimum-cut with respect to a source
and target vertex is equal to the max flow between them. Given a connected graph H and source
and target vertices s, t the Ford-Fulkerson algorithm [18] can find k edge-disjoint paths from s to t
in time O(k|E(H)|) where k is the value of the max flow.
2.1
The Picard-Queyranne graph
Given a set of labels there may be multiple label-consistent minimum-cuts as well as multiple maximum flows in a graph. The Picard-Queyranne (PQ) graph [9] reduces this multiplicity as far as is
possible with respect to the indeterminacy of the maximum flow. The vertices of the PQ-graph are
defined as a super-vertex set on a partition of the original graph?s vertex set. Two vertices are contained in the same super-vertex iff they have the same label in every label-consistent minimum-cut.
An edge between two vertices defines an analogous edge between two super-vertices iff that edge is
conserved in every maximum flow. Furthermore the edges between super-vertices strictly orient the
labels in any label-consistent minimum-cut as may be seen in the formal definition that follows.
First we introduce the following useful notations: let kG,S := min{?G (u|S) : u ? {0, 1}n } denote
S
the minimum-cutsize of G with respect to S; let i?j denote an equivalence relation between vertices
S
in V (G) where i?j iff ?u ? UG? (S) : ui = uj ; and then we define,
Definition 1 ([9]). The Picard-Queyranne graph G(G, S) is derived from graph G and non-trivial
example sequence S. The graph is an orientation of the quotient graph derived from the partition
S
{?, I2 , . . . , IN ?1 ,>} of V (G) induced by ?. The edge set of G is constructed of kG,S edge-disjoint
paths starting at source vertex ? and terminating at target vertex >. A labeling u ? {0, 1}n is in
UG? (S) iff
1. i ? ? implies ui = 0 and i ? > implies ui = 1
3
2. i, j ? H implies ui = uj
3. i ? I, j ? J, (I, J) ? E(G), and ui = 1 implies uj = 1
where ? and > are the source and target vertices and H, I, J ? V (G).
As G(G, S) is a DAG it naturally defines a partial order (V (G), ?G ) on the vertex set where I ?G J
if there exists a path starting at I and ending at J. The least and greatest elements of the partial order
are ? and >. The notation ?R and ?R denote the up set and down set of R. Given the set U ? of all
label-consistent minimum-cuts then if u ? U ? there exists an antichain A ? V (G) \ {>} such that
ui = 0 when i ? I ? ?A otherwise ui = 1; furthermore for every antichain there exists a labelconsistent minimum-cut. The simple structure of G(G, S) was utilized by [9] to enable the efficient
algorithmic enumeration of minimum-cuts. However, the cardinality of this set of all label-consistent
minimum-cuts is potentially exponential in the size of the PQ-graph and the exact computation of
the cardinality was later shown #P-complete in [10]. In Figure 1 we give the algorithm from [9, 19]
PicardQueyranneGraph(graph: G; example sequence: S = (vk , yk )tk=1 )
1. (H, s, t) ? SourceTargetMerge(G, S)
2. F ? MaxFlow(H, s, t)
3. I ? (V (I), E(I)) where V (I) := V (H) and E(I) := {(i, j) : (i, j) ? E(H), (j, i) 6? F}
4.
G0 ? QuotientGraph(StronglyConnectedComponents(I), H)
E(G) ? E(G0 ); V (G) ? V (G0 ) except ?(G) ? ?(G0 ) ? {vk : k ? INt , yk = 0}
and >(G) ? >(G0 ) ? {vk : k ? INt , yk = 1}
Return: directed graph: G
5.
Figure 1: Computing the Picard-Queyranne graph
2
7
9
2
5
1
7
6/9
5
3
10
12
1
3
10
12
6
8
11
4
(a) Graph G and S = h(v1 , 0), (v6 , 1), (v9 , 1)i
2
7
8
11
(b) Graph H (step 1 in Figure 1)
6/9
A
?
4
B
C
5
1
3
4
10
8
12
11
(c) Graph I (step 3 in Figure 1)
?
(d) PQ Graph G (step 4 in Figure 1)
Figure 2: Building a Picard-Queyranne graph
to compute a PQ-graph. We illustrate the computation in Figure 2. The algorithm operates first on
(G, S) (step 1) by ?merging? all vertices which share the same label in S to create H. In step 2 a
max flow graph F ? H is computed by the Ford-fulkerson algorithm. It is well-known in the case
of unweighted graphs that a max flow graph F may be output as a DAG of k edge-disjoint paths
where k is the value of the flow. In step 3 all edges in the flow become directed edges creating I.
The graph G0 is then created in step 4 from I where the strongly connected components become the
super-vertices of G0 and the super-edges correspond to a subset of flow edges from F. Finally, in
4
step 5, we create the PQ-graph G by ?fixing? the source and target vertices so that they also have as
elements the original labeled vertices from S which were merged in step 1. The correctness of the
algorithm follows from arguments in [9]; we provide an independent proof in Appendix B.
Theorem 2 ([9]). The algorithm in Figure 1 computes the unique Picard-Queyranne graph G(G, S)
derived from graph G and non-trivial example sequence S.
3
Mistake Bounds Analysis
In this section we analyze the mistakes incurred by the intractable 0-Ising strategy (see (1)) and
the strategy longest-path (see Figure 3). Our analysis splits into two parts. Firstly, we show
(Section 3.1, Theorem 4) for a sufficiently regular graph label prediction algorithm, that we may
analyze independently the mistake bound of each uniformly-labeled cluster (connected subgraph).
Secondly, the per-cluster analysis then separates into three cases, the result of which is summarized
in Theorem 10. For a given cluster C when its internal connectivity is larger than the number of
edges in the boundary (?(C) > |?eE (C)|) we will incur no more than one mistake in that cluster. On
the other hand for smaller connectivity clusters (?(C) ? |?eE (C)|) we incur up to quadratically in
mistakes via the edge boundary size. When C is a tree we incur O(|?eE (C)| log D(C)) mistakes.
The analysis of smaller connectivity clusters separates into two parts. First, a sequence of trials in
which the label-consistent minimum-cut does not increase, we call a PQ-game (Section 3.2) as in
essence it is played on a PQ-graph. We give a mistake bound for a PQ-game for the intractable
0-Ising prediction and a comparable bound for the strategy longest-path in Theorem 8.
Second, when the label-consistent minimum-cut increases the current PQ-game ends and a new one
begins, leading to a sequence of PQ-games. The mistakes incurred over a sequence of PQ-games is
addressed in the aforementioned Theorem 10 and finally Section 3.3 concludes with a discussion of
the combined bounds of Theorems 4 and 10 with respect to other graph label prediction algorithms.
3.1
Per-cluster mistake bounds for regular graph label prediction algorithms
An algorithm is called regular if it is permutation-invariant, label-monotone, and Markov. An
algorithm is permutation-invariant if the prediction at any time t does not depend on the order of
the examples up to time t; label-monotone if for every example sequence if we insert an example
?between? examples t and t+1 with label y then the prediction at time t+1 is unchanged or changed
to y; and Markov with respect to a graph G if for any disjoint vertex sets P and Q and separating set
R then the predictions in P are independent of the labels in Q given the labels of R. A subgraph is
uniformly-labeled with respect to an example sequence iff the label of each vertex is the same and
these labels are consistent with the example sequence. The following definition characterizes the
?worst-case? example sequences for regular algorithms with respect to uniformly-labeled clusters.
Definition 3. Given an online algorithm A and a uniformly-labeled subgraph C ? G, then BA (C; G)
denotes the maximal mistakes made only in C for the presentation of any permutation of examples
in ?e (C), each with label y, followed by any permutation of examples in C, each with label 1?y.
The following theorem enables us to analyze the mistakes incurred in each uniformly-labeled subgraph C independently of each other and independently of the remaining graph structure excepting
the subgraph?s exterior border ?e (C).
Theorem 4 (Proof in Appendix D). Given an online permutation-invariant label-monotone Markov
algorithm A and a graph G which is covered by uniformly-labeled
subgraphs C1 , . . . , Cc the mistakes
P
incurred by the algorithm may be bounded by M ? ci=1 BA (Ci ; G) .
The above theorem paired with Theorem 10 completes the mistake bound analysis of our algorithms.
3.2
PQ-games
Given a PQ-graph G = G(G, S), the derived online PQ-game is played between a player and an
adversary. The aim of the player is to minimize their mistaken predictions; for the adversary
it is to maximize the player?s mistaken predictions. Thus to play the adversary proposes a vertex
z ? Z ? V (G), the player then predicts a label y? ? {0, 1}, then the adversary returns a label
y ? {0, 1} and either a mistake is incurred or not. The only restriction on the adversary is to not
return a label which increases the label-consistent minimum-cut. As long as the adversary does not
give an example (z ? ?, 1) or (z ? >, 0), the label-consistent minimum-cut does not increase
5
no matter the value of y; which also implies the player has a trivial strategy to predict the label of
z ? ? ? >. After the example is given, we have an updated PQ-graph with new source and target
super-vertices as seen in the proposition below.
Proposition 5. If G(G, S) is a PQ-graph and (z, y = 0) ((z, y = 1)) is an example with
z ? Z ? V (G) and z 6? > (z 6? ?) then let Z = ?{Z} (Z = ?{Z}) then G(G, hS, (z, y)i) =
merge(G(G, S), Z).
Thus given the PQ-graph G the PQ-game is independent of G and S, since a ?play? z ? V (G)
induces a ?play? Z ? V (G) (with z ? Z).
Mistake bounds for PQ-games. Given a single PQ-game, in the following we will discuss the
three strategies fixed-paths, 0-Ising, and longest-path that the player may adopt
for which we prove online mistake bounds. The first strategy fixed-paths is merely motivational: it can be used to play a single PQ-game, but not a sequence. The second strategy 0-Ising
is computationally infeasible. Finally, the longest-path strategy is ?dynamically? similar to
fixed-paths but is also permutation-invariant. Common to all our analyses is a k-path cover
P of PQ-graph G which is a partitioning of the edge-set of G into k edge-disjoint directed paths
P := {p1 , . . . , pk } from ? to >. Note that the cover is not necessarily unique; for example, in
Figure 2d, we have the two unique path covers P1 := {(?, A, >), (?, A, B, >), (?, B, C, >)} and
P2 := {(?, A, >), (?, A, B, C, >), (?, B, >)}. We denote the set of all path covers as P and thus
we have for Figure 2d that P := {P1 , P2 }. This cover motivates a simple mistake bound and strategy. Suppose we had a single path of length |p| where the first and last vertex are the ?source?
and ?target? vertices. So the minimum label-consistent cut-size is ?1? and a natural strategy is simply to predict with the ?nearest-neighbor? revealed label and trivially our mistake bound is log |p|.
Generalizing to multiple paths we have the following strategy.
?): Given a PQ-graph choose a path cover {?
? ? P(G).
Strategy fixed-paths(P
p1 , . . . , p?k } = P
If the path cover is also vertex-disjoint except for the source and target vertex we may directly use the
P
?nearest-neighbor? strategy detailed above, achieving the mistake upper bound M ? ki=1 log |?
pi |.
Unsurprisingly, in the vertex-disjoint case it is a mistake-bound optimal [11] algorithm. If, however,
? is not vertex-disjoint and we need to predict a vertex V we may select a path in P
? containing V
P
and predict with the nearest neighbour and also obtain the bound above. In this case, however, the
bound may not be ?optimal.? Essentially the same technique was used in [20] in a related setting
for learning ?directed cuts.? A limitation of the fixed-paths strategy is that it does not seem
possible to extend into a strategy that can play a sequence of PQ-games and still meet the regularity
properties, particularly permutation-invariance as required by Theorem 4.
Strategy 0-Ising: The prediction of the Ising model in the limit of zero temperature (cf. (1)),
is equivalent to those of the well-known Halving algorithm [21, 22] where the hypothesis class U ?
is the set of label-consistent minimum-cuts. The mistake upper bound of the Halving algorithm
is just M ? log |U ? | where this bound follows from the observation that whenever a mistake is
made at least ?half? of concepts in U ? are no longer consistent. We observe that we may upper
Q
bound |U ? | ? argminP ?P(G) ki=1 |pi | since the product of path lengths from any path cover P
is an upper bound on the cardinality of U ? and hence we have the bound in (2). And in fact this
bound may be a significant improvement over the fixed-paths strategy?s bound as seen in the
following proposition.
Proposition 6 (Proof in Appendix C). For every c ? 2 there exists a PQ-graph Gc , with a path cover
P 0 ? P(Gc ) and a PQ-game example sequence such that the mistakes Mfixed-paths(P 0 ) = ?(c2 ),
while for all PQ-game example sequences on Gc the mistakes M0-Ising = O(c).
Unfortunately the 0-Ising strategy has the drawback that counting label-consistent minimum-cuts
is #P-complete and computing the prediction (see (1)) is NP-hard (see Appendix G).
Strategy longest-path: In our search for an efficient and regular prediction strategy it seems
natural to attempt to ?dynamize? the fixed-paths approach and predict with a nearest neighbor along a dynamic path. Two such permutation-invariant methods are the longest-path and
shortest-path strategies. The strategy shortest-path predicts the label of a super-vertex
Z in a PQ-game G as 0 iff the shortest directed path (?, . . . , Z) is shorter than the shortest directed
path (Z, . . . , >). The strategy longest-path predicts the label of a super-vertex Z in a PQ-game
G as 0 iff the longest directed path (?, . . . , Z) is shorter than the longest directed path (Z, . . . , >).
The strategy shortest-path seems to be intuitively favored over longest-path as it is just
6
Input: Graph: G, Example sequence: S = h(i1 , 0), (i2 , 1), (i3 , y3 ), . . . , (i` , y` )i ? (INn ? {0, 1})`
Initialization: G3 = PicardQueyranneGraph(G, S2 )
for t = 3, . . . , ` do
Receive: it ? {1, . . . , n}
It = V ? V (Gt ) with it ? V
?
0 |longest-path(Gt , ?t , It )| ? |longest-path(Gt , It , >t )|
Predict (longest-path): y?t =
1 otherwise
Predict (0-Ising):
y?t = y?I0 (it |St?1 )
Receive: yt
if (it 6? ?t or yt 6= 1) and (it 6? >t or yt 6= 0) then
?
merge(Gt , ?{It }) yt = 0
Gt+1 =
merge(Gt , ?{It }) yt = 1
else
Gt+1 = PicardQueyranneGraph(G, St )
% as per equation (1)
% cut unchanged
% cut increases
end
Figure 3: Longest-path and 0-Ising online prediction
the ?nearest-neighbor? prediction with respect to the geodesic distance. However, the following
proposition shows that it is strictly worse than any fixed-paths strategy in the worst case.
Proposition 7 (Proof in Appendix C). For every c ? 4 there exists a PQ-graph Gc and a PQ-game
example sequence such that the mistakes Mshortest-path = ?(c2 log(c)), while for every path cover
P ? P(Gc ) and for all PQ-game example sequences on Gc the mistakes Mfixed-paths(P ) = O(c2 ).
In contrast, for the strategy longest-paths in the proof of Theorem 8 we show that there alP
ways exists some retrospective path cover Plp ? P(G) such that Mlongest-paths ? ki=1 log |pilp |.
Computing the ?longest-path? has time complexity linear in the number of edges in a DAG.
Summarizing the mistake bounds for the three PQ-game strategies for a single PQ-game we have
the following theorem.
Theorem 8 (Proof in Appendix C). The mistakes, M , of an online PQ-game for player strategies
?), 0-Ising, and longest-path on PQ-graph G and k-path cover P
? ?
fixed-paths(P
P(G) is bounded by
?P
?)
? ki=1 log |?
pi |
fixed-paths(P
?
Pk
i
M ? argminP ?P(G) i=1 log |p |
.
(2)
0-Ising
?
Pk
?argmax
i
longest-path
P ?P(G)
i=1 log |p |
3.3
Global analysis of prediction at zero temperature
In Figure 3 we summarize the prediction protocol for 0-Ising and longest-path. We claim
the regularity properties of our strategies in the following theorem.
Theorem 9 (Proof in Appendix E). The strategies 0-Ising and longest-path are
permutation-invariant, label-monotone, and Markov.
The technical hurdle here is to prove that label-monotonicity holds over a sequence of PQ-games.
For this we need an analog of Proposition 5 to describe how the PQ-graph changes when the labelconsistent minimum-cut increases (see Proposition 19). The application of the following theorem
along with Theorem 4 implies we may bound the mistakes of each uniformly-labeled cluster in
potentially three ways.
Theorem 10 (Proof in Appendix D). Given either the 0-Ising or longest-path strategy A
the mistakes on uniformly-labeled subgraph C ? G are bounded by
?
?(C) > |?eE (C)|
?O(1)
BA (C; G) ? O |?eE (C)|(1 + |?eE (C)| ? ?(C)) log N (C) ?(C) ? |?eE (C)|
(3)
?
E
O(|?e (C)| log D(C))
C is a tree
with the atomic number N (C) := N|?eE (C)|+1 (C) ? |V (C)|.
7
First, if the internal connectivity of the cluster is high we will only make a single mistake in that cluster. Second, if the cluster is a tree then we pay the external connectivity of the cluster |?eE (C)| times
the log of the cluster diameter. Finally, in the remaining case we pay quadratically in the external
connectivity and logarithmically in the ?atomic number? of the cluster. The atomic number captures
the fact that even a poorly connected cluster may have sub-regions of high internal connectivity.
Computational complexity. If G is a graph and S an example sequence with a label-consistent
minimum-cut of ? then we may implement the longest-path strategy so that it has a cumulative
computational complexity of O(max(?, n) |E(G)|). This follows because if on a trial the ?cut? does
not increase we may implement prediction and update in O(|E(G)|) time. On the other hand if the
?cut? increases by ?0 we pay O(?0 |E(G)|) time. To do so we implement an online ?Ford-Fulkerson?
algorithm [18] which starts from the previous ?residual? graph to which it then adds the additional
?0 flow paths with ?0 steps of size O(|E(G)|).
Discussion. There are essentially five dominating mistake bounds for the online graph labeling problem: (I) the bound of treeOpt [1] on trees, (II) the bound in expectation of treeOpt on a random
spanning tree sampled from a graph [1], (III) the bound of p-seminorm interpolation [14]
tuned for ?sparsity? (p < 2), (IV) the bound of p-seminorm interpolation as tuned to be
equivalent to online label propagation [5] (p = 2), (V) this paper?s longest-path strategy.
The algorithm treeOpt was shown to be optimal on trees. In Appendix F we show that
longest-path also obtains the same optimal bound on trees. Algorithm (II) applies to generic
graphs and is obtained from (I) by sampling a random spanning tree (RST). It is not directly comparable to the other algorithms as its bound holds only in expectation with respect to the RST.
We use [14, Corollary 10] to compare (V) to (III) and (IV). We introduce the following simplifying notation to compare bounds. Let C1 , . . . , Cc denote uniformly-labeled clusters (connected
subgraphs) which cover the graph and set ?r := ?(Cr ) and ?r := |?eE (Cr )|. We define Dr(i) to
be the wide diameter at connectivity level i of cluster Cr . The wide diameter Dr(i) is the minimum
value such that for all pairs of vertices v, w ? Cr there exists i edge-disjoints of paths from v to w of
length at least Dr(i) in Cr (and if i > ?r then Dr(i) := +?). Thus Dr(1) is the diameter of cluster
Cr and Dr(1) ? Dr(2) ? ? ? ? . Let ? denote the minimum label-consistent cutsize and observe that
P
if the cardinality of the cover |{C1 , . . . , Cc }| is minimized then we have that 2? = cr=1 ?r .
Thus using [14, Corollary 10] we have the following upper bounds of (III): (?/?? )2 log D? + c and
?
?
?
?
?
(IV):
Pc (?/? )D + c where ? := minr ?r and D := maxr Dr (? ). In comparison we have (V):
[ r=1 max(0, ?r ? ?r + 1)?r log Nr ] + c with atomic numbers Nr := N?r +1 (Cr ). To contrast
the bounds, consider a double lollipop labeled-graph: first create a lollipop which is a path of n/4
vertices attached to a clique of n/4 vertices. Label these vertices 1. Second, clone the lollipop
except with labels 0. Finally join the two cliques with n/8 edges arbitrarily. For (III) and (IV)
the bounds are O(n) independent of the choice of clusters. Whereas an upper bound for (V) is the
exponentially smaller O(log n) which is obtained by choosing a four cluster cover consisting of the
two paths and the two cliques. This emphasizes the generic problem of (III) and (IV): parameters
?? and D? are defined by the worst clusters; whereas (V) is truly a per-cluster bound. We consider
the previous ?constructed? example to be representative of a generic case where the graph contains
clusters of many resistance diameters as well as sparse interconnecting ?background? vertices.
On the other hand, there are cases in which (III,IV) improve on (V). For a graph with only small
diameter clusters and if the cutsize exceeds the cluster connectivity then (IV) improves on (III,V)
given the linear versus quadratic dependence on the cutsize. The log-diameter may be arbitrarily
smaller than log-atomic-number ((III) improves on (V)) and also vice-versa. Other subtleties not
accounted for in the above comparison include the fact a) the wide diameter is a crude upper bound
for resistance diameter (cf. [14, Theorem 1]) and b) the clusters of (III,IV) are not required to be
uniformly-labeled. Regarding ?a)? replacing ?wide? with ?resistance? does not change the fact
the bound now holds with respect to the worst resistance diameter and the example above is still
problematic. Regarding ?b)? it is a nice property but we do not know how to exploit this to give
an example that significantly improves (III) or (IV) over a slightly more detailed analysis of (V).
Finally (III,IV) depend on a correct choice of tunable parameter p.
Thus in summary (V) matches the optimal bound of (I) on trees, and can often improve on (III,IV)
when a graph is naturally covered by label-consistent clusters of different diameters. However
(III,IV) may improve on (V) in a number of cases including when the log-diameter is significantly
smaller than log-atomic-number of the clusters.
8
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9
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5,182 | 5,691 | Lifted Symmetry Detection and
Breaking for MAP Inference
Tim Kopp
University of Rochester
Rochester, NY
Parag Singla
I.I.T. Delhi
Hauz Khas, New Delhi
Henry Kautz
University of Rochester
Rochester, NY
[email protected]
[email protected]
[email protected]
Abstract
Symmetry breaking is a technique for speeding up propositional satisfiability testing by adding constraints to the theory that restrict the search space while preserving satisfiability. In this work, we extend symmetry breaking to the problem of
model finding in weighted and unweighted relational theories, a class of problems
that includes MAP inference in Markov Logic and similar statistical-relational
languages. We introduce term symmetries, which are induced by an evidence
set and extend to symmetries over a relational theory. We provide the important
special case of term equivalent symmetries, showing that such symmetries can
be found in low-degree polynomial time. We show how to break an exponential
number of these symmetries with added constraints whose number is linear in the
size of the domain. We demonstrate the effectiveness of these techniques through
experiments in two relational domains. We also discuss the connections between
relational symmetry breaking and work on lifted inference in statistical-relational
reasoning.
1
Introduction
Symmetry-breaking is an approach to speeding up satisfiability testing by adding constraints, called
symmetry-breaking predicates (SBPs), to a theory [7, 1, 16]. Symmetries in the theory define a
partitioning over the space of truth assignments, where the assignments in a partition either all satisfy
or all fail to satisfy the theory. The added SBPs rule out some but not all of the truth assignments in
the partitions, thus reducing the size of the search space while preserving satisfiability.
We extend the notion of symmetry-breaking to model-finding in relational theories. A relational
theory is specified by a set of first-order axioms over finite domains, optional weights on the axioms
or predicates of the theory, and a set of ground literals representing evidence. By model finding
we mean satisfiability testing (unweighted theories), weighted MaxSAT (weights on axioms), or
maximum weighted model finding (weights on predicates). The weighted versions of model finding
encompass MAP inference in Markov Logic and similar statistical-relational languages.
We introduce methods for finding symmetries in a relational theory that do not depend upon solving
graph isomorphism over its full propositional grounding. We show how graph isomorphism can be
applied to just the evidence portion of a relational theory in order to find the set of what we call term
symmetries. We go on to define the important subclass of term equivalent symmetries, and show that
they can be found in O(nM log M ) time where n is the number of constants and M is the size of
the evidence.
Next we provide the formulation for breaking term and term equivalent symmetries. An inherent
problem in symmetry-breaking is that a propositional theory may have an exponential number of
symmetries, so breaking them individually would increase the size of the theory exponentially. This
is typically handled by breaking only a portion of the symmetries. We show that term equivalent
symmetries provide a compact representation of exponentially many symmetries, and an exponen1
tially large subset of these can be broken by a small (linear) number of SBPs. We demonstrate these
ideas on two relational domains and compare our approach to other methods for MAP inference in
Markov Logic.
2
Background
Symmetry Breaking for SAT Symmetry-breaking for satisfiability testing, introduced by Crawford et. al.[7], is based upon concepts from group theory. A permutation ? is a mapping from a set L
to itself. A permutation group is a set of permutations that is closed under composition and contains
the identity and a unique inverse for every element. A literal is an atom or its negation. A clause
is a disjunction over literals. A CNF theory T is a set (conjunction) of clauses. Let L be the set of
literals of T . We consider only permutations that respect negation, that is ?(?l) = ??(l) (l ? L).
The action of a permutation on a theory, written ?(T ), is the CNF formula created by applying ? to
each literal in T . We say ? is a symmetry of T if it results in the same theory i.e. ?(T ) = T .
A model M is a truth assignment to the atoms of a theory. The action of ? on M , written ?(M ),
is the model where ?(M )(P ) = M (?(P )). The key property of ? being a symmetry of T is that
M |= T iff ?(M ) |= T . The orbit of a model M under a symmetry group ? is the set of models
that can be obtained by applying any of the symmetries in ?. A symmetry group divides the space
of models into disjoint sets, where the models in an orbit either all satisfy or all do not satisfy the
theory. The idea of symmetry-breaking is to add clauses to T rule out many of the models, but are
guaranteed to not rule out at least one model in each orbit. Note that symmetry-breaking preserves
satisfiability of a theory.
Symmetries can be found in CNF theories using a reduction to graph isomorphism, a problem that is
thought to require super-polynomial time in the worst case, but which can often be efficiently solved
in practice [18]. The added clauses are called symmetry-breaking predicates (SBPs). If we place
a fixed order on the atoms of theory, then a model can be associated with a binary number, where
the i-th digit, 0 or 1, specifies the value of the i-th atom, false or true. Lex-leader SBPs rule out
models that are not the lexicographically-smallest members of their orbits. The formulation below
is equivalent to the lex-leader SBP given by Crawford et. al. in [7]:
^ ^
SBP (?) =
vj ? ?(vj ) ? vi ? ?(vi )
(1)
1?i?n
1?j<i
where vi is the ith variable in the ordering of n variables, and ? is a symmetry over variables.
Even though graph isomorphism is relatively fast in practice, a theory may have exponentially many
symmetries. Therefore, breaking all the symmetries is often impractical, though partial symmetrybreaking is still useful. It is possible to devise new SBPs that can break exponentially more symmetries than the standard form described above; we do so in Section 5.2.
Relational Theories We define a relational theory as a tuple T = (F, W, E), where F is a set
of first-order formulas, W a mapping of predicates and negated predicates to strictly positive real
numbers (weights), and E is a set of evidence. We restrict the formulas in F to be built from
predicates, variables, quantifiers, and logical connectives, but no constants or function symbols. E
is a set of ground literals; that is, literals built from predicates and constant symbols. The predicate
arguments and constant symbols are typed. Universal and existential quantification is over the set of
the theory?s constants D (i.e. the constants that appear in its evidence). Any constants not appearing
explicitly in the evidence can be incorporated by introducing a unary predicate for each constant
type and adding the groundings for those constants to the evidence. Any formula containing a
constant can be made constant-free, by introducing a new unary predicate for each constant, and
then including that predicate applied to that constant in the evidence. A ground theory can be seen
as a special case of a relational theory where each predicate is argument free.
We define the weight of a positive ground literal P (C1 , . . . , Ck ) of a theory as W (P ), and the weight
of negative ground literal ?P (C1 , . . . , Ck ) as W (?P ). In other words, all positive groundings of a
literal have the same weight, as do all negative groundings. The weight of model M with respect
to a theory (F, W, E) is 0 if M fails to satisfy any part of F or E; otherwise, it is the product of the
weights of the ground atoms that are true in M . Maximum weighted model-finding is the task for
finding a model of maximum weight with respect to T . A relational theory can be taken to define a
probability distribution over the set of models, where the probability of model is proportional to its
2
weight. Maximum weighted model-finding thus computes MAP (most probable explanation) for a
given theory. Ordinary satisfiability corresponds to the case where W simply sets the weights of all
literals to 1.
Languages such as Markov Logic [12] use an alternative representation and specify real-valued
weights on formulas rather than positive weights on predicates and their negations. The MAP problem can be formulated as the weighted-MaxSAT problem, i.e. finding a model maximizing the sum
of the weights of satisfied clauses. This can be translated to our notation by introducing a new predicate for each original formula, whose arguments are the free variables in the original formula. F
asserts that the predicate is equivalent to the original formula, and W asserts that the weight of the
new predicate is e raised to the weight of the original formula. Solving weighted MaxSAT in the
alternate representation is thus identical to solving maximum weighted model-finding in the translated theory. For the rest of the discussion in this paper, we will assume that the theory is specified
with weights on predicates (and their negations).
3
Related Work
Our work has connections to research in both the machine learning and constraint-satisfaction research communities. Most research in statistical-relational machine learning has concentrated on
modifying or creating novel probabilistic inference algorithms to exploit symmetries, as opposed to
symmetry-breaking?s solver-independent approach. Developments include lifted versions of variable elimination [27, 8], message passing [29, 30, 23], and DPLL [14]. Our approach of defining
symmetries using group theory and detecting them by graph isomorphism is shared by Bui et al.?s
work on lifted variational inference [5] and Apsel et al.?s work on cluster signatures [2]. Bui notes
that symmetry groups can be defined on the basis of unobserved constants in the domain, while we
have developed methods to explicitly find symmetries among constants that do appear in the evidence. Niepert also gives a group-theoretic formalism of symmetries in relational theories [24, 25],
applying them to MCMC methods. Another line of work is to make use of problem transformations.
First-order knowledge compilation [10, 11] transforms a relational problem into a form for which
MAP and marginal inference is tractable. This is a much more extensive and computationally complex transformation than symmetry-breaking. Mladenov et al. [22] propose an approach to translate
a linear program over a relational domain into an equivalent lifted linear program based on message
passing computations, which can then be solved using any off-the-shelf LP solver. Recent work
on MAP inference in Markov Logic has identified special cases where a relational formula can be
transformed by replacing a quantified formula with a single grounding of the formula [21].
Relatively little work in SRL has explicitly examined the role of evidence, separate from the firstorder part of a theory, on symmetries. One exception is [31], which presents a heuristic method
for approximating an evidence set in order to increase the number of symmetries it induces. Bui et
al. [4] consider the case of a theory plus evidence pair, where the theory has symmetries that are
broken by the evidence. They show that if the evidence is soft and consists only of unary predicates,
then lifting based on the theory followed by incorporation of evidence enables polynomial time
inference. Extending the results of [4], Van Den Broeck and Darwiche [9] show that dealing with
binary evidence is NP-hard in general but can be done efficiently if there is a corresponding low
rank Boolean matrix factorization. They also propose approximation schemes based on this idea.
We briefly touch upon the extensive literature that has grown around the use of symmetries in constraint satisfaction. Symmetry detection has been based either on graph isomorphism on propositional theories as in the original work by by Crawford et. al [7]; by interchangeability of row and/or
columns in CSPs specified in matrix form [20]; by checking for other special cases of geometric
symmetries [28], or by determining that domain elements for a variable are exchangeable [3]. (The
last is a special case of our term equivalent symmetries.) Researchers have suggested symmetryaware modifications to backtracking CSP solvers for variable selection, branch pruning, and nogood learning [20, 13]. A recent survey of symmetry breaking for CSP [32] described alternatives
to the lex-leader formulation of SBPs, including one based on Gray codes.
4
Symmetries in Relational Theories
In this section, we will formally introduce the notion of symmetries over relational theories and
give efficient algorithms to find them. Symmetries of a relational theory can be defined in terms of
symmetries over the corresponding ground theory.
3
Definition 4.1. Let T denote a relational theory. Let the T G denote the theory obtained by grounding
the formulas in T . Let L denote the set of (ground) literals in T G . We say that a permutation ? of
the set L is a symmetry of the relational theory T if ? maps the ground theory T G back to itself
i.e. ?(T G ) = T G . We denote the action of ? on the original theory as ?(T ) = T .
A straightforward way to find symmetries over a relational theory T is to first map it to corresponding ground theory T G and then find symmetries over it using reduction to graph isomorphism. The
complexity of finding symmetries in this way is the same as that of graph isomorphism, which is
believed to be worst-case super-polynomial. Further, the number of symmetries found is potentially
exponential in the number of ground literals. This is particularly significant for relational theories
since the number of ground literals itself is exponential in the highest predicate arity. Computing
symmetries at the ground level can therefore be prohibitively expensive for theories with high predicate arity and many constants. In our work below, we exploit the underlying template structure of
the relational theory to directly generate symmetries based on the evidence.
4.1
Term Symmetries
We introduce the notion of symmetries defined over terms (constants) appearing in a theory T , called
term symmetries.
Definition 4.2. Let T be a relational theory. Let D be the set of constants appearing in the theory.
Then, a permutation ? over the term set D is said to be a term symmetry with respect to evidence E
if application of ? on the terms appearing in E, denoted by ?(E), maps E back to itself. We will also
refer to ? as an evidence symmetry for the set E.
The problem of finding term symmetries can be reduced to colored graph isomorphism. We construct a graph G as follows: for each predicate P and it?s negation, G has a node and a unique color
is assigned to every such node. G also has a unique color for each type in the domain. There is a
node for every term which takes the color of its type. We call an ordered list of terms an argument
list, e.g., given the literal P (C1 , C2 ), where C1 , C2 ? D, the argument list is (C1 , C2 ). The type of
an argument list is simply the cross-product of the types of the terms appearing in it. G has a node
for every argument list appearing in the evidence, which takes the color of its type.
For every evidence literal, there is an edge between the predicate node (or its negation) and the
corresponding argument list node. There is also an edge between the argument list node and each of
the terms appearing in the list. Thus, for the previous example, an edge will be placed between the
node for P and the node for (C1 , C2 ), as well as an edge each between (C1 , C2 ) and the nodes for
C1 and C2 .
Any automorphism of G will map (negated) predicate nodes to themselves and terms will be mapped
in a manner that their association with the corresponding predicate node in the evidence is preserved.
Hence, automorphisms of G will correspond to term symmetries in evidence E. Next, we will
establish a relationship between permutation of terms in the evidence to the permutations of literals
in the ground theory.
Definition 4.3. Let T be a relational theory. Let E be the evidence set and let D be the set of
terms appearing in E. Given a permutation ? of the terms in the set D, we associate a corresponding permutation ?T over the ground literals of the form P (C1 , C2 , . . . , Ck ) in T , such that
?T (P (C1 , C2 , . . . , Ck )) = P (?(C1 ), ?(C2 ), . . . , ?(Ck )) (and similarly for negated literals).
We can now associate a symmetry ? over the terms with a symmetry of the theory T . The following
lemma is proven in the supplementary material:
Lemma 4.1. Let T be a relational theory. Let E denote the evidence set. Let D be the set of terms
appearing in E. If ? is term symmetry for the evidence E, then, the associated theory permutation
?T is also a symmetry of T .
In order to find the term symmetries, we can resort to solving a graph isomorphism problem of
size O(|E|), where |E| is the number of literals in the evidence. Directly finding symmetries over the
ground literals requires solving a problem of size O(|D|k ), |D| being the set of constants and k being
the highest predicate arity. In the worst case where everything is fully observed, |E| = O(|D|k ), but
in practice it is much smaller. Next, we present an important subclass of term symmetries, called
term equivalent symmetries, which capture a wide subset of all the symmetries present in the theory,
and can be efficiently detected and broken.
4
4.2
Term Equivalent Symmetries
A term equivalent symmetry is a set of term symmetries that divides the set of terms into equivalence
classes such that any permutation which maps terms within the same equivalence class is a symmetry
of the evidence set. Let Z = {Z1 , Z2 , . . . , Zm } denote a partition of the term set D into m disjoint
subsets. Given a partition Z, we say that two terms are term equivalent (with respect to Z) if they
occur in the same component of Z. We define a partition preserving permutation as follows.
Definition 4.4. Given a set of terms D and its disjoint partition Z, we say that a permutation ? of the
terms in D is a partition preserving permutation of D with respect to the partition Z if ?Cj , Ck ?
D, ?(Cj ) = Ck implies that ? Zi ? Z st Cj , Ck ? Zi . In other words, ? is partition preserving if it
permutes terms within the same component of Z.
The set of all partition preserving permutations (with respect to a partition Z) forms a group. We
will denote this group by ?Z . It is easy to see that ?Z divides the set of terms in equivalence classes.
Next, we define the notion of term equivalent symmetries.
Definition 4.5. Let T be a relational theory and E denote the evidence set. Let D be the set of terms
in E and Z be a disjoint partition of terms in D. Then, given the partition preserving permutation
?Z , we say that ?Z is a term equivalent symmetry group of D, if ?? ? ?Z , ? is a symmetry of E.
We will refer to each symmetry ? ? ?Z as a term equivalent symmetry of E.
A partition Z of term set D is a term equivalent partition if the partition preserving group ?Z is
a symmetry group of D. We refer to each partition element Zi as a term equivalent subset. The
term equivalent symmetry group can be thought of as a set of symmetry subgroups ?Zi ?s, one for
each term subset Zi , such that, Zi allows for all possible permutations of terms within the set Zi
and defines an identity mapping for terms in other subsets. Note that the size of term equivalent
symmetry group is given by ?m
i=1 |Zi |!. Despite its large size, it can be very efficiently represented
by simply storing the partition Z over the term set D. Note that this formulation works equally well
for typed as well untyped theories; in a typed theory no two terms of differing types can appear in the
same subset. Next, we will look at an efficient algorithm for finding a partition Z which corresponds
to a term equivalent symmetry group over D.
Let the evidence be given by E = {l1 , l2 , . . . , lk }, where each li is a ground literal. Intuitively,
two terms are term equivalent if they occur in exactly the same context in the evidence. For
example, if evidence for constant A is {P1 (A, X), P2 (A, Y, A)}, then the context for term A is
P1 (?, X), P2 (?, Y, ?). Note that here the positions where A occurred in the evidence has been
marked by a ?. Any other term sharing the same context would be term equivalent to A. To find
the set of all the equivalent terms, we first compute the context for each term. Then, we sort each
context based on some lexicographic order defined over predicate symbols and term symbols. Once
the context has been sorted, we can simply hash the context for each term and put those which have
the same context in the same equivalence class. If the evidence size is given by |E| = M and number
of terms in evidence is n, then, the above procedure will take O(nM log(M )) time. The M log(M )
factor is for sorting the context for a single term. Hashing the sorted term takes constant time. This
is done for each term, hence the factor of n. See the supplement for more details and an example.
5
Breaking Symmetries Over Terms
In this section, we provide the SBP formulation for term as well as term equivalent symmetries.
5.1
Breaking Term Symmetries
Consider a relational theory T that has term symmetries {?1 , . . . , ?k }. Fix an ordering P1 , . . . , Pr
over the predicates, an ordering over the predicate positions, and an ordering C1 , . . . , C|D| over the
terms. If T is typed, fix an ordering over types and an ordering over the terms within each type.
This induces a straightforward ordering over the ground atoms of the theory G1 , . . . , Gn .
Let ? be a term symmetry. Consider the following SBP (based on Equation 1) to break ?:
^ ^
SBP (?) =
Gj ? ?(Gj ) ? Gi ? ?(Gi )
1?i?n
1?j<i
Theorem 5.1. If T is weighted, the max model for T ? SBP (?) is a max model for T , and it has
the same weight in both theories.
5
The proof of this theorem follows from [7]. Essentially, an ordering is placed on the atoms of the
theory, which induces an ordering on the models. The SBP constraints ensure that if a set of models
are in the same orbit, i.e. symmetric under ?, then only the first of those models in the ordering is
admitted.
5.2
Breaking Term Equivalent Symmetries
i
Let Z = {Z1 , . . . , Z|Z| } be the term equivalent partitioning over the terms. Let ?j,k
be the term
symmetry that swaps Cj and Ck , the jth and kth constants in an ordering over the term equivalent
subset Zi , and maps everything else to identity. We next show how to break exponentially many
symmetries that respect the term equivalent partition Z, using a linear-sized SBP. Consider the
following SBP (CSBP stands for Composite SBP):
CSBP (Z) =
|Z| |Zi |?1
^
^
i
SBP (?j,j+1
)
i=1 j=1
i
For each term (equivalent) symmetry of the form ?j,j+1
, we apply the SBP formulation from Section 5.1 to break it. The formulation is cleverly choosing which symmetries to explicitly break, so
that exponentially many symmetries are broken. The inner conjunction iterates over all of the terms
in a term equivalent class. An ordering is placed on the terms, and the only symmetries that are
explicitly broken are those that swap two adjacent terms and map everything else to identity. All
of the adjacent pairs of terms in the ordering are broken in this way, with a linear number of calls
to SBP. As we show below, this excludes ?(2|Zi | ) models from the search space, while preserving
at least one model in each orbit. The outer conjunction iterates over the different term equivalent
subsets, breaking each one of them in turn. The following theorem states this in formal terms (see
the supplement for a proof):
P|Z|
Theorem 5.2. CSBP (Z) removes ?( i=1 2|Zi | ) models from the search space while preserving
at least one model in each orbit induced by the term equivalent symmetry group ?Z .
Corollary 5.1. If T is weighted, the max model for T ? CSBP (Z) is a max model for T , and it
has the same weight in both theories.
6
Experiments
We pitted our term and term equivalent SBP formulations against other inference systems to demonstrate their efficacy. For each domain we tested on, we compared the following algorithms:
(1) Vanilla: running an exact MaxSAT solver on the grounded instance (2) Shatter: running an
exact MaxSAT solver on the grounded instance with SBPs added by the Shatter tool [1], (3) Term:
running an exact MaxSAT solver on the grounded instance with SBPs added by our term symmetry
detection algorithm, (4) Tequiv: running an exact MaxSAT solver on the grounded instance with
SBPs added by our term equivalent symmetry detection algorithm (5) RockIt: running RockIt [26]
on the MLN, (6) PTP: running the PTP algorithm [14] for marginal inference 1 on the MLN and
(7) MWS: running the approximate MaxWalkSAT algorithm used by Alchemy-22 on the grounded
instance. These algorithms were chosen so that our methods would be compared against a variety of
other techniques: ground techniques including regular MaxSAT (Vanilla), propositional symmetrybreaking (Shatter), local search (MWS), and lifted techniques including cutting planes (RockIt) and
lifted rules (PTP). It should be noted that all the algorithms except MWS and RockIt are exact.
RockIt solves an LP relaxation but it always gave us an exact solution (whenever it could run).
Therefore, we report the solution quality only for MWS. All the experiments were run on a system
with a 2.2GHz Xeon processor and 62GB of memory.
The algorithms for Term and Tequiv were implemented as a pre-processing step before running a
MaxSAT solver. The MWS runs were allowed a varying number of flips and up to five restarts.
For MWS, the reported results are with equal probability (0.5) of making a random or a greedy
move; other settings were tried, and the results were similar. Both PTP and RockIt were run using
the default setting of their parameters. We experimented on two relational domains: the classic
pigeonhole problem (PHP) with two different variants, and an Advisor domain (details below).
1
The publicly available implementation of PTP only supports marginal inference. Though marginal inference is somewhat harder problem than MAP, PTP failed to run even on the smallest of instances.
2
https://code.google.com/p/alchemy-2/
6
For ground instances that required an exact MaxSAT solver, we experimented with three different
solvers: MiniMaxSAT [15], Open-WBO [19], and Sat4j [17]. This was done because different
solvers and heuristics tend to work better on different application domains. We found that for PHP
(variant 1) Sat4j worked best, for PHP (variant 2) Open-WBO was the best and for Advisor MiniMaxSAT worked the best. We used the best MaxSAT solver for each domain.
PHP: In the PHP problem, we are given n pigeons and n?1 holes. The goal is to fit each pigeon in a
unique hole (the problem is unsatisfiable). We tried two variants of this problem. In the first variant,
there are hard constraints that ensure that every hole has at most one pigeon and every pigeon is
in at most one hole. There is a soft constraint that states that every pigeon is in every hole. The
goal is to find a max-model of the problem. The comparison with various algorithms is given in
Table 1a. Vanilla times out except for the smallest instance. PTP times out on all instances. Our
algorithms consistently outperform RockIt but are not able to outperform shatter (except for smaller
instances where they are marginally better). MWS was run on the largest instance (n=60) and results
are shown in Table 1d. MWS is unable to find the optimal solution after 10 million flips.
In the second PHP variant, there are hard constraints that ensure that every pigeon is in precisely one
hole. There is a soft constraint that each hole has at most one pigeon. The comparison with various
algorithms is given in in Table 1b. Vanilla times out except for the two smallest instances, and PTP
times out for all the instances. Our systems consistently outperform RockIt, and outperform shatter
for the larger instances. Tequiv outperforms Term because the detection step is much faster. MWS
(Table 1d) is able to find the optimal within 100,000 flips and is significantly faster than all other
algorithms. Though MWS does better on this variant of PHP, it fails badly on the first one. Further,
there is no way for MWS to know if it has reached the optimal unlike our (exact) approaches.
Advisor Domain: The advisor-advisee domain has three types: student, prof, and a research area.
There are predicates to indicate student/prof interest in an area and an advisor relationship between
students and profs. The theory has hard constraints that ensure that a) each student has precisely one
advisor b) students and their advisors share an interest. It also has a small negatively-weighted (soft)
constraint saying that a prof advises two different students. The interests of all students and profs
are fully observed. Students have one interest, profs have one or more (chosen randomly). There
is at least one prof interested in each area. The results for this domain are given in Table 1c gives
the comparisons. Vanilla is able to run only on the two smallest instances. PTP times out on every
instance as before. Our algorithms outpeform shatter but is outperformed by RockIt. MWS was run
on the two larger sized instances (see Table 1c) and it is able to outperform our system.
In order to make sure that poor performance of PTP is not due to evidence, we modified the PHP
formulation to work with a no evidence formulation. This did not help PTP either. Based on these
results, there is no clear winner among the algorithms which performs the best on all of these domains. Among all the algorithms, Term, Tequiv and Shatter are the only ones which could run to
completion on all the instances that we tried. Among these, our algorithms won on PHP variant
2 (large instances) and the Advisor domain. Tequiv outperformed Term on PHP variant 2 (large
instances), and they performed similarly in the Advisor domain.
In the PHP variants, all of the pigeons are in one term equivalent symmetry group, and all of the
holes are in another. The PHP is of special interest, because pigeonhole structure was shown to be
present in hard random SAT formulas with very high probability [6]. In the advisor domain, there
is a term equivalent symmetry group of students for each area, and term equivalent symmetry group
of professors for each set of areas.
7
Conclusion and Future Work
In this work, we have provided the theoretical foundation for using symmetry-breaking techniques
from satisfiability testing for MAP inference in weighted relational theories. We have presented the
class of term symmetries, and a subclass of term equivalent symmetries, both of which are found in
the evidence given with the theory. We have presented ways to detect and break both these classes.
For the class of term equivalent symmetries, the detection method runs in low-polynomial time, and
the number of SBPs required to break the symmetries is linear in the domain size. the algorithms
presented, and compared them against other systems on a set of relational domains. Future work
includes carefully characterizing the cases where our algorithms perform better than other systems,
experimenting on additional real-world domains, comparing with more lifted inference approaches,
and the application of similar techniques to the problem of marginal inference.
7
Table 1: Experiments with Lifted Symmetries for MAP Inference
(a) Soft Pigeon Hole Domain ? Variant 1
Vanilla
0.246
TO
TO
TO
TO
TO
TO
#Pigeon
5
10
15
20
30
40
60
Shatter
0.01 + 0.39
0.02 + 0.88
0.08 + 1.85
0.22 + 3.1
1.6 + 12.7
7 + 72
54 + 889
Tequiv
0.01 + 0.24
0.04 + 0.74
0.13 + 1.37
0.36 + 3.9
1.4 + 20
4.06 + 93.3
18 + 1128
Term
0.03 + 0.26
0.09 + 0.68
0.21 + 1.27
0.51 + 3.7
2.2 + 14
6.30 + 88.3
28 + 1128
RockIt
1.15
1.47
2.46
2.70
3.88
F
F
PTP
TO
TO
TO
TO
TO
TO
TO
(b) Soft Pigeon Hole Domain ? Variant 2
#Pigeon
5
10
15
40
80
125
Vanilla
0.002
37.44
TO
TO
TO
TO
Shatter
0.003 + 0.001
0.01 + 0.006
0.06 + 0.01
3.7 + 0.59
115 + 8.36
1090 + 20.98
Tequiv
1.28 + 0.002
1.19 + 0.004
1.29 + 0.01
4.9 + 0.17
50 + 1.7
269 + 10
Term
0.04 + 0.001
0.07 + 0.005
0.19 + 0.01
6.2 + 0.21
86 + 2.9
486 + 18
RockIt
1.1
6.6
TO
F
F
F
PTP
TO
TO
TO
TO
TO
TO
(c) Advisor Domain
#Prof?#Student?#Area
4?12?2
4?16?2
6?18?3
6?24?3
8?24?4
Vanilla
2.3
168
TO
TO
TO
Shatter
0.01 + 0.10
0.01 + 0.17
0.02 + 0.90
0.03 + 105
0.07 + 62
Tequiv
0.02 + 0.01
0.03 + 0.05
0.05 + 0.84
0.07 + 45
0.10 + 56
Term
0.11 + 0.01
0.08 + 0.04
0.09 + 0.80
0.15 + 47
0.16 + 55
RockIt
1.25
2.06
6.56
1.47
13.4
PTP
TO
TO
TO
TO
TO
(d) MaxWalkSAT Results
Problem
Size
Optimal
pigeon1
pigeon2
advisor
advisor
60
125
6?18?3
8?24?4
3481
1
2094
3552
100,000
best time
3496
4.4
1
3.8
2094
0.4
3552
0.5
1,000,000
best time
3494
39
?
?
?
?
?
?
10,000,000
best time
3491 383
?
?
?
?
?
?
All times are given in seconds. ?TO? indicates the program timed out (30min). ?F? indicates the program failed
without timeout (instance too large). The vanilla column is the time to find the max model of the ground theory.
The first three subtables give results for exact algorithms. The Shatter, Term, and Tequiv columns are the
same with propositional, term and term equivalent symmetries broken, respectively. These columns are given
as x + y, where x is the time to detect and break the symmetries, and y is the time to find the max model.
The RockIt column is the time it takes RockIt to find the max model. The lifted PTP algorithm was run on
every instance, and timed out on each instance. The last table gives the results for the approximate algorithm
MaxWalkSAT, giving the time it takes to perform a certain number of iterations, as well as the comparison
between the best model found on that run and the optimal model.
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9
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5,183 | 5,692 | Bandits with Unobserved Confounders:
A Causal Approach
Andrew Forney?
Department of Computer Science
University of California, Los Angeles
[email protected]
Elias Bareinboim?
Department of Computer Science
Purdue University
[email protected]
Judea Pearl
Department of Computer Science
University of California, Los Angeles
[email protected]
Abstract
The Multi-Armed Bandit problem constitutes an archetypal setting for sequential
decision-making, permeating multiple domains including engineering, business,
and medicine. One of the hallmarks of a bandit setting is the agent?s capacity
to explore its environment through active intervention, which contrasts with the
ability to collect passive data by estimating associational relationships between
actions and payouts. The existence of unobserved confounders, namely unmeasured variables affecting both the action and the outcome variables, implies that
these two data-collection modes will in general not coincide. In this paper, we
show that formalizing this distinction has conceptual and algorithmic implications
to the bandit setting. The current generation of bandit algorithms implicitly try to
maximize rewards based on estimation of the experimental distribution, which
we show is not always the best strategy to pursue. Indeed, to achieve low regret
in certain realistic classes of bandit problems (namely, in the face of unobserved
confounders), both experimental and observational quantities are required by the
rational agent. After this realization, we propose an optimization metric (employing both experimental and observational distributions) that bandit agents should
pursue, and illustrate its benefits over traditional algorithms.
1
Introduction
The Multi-Armed Bandit (MAB) problem is one of the most popular settings encountered in the
sequential decision-making literature [Rob52, LR85, EDMM06, Sco10, BCB12] with applications
across multiple disciplines. The main challenge in a prototypical bandit instance is to determine
a sequence of actions that maximizes payouts given that each arm?s reward distribution is initially
unknown to the agent. Accordingly, the problem revolves around determining the best strategy for
learning this distribution (exploring) while, simultaneously, using the agent?s accumulated samples
to identify the current ?best? arm so as to maximize profit (exploiting). Different algorithms employ
different strategies to balance exploration and exploitation, but a standard definition for the ?best?
arm is the one that has the highest payout rate associated with it. We will show that, perhaps
surprisingly, the definition of ?best? arm is more involved when unobserved confounders are present.
This paper complements the vast literature of MAB that encompasses many variants including adversarial bandits (in which an omnipotent adversary can dynamically shift the reward distributions
to thwart the player?s best strategies) [BFK10, AS95, BS12] contextual bandits (in which the payout,
?
The authors contributed equally to this paper.
1
and therefore the best choice of action, is a function of one or more observed environmental variables) [LZ08, DHK+ 11, Sli14], and many different constraints and assumptions over the underlying
generative model and payout structure [SBCAY14]. For a recent survey, see [BCB12].
This work addresses the MAB problem when unobserved confounders are present (called MABUC,
for short), which is arguably the most sensible assumption in real-world, practical applications (obviously weaker than assuming the inexistence of confounders). To support this claim, we should first
note that in the experimental design literature, Fisher?s very motivation for considering randomizing
the treatment assignment was to eliminate the influence of unobserved confounders ? factors that
simultaneously affect the treatment (or bandit arm) and outcome (or bandit payout), but are not accounted for in the analysis. In reality, the reason for not accounting for such factors explicitly in the
analysis is that many of them are unknown a priori by the modeller [Fis51].
The study of unobserved confounders is one of the central themes in the modern literature of causal
inference. To appreciate the challenges posed by these confounders, consider the comparison between a randomized clinical trial conducted by the Food and Drug Administration (FDA) versus
physicians prescribing drugs in their offices. A key tenet in any FDA trial is the use of randomization for the treatment assignment, which precisely protects against biases that might be introduced
by physicians. Specifically, physicians may prescribe Drug A for their wealthier patients who have
better nutrition than their less wealthy ones, when unknown to the doctors, the wealthy patients
would recover without treatment. On the other hand, physicians may avoid prescribing the expensive Drug A to their less privileged patients, who (again unknown to the doctors) tend to suffer less
stable immune systems causing negative reactions to the drug. If a naive estimate of the drug?s causal
effect is computed based on physicians? data (obtained through random sampling, but not random
assignment), the drug would appear more effective than it is in practice ? a bias that would otherwise
be avoided by random assignment. Confounding biases (of variant magnitude) appear in almost any
application in which the goal is to learn policies (instead of statistical associations), and the use of
randomization of the treatment assignment is one established tool to combat them [Pea00].
To the best of our knowledge, no method in the bandit literature has studied the issue of unobserved
confounding explicitly, in spite of its pervasiveness in real-world applications. Specifically, no MAB
technique makes a clear-cut distinction between experimental exploration (through random assignment as required by the FDA) and observational data (as given by random sampling in the doctors?
offices). In this paper, we explicitly acknowledge, formalize, and then exploit these different types
of data-collection. More specifically, our contributions are as follow:
? We show that the current bandit algorithms implicitly attempt to maximize rewards by estimating the experimental distribution, which does not guarantee an optimal strategy when
unobserved confounders are present (Section 2).
? Based on this observation, we translate the MAB problem to causal language, and then
suggest a more appropriate metric that bandit players should optimize for when unobserved
confounders are present. This leads to a new exploitation principle that can take advantage
of data collected under both observational and experimental modes (Section 3).
? We empower Thompson Sampling with this new principle and run extensive simulations.
The experiments suggest that the new strategy is stats. efficient and consistent (Sec. 4).
2
Challenges due to Unobserved Confounders
In this section, we discuss the mechanics of how the maximization of rewards is treated based
on a bandit instance with unobserved confounders. Consider a scenario in which a greedy casino
decides to demo two new models of slot machines, say M1 and M2 for simplicity, and wishes to
make them as lucrative as possible. As such, they perform a battery of observational studies (using
random sampling) to compare various traits of the casino?s gamblers to their typical slot machine
choices. From these studies, the casino learns that two factors well predict the gambling habits of
players when combined (unknown by the players): player inebriation and machine conspicuousness
(say, whether or not a machine is blinking). Coding both of these traits as binary variables, we let
B ? {0, 1} denote whether or not a machine is blinking, and D ? {0, 1} denote whether or not the
gambler is drunk. As it turns out, a gambler?s ?natural? choice of machine, X ? {M1 , M2 }, can be
modelled by the structural equation indicating the index of their chosen arm (starting at 0):
X ? fX (B, D) = (D ? ?B) ? (?D ? B) = D ? B
2
(1)
Figure 1: Performance of different bandit strategies in the greedy casino example. Left panel:
no algorithm is able to perform better than random guessing. Right panel: Regret grows without
bounds.
Moreover, the casino learns that every gambler has an equal chance of being intoxicated and each
machine has an equal chance of blinking its lights at a given time, namely, P (D = 0) = P (D =
1) = 0.5 and P (B = 0) = P (B = 1) = 0.5.
The casino?s executives decide to take advantage of these propensities by introducing a new type
of reactive slot machine that will tailor payout rates to whether or not it believes (via sensor input,
assumed to be perfect for this problem) a gambler is intoxicated. Suppose also that a new gambling
law requires that casinos maintain a minimum attainable payout rate for slots of 30%. Cognizant of
this new law, while still wanting to maximize profits by exploiting gamblers? natural arm choices,
the casino executives modify their new slots with the payout rates depicted in Table 1a.
(a)
X = M1
X = M2
D=0
B=0 B=1
*0.10
0.50
0.50 *0.10
D=1
B=0 B=1
0.40 *0.20
*0.20
0.40
(b)
X = M1
X = M2
P (y|X)
0.15
0.15
P (y|do(X))
0.3
0.3
Table 1: (a) Payout rates decided by reactive slot machines as a function of arm choice, sobriety,
and machine conspicuousness. Players? natural arm choices under D, B are indicated by asterisks.
(b) Payout rates according to the observational, P (Y = 1|X), and experimental P (Y = 1|do(X)),
distributions, where Y = 1 represents winning (shown in the table), and 0 otherwise.
The state, blind to the casino?s payout strategy, decides to perform a randomized study to verify
whether the win rates meet the 30% payout requisite. Wary that the casino might try to inflate
payout rates for the inspectors, the state recruits random players from the casino floor, pays them to
play a random slot, and then observes the outcome. Their randomized experiment yields a favorable
outcome for the casino, with win rates meeting precisely the 30% cutoff. The data looks like Table
1b (third column), assuming binary payout Y ? {0, 1}, where 0 represents losing, and 1 winning.
As students of causal inference and still suspicious of the casino?s ethical standards, we decide to go
to the casino?s floor and observe the win rates of players based on their natural arm choices (through
random sampling). We encounter a distribution close to Table 1b (second column), which shows
that the casino is actually paying ordinary gamblers only 15% of the time.
In summary, the casino is at the same time (1) exploiting the natural predilections of the gamblers?
arm choices as a function of their intoxication and the machine?s blinking behavior (based on eq. 1),
(2) paying, on average, less than the legally allowed (15% instead of 30%), and (3) fooling state?s
inspectors since the randomized trial payout meets the 30% legal requirement.
As machine learning researchers, we decide to run a battery of experiments using the standard bandit
algorithms (e.g., -greedy, Thompson Sampling, UCB1, EXP3) to test the new slot machines on the
casino floor. We obtain data encoded in Figure 1a, which shows that the probability of choosing
the correct action is no better than a random coin flip even after a considerable number of steps.
We note, somewhat surprised, that the cumulative regret (Fig. 1b) shows no signs of abating, and
3
that we are apparently unable to learn a superior arm. We also note that the results obtained by the
standard algorithms coincide with the randomized study conducted by the state (purple line).
Under the presence of unobserved confounders such as in the casino example, however, P (y|do(X))
does not seem to capture the information required to maximize payout, but rather the average payout
akin to choosing arms by a coin flip. Specifically, the payout given by coin flipping is the same for
both machines, P (Y = 1|do(X = M1 )) = P (Y = 1|do(X = M2 )) = 0.3, which means that
the arms are statistically indistinguishable in the limit of large sample size. Further, if we consider
using the observational data from watching gamblers on the casino floor (based on their natural
predilections), the average payoff will also appear independent of the machine choice, P (Y =
1|X = M1 ) = P (Y = 1|X = M2 ) = 0.15, albeit with an even lower payout. 1
Based on these observations, we can see why no arm choice is better than the other under either distribution alone, which explains the reason any algorithm based on these distributions will certainly
fail to learn an optimal policy. More fundamentally, we should be puzzled by the disagreement
between observational and interventional distributions. This residual difference may be encoding
knowledge about the unobserved confounders, which may lead to some indication on how to differentiate the arms. This indeed may lead to some indication on how to differentiate the arms as well as
a sensible strategy to play better than pure chance. In the next section, we will use causal machinery
to realize this idea.
3
Bandits as a Causal Inference Problem
We will use the language of structural causal models [Pea00, Ch. 7] for expressing the bandit datagenerating process and for allowing the explicit manipulation of some key concepts in our analysis ?
i.e., confounding, observational and experimental distributions, and counterfactuals (to be defined).
Definition 3.1. (Structural Causal Model) ([Pea00, Ch. 7]) A structural causal model M is a
4-tuple h U, V, f, P (u) i where:
1. U is a set of background variables (also called exogenous), that are determined by factors
outside of the model,
2. V is a set {V1 , V2 , ..., Vn } of observable variables (also called endogenous), that are determined by variables in the model (i.e., determined by variables in U ? V ),
3. F is a set of functions {f1 , f2 , ..., fn } such that each fi is a mapping from the respective
domains of Ui ? P Ai to Vi , where Ui ? U and P Ai ? V \ Vi and the entire set F forms a
mapping from U to V . In other words, each fi in vi ? fi (pai , ui ), i = 1, ..., n, assigns a
value to Vi that depends on the values of the select set of variables (Ui ? P Ai ), and
4. P (u) is a probability distribution over the exogenous variables.
Each structural model M is associated with a directed acyclic graph G, where nodes correspond to
endogenous variables V and edges represent functional relationships ? i.e., there exists an edge from
X to Y whenever X appears in the argument of Y ?s function. We define next the MABUC problem
within the structural semantics.
Definition 3.2. (K-Armed Bandits with Unobserved Confounders) A K-Armed bandit problem
with unobserved confounders is defined as a model M with a reward distribution over P (u) where:
1. Xt ? {x1 , ..., xk } is an observable variable encoding player?s arm choice from one of k
arms, decided by Nature in the observational case, and do(Xt = ?(x0 , y0 , ..., xt?1 , yt?1 )),
for strategy ? in the experimental case (i.e., when the strategy decides the choice),
2. Ut represents the unobserved variable encoding the payout rate of arm xt as well as the
propensity to choose xt , and
3. Yt ? 0, 1 is a reward (0 for losing, 1 for winning) from choosing arm xt under unobserved
confounder state ut decided by yt = fy (xt , ut ).
1
One may surmise that these ties are just contrived examples, or perhaps numerical coincidences, which
do not appear in realistic bandit instances. Unfortunately, that?s not the case as shown in the other scenarios
discussed in the paper. This phenomenon is indeed a manifestation of the deeper problem arising due to the
lack of control for the unobserved confounders.
4
Figure 2: (a) Model for the standard MAB sequential decision game. (b) Model for the MABUC
sequential decision game. In each model, solid nodes denote observed variables and open nodes
represent unobserved variables. Square nodes denote the players strategys arm choice at time t.
Dashed lines illustrate influences on future time trials that are not pictured.
First note that this definition also applies to the MAB problem (without confounding) as shown
in Fig. 2a. This standard MAB instance is defined by constraining the MABUC definition such
that Ut affects only the outcome variable Yt ? there is no edge from Ut to Xt (Def. 3.2.2). In the
unconfounded case, it is clear that P (y|do(x)) = P (y|x) [Pea00, Ch. 3], which means that that
payouts associated with flipping a coin to randomize the treatment or observing (through random
sampling) the player gambling on the casino?s floor based on their natural predilections will yield
the same answer. The variable U carries the unobserved payout parameters of each arm, which is
usually the target of analysis. 2 3
Fig. 2b provides a graphical representation of the MABUC problem. Note that ?t represents the
system?s choice policy, which is affected by the unobserved factors encoded through the arrow from
Ut to ?t . One way to understand this arrow is through the idea of players? natural predilections.
In the example from the previous section, the predilection would correspond to the choices arising
when the gambler is allowed to play freely on the casino?s floor (e.g., drunk players desiring to play
on the blinking machines) or doctors prescribing drugs based on their gut feeling (e.g., physicians
prescribing the more expensive drug to their wealthier patients). These predilections are encoded in
the observational distribution P (y|x). On the other hand, the experimental distribution P (y|do(x))
encodes the process in which the natural predilections are overridden, or ceased by external policies. In our example, this distribution arises when the government?s inspectors flip a coin and send
gamblers to machines based on the coin?s outcome, regardless of their predilections.
Remarkably, it is possible to use the information embedded in these distinct data-collection modes
(and their corresponding distributions) to understand players? predilections and perform better than
random guessing in these bandit instances. To witness, assume there exists an oracle on the casino?s
floor operating by the following protocol. The oracle observes the gamblers until they are about to
play a given machine. The oracle intercepts each gambler who is about to pull the arm of machine
M1 , for example, and suggests the player to contemplate whether following his predilection (M1 )
or going against it (playing M2 ) would lead to a better outcome. The drunk gambler, who is a clever
machine learning student and familiar with Fig. 1, says that this evaluation cannot be computed a
priori. He affirms that, despite spending hours on the casino estimating the payoff distribution based
on players? natural predilections (namely, P (y|x)), it is not feasible to relate this distribution with
the hypothetical construction what would have happened had he decided to play differently. He also
acknowledges that the experimental distribution P (y|do(x)), devoid of the gamblers? predilections,
does not support any clear comparison against his personal strategy. The oracle says that this type
of reasoning is possible, but first one needs to define the concept of counterfactual.
Definition 3.3. (Counterfactual) ([Pea00, pp. 204]) Let X and Y be two subsets of exogenous
variables in V . The counterfactual sentence ?Y would be y (in situation u), had X been x? is
interpreted as the equality with Yx (u) = y, with Yx (u) being the potential response of Y to X = x.
2
On a more fundamental level, it is clear that unconfoundedness is (implicitly) assumed not to hold in the
general case. Otherwise, the equality between observational and experimental distributions would imply that
no randomization of the action needs to be carried out since standard random sampling would recover the same
distribution. In this case, this would imply that many works in the literature are acting in a suboptimal way
since, in general, experiments are more expensive to perform than collecting data through random sampling.
3
The interventional nature of the MAB problem is virtually not discussed in the literature, one of the few
exceptions is the causal interpretation of Thompson Sampling established in [OB10].
5
This definition naturally leads to the judgement suggested by the oracle, namely, ?would I (the agent)
win (Y = 1) had I played on machine M1 (X = 1)?, which can be formally written as YX=1 = 1
(we drop the M for simplicity). Assuming that the agent?s natural predilection is to play machine
1, the oracle suggests an introspection comparing the odds of winning following his intuition or
going against it. The former statement can be written in counterfactual notation, probabilistically,
as E(YX=1 = 1|X = 1), which reads as ?the expected value of winning (Y = 1) had I play
machine 1 given that I am about to play machine 1?, which contrasts with the alternative hypothesis
E(YX=0 = 1|X = 1), which reads as ?the expected value of winning (Y = 1) had I play machine
1 given that I am about to play machine 0?. This is also known in the literature as the effect of the
treatment on the treated (ETT) [Pea00]. So, instead of using a decision rule comparing the average
payouts across arms, namely (for action a),
argmax E(Y |do(X = a)),
(2)
a
which was shown in the previous section to be insufficient to handle the MABUC, we should consider the rule using the comparison between the average payouts obtained by players for choosing
in favour or against their intuition, respectively,
argmax E(YX=a = 1|X = x),
(3)
a
where x is the player?s natural predilection and a is their final decision. We will call this procedure
RDC (regret decision criterion), to emphasize the counterfactual nature of this reasoning step and
the idea of following or disobeying the agent?s intuition, which is motivated by the notion of regret.
Remarkably, RDC accounts for the agents individuality and the fact that their natural inclination
encodes valuable information about the confounders that also affect the payout. In the binary case,
for example, assuming that X = 1 is the player?s natural choice at some time step, if E(YX=0 =
1|X = 1) is greater than E(YX=1 = 1|X = 1), this would imply that the player should refrain of
playing machine X = 1 to play machine X = 0.
Assuming one wants to implement an algorithm based on RDC, the natural question that arises
is how the quantities entailed by Eq. 3 can be computed from data. For the factors in the form
E(YX=1 = 1|X = 1), the consistency axiom [Pea00, pp. 229] implies that E(YX=1 = 1|X =
1) = E(Y = 1|X = 1), where the l.h.s. is estimable from observational data. Counterfactuals
in the form E(YX=a = 1|X = x), where a 6= x, can be computed in the binary case through
algebraic means [Pea00, pp. 396-7]. For the general case, however, ETT is not computable without
knowledge of the causal graph. 4 Here, ETT will be computed in an alternative fashion, based on the
idea of intention-specific randomization. The main idea is to randomize intention-specific groups,
namely, interrupt any reasoning agent before they execute their choice, treat this choice as intention,
delibarte, and then act. We discuss next about the algorithmic implementation of this randomization.
4
Applications & Experiments
Based on the previous discussion, we can revisit the greedy casino example from Section 2, apply
RDC and use the following inequality to guide agent?s decisions:
E(YX=0 |X = 1) > E(YX=1 |X = 1) ? E(YX=0 |X = 1) > P (Y |X = 1)
(4)
There are different ways of incorporating this heuristic into traditional bandit algorithms, and we
describe one such approach taking the Thompson Sampling algorithm as the basis [OB10, CL11,
AG11]. (For simulation source code, see https://github.com/ucla-csl/mabuc )
Our proposed algorithm, Causal Thompson Sampling (T S C ) takes the following steps: (1) T S C
first accepts an observational distribution as input, which it then uses to seed estimates of ETT quantities; i.e., for actions a and intuition x, by consistency we may seed knowledge of E(YX=a |X =
x) = Pobs (y|x), ?a = x. With large samples from an input set of observations, this seeding reduces
(and possibly eliminates) the need to explore the payout rates associated with following intuition,
leaving only the ?disobeying intuition? payout rates left for the agent to learn. As such, (2) at each
time step, our oracle observes the agent?s arm-choice predilection, and then uses RDC to deter4
Graphical conditions for identifying ETT [Pea00, SP07] are orthogonal to the bandit problem studied in
this paper, since no detailed knowledge about the causal graph (as well as infinite samples) is assumed here.
6
mine their best choice.5 Lastly, note that our seeding in (2) immediately improves the accuracy of
our comparison between arms, viz. that a superior arm will emerge more quickly than had we not
seeded. We can exploit this early lead in accuracy by weighting the more favorable arm, making it
more likely to be chosen earlier in the learning process (which empirically improves the convergence
rate as shown in the simulations).
Algorithm 1 Causal Thompson Sampling (T S C )
1: procedure TSC (Pobs , T)
2:
E(YX=a |X) ? Pobs (y|X)
(seed distribution)
3:
for t = [1, ..., T ] do
4:
x ? intuition(t)
(get intuition for trial)
5:
Q1 ? E(YX=x0 |X = x)
(estimated payout for counter-intuition)
6:
Q2 ? P (y|X = x)
(estimated payout for intuition)
7:
w ? [1, 1]
(initialize weights)
8:
bias ? 1 ? |Q1 ? Q2 |
(compute weighting strength)
9:
if Q1 > Q2 then w[x] ? bias else w[x0 ] ? bias
(choose arm to bias)
10:
11:
12:
a ? max(?(sM1 ,x , fM1 ,x ) ? w[1], ?(sM2 ,x , fM2 ,x ) ? w[2])
y ? pull(a)
E(YX=a |X = x) ? y|a, x
(choose arm) 6
(receive reward)
(update)
In the next section, we provide simulations to support the efficacy of T S C in the MABUC context.
For simplicity, we present two simulation results for the model described in Section 2.7 Experiment
1 employs the ?Greedy Casino? parameterization found in Table 1, whereas Experiment 2 employs
the ?Paradoxical Switching? parameterization found in Table 2. Each experiment compares the
performance of traditional Thompson Sampling bandit players versus T S C .
(a)
X = M1
X = M2
D=0
B=0 B=1
*0.40
0.30
0.60 *0.10
D=1
B=0 B=1
0.30 *0.40
*0.20
0.60
(b)
X = M1
X = M2
P (y|X)
0.4
0.15
P (y|do(X))
0.35
0.375
Table 2: ?Paradoxical Switching? parameterization. (a) Payout rates decided by reactive slot machines as a function of arm choice, sobriety, and machine conspicuousness. Players? natural arm
choices under D, B are indicated by asterisks. (b) Payout rates associated with the observational
and experimental distributions, respectively.
Procedure. All reported simulations are partitioned into rounds of T = 1000 trials averaged over
N = 1000 Monte Carlo repetitions. At each time step in a single round, (1) values for the unobserved confounders (B, D) and observed intuitive arm choice (x) are selected by their respective
structural equations (see Section 2), (2) the player observes the value of x, (3) the player chooses
an arm based on their given strategy to maximize reward (which may or may not employ x), and
finally, (4) the player receives a Bernoulli reward Y ? {0, 1} and records the outcome.
Furthermore, at the start of every round, players possess knowledge of the problem?s observational
distribution, i.e., each player begins knowing P (Y |X) (see Table 2b). However, only causallyempowered strategies will be able to make use of this knowledge, since this distribution is not, as
we?ve seen, the correct one to maximize.
Candidate algorithms. Standard Thompson Sampling (T S) attempts to maximize rewards based
on P (y|do(X)), ignoring the intuition x. Z-Empowered Thompson Sampling (T S Z ) treats the
5
Note that using predilection as a criteria for the inequality does not uniquely map to the contextual bandit
problem. To understand this point, note that not all variables are equally legitimate for confounding control in
causal settings, while the agent?s predilection is certainly one of such variables in our setup. Specifically when
considering whether a variable qualifies as a causal context requires a much deeper understanding of the data
generating model, which is usually not available in the general case.
6
The notation: ?(sMk ,x , fMk ,x ) means to sample from a Beta distribution with parameters equal to the
successes encountered choosing action x on machine Mk (sMk ,x ) and the failures encountered choosing action
x on that machine (fMk ,x ).
7
For additional experimental results and parameterizations, see Appendix [BFP15].
7
Figure 3: Simulation results for Experiments 1 and 2 comparing standard Thompson Sampling
(T S), Z-Empowered Thompson Sampling (T S Z ), and Causal Thompson Sampling (T S ? )
predilection as a new context variable, Z, and attempts to maximize based on P (y|do(X), Z) at
each round. Causal Thompson Sampling (T S C ), as described above, employs the ETT inequality
and input observational distribution.
Evaluation metrics. We assessed each algorithms? performances with standard bandit evaluation
metrics: (1) the probability of choosing the optimal arm and (2) cumulative regret. As in traditional
bandit problems, these measures are recorded as a function of the time step t averaged over all
N round repetitions. Note, however, that traditional definitions of regret are not phrased in terms
of unobserved confounders; our metrics, by contrast, compare each algorithm?s chosen arm to the
optimal arm for a given instantiation of Bt and Dt , even though these instantiations are never directly
available to the players. We believe that this is a fair operationalization for our evaluation metrics
because it allows us to compare regret experienced by our algorithms to a truly optimal (albeit
hypothetical) policy that has access to the unobserved confounders.
Experiment 1: ?Greedy Casino.? The Greedy Casino parameterization (specified in Table 1) illustrates the scenario where each arm?s payout appears to be equivalent under the observational and
experimental distributions alone. Only when we concert the two distributions and condition on a
player?s predilection can we obtain the optimal policy. Simulations for Experiment 1 support the efficacy of the causal approach (see Figure 3). Analyses revealed a significant difference in the regret
experienced by T S Z (M = 11.03, SD = 15.96) compared to T S C (M = 0.94, SD = 15.39),
t(999) = 14.52, p < .001. Standard T S was, predictably, not a competitor experiencing high regret
(M = 150.47, SD = 14.09).
Experiment 2: ?Paradoxical Switching.? The Paradoxical Switching parameterization (specified
in Table 2a) illustrates the scenario where one arm (M1 ) appears superior in the observational
distribution, but the other arm (M2 ) appears superior in the experimental. Again, we must use
causal analyses to resolve this ambiguity and obtain the optimal policy. Simulations for Experiment 2 also support the efficacy of the causal approach (see Figure 3). Analyses revealed a significant difference in the regret experienced by T S Z (M = 13.39, SD = 17.15) compared to T S C
(M = 4.71, SD = 17.90), t(999) = 11.28, p < .001. Standard T S was, again predictably, not a
competitor experiencing high regret (M = 83.56, SD = 15.75).
5
Conclusions
In this paper, we considered a new class of bandit problems with unobserved confounders (MABUC)
that are arguably more realistic than traditional formulations. We showed that MABUC instances
are not amenable to standard algorithms that rely solely on the experimental distribution. More
fundamentally, this lead to an understanding that in MABUC instances the optimization task is not
attainable through the estimation of the experimental distribution, but relies on both experimental
and observational quantities rooted in counterfactual theory and based on the agents? predilections.
To take advantage of our findings, we empowered the Thompson Sampling algorithm in two different
ways. We first added a new rule capable of improving the efficacy of which arm to explore. We
then jumpstarted the algorithm by leveraging non-experimental (observational) data that is often
available, but overlooked. Simulations demonstrated that in general settings these changes lead to a
more effective decision-making with faster convergence and lower regret.
8
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9
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5,184 | 5,693 | Sample Complexity Bounds for Iterative Stochastic
Policy Optimization
Marin Kobilarov
Department of Mechanical Engineering
Johns Hopkins University
Baltimore, MD 21218
[email protected]
Abstract
This paper is concerned with robustness analysis of decision making under uncertainty. We consider a class of iterative stochastic policy optimization problems
and analyze the resulting expected performance for each newly updated policy
at each iteration. In particular, we employ concentration-of-measure inequalities to compute future expected cost and probability of constraint violation using
empirical runs. A novel inequality bound is derived that accounts for the possibly unbounded change-of-measure likelihood ratio resulting from iterative policy
adaptation. The bound serves as a high-confidence certificate for providing future
performance or safety guarantees. The approach is illustrated with a simple robot
control scenario and initial steps towards applications to challenging aerial vehicle
navigation problems are presented.
1
Introduction
We consider a general class of stochastic optimization problems formulated as
? ? = arg min E? ?p(?|?) [J(? )],
?
(1)
where ? defines a vector of decision variables, ? represents the system response defined through
the density p(? |?), and J(? ) defines a positive cost function which can be non-smooth and nonconvex. It is assumed that p(? |?) is either known or can be sampled from, e.g. in a black-box
manner. The objective is to obtain high-confidence sample complexity bounds on the expected cost
for a given decision strategy by observing past realizations of possibly different strategies. Such
bounds are useful for two reasons: 1) for providing robustness guarantees for future executions, and
2) for designing new algorithms that directly minimize the bound and therefore are expected to have
built-in robustness.
Our primary motivation arises from applications in robotics, for instance when a robot executes
control policies to achieve a given task such as navigating to a desired state while perceiving the
environment and avoiding obstacles. Such problems are traditionally considered in the framework
of reinforcement learning and addressed using policy search algorithms, e.g. [1, 2] (see also [3] for a
comprehensive overview with a focus on robotic applications [4]). When an uncertain system model
is available the problem is equivalent to robust model-predictive control (MPC) [5].
Our specific focus is on providing formal guarantees on future executions of control algorithms in
terms of maximum expected cost (quantifying performance) and maximum probability of constraint
violation (quantifying safety). Such bounds determine the reliability of control in the presence of
process, measurement and parameter uncertainties, and contextual changes in the task. In this work
we make no assumptions about nature of the system structure, such as linearity, convexity, or Gaussianity. In addition, the proposed approach applies either to a physical system without an available
1
model, to an analytical stochastic model, or to a white-box model (e.g. from a high-fidelity opensource physics engine). In this context, PAC bounds have been rarely considered but could prove
essential for system certification, by providing high-confidence guarantees for future performance
and safety, for instance ?with 99% chance the robot will reach the goal within 5 minutes?, or ?with
99% chance the robot will not collide with obstacles?.
Approach. To cope with such general conditions, we study robustness through a statistical learning viewpoint [6, 7, 8] using finite-time sample complexity bounds on performance based on empirical runs. This is accomplished using concentration-of-measure inequalities [9] which provide
only probabilistic bounds , i.e. they certify the algorithm execution in terms of statements such as:
?in future executions, with 99% chance the expected cost will be less than X and the probability of
collision will be less than Y?. While such bounds are generally applicable to any stochastic decision
making process, our focus and initial evaluation is on stochastic control problems.
Randomized methods in control analysis. Our approach is also inspired by existing work on
randomized algorithms in control theory originally motivated by robust linear control design [10].
For example, early work focused on probabilistic root-locus design [11] and later applied to constraint satisfaction [12] and general cost functions [13]. High-confidence bounds for decidability of
linear stability were refined in [14]. These are closely related to the concepts of randomized stability robustness analysis (RSRA) and randomized performance robustness analysis (RPRA) [13].
Finite-time probabilistic bounds for system identification problems have also been obtained through
a statistical learning viewpoint [15].
2
Iterative Stochastic Policy Optimization
Instead of directly searching for the optimal ? to solve (1) a common strategy in direct policy search
and global optimization [16, 17, 18, 19, 20, 21] is to iteratively construct a surrogate stochastic
model ?(?|?) with hyper-parameters ? ? V, such as a Gaussian Mixture Model (GMM), where V
is a vector space. The model induces a joint density p(?, ?|?) = p(? |?)?(?|?) encoding natural
stochasticity p(? |?) and artificial control-exploration stochasticity ?(?|?). The problem is then to
find ? to minimize the expected cost
J (v) , E ?,??p(?|?) [J(? )],
iteratively until convergence, which in many cases also corresponds to ?(?|?) shrinking close to a
delta function around the optimal ? ? (or to multiple peaks when multiple disparate optima exist as
long as ? is multi-modal).
The typical flow of the iterative policy optimization algorithms considered in this work is:
0.
1.
2.
3.
Iterative Stochastic Policy Optimization (ISPO)
Start with initial hyper-parameters ?0 (i.e. a prior), set i = 0
Sample M trajectories (?j , ?j ) ? p(?|?i ) for j = 1, . . . , M
Compute new policy ?i+1 using observed costs J(?j )
Compute bound on expected cost and Stop if below threshold, else set i = i+1 and Goto 1
The purpose of computing probably-approximate bounds is two-fold: a) to analyze the performance
of such standard policy search algorithms; b) to design new algorithms by not directly minimizing
an estimate of the expected cost, but by minimizing an upper confidence bound on the expected
cost instead. The computed policy will thus have ?built-in? robustness in the sense that, with highprobability, the resulting cost will not exceed an a-priori known value. The present paper develops
bounds applicable to both (a) and (b), but only explores their application to (a), i.e. to the analysis
of existing iterative policy search methods.
Cost functions. We consider two classes of cost functions J. The first class encodes system performance and is defined as a bounded real-valued function such that 0 ? J(? ) ? b for any ? .
The second are binary-valued indicator functions representing constraint violation. Assume that the
variable ? must satisfy the condition g(? ) ? 0. The cost is then defined as J(? ) = I{g(? )>0} and its
expectation can be regarded as the probability of constraint violation, i.e.
P(g(? ) > 0) = E? ?p(?|?) I{g(? )>0} .
In this work, we will be obtain bounds for both classes of cost functions.
2
3
A Specific Application: Discrete-time Stochastic Control
We next illustrate the general stochastic optimization setting using a classical discrete-time nonlinear optimal control problem. Specific instances of such control problems will later be used for
numerical evaluation. Consider a discrete-time dynamical model with state xk ? X, where X is
an n-dimensional manifold, and control inputs uk ? Rm at time tk ? [0, T ] where k = 0, . . . , N
denotes the time stage. Assume that the system dynamics are given by
xk+1 = fk (xk , uk , wk ),
subject to
gk (xk , uk ) ? 0, gN (xN ) ? 0,
where fk and gk correspond either to the physical plant, to an analytical model, or to a ?white-box?
high-fidelity physics-engine update step. The terms wk denotes process noise. Equivalently, such a
formulation induces the process model density p(xk+1 |xk , uk ). In addition, consider the cost
J(x0:N , u0:N ?1 ) ,
N
?1
X
Lk (xk , uk ) + LN (xN ),
k=0
where x0:N , {x0 , . . . , xN } denotes the complete trajectory and Lk are given nonlinear functions.
Our goal is to design feedback control policies to optimize the expected value of J. For simplicity,
we will assume perfect measurements although this does not impose a limitation on the approach.
Assume that any decision variables in the problem (such as feedforward or feedback gains, obstacle
avoidance terms, mode switching variables) are encoded using a finite-dimensional vector ? ? Rn?
and define the control law uk = ?k (xk )? using basis functions ?k (x) ? Rm?n? for all k =
0, . . . , N ? 1. This representation captures both static feedback control laws as well as time-varying
optimal control laws of the form uk = u?k + KkLQR (xk ? x?k ) where u?k = B(tk )? is an optimized
feedforward control (parametrized using basis functions B(t) ? Rm?z such as B-splines), KkLQR
is the optimal feedback gain matrix of the LQR problem based on the linearized dynamics and
second-order cost expansion around the optimized nominal reference trajectory x? , i.e. such that
x?k+1 = fk (x?k , u?k , 0).
The complete trajectory of the system is denoted by the random variable ? = (x0:N , u0:N ?1 ) and
?1
has density p(? |?) = p(x0 )?N
? ?k (xk )?), where ?(?) is the Dirac delta.
k=0 p(xk+1 |xk , uk )?(uk V
N ?1
The trajectory constraint takes the form {g(? ) ? 0} , k=0 {gk (xk , uk ) ? 0} ? {gN (xN ) ? 0}.
A simple example. As an example, consider a point-mass robot modeled as a double-integrator
system with state x = (p, v) where p ? Rd denotes position and v ? Rd denotes velocity with d = 2
for planar workspaces and d = 3 for 3-D workspaces. The dynamics is given, for ?t = T /N , by
1
pk+1 = pk + ?tvk + ?t2 (uk + wk ),
2
vk+1 = vk + ?t(uk + wk ),
where uk are the applied controls and wk is zero-mean white noise. Imagine that the constraint
gk (x, u) ? 0 defines circular obstacles O ? Rd and control norm bounds defined as
ro ? kp ? po k ? 0,
kuk ? umax ,
where ro is the radius of an obstacle at position po ? Rd . The cost J could be arbitrary but a
typical choice is L(x, u) = 21 kuk2R + q(x) where R > 0 is a given matrix and q(x) is a nonlinear
function defining a task. The final cost could force the system towards a goal state xf ? Rn (or a
region Xf ? Rn ) and could be defined according to LN (x) = 21 kx ? xf k2Qf for some given matrix
Qf ? 0. For such simple systems one can choose a smooth feedback control law uk = ?k (x)? with
static positive gains ? = (kp , kd , ko ) ? R3 and basis function
?(x) = [ pf ? p
vf ? v
?(x, O) ] ,
where ?(x, O) is an obstacle-avoidance force, e.g. defined as the gradient of a potential field or as a
gyroscopic ?steering? force ?(x, O) = s(x, O) ? v that effectively rotates the velocity vector [22] .
Alternatively, one could employ a time-varying optimal control law as described in ?3.
3
4
PAC Bounds for Iterative Policy Adaptation
We next compute probabilistic bounds on the expected cost J (?) resulting from the execution of
a new stochastic policy with hyper-parameters ? using observed samples from previous policies
?0 , ?1 , . . . . The bound is agnostic to how the policy is updated (i.e. Step 2 in the ISPO algorithm).
4.1
A concentration-of-measure inequality for policy adaptation
The stochastic optimization setting naturally allows the use of a prior belief ? ? ?(?|?0 ) on what
good control laws could be, for some known ?0 ? V. After observing M executions based on such
prior, we wish to find a new improved policy ?(?|?) which optimizes the cost
?(?|?)
J (?) , E?,??p(?|?) [J(? )] = E?,??p(?|?0 ) J(? )
,
(2)
?(?|?0 )
which can be approximated using samples ?j ? ?(?|?0 ) and ?j ? p(? |?j ) by the empirical cost
M
1 X
?(?j |?)
J(?j )
.
(3)
M j=1
?(?j |?0 )
The goal is to compute the parameters ? using the sampled decision variables ?j and the corresponding observed costs J(?j ). Obtaining practical bounds for J (?) becomes challenging since the
?(?|?)
can be unbounded (or have very large values) [23] and a
change-of-measure likelihood ratio ?(?|?
0)
standard bound, e.g. such as Hoeffding?s or Bernstein?s becomes impractical or impossible to apply.
To cope with this we will employ a recently proposed robust estimation [24] technique stipulating
that instead of estimating the expectation m = E[X] of a random variable X ? [0, ?) using its
PM
1
empirical mean m
b = M
j=1 Xj , a more robust estimate can be obtained by truncating its higher
PM
1
moments, i.e. using m
b ? , ?M
j=1 ?(?Xj ) for some ? > 0 where
1
(4)
?(x) = log(1 + x + x2 ).
2
What makes this possible is the key assumption that the (possibly unbounded) random variable must
have bounded second moment. We employ this idea to deal with the unboundedness of the policy
adaptation ratio by showing that in fact its second moment can be bounded and corresponds to an
information distance between the current and previous stochastic policies.
To obtain sharp bounds though it is useful to employ samples over multiple iterations of the ISPO
algorithm, i.e. from policies ?0 , ?1 , . . . , ?L?1 computed in previous iterations. To simplify notation
?(?|?)
let z = (?, ?) and define `i (z, ?) , J(? ) ?(?|?
. The cost (2) of executing ? can now be equivalently
i)
expressed as
L?1
1 X
Ez?p(?|?i ) `i (z, ?)
J (?) ?
L i=0
using the computed policies in previous iterations i = 0, . . . , L ? 1. We next state the main result:
Proposition 1. With probability 1 ? ? the expected cost of executing a stochastic policy with parameters ? ? ?(?|?) is bounded according to:
(
)
L?1
X
?
1
1
J (?) ? inf Jb? (?) +
b2 eD2 (?(?|?)||?(?|?i )) +
log
,
(5)
?>0
2L i=0 i
?LM
?
where Jb? (?) denotes a robust estimator defined by
Jb? (?) ,
L?1 M
1 XX
? (?`(zij , ?)) ,
?LM i=0 j=1
computed after L iterations, with M samples zi1 , . . . , ziM ? p(?|?i ) obtained at iterations i =
0, . . . , L ? 1, where D? (p||q) denotes the Renyii divergence between p and q defined by
Z
1
p? (x)
D? (p||q) =
log
dx.
??1
q ??1 (x)
The constants bi are such that 0 ? J(? ) ? bi at each iteration i = 0, . . . , L ? 1.
4
Proof. The bound is obtained by relating the mean to its robust estimate according to
P LM (J (?) ? Jb? (?)) ? t
b
= P e?LM (J (?)?J? (?)) ? e?t ,
i
h
b
? E e?LM (J (?)?J? (?)) e??t ,
h PL?1 PM
i
= e??t+?LM J (?) E e i=0 j=1 ??(?`i (zij ,?))
?
?
L?1
M
YY
= e??t+?LM J E ?
e??(?`i (zij ,?)) ?
(6)
i=0 j=1
= e??t+?LM J
L?1
M
YY
i=0
= e??t+?LM J (?)
?2
`i (z, ?)2
E z?p(?|?i ) 1 ? ?`i (z, ?) +
2
j=1
L?1
M
YY
1 ? ?J (?) +
i=0 j=1
? e??t+?LM J (?)
M
L?1
YY
e??J (?)+
?2
2
(7)
?2
E z?p(?|?i ) [`i (z, ?)2 ]
2
Ez?p(?|?i ) [`i (z,?)2 ]
(8)
i=0 j=1
? e??t+M
?2
2
PL?1
i=0
Ez?p(?|?i ) [`i (z,?)2 ]
,
using Markov?s inequality to obtain (6), the identities ?(x) ? ? log(1 ? x + 12 x2 ) in (7) and
1 + x ? ex in (8). Here, we adapted the moment-truncation technique proposed by Catoni [24] for
general unbounded losses to our policy adaptation setting in order to handle the possibly unbounded
likelihood ratio. These results are then combined with
?(?|?)2
2
2
= b2i eD2 (?||?i ) ,
E [`i (z, ?) ] ? bi E?(?|?i )
?(?|?i )2
where the relationship between the likelihood ratio variance and the Renyii divergence was established in [23].
Note that the Renyii divergence can be regarded as a distance between two distribution and can be
computed in closed bounded form for various distributions such as the exponential families; it is
also closely related to the Kullback-Liebler (KL) divergence, i.e. D1 (p||q) = KL(p||q).
4.2
Illustration using simple robot navigation
We next illustrate the application of these bounds using the simple scenario introduced in ?3. The
stochasticity is modeled using a Gaussian density on the initial state p(x0 ), on the disturbances wk
and on the goal state xf . Iterative policy optimization is performed using a stochastic model ?(?|?)
encoding a multivariate Gaussian, i.e.
?(?|?) = N (?|?, ?)
which is updated through reward-weighted-regression (RWR) [3], i.e. in Step 2 of the ISPO algorithm we take M samples, observe their costs, and update the parameters according to
?=
M
X
j=1
w(?
? j )?j ,
?=
M
X
w(?
? j )(?j ? ?)(?j ? ?)T ,
(9)
j=1
using the tilting weights w(? ) = e??J(? ) for some adaptively chosen ? > 0 and where w(?
? j) ,
PM
w(?j )/ `=1 w(?` ) are the normalized weights.
At each iteration i one can compute a bound on the expected cost using the previously computed
?0 , . . . , ?i?1 . We have computed such bounds using (5) for both the expected cost and probability of
5
5
5
5
obstacles
goal
0
0
0
-5
-5
-5
-10
-10
-10
obstacles
sampled start states
-15
-15
-10
-5
0
5
-15
-15
-10
iteration #1
-5
0
5
-15
-15
-10
iteration #4
-5
0
5
iteration #9
iteration #28
Expected Cost
Probability of Collision
8
empirical Jb
robust Jb?
PAC bound J +
7
6
0.7
empirical P
robust P?
PAC bound P +
0.6
0.5
5
0.4
4
0.3
3
2
0.2
1
0.1
0
0
0
5
10
15
iterations
a)
b)
20
25
30
0
5
10
15
iterations
20
25
30
c)
Figure 1: Robot navigation scenario based on iterative policy improvement and resulting predicted performance: a) evolution of the density p(?|?) over the decision variables (in this case the control gains); b) cost
function J and its computed upper bound J + for future executions; c) analogous bounds on probability-ofcollision P ; snapshots of sampled trajectories (top). Note that the initial policy results in ? 30% collisions.
Surprisingly, the standard empirical and robust estimates are nearly identical.
collision, denoted respectively by J + and P + using M = 200 samples (Figure 1) at each iteration.
We used a window of maximum L = 10 previous iterations to compute the bounds, i.e. to compute
?i+1 all samples from densities ?i?L+1 , ?i?L+2 , . . . , ?i were used. Remarkably, using our robust
statistics approach the resulting bound eventually becomes close to the standard empirical estimate
Jb. The collision probability bound P + decreses to less than 10% which could be further improved
by employing more samples and more iterations. The significance of these bounds is that one can
stop the optimization (regarded as training) at any time and be able to predict expected performance
in future executions using the newly updated policy before actually executing the policy, i.e. using
the samples from the previous iteration.
Finally, the Renyii divergence term used in these computations takes the simple form
D? (N (?|?0 , ?0 )kN (?|?1 , ?1 )) =
?
1
|?? |
k?1 ? ?0 k2??1 +
log
,
?
2
2(1 ? ?)
|?0 |1?? |?1 |?
where ?? = (1 ? ?)?0 + ??1 .
4.3
Policy Optimization Methods
We do not impose any restrictions on the specific method used for optimizing the policy ?(?|?).
When complex constraints are present such computation will involve a global motion planning step
combined with local feedback control laws (we show such an example in ?5). The approach can
be used to either analyze such policies computed using any method of choice or to derive new
algorithms based on minimizing the right-hand side of the bound. The method also applies to modelfree learning. For instance, related to recent methods in robotics one could use reward-weightedregression (RWR) or policy learning by weighted samples with returns (PoWeR) [3], stochastic
optimization methods such as [25, 26], or using the related cross-entropy optimization [16, 27].
6
5
Application to Aerial Vehicle Navigation
Consider an aerial vehicle such as a quadrotor navigating at high speed through a cluttered environment. We are interested in minimizing a cost metric related to the total time taken and control effort
required to reach a desired goal state, while maintaining low probability of collision. We employ an
experimentally identified model of an AscTec quadrotor (Figure 2) with 12-dimensional state space
X = SE(3) ? R6 with state x = (p, R, p,
? ?) where p ? R3 is the position, R ? SO(3) is the
3
rotation matrix, and ? ? R is the body-fixed angular velocity. The vehicle is controlled with inputs
u = (F, M ) ? R4 including the lift force F ? 0 and torque moments M ? R3 . The dynamics is
m?
p = Re3 F + mg + ?(p, p),
?
(10)
R? = Rb
?,
(11)
J?? = J? ? ? + M,
(12)
where m is the mass, J?the inertia tensor, e3 = (0, 0, 1) and the matrix ?
b is such that ?
b? = ? ? ?
for any ? ? R3 . The system is subject to initial localization errors and also to random disturbances,
e.g. due to wind gusts and wall effects, defined as stochastic forces ?(p, p)
? ? R3 . Each component
in ? is zero-mean and has standard deviation of 3 Newtons, for a vehicle with mass m ? 1 kg.
The objective is to navigate through a given urban environment at high speed to a desired goal
state. We employ a two-stage approach consisting of an A*-based global planner which produces
a sequence of local sub-goals that the vehicle must pass through. A standard nonlinear feedback
backstepping controller based on a ?slow? position control loop and a ?fast? attitude control is
employed [28, 29] for local control. In addition, and obstacle avoidance controller is added to avoid
collisions since the vehicle is not expected to exactly follow the A* path. At each iteration M = 200
samples are taken with 1 ? ? = 0.95 confidence level. A window of L = 5 past iterations were
used for the bounds. The control density ?(?|?) is a single Gaussian as specified in ?4.2. The most
sensitive gains in the controller are the position proporitional and derivative terms, and the obstacle
gains, denoted by kp , kd , and ko , which we examine in the following scenarios:
a) fixed goal, wind gusts disturbances, virtual environment: the system is first tested in a cluttered
simulated environment (Figure 2). The simulated vehicle travels at an average velocity of 20
m/s (see video in Supplement) and initially experiences more than 50% collisions. After a few
iterations the total cost stabilizes and the probability of collision reduces to around 15%. The
bound is close to the empirical estimate which indicates that it can be tight if more samples are
taken. The collision probability bound is still too high to be practical but our goal was only
to illustrate the bound behavior. It is also likely that our chosen control strategy is in fact not
suitable for high-speed traversal of such tight environments.
b) sparser campus-like environment, randomly sampled goals: a more general evaluation was performed by adding the goal location to the stochastic problem parameters so that the bound will
apply to any future desired goal in that environment (Figure 3). The algorithm converges to
similar values as before, but this time the collision probability is smaller due to more expansive environment. In both cases, the bounds could be reduced further by employing more than
M = 200 samples or by reusing more samples from previous runs according to Proposition 1.
6
Conclusion
This paper considered stochastic decision problems and focused on a probably-approximate bounds
on robustness of the computed decision variables. We showed how to derive bounds for fixed policies in order to predict future performance and/or constraint violation. These results could then be
employed for obtaining generalization PAC bounds, e.g. through a PAC-Bayesian approach which
could be consistent with the proposed notion of policy priors and policy adaptation. Future work
will develop concrete algorithms by directly optimizing such PAC bounds, which are expected to
have built-in robustness properties.
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9
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5,185 | 5,694 | Basis Refinement Strategies for Linear Value
Function Approximation in MDPs
Gheorghe Comanici
School of Computer Science
McGill University
Montreal, Canada
[email protected]
Doina Precup
School of Computer Science
McGill University
Montreal, Canada
[email protected]
Prakash Panangaden
School of Computer Science
McGill University
Montreal, Canada
[email protected]
Abstract
We provide a theoretical framework for analyzing basis function construction for
linear value function approximation in Markov Decision Processes (MDPs). We
show that important existing methods, such as Krylov bases and Bellman-errorbased methods are a special case of the general framework we develop. We provide a general algorithmic framework for computing basis function refinements
which ?respect? the dynamics of the environment, and we derive approximation
error bounds that apply for any algorithm respecting this general framework. We
also show how, using ideas related to bisimulation metrics, one can translate basis refinement into a process of finding ?prototypes? that are diverse enough to
represent the given MDP.
1
Introduction
Finding optimal or close-to-optimal policies in large Markov Decision Processes (MDPs) requires
the use of approximation. A very popular approach is to use linear function approximation over a set
of features [Sutton and Barto, 1998, Szepesvari, 2010]. An important problem is that of determining
automatically this set of features in such a way as to obtain a good approximation of the problem
at hand. Many approaches have been explored, including adaptive discretizations [Bertsekas and
Castanon, 1989, Munos and Moore, 2002], proto-value functions [Mahadevan, 2005], Bellman error
basis functions (BEBFs) [Keller et al., 2006, Parr et al., 2008a], Fourier basis [Konidaris et al.,
2011], feature dependency discovery [Geramifard et al., 2011] etc. While many of these approaches
have nice theoretical guarantees when constructing features for fixed policy evaluation, this problem
is significantly more difficult in the case of optimal control, where multiple policies have to be
evaluated using the same representation.
We analyze this problem by introducing the concept of basis refinement, which can be used as a general framework that encompasses a large class of iterative algorithms for automatic feature extraction. The main idea is to start with a set of basis which are consistent with the reward function, i.e.
which allow only states with similar immediate reward to be grouped together. One-step look-ahead
is then used to find parts of the state space in which the current basis representation is inconsistent
with the environment dynamics, and the basis functions are adjusted to fix this problem. The process
continues iteratively. We show that BEBFs [Keller et al., 2006, Parr et al., 2008a] can be viewed as
a special case of this iterative framework. These methods iteratively expand an existing set of basis
functions in order to capture the residual Bellman error. The relationship between such features and
augmented Krylov bases allows us to show that every additional feature in these sets is consistently
refining intermediate bases. Based on similar arguments, it can be shown that other methods, such
as those based on the concept of MDP homomorphisms [Ravindran and Barto, 2002], bisimulation
metrics [Ferns et al., 2004], and partition refinement algorithms [Ruan et al., 2015], are also special cases of the framework. We provide approximation bounds for sequences of refinements, as
1
well as a basis convergence criterion, using mathematical tools rooted in bisimulation relations and
metrics [Givan et al., 2003, Ferns et al., 2004].
A final contribution of this paper is a new approach for computing alternative representations based
on a selection of prototypes that incorporate all the necessary information to approximate values
over the entire state space. This is closely related to kernel-based approaches [Ormoneit and Sen,
2002, Jong and Stone, 2006, Barreto et al., 2011], but we do not assume that a metric over the state
space is provided (which allows one to determine similarity between states). Instead, we use an
iterative approach, in which prototypes are selected to properly distinguish dynamics according to
the current basis functions, then a new metric is estimated, and the set of prototypes is refined again.
This process relies on using pseudometrics which in the limit converge to bisimulation metrics.
2
Background and notation
We will use the framework of Markov Decision Processes, consisting of a finite state space S, a
finite action space A, a transition function P : (S ? A) ? P(S)1 , where P (s, a) is a probability
distribution over the state space S, a reward function2 R : (S ?A) ? R. For notational convenience,
P a (s), Ra (s) will be used to denote P (s, a) and R(s, a), respectively. One of the main objectives
of MDP solvers is to determine a good action choice, also known as a policy, from every state that
the system would visit. APpolicy ? : S ? P(A) determines the probability of choosing each action
a given the state s (with a?A ?(s)(a) = 1). The value of a policy ? given a state s0 is defined as
P? i a
ai
i
V ? (s0 ) = E
i=0 ? R (si ) si+1 ? P (si ), ai ? ?(si ) .
Note that V ? is a real valued function [[S ? R]]; the space of all such functions will be denoted
by FS . We will also call such functions features. Let R? and P ? denote the reward and transition probabilities corresponding to choosing actions according to ?. Note that R? ? FS and
P ? ? [[FS ? FS ]], where3 R? (s) = Ea??(s) [Ra (s)] and P ? (f )(s) = Ea??(s) EP a (s) [f ] . Let
T ? ? [[FS ? FS ]] denote the Bellman operator: T ? (f ) = R? + ?P ? (f ). This operator is linear
and V ? is its fixed point, i.e. T ? (V ? ) = V ? . Most algorithms for solving MDPs will either use
the model (R? , P ? ) to find V ? (if this model is available and/or can be estimated efficiently), or
?
they will estimate V ? directly using samples of the model, {(si , ai , ri , si+1 )}?
i=0 . The value V
?
?
associated with the best policy ? is the fixed point of the Bellman optimality operator T (not a
linear operator), defined as: T ? (f ) = maxa?A (Ra + ?P a (f )).
The main problem we address in this paper is that of finding alternative representations for a given
MDP. In particular, we look for finite, linearly independent subsets ? of FS . These are bases for subspaces that will be used to speed up the search for V ? , by limiting it to span(?). We say that a basis
B is a partition if there exists an equivalence relation ? on S such that B = {?(C) | C ? S/?},
where ? is the characteristic function (i.e. ?(X)(x) = 1 if x ? X and 0 otherwise). Given any
equivalence relation ?, we will use the notation ?(?) for the set of characteristic functions on the
equivalence classes of ?, i.e. ?(?) = {?(C) | C ? S/?}.4 .
Our goal will be to find subsets ? ? FS which allow a value function approximation with strong
quality guarantees. More precisely, for any policy ? we would like to approximate V ? with
Pk
V?? =
i=1 wi ?i for some choice of wi ?s, which amounts to finding the best candidate inside
the space spanned by ? = {?1 , ?2 , ..., ?k }. A sufficient condition for V ? to be an element of
span(?) (and therefore representable exactly using the chosen set of bases), is for ? to span
the reward function and be an invariant subspace of the transition function: R? ? span(?) and
?f ? ?, P ? (f ) ? span(?). Linear fixed point methods like TD, LSTD, LSPE [Sutton, 1988,
Bradtke and Barto, 1996, Yu and Bertsekas, 2006] can be used to find the least squares fixed point
approximation V?? of V ? for a representation ?; these constitute proper approximation schemes, as
1
We will use P(X) to denote the set of probability distributions on a given set X.
For simplicity, we assume WLOG that the reward is deterministic and independent of the state into which
the system arrives.
P
3
We will use E? [f ] =
of a function f wrt distribution ?. If the
x f (x)?(x) to mean the expectation
P
function f is multivariate, we will use Ex?? [f (x, y)] = x f (x, y)?(x) to denote expectation of f when y is
fixed.
4
The equivalence class of an element s ? S is {s0 ? S | s ? s0 }. S/? is used for the quotient set of all
equivalence classes of ?.
2
2
one can determine the number of iterations required to achieve a desired approximation error. Given
a representation ?, the approximate value function V?? is the fixed point of the operator T?? , defined
as: T?? f := ?? (R? + ?P ? (f )), where ?? is the orthogonal projection operator on ?. Using the
linearity of ?? , it directly follows that T?? (f ) = ?? R? + ??? P ? (f ) and V?? is the fixed point of
?
the Bellman operator over the transformed linear model (R?
, P?? ) := (?? R? , ?? P ? ). For more
details, see [Parr et al., 2008a,b].
The analysis tools that we will use to establish our results are based on probabilistic bisimulation and
its quantitative analogues. Strong probabilistic bisimulation is a notion of behavioral equivalence
between the states of a probabilistic system, due to [Larsen and Skou, 1991] and applied to MDPs
with rewards by [Givan et al., 2003]. The metric analog is due to [Desharnais et al., 1999, 2004] and
the extension of the metric to include rewards is due to [Ferns et al., 2004]. An equivalence relation
? is a a bisimulation relation on the state space S if for every pair (s, s0 ) ? S ?S, s ? s0 if and only
if ?a ? A, ?C ? S/?, Ra (s) = Ra (s0 ) and P a (s)(C) = P a (s0 )(C) (we use here P a (s)(C) to denote the probability of transitioning into C, under transition s, a). A pseudo-metric is a bisimulation
metric if there exists some bisimulation relation ? such that ?s, s0 , d(s, s0 ) = 0 ?? s ? s0 .
The bisimulation metrics described by [Ferns et al., 2004] are constructed using the Kantorovich
metric for comparing two probability distributions. Given a ground metric d over S, the Kantorovich
metric over P(S) takes the largest difference in the expected value of Lipschitz-1 functions with
respect to d: ?(d) := {f ? FS | ?s, s0 , f (s) ? f (s0 ) ? d(s, s0 )}. The distance between two
probabilities ? and ? is computed as: K(d) : (?, ?) 7? sup???(d) E? [?] ? E? [?]. For more details
on the Kantorovich metric, see [Villani, 2003]. The following approximation scheme converges to
a bisimulation metric (starting with d0 = 0, the metric that associates 0 to all pairs):
dk+1 (s, s0 ) = T (dk )(s, s0 ) := max (1 ? ?) Ra (s) ? Ra (s0 ) + ?K(dk ) P a (s), P a (s0 ) . (1)
a
?
The operator T has a fixed point d , which is a bisimulation metric, and dk ? d? as k ? ?. [Ferns
et al., 2004] provide bounds which allow one to assess the quality of general state aggregations using
this metric. Given a relation ? and its corresponding partition ?(?), one can define an MDP model
? a = ??(?) Ra and P? a = ??(?) P a , ?a ? A. The approximation error between the
over ?(?) as: R
true MDP optimal value function V ? and its approximation using this reduced MDP model, denoted
?
by V?(?)
, is bounded above by:
1
?
?
d?? (s) + max
d?? (s0 ).
(2)
V?(?) (s) ? V ? (s) ?
0
1??
s ?S (1 ? ?)2
where d?? (s) is average distance from a state s to its ?-equivalence class, defined as an expectation
over the uniform distribution U: d?? (s) = Es??U [d? (s, s?) | s ? s?]. Similar bounds for representations that are not partitions can be found in [Comanici and Precup, 2011]. Note that these bounds
are minimized by aggregating states which are ?close? in terms of the bisimulation distance d? .
3
Basis refinement
In this section we describe the proposed basis refinement framework, which relies on ?detecting?
and ?fixing? inconsistencies in the dynamics induced by a given set of features. Intuitively, states
are dynamically consistent with respect to a set of basis functions if transitions out of these states
are evaluated the same way by the model {P a | a ? A}. Inconsistencies are ?fixed? by augmenting
a basis with features that are able to distinguish inconsistent states, relative to the initial basis. We
are now ready to formalize these ideas.
Definition 3.1. Given a subset F ? FS , two states s, s0 ? S are consistent with respect to F ,
denoted s ?F s0 , if ?f ? F, ?a ? A, f (s) = f (s0 ) and EP a (s) [f ] = EP a (s0 ) [f ].
Definition 3.2. Given two subspaces F, G ? FS , G refines F in an MDP M , and write F n G, if
F ? G and
?s, s0 ? S, s ?F s0 ?? [?g ? G, g(s) = g(s0 )].
Using the linearity of expectation, one can prove that, given
two probability distributions
?, ?,
and
a
finite
subset
?
?
F
,
if
span(?)
=
F
,
then
?f
?
F,
E
[f
]
=
E
[f
]
??
?
?
?b ? ?, E? [b] = E? [b] . For the special case of Dirac distributions ?s and ?s0 , for which
3
E?s [f ] = f (s), it also holds that ?f ? F, f (s) = f (s0 ) ?? ?b ? ?, b(s) = b(s0 ) .
Therefore, Def. 3.2 gives a relation between two subspaces, but the refinement conditions could be
checked on any basis choice. It is the subspace itself rather than a particular basis that matters, i.e.
? n ?0 if span(?) n span(?0 ). To fix inconsistencies on a pair (s, s0 ), for which we can find f ? ?
and a ? A such that either f (s) 6= f (s0 ) or EP a (s) [f ] 6= EP a (s0 ) [f ], one should construct a new
function ? with ?(s) 6= ?(s0 ) and add it to ?0 . To guarantee that all inconsistencies have been
addressed, if ?(s) 6= ?(s0 ) for some ? ? ?0 , ? must contain a feature f such that, for some a ? A,
either f (s) 6= f (s0 ) or EP a (s) [f ] 6= EP a (s0 ) [f ].
In Sec. 5 we present an algorithmic framework consisting of sequential improvement steps, in which
a current basis ? is refined into a new one, ?0 , with span(?) n span(?0 ). Def 3.2 guarantees that
following such strategies expands span(?) and that the approximation error for any policy will be
decreased as a result. We now discuss bounds that can be obtained based on these definitions.
3.1
Value function approximation results
One simple way to create a refinement is to add to ? a single element that would address all inconsistencies: a feature that is valued differently for every element
nP of ?(?? ). Given
o ? : ?(?? ) ? R,
?b, b0 ? ?(?? ), b 6= b0 ? ?(b) 6= ?(b0 ) ? ? n ? ?
?(b)b
. On the other hand,
b??(?? )
such a construction provides no approximation guarantee for the optimal value function (unless we
make additional assumptions on the problem - we will discuss this further in Section 3.2). Although
it addresses inconsistencies in the dynamics over the set of features spanned by ?, it does not necessarily provide the representation power required to properly approximate the value of the optimal
policy. The main theoretical result in this section provides conditions for describing refining sequences of bases, which are not necessarily accurate, but have approximation errors bounded by
an exponentially decreasing function. These results are based on ?(?? ), the largest basis refining
subspace: any feature that is constant over equivalence classes of ?? will be spanned by ?(?), i.e.
for any refinement V n W , V ? W ? span(?(?V )). These subsets are convenient as they can be
analyzed using the bisimulation metric introduced in [Ferns et al., 2004].
Lemma 3.1. The bisimulation operator in Eq. 1) is a contraction with constant ?. That is, for any
metric d over S, sups,s0 ?S |T (d)(s, s0 )| ? ? sups,s0 ?S |d(s, s0 )|.
The proof relies on the Monge-Kantorovich duality (see [Villani, 2003]) to check that T satisfies
sufficient conditions to be a contraction operator. An operator Z is a contraction (with constant
? < 1) if Z(x) ? Z(x0 ) whenever x ? x0 , and if Z(x + c) = Z(x) + ?c for any constant
c ? R [Blackwell, 1965]. One could easily check these conditions on the operator in Equation 1.
Theorem 3.1. Let ?0 represent reward consistency, i.e. s ?0 s0 ?? ?a ? A, Ra (s) = Ra (s0 ),
and ?1 = ?(?0 ). Additionally, assume {?n }?
n=1 is a sequence of bases such that for all n ? 1,
?n n ?n+1 and ?n+1 is as large as the partition corresponding to consistency over ?n , i.e.
?
|?n+1 | = |S/?
optimal value function computed with respect to representa ??n |. If? V ?n is the
tion ?n , then V?n ? V ? ? ? n+1 sups,s0 ,a |Ra (s) ? Ra (s0 )|/(1 ? ?)2 .
Proof. We will use the bisimulation metric defined in Eq. 1 and Eq. 2 applied to the special case of
reduced models over bases {?n }?
n=1 .
First, note that Monge-Kantorovich duality is crucial in this proof. It basically states that the Kantorovich metric is a solution to the Monge-Kantorovich problem, when its cost function is equal
to the base metric for the Kantorovich metric. Specifically, for two measures ? and ?, and a cost
function f ? [S ? S ? R], the Monge-Kantorovich problem computes:
J (f )(?, ?) = inf{E? [f (x, y)] | ? ? P(S ? S) s.t. ?, ? are the marginals corresponding to x and y}
The set of measures ? with marginals ? and ? is also known as the set of couplings of ? and ?. For
any metric d over S, J (d)(?, ?) = K(d)(?, ?) (for proof, see [Villani, 2003]).
Next, we describe a relation between the metric T n (0) and ?n .
Since
|?n+1 | = |S/??n | = |?(??n )| and ?n+1 ? span(?(??n )), it must be the case that
span(?n+1 ) = span(?(??n )). It is not hard to see that for the special case of partitions, a refinement can be determined based on transitions into equivalence classes. Given
4
two equivalence relations ?1 and ?2 , the refinement ?(?1 ) n ?(?2 ) holds if and only if
s ?2 s0 ? s ?1 s0 and s ?2 s0 ? ?a ? A, ?C ? S/?1 P a (s)(C) = P a (s0 )(C) . In particular,
?s, s0 with s ??n+1 s0 , and ?C ? S/??n , P a (s)(C) = P a (s0 )(C). This equality is crucial in
defining the following coupling for J (f )(P a (s), P a (s0 )): let ?C ? P(S ? S) be any coupling of
P a (s)|C and P a (s0 )|C , the restrictions of P a (s) and P a (s0 ) to C; thePlatter is possible as the two
distributions are equal. Next, define the coupling ? of ? and ? as ? = C?S/?? ?C . For any cost
n
P
function f , if s ??n+1 s0 , then J (f )(P a (s), P a (s0 )) ? C?S/??n E?C [f ].
Using an inductive argument, we will now show that ?n, s ??n s0 ? T n (0)(s, s0 ) = 0. The base
case is clear from the definition: s ?0 s0 ? T (0)(s, s0 ) = 0. Now, assume the former holds for n;
that is, ?C ? S/??n , ?s, s0 ? C, T n (0)(s, s0 ) = 0. But ?C is zero everywhere except on the set
C ? C, so E?C [T n (0)] = 0. Combining the last two results, we get the following upper bound:
P
s ??n+1 s0 ? J (T n (0))(P a (s), P a (s0 )) ? C?S/??n E?C [T n (0)] = 0.
Since T n (0) is a metric, it also holds that J (T n (0))(P a (s), P a (s0 )) ? 0. Moreover, as s and
s0 are consistent over ?n ? ?(?0 ), this pair of states agree on the reward function. Therefore,
T n+1 (0)(s, s0 ) = maxa ((1 ? ?)|Ra (s) ? Ra (s0 )| + ?J (T n (0))(P a (s), P a (s0 ))) = 0.
Finally, for any b ? ?(??n ) and s ? S with b(s) = 1, and any other state s? with b(?
s) = 1, it must
be the case that s ??n s? and T n (0)(s, s?) = 0. Therefore,
E [d? (s, s?) | s ??n s?] = E [d? (s, s?) ? T n (0)(s, s?) | s ??n s?] ? ||d? ? T n (0)||? .
s??U
s??U
(3)
As span(?n ) = span(?(?n )), V??n is the optimal value function for the MDP model over ?(?n ).
Based on (2) and (3), we can conclude that
?
V? ? V ? ? ?||d? ? T n (0)||? /(1 ? ?)2 .
(4)
n
?
But we already know from Lemma 3.1 that d? (defined in Eq. 1) is the fixed point of a contraction
operator with constant ?. As J (0)(?, ?) = 0, the following holds for all n ? 1
||d? ? T n (0)||? ? ? n ||T (0) ? 0||? /(1 ? ?) ? ? n sup |Ra (s) ? Ra (s0 )|.
(5)
s,s0 ,a
The final result is easily obtained by putting together Equations 4 and 5.
The result of the theorem provides a strategy for constructing refining sequences with strong approximation guarantees. Still, it might be inconvenient to generate refinements as large as S/??n , as
this might be over-complete; although faithful to the assumptions of the theorem, it might generate
features that distinguish states that are not often visited, or pairs of states which are only slightly
different. To address this issue, we provide a variation on the concept of refinement that can be used
to derive more flexible refining algorithms: refinements that concentrate on local properties.
Definition 3.3. Given a subset F ? FS , and a subset ? ? S, two states s, s0 ? S are consistent on ? with respect to F , denoted s ?F,? s0 , if ?f ? F, ?a ? A, f (s) = f (s0 ) and
??
s ? ?, EP a (?s) [f ] = EP a (s) [f ] ?? EP a (?s) [f ] = EP a (s0 ) [f ].
Definition 3.4. Given two subspaces F, G ? FS , G refines F locally with respect to ?, denoted
F n? G, if F ? G and ?s, s0 ? S, s ?F,? s0 ?? [?g ? G, g(s) = g(s0 )].
Definition 3.2 is the special case of Definition 3.4 corresponding to a refinement with respect to the
whole state space S, i.e. F n G ? F nS G. When the subset ? is not important, we will use
the notation V n? W to say that W refines V locally with respect to some subset of S. The result
below states that even if one provides local refinements n? , one will eventually generate a pair of
subspaces which are related through a global refinement property n.
Proposition 3.1. Let {?i }ni=0 be a set of bases over S with ?i?1 n?i ?i , i = 1, ..., n, for some
{?i }ni=1 . Assume that ?n is the maximal refinement (i.e. |?n | = |S/??n?1 ,?n |). Let ? = ?i ?i .
Then ?(??0 ,? ) ? span(?n ).
Proof. Assume s ??n?1 ,?n s0 . We will check below all conditions necessary to conclude that
s ??0 ,? s0 . First, let f ? ?0 . It is immediate from the definition of local refinements that
?j ? n ? 1, ?j ? ?n?1 , so that s ??0 ,?n s0 . It follows that ?f ? ?0 , f (s) = f (s0 ).
5
Next, fix f ? ?0 , a ? A and s? ? ?. If s? ? ?n , then EP a (?s) [f ] = EP a (s) [f ] ??
EP a (?s) [f ] = EP a (s0 ) [f ], by the assumption above on the pair s, s0 . Otherwise, ?j < n such that
s? ? ?j and ?j?1 n?j ?j . But we already know that ?f ? ?j , f (s) = f (s0 ), as ?j ? ?n?1 . We
can use this result in the definition of local refinement ?j?1 n?j ?j to conclude that s ??j?1 ,?j s0 .
Moreover, as s? ? ?j , f ? ?0 ? ?j?1 , EP a (?s) [f ] = EP a (s) [f ] ?? EP a (?s) [f ] = EP a (s0 ) [f ]. This
completes the definition of consistency on ?, and it becomes clear that s ??n?1 ,?n s0 ? s ??0 ,? s0 ,
or ?(??0 ,? ) ? span(?(??n?1 ,?n )).
Finally, both ?n and ?(??n?1 ,? ) are bases of the same size, and both refine ?n?1 . It must be that
span(?n ) = span(?(??n?1 ,?n )) ? ?(??0 ,? ).
3.2
Examples of basis refinement for feature extraction
The concept of basis refinement is not only applicable to the feature extraction methods we will
present later, but to methods that have been studied in the past. In particular, methods based on
Bellman error basis functions, state aggregation strategies, and spectral analysis using bisimulation
metrics are all special cases of basis refinement. We briefly describe the refinement property for
the first two cases, and, in the next section, we elaborate on the connection between refinement and
bisimulation metrics to provide a new condition for convergence to self-refining bases.
Krylov bases: Consider the uncontrolled (policy evaluation) case, in which one would like to
find a set of features that is suited to evaluating a single policy of interest. A common approach to
automatic feature generation in this context computes Bellman error basis functions (BEBFs), which
have been shown to generate a sequence of representations known as Krylov bases. Given a policy
?, a Krylov basis ?n of size n is built using the model (R? , P ? ) (defined in Section 2 as elements
of FS and [[FS ? FS ]], respectively): ?n = span{R? , P ? R? , (P ? )2 R? , ..., (P ? )n R? }. It is not
hard to check that ?n n ?n+1 , where n is the refinement relational property in Def 3.2. Since the
initial feature R? ? ?(?0 ), the result in Theorem 3.1 holds for the Krylov bases.
Under the assumption of a finite-state MDP (i.e. |S| < ?), ?? := {?({s}) | s ? S} is a basis for
FS , therefore this set of features is finite dimensional. It follows that one can find N ? |S| such
that one of the Krylov bases is a self-refinement, i.e. ?N n ?N . This would by no means be the
only self-refining basis. In fact this property holds for the basis of characteristic functions, ?? n ?? .
The purpose our framework is to determine other self-refining bases which are suited for function
approximation methods in the context of controlled systems.
State aggregation: One popular strategy used for solving MDPs is that of computing state aggregation maps. Instead of working with alternative subspaces, these methods first compute equivalence relations on the state space. An aggregate/collapsed model is then derived, and the solution to this model is translated to one for the original problem: the resulting policy provides
the same action choice for states that have originally been related. Given any equivalence relation ? on S, a state aggregation map is a function from S to any set X, ? : S ? X, such that
?s, s0 , ?(s0 ) = ?(s) ?? s ? s0 . In order to obtain a significant computational gain, one would
like to work with aggregation maps ? that reduce the size of the space for which one looks to provide
action choices, i.e. |X| |S|. As discussed in Section 3.1, one could work with features that are
defined on an aggregate state space instead of the original state space. That is, instead of computing
a set of state features ? ? FS , we could work instead with an aggregation map ? : S ? X and a
? ? FX . If ? is the relation such that s ? s0 ?? ?(s) = ?(s0 ), then
set of features over X, ?
?
?? ? ?, ? ? ? ? span(?(?)).
4
Using bisimulation metrics for convergence of bases
In Section 3.2 we provide two examples of self-refining subspaces: the Krylov bases and the characteristic functions on single states. The latter is the largest and sparsest basis; it spans the entire state
space and the features share no information. The former is potentially smaller and it spans the value
of the fixed policy for which it was designed. In this section we will present a third self-refining
construction, which is designed to capture bisimulation properties. Based on the results presented
in Section 3.1, it can be shown that given a bisimulation relation ?, the partition it generates is
self-refining, i.e. ?(?) n ?(?).
6
Desirable self-refining bases might be be computationally demanding and/or too complex to use or
represent. We propose iterative schemes which ultimately provide a self-refining result - albeit we
would have the flexibility of stopping the iterative process before reaching the final result. At the
same time, we need a criterion to describe convergence of sequences of bases. That is, we would
want to know how close an iterative process is to obtaining a self-refining basis. Inspired by the fixed
point theory used to study bisimulation metrics [Desharnais et al., 1999], instead of using a metric
over the set of all bases to characterize convergence of such sequences, we will use corresponding
metrics over the original state space. This choice is better suited for generalizing previously existing
methods that compare pairs of states for bisimilarity through their associated reward models and
expected realizations of features over the next state distribution model associated with these states.
We will study metric construction strategies based on a map D, defined below, which takes an
element of the powerset P(FS ) of FS and returns an element of all pseudo-metrics M (S) over S.
D(?) : (s, s0 ) 7? maxa (1 ? ?) |Ra (s) ? Ra (s0 )| + ? sup??? EP a (s) [?] ? EP a (s0 ) [?]
(6)
? is a set of features whose expectation over next-state distributions should be matched. It is not hard
to see that bases ? for which D(?) is a bisimulation metric are by definition self-refining. For example, consider the largest bisimulation relation ? on a given MDP. It is not hard to see that D(?(?))
is a bisimulation. A more elaborate example involves the set ?(d) of Lipschitz-1 continuous functions on [[(S, d) ? (R, L1 )]] (recall definition and computation details from Section 2). Define d?
to be the fixed point of the operator T : d 7? D(?(d)), i.e. d? = supn?N T n (0). d? has the same
property as the bisimulation metric defined in Equation 1. Moreover, given any bisimulation metric
d, D(?(d)) is a bisimulation metric.
Definition 4.1. We say a sequence {?n }?
n=1 is a a bisimulation sequence of bases if D(?n ) converges uniformly from below to a bisimulation metric. If one has the a sequence of refining bases
with ?n n ?n+1 , ?n, then {D(?n )}?
n=1 is an increasing sequence, but not necessarily a bisimulation
sequence.
A bisimulation sequence of bases provide an approximation scheme for bases that satisfy two important properties studied in the past: self-refinement and bisimilarity. One could show that the
approximation schemes presented in [Ferns et al., 2004], [Comanici and Precup, 2011], and [Ruan
et al., 2015] are all examples of bisimulation sequences. We will present in the next section a
framework that generalizes all these examples, but which can be easily extended to a broader set of
approximation schemes that incorporate both refining and bisimulation principles.
5
Prototype based refinements
In this section we propose a strategy that iteratively builds sequences of refineing sets of features, based on the concepts described in the previous sections. This generates layered sets of
features, where the nth layer in the construction will be dependent only on the (n ? 1)th layer.
Additionally, each feature will be associated with a reward-transition prototype: elements of
Q := [[A ? (R ? P(S))]], associating to each action a reward and a next-state probability distribution. Prototypes can be viewed as ?abstract? or representative states, such as used in KBRL
methods [Ormoneit and Sen, 2002]. In the layered structure, the similarity between prototypes at
the nth layer is based on a measure of consistency with respect to features at the (n ? 1)th layer. The
same measure of similarity is used to determine whether the entire state space is ?covered? by the
set of prototypes/features chosen for the nth layer. We say that a space is covered if every state of
the space is close to at least one prototype generated by the construction, with respect to a predefined measure of similarity. This measure is designed to make sure that consecutive layers represent
refining sets of features. Note that for any given MDP, the state space S is embedded into Q (i.e.
S ? Q), as (Ra (s), P a (s)) ? Q for every state s ? S. Additionally, The metric generator D, as
defined in Equation 6, can be generalized to a map from P(FS ) to M (Q).
The algorithmic strategy will look for a sequence {Jn , ?n }?
n=1 , where Jn ? Q is a set of covering
prototypes, and ?n : Jn ? FS is a function that associates a feature to every prototype in Jn .
Starting with J0 = ? and ?0 = ?, the strategy needs to find, at step n > 0, a cover J?n for S,
based on the distance metric D(?n?1 ). That is, it has to guarantee that ?s ? S, ?? ? J?n with
D(?n?1 )(s, ?) = 0. With Jn = J?n ? Jn?1 and using a strictly decreasing function ? : R?0 ? R
(e.g. the energy-based Gibbs measure ? (x) = exp(??x) for some ? > 0), the framework constructs
?n : Jn ? FS , a map that associates prototypes to features as ?n (?)(s) = ? (D(?n?1 )(?, s)).
7
Algorithm 1 Prototype refinement
1: J0 = ? and ?0 = ?
2: for n = 1 to ? do
3:
choose a representative subset ?n ? S and a cover approximation error n ? 0
4:
find an n -cover J?n for ?n
5:
define Jn = J?n ? Jn?1
6:
choose a strictly decreasing function ? : R?0 ? R
s 7? ? (D(?n?1 )(?, s)) if ??
s ? ?n , such that D(?n?1 )(?, s?) ? n
7:
define ?n (?) =
?n?1 (?)
otherwise
8:
define ?n = {?n (?) | ? ? Jn } (note that ?n is a local refinement, ?n?1 n?n ?n )
It is not hard to see that the refinement property holds at every step, i.e. ?n n ?n+1 . First, every
equivalence class of ??n is represented by some prototype in Jn . Second, ?n is purposely defined
to make sure that a distinction is made between each prototype in Jn+1 . Moreover, {?n }?
n=1 is a
bisimulation sequence of bases, as the metric generator D is the main tool used in ?covering? the
state space with the set of prototypes Jn . Two states will be represented by the same prototype (i.e.
they will be equivalent with respect to ??n ) if and only if the distance between their corresponding
reward-transition models is 0.
Algorithm 1 provides pseudo-code for the framework described in this section. Note that it also contains two additional modifications, used to illustrate the flexibility of this feature extraction process.
Through the first modification, one could use the intermediate results at time step n to determine
a subset ?n ? S of states which are likely to have a model with significantly distinct dynamics
over ?n?1 . As such, the prototypes J?n?1 can be specialized to cover only the significant subset ?n .
Moreover Theorem 3.1 guarantees that if every state in S is picked in ?n infinitely often, as n ? ?,
then the approximation power of the final result is not be compromised. The second modification
is based on using the values in the metric D(?n?1 ) for more than just choosing feature activations:
one could set at every step constants n ? 0 and then find Jn such that ?n is covered using n -balls,
i.e. for every state in ?n , there exists a prototype ? ? Jn with D(?n?1 )(?, s) ? n . One can easily
show that the refinement property can be maintained using the modified defition of ?n described in
Algorithm 1.
6
Discussion
We proposed a general framework for basis refinement for linear function approximation. The theoretical results show that any algorithmic scheme of this type satisfies strong bounds on the quality
of the value function that can be obtained. In other words, this approach provides a ?blueprint? for
designing algorithms with good approximation guarantees. As discussed, some existing value function construction schemes fall into this category (such as state aggregation refinement, for example).
Other methods, like BEBFs, can be interpreted in this way in the case of policy evaluation; however,
the ?traditional? BEBF approach in the case of control does not exactly fit this framework. However,
we suspect that it could be adapted to exactly follow this blueprint (something we leave for future
work).
We provided ideas for a new algorithmic approach to this problem, which would provide strong
guarantees while being significantly cheaper than other existing methods with similar bounds (which
rely on bisimulation metrics). We plan to experiment with this approach in the future. The focus
of this paper was to establish the theoretical underpinnings of the algorithm. The algorithm structure we propose is close in spirit to [Barreto et al., 2011], which selects prototype states in order
to represent well the dynamics of the system by means of stochastic factorization. However, their
approach assumes a given metric which measures state similarity, and selects representative states
using k-means clustering based on this metric. Instead, we iterate between computing the metric
and choosing prototypes. We believe that the theory presented in this paper opens up the possibility of further development of algorithms for constructive function approximation that have quality
guarantees in the control case, and which can be effective also in practice.
8
References
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9
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5,186 | 5,695 | Probabilistic Variational Bounds
for Graphical Models
Qiang Liu
Computer Science
Dartmouth College
[email protected]
John Fisher III
CSAIL
MIT
[email protected]
Alexander Ihler
Computer Science
Univ. of California, Irvine
[email protected]
Abstract
Variational algorithms such as tree-reweighted belief propagation can provide deterministic bounds on the partition function, but are often loose and difficult to use
in an ?any-time? fashion, expending more computation for tighter bounds. On the
other hand, Monte Carlo estimators such as importance sampling have excellent
any-time behavior, but depend critically on the proposal distribution. We propose
a simple Monte Carlo based inference method that augments convex variational
bounds by adding importance sampling (IS). We argue that convex variational
methods naturally provide good IS proposals that ?cover? the target probability,
and reinterpret the variational optimization as designing a proposal to minimize an
upper bound on the variance of our IS estimator. This both provides an accurate
estimator and enables construction of any-time probabilistic bounds that improve
quickly and directly on state-of-the-art variational bounds, and provide certificates
of accuracy given enough samples relative to the error in the initial bound.
1
Introduction
Graphical models such as Bayesian networks, Markov random fields and deep generative models
provide a powerful framework for reasoning about complex dependency structures over many variables [see e.g., 14, 13]. A fundamental task is to calculate the partition function, or normalization
constant. This task is #P-complete in the worst case, but in many practical cases it is possible to
find good deterministic or Monte Carlo approximations. The most useful approximations should
give not only accurate estimates, but some form of confidence interval, so that for easy problems
one has a certificate of accuracy, while harder problems are identified as such. Broadly speaking,
approximations fall into two classes: variational optimization, and Monte Carlo sampling.
Variational inference [29] provides a spectrum of deterministic estimates and upper and lower
bounds on the partition function; these include loopy belief propagation (BP), which is often quite
accurate; its convex variants, such as tree reweighted BP (TRW-BP), which give upper bounds on the
partition function; and mean field type methods that give lower bounds. Unfortunately, these methods often lack useful accuracy assessments; although in principle a pair of upper and lower bounds
(such as TRW-BP and mean field) taken together give an interval containing the true solution, the
gap is often too large to be practically useful. Also, improving these bounds typically means using
larger regions, which quickly runs into memory constraints.
Monte Carlo methods, often based on some form of importance sampling (IS), can also be used
to estimate the partition function [e.g., 15]. In principle, IS provides unbiased estimates, with the
potential for a probabilistic bound: a bound which holds with some user-selected probability 1 ? ?.
Sampling estimates can also easily trade time for increased accuracy, without using more memory.
Unfortunately, choosing the proposal distribution in IS is often both crucial and difficult; if poorly
chosen, not only is the estimator high-variance, but the samples? empirical variance estimate is also
misleading, resulting in both poor accuracy and poor confidence estimates; see e.g., [35, 1].
1
We propose a simple algorithm that combines the advantages of variational and Monte Carlo methods. Our result is based on an observation that convex variational methods, including TRW-BP and
its generalizations, naturally provide good importance sampling proposals that ?cover? the probability of the target distribution; the simplest example is a mixture of spanning trees constructed by
TRW-BP. We show that the importance weights of this proposal are uniformly bounded by the convex upper bound itself, which admits a bound on the variance of the estimator, and more importantly,
allows the use of exponential concentration inequalities such as the empirical Bernstein inequality
to provide explicit confidence intervals. Our method provides several important advantages:
First, the upper bounds resulting from our sampling approach improve directly on the initial variational upper bound. This allows our bound to start at a state-of-the-art value, and be quickly and
easily improved in an any-time, memory efficient way. Additionally, using a two-sided concentration
bound provides a ?certificate of accuracy? which improves over time at an easily analyzed rate. Our
upper bound is significantly better than existing probabilistic upper bounds, while our corresponding lower bound is typically worse with few samples but eventually outperforms state-of-the-art
probabilistic bounds [11].
Our approach also results in improved estimates of the partition function. As in previous work [32,
34, 31], applying importance sampling serves as a ?bias correction? to variational approximations.
Here, we interpret the variational bound optimization as equivalent to minimizing an upper bound
on the IS estimator?s variance. Empirically, this translates into estimates that can be significantly
more accurate than IS using other variational proposals, such as mean field or belief propagation.
Related Work.
Importance sampling and related approaches have been widely explored in the
Bayesian network literature, in which the partition function corresponds to the probability of observed evidence; see e.g., [8, 26, 33, 11] and references therein. Dagum and Luby [4] derive a
sample size to ensure a probabilistic bound with given relative accuracy; however, they use the
normalized Bayes net distribution as a proposal, leading to prohibitively large numbers of samples
when the partition function is small, and making it inapplicable to Markov random fields. Cheng [2]
refines this result, including a user-specified bound on the importance weights, but leaves the choice
of proposal unspecified.
Some connections between IS and variational methods are also explored in Yuan and Druzdzel
[32, 34], Wexler and Geiger [31], Gogate and Dechter [11], in which proposals are constructed
based on loopy BP or mean field methods. While straightforward in principle, we are not aware of
any prior work which uses variational upper bounds to construct a proposal, or more importantly,
analyzes their properties. An alternative probabilistic upper bound can be constructed using ?perturb and MAP? methods [23, 12] combined with recent concentration results [22]; however, in our
experiments the resulting bounds were quite loose. Although not directly related to our work, there
are also methods that connect variational inference with MCMC [e.g., 25, 6].
Our work is orthogonal to the line of research on adaptive importance sampling, which refines the
proposal as more samples are drawn [e.g., 21, 3]; we focus on developing a good fixed proposal
based on variational ideas, and leave adaptive improvement as a possible future direction.
Outline. We introduce background on graphical models in Section 2. Our main result is presented
in Section 3, where we construct a tree reweighted IS proposal, discuss its properties, and propose
our probabilistic bounds based on it. We give a simple extension of our method to higher order
cliques based on the weighted mini-bucket framework in Section 4. We then show experimental
comparisons in Section 5 and conclude with Section 6.
2
2.1
Background
Undirected Probabilistic Graphical Models
def
Let x = [x1 , . . . , xp ] be a discrete random vector taking values in X = X1 ? ? ? ? ? Xp ; a probabilistic graphical model on x, in an over-complete exponential family form, is
X
X
f (x; ?)
p(x; ?) =
,
with
f (x; ?) = exp
?? (x? ) ,
Z(?) =
f (x; ?),
(1)
Z(?)
??I
2
x?X
where I = {?} is a set of subsets of variable indices, and ?? : X? ? R are functions of x? ; we
denote by ? = {?? (x? ) : ?? ? I, x? ? X? } the vector formed by the elements of ?? (?), called
the natural parameters. Our goal is to calculate the partition function Z(?) that normalizes the
distribution; we often drop the dependence on ? and write p(x) = f (x)/Z for convenience.
The factorization of p(x; ?) can be represented by an undirected graph G = (V, EG ), called its
Markov graph, where each vertex k ? V is associated with a variable xk , and nodes k, l ? V are
connected (i.e., (kl) ? EG ) iff there exists some ? ? I that contains both k and l; then, I is a set of
cliques of G. A simple special case of (1) is the pairwise model, in which I = V ? E:
X
X
f (x; ?) = exp
?k (xk ) +
?kl (xk , xl ) .
(2)
i?V
2.2
(kl)?EG
Monte Carlo Estimation via Importance Sampling
Importance sampling (IS) is at the core of many Monte Carlo methods for estimating the partition
function. The idea is to take a tractable, normalized distribution q(x), called the proposal, and
estimate Z using samples {xi }ni=1 ? q(x):
n
1X
Z? =
w(xi ),
n i=1
with
w(xi ) =
f (xi )
,
q(xi )
where w(x) is called the importance weight. It is easy to show that Z? is an unbiased estimator of Z,
in that EZ? = Z, if q(x) > 0 whenever p(x) > 0, and has a MSE of E(Z? ? Z)2 = n1 var(w(x)).
Unfortunately, the IS estimator often has very high variance if the choice of proposal distribution is
very different from the target, especially when the proposal is more peaked or has thinner tails than
the target. In these cases, there exist configurations x such that q(x) p(x), giving importance
weights w(x) = f (x)/q(x) with extremely large values, but very small probabilities. Due to
the low probability of seeing these large weights, a ?typical? run of IS often underestimates Z in
practice, that is, Z? ? Z with high probability, despite being unbiased.
Similarly, the empirical variance of {w(xi )} can also severely underestimate the true variance
var(w(x)), and so fail to capture the true uncertainty of the estimator. For this reason, concentration inequalities that make use of the empirical variance (see Section 3) also require that w, or its
variance, be bounded. It is thus desirable to construct proposals that are similar to, and less peaked
than, the target distribution p(x). The key observation of this work is to show that tree reweighted
BP and its generalizations provide a easy way to construct such good proposals.
2.3
Tree Reweighted Belief Propagation
Next we describe the tree reweighted (TRW) upper bound on the partition function, restricting to
pairwise models (2) for notational ease. In Section 4 we give an extension that includes both more
general factor graphs, and more general convex upper bounds.
Let T = {T } be a set of spanning trees T = (V, ET ) of G that covers G: ?T ET = EG . We assign
P
a set of nonnegative weights {?T : T ? T } on T such that T ?T = 1. Let ? T = {? T : T ? T }
P
be a set of natural parameters that satisfies T ?T ? T = ?, and each ? T respects the structure of T
T
(so that ?kl
(xk , xl ) ? 0 for ?(kl) 6? ET ). Define
def
pT (x) = p(x; ? T ) =
X
X
f (x; ? T )
T
, with f (x; ? T ) = exp
?kT (xk ) +
?kl
(xk , xl ) ;
T
Z(? )
k?V
(kl)?E
T
then pT (x) is a tree structured graphical model with Markov graph T . Wainwright et al. [30] use
the fact that log Z(?) is a convex function of ? to propose to upper bound log Z(?) by
X
X
log Ztrw (? T ) =
?T log Z(? T ) ? log Z(
?T ? T ) = log Z(?),
T ?T
T ?T
3
via Jensen?s inequality. Wainwright et al. [30] find the tightest bound via a convex optimization:
X
?
log Ztrw
(?) = min log Ztrw (? T ),
s.t.
?T ? T = ? .
(3)
?T
T
Wainwright et al. [30] solve this optimization by a tree reweighted belief propagation (TRW-BP)
algorithm, and note that the optimality condition of (3) is equivalent to enforcing a marginal consistency condition on the trees ? a ? T optimizes (3) if and only if there exists a set of common
singleton and pairwise ?pseudo-marginals? {bk (xk ), bkl (xk , xl )}, corresponding to the fixed point
of TRW-BP in Wainwright et al. [30], such that
b(xk , xl ) = pT (xk , xl ), ?(kl) ? T,
b(xk ) = pT (xk ), ?k ? V,
and
where pT (xk ) and pT (xk , xl ) are the marginals of pT (x). Thus, after running TRW-BP, we can
calculate pT (x) via
Y
Y bkl (xk , xl )
pT (x) = p(x ; ? T ) =
bk (xk )
.
(4)
bk (xk )bl (xl )
k?V
kl?ET
Because TRW provides a convex upper bound, it is often well-suited to the inner loop of learning
algorithms [e.g., 28]. However, it is often far less accurate than its non-convex counterpart, loopy
BP; in some sense, this can be viewed as the cost of being a bound. In the next section, we show
that our importance sampling procedure can ?de-bias? the TRW bound, to produce an estimator
that significantly outperforms loopy BP; in addition, due to the nice properties of our TRW-based
proposal, we can use an empirical Bernstein inequality to construct a non-asymptotic confidence
interval for our estimator, turning the deterministic TRW bound into a much tighter probabilistic
bound.
3
Tree Reweighted Importance Sampling
We propose to use the collection of trees pT (x) and weights ?T in TRW to form an importance
sampling proposal,
X
q(x; ? T ) =
?T pT (x),
(5)
T ?T
Pn
which defines an estimator Z? = n1 i=1 w(xi ) with xi drawn i.i.d. from q(x; ? T ). Our observation
is that this proposal is good due toP
the special convex construction of TRW. To see this, we note that
the reparameterization constraint T ?T ? T = ? can be rewritten as
Y
?T
f (x; ?) = Ztrw (? T )
pT (x) ,
(6)
T
T
that is, f (x; ?) is the {? }-weighted geometric mean of pT (x) up to a constant Ztrw ; on the other
hand, q(x; ? T ), by its definition, is the arithmetic mean of pT (x), and hence will always be larger
than the geometric mean by the AM-GM inequality, guaranteeing good coverage of the target?s probability. To be specific, we have q(x; ? T ) is always no smaller than f (x; ?)/Ztrw (? T ), and hence
the importance weight w(x) is always upper bounded by Ztrw (? T ). Note that (5)?(6) immediately
implies that q(x; ? T ) > 0 whenever f (x; ?) > 0. We summarize our result as follows.
P
P
Proposition 3.1. i) If T ?T ? T = ?, ?T ? 0, T ?T = 1, then the importance weight w(x) =
f (x; ?)/q(x; ? T ), with q(x; ? T ) defined in (5), satisfies
w(x) ? Ztrw (? T ), ?x ? X ,
(7)
that is, the importance weights of (5) are always bounded by the TRW upper bound; this reinterprets
the TRW optimization (3) as finding the mixture proposal in (5) that has the smallest upper bound
on the importance weights.
2
ii) As a result, we have max{var(w(x)), var(w(x))}
c
? 14 Ztrw
for x ? q(x; ? T ), where var(w(x))
c
1
2
2
?
is the empirical variance of the weights. This implies that E(Z ? Z) ? 4n Ztrw .
4
Proof. i) Directly apply AM-GM inequality on (5) and (6). ii) Note that E(w(x)) = Z and hence
2
var(w(x)) = E(w(x)2 ) ? E(w(x))2 ? Ztrw Z ? Z 2 ? 14 Ztrw
.
Note that the TRW reparameterization (6) is key to establishing our results. Its advantage is two-fold:
First, it provides a simple upper bound on w(x); for an arbitrary q(?), establishing such an upper
bound may require a difficult combinatorial optimization over x. Second, it enables that bound to
be optimized over q(?), resulting in a good proposal.
Empirical Bernstein Confidence Bound. The upper bound of w(x) in Proposition 3.1 allows us
to use exponential concentration inequalities and construct tight finite-sample confidence bounds.
Based on the empirical Bernstein inequality in Maurer and Pontil [19], we have
Corollary 3.2 (Maurer and Pontil [19]). Let Z? be the IS estimator resulting from q(x) in (5). Define
r
2var(w(x))
c
log(2/?) 7Ztrw (? T ) log(2/?)
?=
+
,
(8)
n
3(n ? 1)
where var(w(x)
c
is the empirical variance of the weights, then Z?+ = Z? + ? and Z? = Z? ? ? are
upper and lower bounds of Z with at least probability (1 ? ?), respectively, that is, Pr(Z ? Z?+ ) ?
1 ? ? and Pr(Z?? ? Z) ? 1 ? ?.
The quantity ? is quite intuitive,
? with the first term proportional to the empirical standard deviation
and decaying at the classic 1/ n rate. The second term captures the possibility that the empirical variance is inaccurate; it depends on the boundedness of w(x) and decays at rate 1/n. Since
2
, the second term typically dominates for small n, and the first term for large n.
var(w)
c
< Ztrw
When ? is large, the lower bound Z? ? ? may be negative; this is most common when n is small and
Ztrw is much larger than Z. In this case, we may replace Z?? with any deterministic lower bound, or
? which is a (1 ? ?) probabilistic bound by the Markov inequality; see Gogate and Dechter
with Z?,
[11] for more Markov inequality based lower bounds. However, once n is large enough, we expect
Z?? should be much tighter than using Markov?s inequality, since Z?? also leverages boundedness
and variance information.1 On the other hand, the Bernstein upper bound Z?+ readily gives a good
upper bound, and is usually much tighter than Ztrw even with a relatively small n.
For example, if Z? Ztrw (e.g., the TRW bound is not tight), our upper bound Z?+ improves rapidly
on Ztrw at rate 1/n and passes Ztrw when n ? 37 log(2/?) + 1 (for example, for ? = 0.025 used
in our experiments, we have Z?+ ? Ztrw by n = 12). Meanwhile, one can show that the lower
? log(2/?) + 1. During sampling, we can
bound must be non-trivial (Z?? > 0) if n > 6(Ztrw /Z)
roughly estimate the point at which it will become non-trivial, by finding n such that Z? ? ?. More
rigorously, one can apply a stopping criterion [e.g., 5, 20] on n to guarantee a relative error with
probability at least 1 ? ?, using the bound on w(x); roughly, the expected number of samples will
depend on Ztrw /Z, the relative accuracy of the variational bound.
4
Weighted Mini-bucket Importance Sampling
We have so far presented our results for tree reweighted BP on pairwise models, which approximates
the model using combinations of trees. In this section, we give an extension of our results to general
higher order models, and approximations based on combinations of low-treewidth graphs. Our
extension is based on the weighted mini-bucket framework [7, 17, 16], but extensions based on
other higher order generalizations of TRW, such as Globerson and Jaakkola [9], are also possible.
We only sketch the main idea in this section.
We start by rewriting the distribution using the chain rule along some order o = [x1 , . . . , xp ],
Y
f (x) = Z
p(xk |xpa(k) ).
(9)
k
1
The Markov lower bounds by Gogate and Dechter [11] have the undesirable property that they may not
become tighter with increasing n, and may even decrease.
5
where pa(k), called the induced parent set of k, is the set of variables adjacent to xk when it is
eliminated along order o. The largest parent size ? := maxk?V |pa(k)| is called the induced width
of G along order o, and the computational complexity of exact variable elimination along order o is
O(exp(?)), which is intractable when ? is large.
Weighted mini-bucket is an approximation method that avoids the O(exp(?)) complexity by splitting each pa(k) into several smaller ?mini-buckets? pa` (k), such that ?` pa` (k) = pa(k), where
the size of the pa` (k) is controlled by a predefined number ibound ? |pa` (k)|, so that the ibound
trades off the computational complexity
P with approximation quality. We associate each pa` (k) with
a nonnegative weight ?k` , such that ` ?k` = 1. The weighted mini-bucket algorithm in Liu [16]
then frames a convex optimization to output an upper bound Zwmb ? Z together with a set of
?pseudo-? conditional distributions bk` (xk |xpa` (k) ), such that
YY
f (x) = Zwmb
(10)
bk` (xk |xpa` (k) )?k` ,
k
`
which, intuitively speaking, can be treated as approximating each conditional distribution
p(xk |xpa(k) ) with a geometric mean of the bk` (xk |xpa` (k) ); while we omit the details of weighted
mini-bucket [17, 16] for space, what is most important for our purpose is the representation (10).
Similarly to with TRW, we define a proposal distribution by replacing the geometric mean with an
arithmetic mean:
YX
(11)
q(x) =
?k` bk` (xk |xpa` (k) ).
k
`
We can again use the AM-GM inequality to obtain a bound on w(x), that w(x) ? Zwmb .
Proposition
4.1. Let w(x) = f (x)/q(x), where f (x) and q(x) satisfy (10) and (11), with
P
` ?k` = 1, ?k` ? 0, ?k, `. Then,
w(x) ? Zwmb ,
Proof. Use the AM-GM inequality,
?x ? X .
Q
` bk` (xk |xpa` (k) )
?k`
?
P
`
?k` bk` (xk |xpa` (k) ), for each k.
Note that the formP
of q(x) makes it convenient to sample by sequentially drawing each variable xk
from the mixture ` ?k` bk` (xk |xpa` (k) ) along the reverse order [xp , . . . , x1 ]. The proposal q(x)
also can be viewed as a mixture of a large number of models with induced width controlled by
ibound; this can be seen by expanding the form in (11),
X
Y
Y
q(x) =
?`1 ???`p q`1 ???`p (x), where ?`1 ???`p =
?k`k , q`1 ???`p (x) =
bk`k (xk |xpa` (k) ).
`1 ???`p
5
k
k
Experiments
We demonstrate our algorithm using synthetic Ising models, and real-world models from recent
UAI inference challenges. We show that our TRW proposal can provide better estimates than other
proposals constructed from mean field or loopy BP, particularly when it underestimates the partition
function; in this case, the proposal may be too peaked and fail to approach the true value even for
extremely large sample sizes n. Using the empirical Bernstein inequality, our TRW proposal also
provides strong probabilistic upper and lower bounds. When the model is relatively easy or n is
large, our upper and lower bounds are close, demonstrating the estimate has high confidence.
5.1
MRFs on 10 ? 10 Grids
We illustrate our method using pairwise Markov random fields (2) on a 10 ? 10 grid. We start with
a simple Ising model with ?k (xk ) = ?s xk and ?kl (xk , xl ) = ?p xk xl , xk ? {?1, 1}, where ?s
represents the external field and ?p the correlation. We fix ?s = 0.01 and vary ?p from ?1.5 (strong
negative correlation) to 1.5 (strong positive correlation). Different ?p lead to different inference
hardness: inference is easy when the correlation is either very strong (|?p | large) or very weak (|?p |
small), but difficult for an intermediate range of values, corresponding to a phase transition.
6
log Z? ? log Z
0
TRW/MF Interval
0.1
IS(TRW) Bernstein
10
?1
5
5
IS(TRW)
IS(MF)
IS(LBP)
Loopy BP
?2
?3
?1
0
1
Pairwise Strength ?p
(a) Fixed n = 104
TRW
10
0
0
0
LBP
MF
?5 IS(TRW)
-5
Markov (TRW)
?0.1
-1
0
1
?1
Pairwise Strength ?p
(b) Fixed n = 104
0
1
1
10
Pairwise Strength ?p
(c) Fixed n = 107
10
3
5
10
7
10
Sample Size n
(d) Fixed ?p = ?0.5
Figure 1: Experiments on 10 ? 10 Ising models with interaction strength ?p ranging from strong
negative (-1.5) to strong positive (1.5).
We first run the standard variational algorithms, including loopy BP (LBP), tree reweighted BP
(TRW), and mean field (MF). We then calculate importance sampling estimators based on each
of the three algorithms. The TRW trees are chosen by adding random spanning trees until their
union covers the grid; we assign uniform probability ?T to each tree. The LBP proposal follows
Gogate [10], constructing a (randomly selected)
tree structured proposal based on the LBP pseudoQ
marginals. The MF proposal is q(x) = k?V qk (xk ), where the qk (xk ) are the mean field beliefs.
Figure 1(a) shows the result of the IS estimates based on a relatively small number of importance
samples (n = 104 ). In this case the TRW proposal outperforms both the MF and LBP proposals;
all the methods degrade when ?p ? ?.5, corresponding to inherently more difficult inference.
However, the TRW proposal converges to the correct values when the correlation is strong (e.g.,
|?p | > 1), while the MF and LBP proposals underestimate the true value, indicating that the MF and
LBP proposals are too peaked, and miss a significant amount of probability mass of the target.
Examining the deterministic estimates, we note that the LBP approximation, which can be shown to
be a lower bound on these models [27, 24], is also significantly worse than IS with the TRW proposal, and slightly worse than IS based on the LBP proposal. The TRW and MF bounds, of course,
are far less accurate compared to either LBP or the IS methods, and are shown separately in Figure 1(b). This suggests it is often beneficial to follow the variational procedure with an importance
sampling process, and use the corresponding IS estimators instead of the variational approximations
to estimate the partition function.
Figure 1(b) compares the 95% confidence interval of the IS based on the TRW proposal (filled with
red), with the interval formed by the TRW upper bound and the MF lower bound (filled with green).
We can see that the Bernstein upper bound is much tighter than the TRW upper bound, although at
the cost of turning a deterministic bound into a (1 ? ?) probabilistic bound. On the other hand, the
Bernstein interval fails to report a meaningful lower bound when the model is difficult (?p ? ?0.5),
because n = 104 is small relative to the difficulty of the model. As shown in Figure 1(c), our method
eventually produces both tight upper and lower bounds as sample size increases.
In addition to the simple Ising model, we also tested grid
models with normally distributed parameters: ?k (xk ) ?
N (0, ?s2 ) and ?kl (xk , xl ) ? N (0, ?p2 ). Figure 2 shows
the results when ?s = 0.01 and we vary ?p . In this case,
LBP tends to overestimate the partition function, and IS
with the LBP proposal performs quite well (similarly to
our TRW IS); but with the previous example, this illustrates that it is hard to know whether BP will result in a
high- or low-variance proposal. On this model, mean field
IS is significantly worse and is not shown in the figure.
7
log Z? ? log Z
Figure 1(d) shows the Bernstein bound as we increase n on a fixed model with ?p = ?0.5, which
is relatively difficult according to Figure 1. Of the methods, our IS estimator becomes the most
? as
accurate by around n = 103 samples. We also show the Markov lower bound Z?markov = Z?
suggested by Gogate [10]; it provides non-negative lower bounds for all sample sizes, but does not
converge to the true value even with n ? +? (in fact, it converges to Z?).
IS(TRW) Bernstein
Loopy BP
IS (BP)
0.2
0.1
0
?0.1
0.5
1
1.5
2
Pairwise Strength ?p
Figure 2: MRF with mixed interactions.
log Z? ? log Z
5
WMB
0
IS(WMB)
0
Markov (WMB)
?5
1
10
2
3
4
WMB
5
IS(WMB)
Markov (WMB)
?5
5
1
10 10 10 10
Sample Size (n)
2
3
4
5
10 10 10 10 10 10
Sample Size (n)
(a) BN 6, ibound = 1
6
(b) BN 11, ibound = 1
log Z? ? log Z
Figure 3: The Bernstein interval on (a) BN 6 and (b) BN 11 using ibound = 1 and different sample
sizes n. These problems are relatively easy for variational approximations; we illustrate that our
method gives tight bounds despite using no more memory than the original model.
1
2
0
0
GBP
GBP
IS(WMB)
IS(WMB)
?2
1
2
3
4
5
6
10 10 10 10 10 10
Sample Size (n)
?1
1
2
3
4
5
6
10 10 10 10 10 10
Sample Size (n)
(a) ibound = 8
(b) ibound = 15
Figure 4: Results on a harder instance, pedigree20, at ibound = 8, 15 and different n.
5.2
UAI Instances
We test the weighted mini-bucket (WMB) version of our algorithm on instances from past UAI
approximate inference challenges. For space reasons, we only report a few instances for illustration.
BN Instances. Figure 3 shows two Bayes net instances, BN 6 (true log Z = ?58.41) and BN 11
(true log Z = ?39.37). These examples are very easy for loopy BP, which estimates log Z nearly
exactly, but of course gives no accuracy guarantees. For comparison, we run our WMB IS estimator
using ibound = 1, e.g., cliques equal to the original factors. We find that we get tight confidence
intervals by around 104 ?105 samples. For comparison, the method of Dagum and Luby [4], using
the normalized distribution as a proposal, would require samples proportional to 1/Z: approximately
1025 and 1017 , respectively.
Pedigree Instances. We next show results for our method on pedigree20, (log Z = ?68.22,
induced width ? = 21). and various ibounds; Figure 4 shows the results for ibound 8 and 15.
For comparision, we also evaluate GBP, defined on a junction graph with cliques found in the same
way as WMB [18], and complexity controlled by the same ibound. Again, LBP and GBP generally
give accurate estimates; the absolute error of LBP (not shown) is about 0.7, reducing to 0.4 and
0.2 at ibound = 8 and 15, respectively. The initial WMB bounds overestimate by 6.3 and 2.4 at
ibound = 8 and 15, and are much less accurate. However, our method surpasses GBP?s accuracy
with a modest number of samples: for example, with ibound = 15 (Figure 4b), our IS estimator is
more accurate than GBP with fewer than 100 samples, and our 95% Bernstein confidence interval
passes GBP at roughly 1000 samples.
6
Conclusion
We propose a simple approximate inference method that augments convex variational bounds by
adding importance sampling. Our formulation allows us to frame the variational optimization as designing a proposal that minimizes an upper bound on our estimator?s variance, providing guarantees
on the goodness of the resulting proposal. More importantly, this enables the construction of anytime probabilistic bounds that improve quickly and directly on state-of-the-art variational bounds,
and provide certificates of accuracy given enough samples, relative to the error in the initial bound.
One potential future direction is whether one can adaptively improve the proposal during sampling.
Acknowledgement This work is supported in part by VITALITE, under the ARO MURI program
(Award number W911NF-11-1-0391); NSF grants IIS-1065618 and IIS-1254071; and by the United
States Air Force under Contract No. FA8750-14-C-0011 under the DARPA PPAML program.
8
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5,187 | 5,696 | On the Convergence of Stochastic Gradient MCMC
Algorithms with High-Order Integrators
Changyou Chen?
Nan Ding?
Lawrence Carin?
Dept. of Electrical and Computer Engineering, Duke University, Durham, NC, USA
?
Google Inc., Venice, CA, USA
[email protected]; [email protected]; [email protected]
?
Abstract
Recent advances in Bayesian learning with large-scale data have witnessed emergence of stochastic gradient MCMC algorithms (SG-MCMC), such as stochastic gradient Langevin dynamics (SGLD), stochastic gradient Hamiltonian MCMC
(SGHMC), and the stochastic gradient thermostat. While finite-time convergence
properties of the SGLD with a 1st-order Euler integrator have recently been studied, corresponding theory for general SG-MCMCs has not been explored. In this
paper we consider general SG-MCMCs with high-order integrators, and develop
theory to analyze finite-time convergence properties and their asymptotic invariant
measures. Our theoretical results show faster convergence rates and more accurate invariant measures for SG-MCMCs with higher-order integrators. For example, with the proposed efficient 2nd-order symmetric splitting integrator, the mean
square error (MSE) of the posterior average for the SGHMC achieves an optimal
convergence rate of L?4/5 at L iterations, compared to L?2/3 for the SGHMC
and SGLD with 1st-order Euler integrators. Furthermore, convergence results of
decreasing-step-size SG-MCMCs are also developed, with the same convergence
rates as their fixed-step-size counterparts for a specific decreasing sequence. Experiments on both synthetic and real datasets verify our theory, and show advantages of the proposed method in two large-scale real applications.
1
Introduction
In large-scale Bayesian learning, diffusion based sampling methods have become increasingly popular. Most of these methods are based on It?o diffusions, defined as:
d Xt = F (Xt )dt + ?(Xt )dWt .
(1)
n
Here Xt ? R represents model states, t the time index, Wt is Brownian motion, functions F :
Rn ? Rn and ? : Rn ? Rn?m (m not necessarily equal to n) are assumed to satisfy the usual
Lipschitz continuity condition.
In a Bayesian setting, the goal is to design appropriate functions F and ?, so that the stationary
distribution, ?(X), of the It?o diffusion has a marginal distribution that is equal to the posterior
distribution of interest.? For example, 1st-order Langevin dynamics (LD) correspond to X = ?,
F = ??? U and ? = 2 In , with In being the n ? n identity matrix; 2nd-order Langevin dynamics
? 0 0
p
correspond to X = (?, p), F =
, and ? = 2D
for some D > 0.
?D p ??? U
0 In
Here U is the unnormalized negative log-posterior, and p is known as the momentum [1, 2]. Based
on the Fokker-Planck equation [3], the stationary distributions of these dynamics exist and their
marginals over ? are equal to ?(?) ? exp(?U (?)), the posterior distribution we are interested in.
Since It?o diffusions are continuous-time Markov processes, exact sampling is in general infeasible.
As a result, the following two approximations have been introduced in the machine learning liter1
ature [1, 2, 4], to make the sampling numerically feasible and practically scalable: 1) Instead of
analytically integrating infinitesimal increments dt, numerical integration over small step h is used
to approximate the integration of the true dynamics. Although many numerical schemes have been
studied in the SDE literature, in machine learning only the 1st-order Euler scheme is widely applied.
2) During every integration, instead of working with the gradient of the full negative log-posterior
?l (?), is calculated from the l-th minibatch of data, imU (?), a stochastic-gradient version of it, U
portant when considering problems with massive data. In this paper, we call algorithms based on 1)
and 2) SG-MCMC algorithms. To be complete, some recently proposed SG-MCMC algorithms are
briefly reviewed in Appendix A. SG-MCMC algorithms often work well in practice, however some
theoretical concerns about the convergence properties have been raised [5?7].
Recently, [5, 6, 8] showed that the SGLD [4] converges weakly to the true posterior. In [7], the author
studied the sample-path inconsistency of the Hamiltonian PDE with stochastic gradients (but not the
SGHMC), and pointed out its incompatibility with data subsampling. However, real applications
only require convergence in the weak sense, i.e., instead of requiring sample-wise convergence as
in [7], only laws of sample paths are of concern? . Very recently, the invariance measure of an SGMCMC with a specific stochastic gradient noise was studied in [9]. However, the technique is not
readily applicable to our general setting.
In this paper we focus on general SG-MCMCs, and study the role of their numerical integrators. Our
main contributions include: i) From a theoretical viewpoint, we prove weak convergence results for
general SG-MCMCs, which are of practical interest. Specifically, for a Kth-order numerical integrator, the bias of the expected sample average of an SG-MCMC at iteration L is upper bounded by
L?K/(K+1) with optimal step size h ? L?1/(K+1) , and the MSE by L?2K/(2K+1) with optimal
h ? L?1/(2K+1) . This generalizes the results of the SGLD with an Euler integrator (K = 1) in
[5, 6, 8], and is better when K ? 2; ii) From a practical perspective, we introduce a numerically efficient 2nd-order integrator, based on symmetric splitting schemes [9]. When applied to the SGHMC,
it outperforms existing algorithms, including the SGLD and SGHMC with Euler integrators, considering both synthetic and large real datasets.
2
Preliminaries & Two Approximation Errors in SG-MCMCs
In weak convergence analysis, instead of working directly with sample-paths in (1), we study how
the expected value of any suitably smooth statistic of Xt evolves in time. This motivates the introduction of an (infinitesimal) generator. Formally, the generator L of the diffusion (1) is defined for
any compactly supported twice differentiable function f : Rn ? R, such that,
E [f (Xt+h )] ? f (Xt )
1
Lf (Xt ) , lim+
= F (Xt ) ? ? +
?(Xt )?(Xt )T : ??T f (Xt ) ,
h
2
h?0
where a ? b , aT b, A : B , tr(AT B), h ? 0+ means h approaches zero along the positive real
axis. L is associated with an integrated form via Kolmogorov?s backward equation? : E [f (XeT )] =
eT L f (X0 ), where XeT denotes the exact solution of the diffusion (1) at time T . The operator eT L
is called the Kolmogorov operator for the diffusion (1). Since diffusion (1) is continuous, it is
generally infeasible to solve analytically (so is eT L ). In practice, a local numerical integrator is
used for every small step h, with the corresponding Kolmogorov operator Ph approximating ehL .
Let Xnlh denote the approximate sample path from such a numerical integrator; similarly, we have
E[f (Xnlh )] = Ph f (Xn(l?1)h ). Let A ? B denote the composition of two operators A and B, i.e., A is
evaluated on the output of B. For time T = Lh, we have the following approximation
A
A2
E [f (XeT )] =1 ehL ? . . . ? ehL f (X0 ) ' Ph ? . . . ? Ph f (X0 ) = E[f (XnT )],
with L compositions, where A1 is obtained by decomposing T L into L sub-operators, each for
a minibatch of data, while approximation A2 is manifested by approximating the infeasible ehL
with Ph from a feasible integrator, e.g., the symmetric splitting integrator proposed later, such that
?
For completeness, we provide mean sample-path properties of the SGHMC (similar to [7]) in Appendix J.
More details of the equation are provided in Appendix B. Specifically, under mild conditions on F , we can
expand the operator ehL up to the mth-order (m ? 1) such that the remainder terms are bounded by O(hm+1 ).
Refer to [10] for more details. We will assume these conditions to hold for the F ?s in this paper.
?
2
E [f (XnT )] is close to the exact expectation E [f (XeT )]. The latter is the first approximation error
introduced in SG-MCMCs. Formally, to characterize the degree of approximation accuracies for
different numerical methods, we use the following definition.
Definition 1. An integrator is said to be a Kth-order local integrator if for any smooth and bounded
function f , the corresponding Kolmogorov operator Ph satisfies the following relation:
Ph f (x) = ehL f (x) + O(hK+1 ) .
(2)
The second approximation error is manifested when handling large data. Specifically, the SGLD
and SGHMC use stochastic gradients in the 1st and 2nd-order LDs, respectively, by replacing in F
?l , from the l-th minibatch. We
and L the full negative log-posterior U with a scaled log-posterior, U
?
denote the corresponding generators with stochastic gradients as Ll , e.g., the generator in the l-th
? l = L +?Vl , where ?Vl = (?? U
?l ??? U )??p . As a result, in
minibatch for the SGHMC becomes L
?
l
?
SG-MCMC algorithms, we use the noisy operator Ph to approximate ehLl such that E[f (Xn,s
lh )] =
n,s
P?hl f (X(l?1)h ), where Xlh denotes the numerical sample-path with stochastic gradient noise, i.e.,
B1
B2
?
?
E [f (XeT )] ' ehLL ? . . . ? ehL1 f (X0 ) ' P?hL ? . . . ? P?h1 f (X0 ) = E[f (Xn,s
T )].
(3)
Approximations B1 and B2 in (3) are from the stochastic gradient and numerical integrator ap?l
proximations, respectively. Similarly, we say P?hl corresponds to a Kth-order local integrator of L
?l
l
hL
K+1
?
if Ph f (x) = e f (x) + O(h
). In the following sections, we focus on SG-MCMCs which
use numerical integrators with stochastic gradients, and for the first time analyze how the two introduced errors affect their convergence behaviors. For notational simplicity, we henceforth use Xt to
represent the approximate sample-path Xn,s
t .
3
Convergence Analysis
This section develops theory to analyze finite-time convergence properties of general SG-MCMCs
with both fixed and decreasing step sizes, as well as their asymptotic invariant measures.
3.1
Finite-time error analysis
Given an Rergodic? It?o diffusion (1) with an invariant measure ?(x), the posterior average is defined
as: ?? , X ?(x)?(x)d x for some test function ?(x) of interest. For a given numerical method
PL
1
?
?
with generated samples (Xlh )L
l=1 , we use the sample average ? defined as ? = L
l=1 ?(Xlh ) to
? In the analysis, we define a functional ? that solves the following Poisson Equation:
approximate ?.
L
? or equivalently,
L?(Xlh ) = ?(Xlh ) ? ?,
1X
?
L?(Xlh ) = ?? ? ?.
L
(4)
l=1
The solution functional ?(Xlh ) characterizes the difference between ?(Xlh ) and the posterior average ?? for every Xlh , thus would typically possess a unique solution, which is at least as smooth as
? under the elliptic or hypoelliptic settings [12]. In the unbounded domain of Xlh ? Rn , to make
the presentation simple, we follow [6] and make certain assumptions on the solution functional,
?, of the Poisson equation (4), which are used in the detailed proofs. Extensive empirical results
have indicated the assumptions to hold in many real applications, though extra work is needed for
theoretical verifications for different models, which is beyond the scope of this paper.
Assumption 1. ? and its up to 3rd-order derivatives, Dk ?, are bounded by a function? V, i.e.,
kDk ?k ? Ck V pk for k = (0, 1, 2, 3), Ck , pk > 0. Furthermore, the expectation of V on {Xlh }
is bounded: supl EV p (Xlh ) < ?, and V is smooth such that sups?(0,1) V p (s X + (1 ? s) Y) ?
C (V p (X) + V p (Y)), ? X, Y, p ? max{2pk } for some C > 0.
?
See [6, 11] for conditions to ensure (1) is ergodic.
The existence of such function can be translated into finding a Lyapunov function for the corresponding
SDEs, an important topic in PDE literatures [13]. See Assumption 4.1 in [6] and Appendix C for more details.
?
3
We emphasize that our proof techniques are related to those of the SGLD [6, 12], but with significant
distinctions in that, instead of expanding the function ?(Xlh ) [6], whose parameter Xlh does not
endow an explicit form in general SG-MCMCs, we start from expanding the Kolmogorov?s backward equation for each minibatch. Moreover, our techniques apply for general SG-MCMCs, instead
of for one specific algorithm. More specifically, given a Kth-order local integrator with the corresponding Kolmogorov operator P?hl , according to Definition 1 and (3), the Kolmogorov?s backward
equation for the l-th minibatch can be expanded as:
?
E[?(Xlh )] = P?hl ?(X(l?1)h ) = ehLl ?(X(l?1)h ) + O(hK+1 )
K
X
hk ? k
? l ?(X(l?1)h ) +
= I + hL
L ?(X(l?1)h ) + O(hK+1 ) ,
k! l
(5)
k=2
? l = L +?Vl , e.g., ?Vl = (?? U
?l ? ?? U ) ? ?p in SGHMC.
where I is the identity map. Recall that L
By further using the Poisson equation (4) to simplify related terms associated with L, after some
? =
algebra shown in Appendix D, the bias can be derived from (5) as: |E?? ? ?|
K
L
E[?(X )] ? ?(X )
X
1X
hk?1 X ? k
lh
0
?
E[?Vl ?(X(l?1)h )] ?
E[Ll ?(X(l?1)h )] + O(hK ) .
Lh
L
k!L
l
k=2
l=1
All terms in the above equation can be bounded, with details provided in Appendix D. This gives us
a bound for the bias of an SG-MCMC algorithm in Theorem 2.
Theorem 2. Under Assumption 1, let k?k be the operator norm. The bias of an SG-MCMC with a
Kth-order integrator at time T = hL can be bounded as:
P
kE?Vl k
1
? ?
+ l
+ hK .
E? ? ? = O
Lh
L
P
Note the bound above includes the term l kE?Vl k /L, measuring the difference between the expectation of stochastic gradients and the true gradient. It vanishes when the stochastic gradient is
an unbiased estimation of the exact gradient, an assumption made in the SGLD. This on the other
? might diverge when the growth of
hand indicates that if the stochastic gradient is biased, |E?? ? ?|
P
kE?V
k
is
faster
than
O(L).
We
point
this
out
to
show
our
result to be more informative than
l
l
that of the SGLD [6], though this case might not happen in real applications. By expanding the proof
for the bias, we are also able to bound the MSE of SG-MCMC algorithms, given in Theorem 3.
?l is an unbiased estimate of Ul . For a smooth test
Theorem 3. Under Assumption 1, and assume U
function ?, the MSE of an SG-MCMC with a Kth-order integrator at time T = hL is bounded, for
some C > 0 independent of (L, h), as
!
P
2
1
2
1
l E k?Vl k
2K
L
?
?
+
+h
.
E ??? ?C
L
Lh
P
2
Compared to the SGLD [6], the extra term L12 l E k?Vl k relates to the variance of noisy gradients. As long as the variance is bounded, the MSE still converges with the same rate. Specifically,
when optimizing bounds for the bias and MSE, the optimal bias decreases at a rate of L?K/(K+1)
with step size h ? L?1/(K+1) ; while this is L?2K/(2K+1) with step size h ? L?1/(2K+1) for the
MSE? . These rates decrease faster than those of the SGLD [6] when K ? 2. The case of K = 2 for
the SGHMC with our proposed symmetric splitting integrator is discussed in Section 4.
3.2
Stationary invariant measures
The asymptotic invariant measures of SG-MCMCs correspond to L approaching infinity in the above
analysis. According to the bias and MSE above, asymptotically (L ? ?) the sample average ?? is a
K
?
? ?)
? 2 ? E(??
? ?)
? 2 +E(??E
? ?)
?2=
random variable with mean E?? = ?+O(h
), and variance E(??E
2K
?
O(h ), close to the true ?. This section defines distance between measures, and studies more
formally how the approximation errors affect the invariant measures of SG-MCMC algorithms.
?
To compare with the standard MCMC convergence rate of 1/2, the rate needs to be taken a square root.
4
First we note that under mild conditions, the existence of a stationary invariant measure for an SGMCMC can be guaranteed by application of the Krylov?Bogolyubov Theorem [14]. Examining the
conditions is beyond the scope of this paper. For simplicity, we follow [12] and assume stationary
invariant measures do exist for SG-MCMCs. We denote the corresponding invariant measure as ??h ,
and the true posterior of a model as ?. Similar to [12],Rwe assume our numerical
solver is geometric
R
ergodic, meaning that for a test function ?, we have X ?(x)?
?h (d x) = X Ex ?(Xlh )?
?h (d x) for
any l ? 0 from the ergodic theorem, where Ex denotes the expectation conditional on X0 = x. The
geometric ergodicity implies that the integration is independent of the starting point of an algorithm.
Given this, we have the following theorem on invariant measures of SG-MCMCs.
Theorem 4. Assume that a Kth-order integrator is geometric ergodic and its invariance measures ?R?h exist. Define the
the invariant measures ??h and ? as: d(?
?h , ?) ,
R distance between
?h (d x) ? X ?(x)?(d x) . Then any invariant measure ??h of an SG-MCMC is close
sup? X ?(x)?
to ? with an error up to an order of O(hK ), i.e., there exists some C ? 0 such that: d(?
?h , ?) ? ChK .
For a Kth-order integrator with full gradients, the corresponding invariant measure has been shown
to be bounded by an order of O(hK ) [9, 12]. As a result, Theorem 4 suggests only orders of numerical approximations but not the stochastic gradient approximation affect the asymptotic invariant
measure of an SG-MCMC algorithm. This is also reflected by experiments presented in Section 5.
3.3
SG-MCMCs with decreasing step sizes
The original SGLD was first proposed with a decreasing-step-size sequence [4], instead of fixing
step sizes, as analyzed in [6]. In [5], the authors provide theoretical foundations on its asymptotic
convergence properties. We demonstrate in this section that for general SG-MCMC algorithms, decreasing step sizes for each minibatch are also feasible. Note our techniques here are different from
those used for the decreasing-step-size SGLD [5], which interestingly result in similar convergence
patterns. Specifically, by adapting the same techniques used in the previous sections, we establish
conditions on the step size sequence to ensure asymptotic convergence, and develop theory on their
finite-time ergodic error as well. To guarantee asymptotic consistency, the following conditions on
decreasing step size sequences are required.
Assumption 2. The step sizes P
{hl } are decreasingk , i.e., 0 < hl+1 < hl , and satisfy that 1)
K+1
L
P?
l=1 hl
P
= 0.
L
l=1 hl = ?; and 2) limL??
h
l=1
l
PL
Denote the finite sum of step sizes as SL ,
l=1 hl . Under Assumption 2, we need to mod?
ify the sample average ? defined in Section 3.1 as a weighted summation of {?(Xlh )}: ?? =
PL hl
?
l=1 SL ?(Xlh ). For simplicity, we assume Ul to be an unbiased estimate of U such that E?Vl = 0.
Extending techniques in previous sections, we develop the following bounds for the bias and MSE.
Theorem 5. Under Assumptions 1 and 2, for a smooth test function ?, the bias and MSE of a
decreasing-step-size SG-MCMC with a Kth-order integrator at time SL are bounded as:
!
PL
K+1
1
? ?
l=1 hl
BIAS: E? ? ? = O
+
(6)
SL
SL
!
PL
K+1 2
2
X h2
h
)
(
1
2
l
l=1
l
MSE: E ?? ? ?? ? C
E k?Vl k +
+
.
(7)
SL2
SL
SL2
l
As
a result, the asymptotic bias approaches 0 according to the assumptions. If further assuming??
P?
2
l=1 hl
= 0, the MSE also goes to 0. In words, the decreasing-step-size SG-MCMCs are consistent.
S2
L
Among the kinds of decreasing step size sequences, a commonly recognized one is hl ? l?? for
0 < ? < 1. We show in the following corollary that such a sequence leads to a valid sequence.
Corollary 6. Using the step size sequences hl ? l?? for 0 < ? < 1, all the step size assumptions
in Theorem 5 are satisfied. As a result, the bias and MSE approach zero asymptotically, i.e., the
?
sample average ?? is asymptotically consistent with the posterior average ?.
k
??
Actually the sequence
P need2not be decreasing; we assume it is decreasing for simplicity.
The assumption of ?
l=1 hl < ? satisfies this requirement, but is weaker than the original assumption.
5
Remark 7. Theorem 5 indicates the sample average ?? asymptotically converges to the true posterior
? It is possible to find out the optimal decreasing rates for the specific decreasing sequence
average ?.
PL
hl ? l?? . Specifically, using the bounds for l=1 l?? (see the proof of Corollary 6), for the two
PL
terms in the bias (6) in Theorem 5, S1L decreases at a rate of O(L??1 ), whereas ( l=1 hK+1
)/SL
l
decreases as O(L?K? ). The balance between these two terms is achieved when ? = 1/(K + 1),
which agrees with Theorem 2 on the optimal rate of fixed-step-size SG-MCMCs. Similarly, for
the MSE (7), the first term decreases as L?1 , independent of ?, while the second and third terms
decrease as O(L??1 ) and O(L?2K? ), respectively, thus the balance is achieved when ? = 1/(2K+
1), which also agrees with the optimal rate for the fixed-step-size MSE in Theorem 3.
According to Theorem 5, one theoretical advantage of decreasing-step-size SG-MCMCs over fixedstep-size variants is the asymptotically unbiased estimation of posterior averages, though the benefit
might not be significant in large-scale real applications where the asymptotic regime is not reached.
4
Practical Numerical Integrators
Given the theory for SG-MCMCs with high-order integrators, we here propose a 2nd-order symmetric splitting integrator for practical use. The Euler integrator is known as a 1st-order integrator; the
proof and its detailed applications on the SGLD and SGHMC are given in Appendix I.
? l into several
The main idea of the symmetric splitting scheme is to split the local generator L
??
sub-generators that can be solved analytically . Unfortunately, one cannot easily apply a splitting
? l = LA +LB +LO ,
scheme with the SGLD. However, for the SGHMC, it can be readily split into: L
l
where
? (?) ? ?p + 2D In : ?p ?T .
LA = p ??? , LB = ?D p ??p , LO = ??? U
(8)
l
p
These sub-generators correspond to the following SDEs, which are all analytically solvable:
d? = 0
d? = p dt
d? = 0
?
A:
,B :
,O :
?l (?)dt + 2DdW
dp = 0
d p = ?D p dt
d p = ??? U
(9)
Based on these sub-SDEs, the local Kolmogorov operator P?hl is defined as:
h
h
h
h
E[f (Xlh )] = P?hl f (X(l?1)h ), where, P?hl , e 2 LA ? e 2 LB ? ehLOl ? e 2 LB ? e 2 LA ,
so that the corresponding updates for Xlh = (?lh , plh ) consist of the following 5 steps:
(1)
(1)
(2)
(1)
?l (? (1) )h +
?lh = ?(l?1)h + p(l?1)h h/2 ? plh = e?Dh/2 p(l?1)h ? plh = plh ??? U
lh
? plh =
(2)
e?Dh/2 plh
? ?lh =
(1)
?lh
?
2Dh?l
+ plh h/2 ,
(1)
(1)
(2)
(?lh , plh , plh )
where
are intermediate variables. We denote such a splitting method as the
ABOBA scheme. From the Markovian property of a Kolmogorov operator, it is readily seen that
all such symmetric splitting schemes (with different orders of ?A?, ?B? and ?O?) are equivalent [15].
Lemma 8 below shows the symmetric splitting scheme is a 2nd-order local integrator.
Lemma 8. The symmetric splitting scheme is a 2nd-order local integrator, i.e., the corresponding
?
Kolmogorov operator P?hl satisfies: P?hl = ehLl + O(h3 ).
When this integrator is applied to the SGHMC, the following properties can be obtained.
Remark 9. Applying Theorem 2 to the SGHMC
with the symmetric splitting scheme (K = 2), the
P
? = O( 1 + l kE?Vl k + h2 ). The optimal bias decreasing rate is
bias is bounded as: |E?? ? ?|
Lh
L
?2 ?
L?2/3P, compared to L?1/2 for the SGLD [6]. Similarly, the MSE is bounded by: E(?? ? ?)
1
Ek?V k2
1
+ h4 ), decreasing optimally as L?4/5 with step size h ? L?1/5 , compared
C( L l L l + Lh
?2/3
to the MSE of L
for the SGLD [6]. This indicates that the SGHMC with the splitting integrator
converges faster than the SGLD and SGHMC with 1st-order Euler integrators.
Remark 10. For a decreasing-step-size SGHMC, based on Remark 7, the optimal step size decreasing rate for the bias is ? = 1/3, and ? = 1/5 for the MSE. These agree with their fixed-step-size
counterparts in Remark 9, thus are faster than the SGLD/SGHMC with 1st-order Euler integrators.
??
? l is split.
This is different from the traditional splitting in SDE literatures[9, 15], where L instead of L
6
10 1
10 2
, = 0:1
, = 0:2
, = 0:33
, = 0:7
10 0
, = 0:1
, = 0:2
, = 0:3
, = 0:4
MSE
BIAS
10 0
10 -2
10 -1
10 -2
10 1
10 2
10 3
#iterations
10 -4
10 1
10 4
10 2
10 3
10 4
#iterations
Figure 2: Bias of SGHMC-D (left) and MSE of SGHMC-F (right) with different step size rates ?.
Thick red curves correspond to theoretically optimal rates.
5
Experiments
BIAS
We here verify our theory and compare with related algorithms on both synthetic data and large-scale
10 -1
machine learning applications.
Splitting
Euler
Synthetic data We consider a standard Gaussian model
where xi ? N (?, 1), ? ? N (0, 1). 1000 data samples {xi }
10 -2
are generated, and every minibatch in the stochastic gradient
is of size 10. The test function is defined as ?(?) , ?2 , with
explicit expression for the posterior average. To evaluate the
10 -3
expectations in the bias and MSE, we average over 200 runs
with random initializations.
10 -4
First we compare the invariant measures (with L = 106 ) of
0.001
0.005
0.01
0.02 0.05 0.1
step size
the proposed splitting integrator and Euler integrator for the
SGHMC. Results of the SGLD are omitted since they are not Figure 1: Comparisons of symmetas competitive. Figure 1 plots the biases with different step ric splitting and Euler integrators.
sizes. It is clear that the Euler integrator has larger biases
in the invariant measure, and quickly explodes when the step size becomes large, which does not
happen for the splitting integrator. In real applications we also find this happen frequently (shown
in the next section), making the Euler scheme an unstable integrator.
Next we examine the asymptotically optimal step size rates for the SGHMC. From the theory we
know these are ? = 1/3 for the bias and ? = 1/5 for the MSE, in both fixed-step-size SGHMC
(SGHMC-F) and decreasing-step-size SGHMC (SGHMC-D). For the step sizes, we did a grid search
to select the best prefactors, which resulted in h=0.033?L?? for the SGHMC-F and hl= 0.045?l??
for the SGHMC-D, with different ? values. We plot the traces of the bias for the SGHMC-D and the
MSE for the SGHMC-F in Figure 2. Similar results for the bias of the SGHMC-F and the MSE of
the SGHMC-D are plotted in Appendix K. We find that when rates are smaller than the theoretically
optimal rates, i.e., ? = 1/3 (bias) and ? = 1/5 (MSE), the bias and MSE tend to decrease faster
than the optimal rates at the beginning (especially for the SGHMC-F), but eventually they slow down
and are surpassed by the optimal rates, consistent with the asymptotic theory. This also suggests that
if only a small number of iterations were feasible, setting a larger step size than the theoretically
optimal one might be beneficial in practice.
Finally, we study the relative convergence speed of the SGHMC and SGLD. We test both fixedstep-size and decreasing-step-size versions. For fixed-step-size experiments, the step sizes are set
to h = CL?? , with ? chosen according to the theory for SGLD and SGHMC. To provide a fair
comparison, the constants C are selected via a grid search from 10?3 to 0.5 with an interval of
0.002 for L = 500, it is then fixed in the other runs with different L values. The parameter D in
the SGHMC is selected within (10, 20, 30) as well. For decreasing-step-size experiments, an initial
step size is chosen within [0.003, 0.05] with an interval of 0.002 for different algorithms?? , and then
it decreases according to their theoretical optimal rates. Figure 3 shows a comparison of the biases
for the SGHMC and SGLD. As indicated by both theory and experiments, the SGHMC with the
splitting integrator yields a faster convergence speed than the SGLD with an Euler integrator.
Large-scale machine learning applications For real applications, we test the SGLD with an
Euler integrator, the SGHMC with the splitting integrator (SGHMC-S), and the SGHMC with an
??
Using the same initial step size is not fair because the SGLD requires much smaller step sizes.
7
0.15
0.1
SGLD
SGHMC
0.08
SGLD
SGHMC
0.06
BIAS
BIAS
0.2
0.04
0.02
0.05
0
20 100
0
250 400 550 700
20 100 250 400 550 700
#iterations
#iterations
Figure 3: Biases for the fixed-step-size (left) and decreasing-step-size (right) SGHMC and SGLD.
Euler integrator (SGHMC-E). First we test them on the latent Dirichlet allocation model (LDA)
[16]. The data used consists of 10M randomly downloaded documents from Wikipedia, using scripts
provided in [17]. We randomly select 1K documents for testing and validation, respectively. As
in [17, 18], the vocabulary size is 7,702. We use the Expanded-Natural reparametrization trick to
sample from the probabilistic simplex [19]. The step sizes are chosen from {2, 5, 8, 20, 50, 80}?10?5 ,
and parameter D from {20, 40, 80}. The minibatch size is set to 100, with one pass of the whole
data in the experiments (and therefore L = 100K). We collect 300 posterior samples to calculate
test perplexities, with a standard holdout technique as described in [18].
Next a recently studied sigmoid belief network model (SBN) [20] is tested, which is a directed counterpart of the popular RBM model. We use a one layer model where the bottom layer corresponds to
binary observed data, which is generated from the hidden layer (also binary) via a sigmoid function.
As shown in [18], the SBN is readily learned by SG-MCMCs. We test the model on the MNIST
dataset, which consists of 60K hand written digits of size 28 ? 28 for training,
? and 10K for testing.
Again the step sizes are chosen from {3, 4, 5, 6}?10?4 , D from {0.9, 1, 5}/ h. The minibatch is
set to 200, with 5000 iterations for training. Like applied for the RBM [21], an advance technique
called anneal importance sampler (AIS) is adopted for calculating test likelihoods.
Perplexity
2000
We briefly describe the results here, more details
SGHMC-Euler
1800
SGHMC-Splitting
are provided in Appendix K. For LDA with 200
topics, the best test perplexities for the SGHMC-S,
1600
SGHMC-E and SGLD are 1168, 1180 and 2496, re1400
spectively; while these are 1157, 1187 and 2511, respectively, for 500 topics. Similar to the synthetic
1200
experiments, we also observed SGHMC-E crashed
2e-05
5e-05
0.0002
0.0008
when using large step sizes. This is illustrated more
step size
clearly in Figure 4. For the SBN with 100 hidFigure 4: SGHMC with 200 topics. The Euden units, we obtain negative test log-likelihoods of
ler explodes with large step sizes.
103, 105 and 126 for the SGHMC-S, SGHMC-E and
SGLD, respectively; and these are 98, 100, and 110 for 200 hidden units. Note the SGHMC-S on
SBN yields state-of-the-art results on test likelihoods compared to [22], which was 113 for 200 hidden units. A decrease of 2 units in the neg-log-likelihood with AIS is considered to be a reasonable
gain [20], which is approximately equal to the gain from a shallow to a deep model [22]. SGHMC-S
is more accuracy and robust than SGHMC-E due to its 2nd-order splitting integrator.
6
Conclusion
For the first time, we develop theory to analyze finite-time ergodic errors, as well as asymptotic
invariant measures, of general SG-MCMCs with high-order integrators. Our theory applies for
both fixed and decreasing step size SG-MCMCs, which are shown to be equivalent in terms of
convergence rates, and are faster with our proposed 2nd-order integrator than previous SG-MCMCs
with 1st-order Euler integrators. Experiments on both synthetic and large real datasets validate our
theory. The theory also indicates that with increasing order of numerical integrators, the convergence
rate of an SG-MCMC is able to theoretically approach the standard MCMC convergence rate. Given
the theoretical convergence results, SG-MCMCs can be used effectively in real applications.
Acknowledgments Supported in part by ARO, DARPA, DOE, NGA and ONR. We acknowledge
Jonathan C. Mattingly and Chunyuan Li for inspiring discussions; David Carlson for the AIS codes.
8
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9
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5,188 | 5,697 | An Active Learning Framework using Sparse-Graph
Codes for Sparse Polynomials and Graph Sketching
Kannan Ramchandran?
UC Berkeley
[email protected]
Xiao Li
UC Berkeley
[email protected]
Abstract
Let f : {?1, 1}n ? R be an n-variate polynomial consisting of 2n monomials,
in which only s 2n coefficients are non-zero. The goal is to learn the polynomial by querying the values of f . We introduce an active learning framework
that is associated with a low query cost and computational runtime. The significant savings are enabled by leveraging sampling strategies based on modern
coding theory, specifically, the design and analysis of sparse-graph codes, such
as Low-Density-Parity-Check (LDPC) codes, which represent the state-of-the-art
of modern packet communications. More significantly, we show how this design
perspective leads to exciting, and to the best of our knowledge, largely unexplored
intellectual connections between learning and coding.
The key is to relax the worst-case assumption with an ensemble-average setting,
where the polynomial is assumed to be drawn uniformly at random from the ensemble of all polynomials (of a given size n and sparsity s). Our framework succeeds with high probability with respect to the polynomial ensemble with sparsity
up to s = O(2?n ) for any ? ? (0, 1), where f is exactly learned using O(ns)
queries in time O(ns log s), even if the queries are perturbed by Gaussian noise.
We further apply the proposed framework to graph sketching, which is the problem of inferring sparse graphs by querying graph cuts. By writing the cut function
as a polynomial and exploiting the graph structure, we propose a sketching algorithm to learn the an arbitrary n-node unknown graph using only few cut queries,
which scales almost linearly in the number of edges and sub-linearly in the graph
size n. Experiments on real datasets show significant reductions in the runtime
and query complexity compared with competitive schemes.
1
Introduction
One of the central problems in computational learning theory is the efficient learning of polynomials
f (x) : x ? {?1, 1}n ? R. The task of learning an s-sparse polynomial f has been studied
extensively in the literature, often in the context of Fourier analysis for pseudo-boolean functions (a
real-valued function defined on a set of binary variables). Many concept classes, such as ?(1)-juntas,
polynomial-sized circuits, decision trees and disjunctive normative form (DNF) formulas, have been
proven very difficult [1] to learn in the worst-case with random examples. Almost all existing
efficient algorithms are based on the membership query model [1, 6?8, 10, 11, 17], which provides
arbitrary access to the value of f (x) given any x ? {?1, 1}n . This makes a richer set of concept
classes learnable in polynomial time poly(s, n). This is a form of what is now popularly referred to
as active learning, which makes queries using different sampling strategies. For instance, [3,10] use
regular subsampling and [9, 14, 18] use random sampling based on compressed sensing. However,
they remain difficult to scale computationally, especially for large s and n.
?
This work was supported by grant NSF CCF EAGER 1439725.
1
In this paper, we are interested in learning polynomials with s = O(2?n ) for some ? ? (0, 1).
Although this regime is not typically considered in the literature, we show that by relaxing the
?worst-case? mindset to an ensemble-average setting (explained later), we can handle this more
challenging regime and reduce both the number of queries and the runtime complexity, even if the
queries are corrupted by Gaussian noise. In the spirit of active learning, we design a sampling
strategy that makes queries to f based on modern coding theory and signal processing. The queries
are formed by ?strategically? subsampling the input to induce aliasing patterns in the dual domain
based on sparse-graph codes. Then, our framework exploits the aliasing pattern (code structure) to
reconstruct f by peeling the sparse coefficients with an iterative simple peeling decoder. Through a
coding-theoretic lens, our algorithm achieves a low query complexity (capacity-approaching codes)
and low computational complexity (peeling decoding).
Further, we apply our proposed framework to graph sketching, which is the problem of inferring
hidden sparse graphs with n nodes by actively querying graph cuts (see Fig. 1). Motivated by
bioinformatics applications [2], learning hidden graphs from additive or cross-additive queries (i.e.
edge counts within a set or across two sets) has gained considerable interest. This problem closely
pertains to our learning framework because the cut function of any graph can be written as a sparse
polynomial with respect to the binary variables x ? {?1, +1}n indicating a graph partition for the
cut [18]. Given query access to the cut value for an arbitrary partition of the graph, how many cut
queries are needed to infer the hidden graph structure? What is the runtime for such inference?
(a) Unknown Graph
(b) Cut Query
(c) Inferred Graph
Figure 1: Given a set of n nodes, infer the graph structure by querying graph cuts.
Most existing algorithms that achieve the optimal query cost for graph sketching (see [13]) are nonconstructive, except for a few algorithms [4, 5, 9, 18] that run in polynomial time in the graph size
n. Inspired by our active learning framework, we derive a sketching algorithm associated with a
query cost and runtime that are both sub-linear in the graph size n and almost-linear in the number
of edges. To the best of our knowledge, this is the first constructive non-adaptive sketching scheme
with sub-linear costs in the graph size n. In the following, we introduce the problem setup, our
learning model, and summarize our contributions.
1.1
Problem Setup
Our goal is to learn the following polynomial in terms of its coefficients:
X
f (x) =
?[k]?k (x), ? x ? {?1, 1}n , F2 := {0, 1},
(1)
k?Fn
2
Q
k[i]
where k := [k[1], ? ? ? , k[n]]T ? Fn2 is the index of the monomial1 ?k (x) = i?[n] xi , and
?[k] ? R is the coefficient. In this work, we consider an ensemble-average setting for learning.
Definition 1 (Polynomial Ensemble). The polynomial ensemble F(s, n, A) is a collection of polynomials f : {?1, 1}n ? R satisfying the following conditions:
? the vector ? := [? ? ? , ?[k], ? ? ? ]T is s-sparse with s = O(2?n ) for some 0 < ? < 1;
? the support supp (?) := {k : ?[k] 6= 0, k ? Fn2 } is chosen uniformly at random over Fn2 ;
? each non-zero coefficient ?[k] takes values from some set A according to ?[k] ? PA for
all k ? supp (?), and PA is some probability distribution over A.
1
The notation is defined as [n] := {1, ? ? ? , n}.
2
We consider active learning under the membership query model. Each query to f at x ? {?1, 1}n
returns the data-label pair (x, f (x) + ?), where ? is some additive noise. We propose a query frameb of the polynomial
work that leads to a fast reconstruction algorithm, which outputs an estimate ?
coefficients. The performance of our framework is evaluated
by the probability of failing to recover
b 6= ?) = E 1?6
the exact coefficients PF := Pr (?
b =? , where 1(?) is the indicator function and the
expectation is taken with respect to the noise ?, the randomized construction of our queries, as well
as the random polynomial ensemble F(s, n, A).
1.2
Our Approach and Contributions
Particularly relevant to this work are the algorithms on learning decision trees and boolean functions
by uncovering the Fourier spectrum of f [3, 5, 10, 12]. Recent papers further show that this problem
can be formulated and solved as a compressed sensing problem using random queries [14, 18].
Specifically, [14] gives an algorithm using O(s2 n) queries based on mutual coherence, whereas the
Restricted Isometry Property (RIP) is used in [18] to give a query complexity of O(sn4 ). However,
this formulation needs to estimate a length-2n vector and hence the complexity is exponential in n.
To alleviate the computational burden, [9] proposes a pre-processing scheme to reduce the number
of unknowns to 2s , which shortens the runtime to poly(2s , n) using O(n2s ) samples. However, this
method only works with very small s due to the exponential scaling. Under the sparsity regime s =
O(2?n ) for some 0 < ? < 1, existing algorithms [3, 9, 10, 14, 18], irrespective of using membership
queries or random examples, do not immediately apply here because this may require 2n samples
(and large runtime) due to the obscured polynomial scaling in s.
In our framework, we show that f can be learned exactly in time almost-linear in s and strictly-linear
in n, even when the queries are perturbed by random Gaussian noise.
Theorem 1 (Noisy Learning). Let f ? F(s, n, A) where A is some arbitrarily large but finite set.
In the presence of noise ? ? N (0, ? 2 ), our algorithm learns f exactly in terms of the coefficients
b = ?, which runs in time O(ns log s) using O(ns) queries with probability at least 1 ? O(1/s).
?
The proposed algorithm and proofs are given in the supplementary material. Further, we apply this
framework on learning hidden graphs from cut queries. We consider an undirected weighted graph
G = (V, E, W ) with |E| = r edges and weights W ? Rr , where V = {1, ? ? ? , n} is given but the
edge set E ? V ? V is unknown. This generalizes to hypergraphs, where an edge can connect at
most d nodes, called the rank of the graph. For a d-rank hypergraph with r edges, the cut function is a
s-sparse d-bounded pseudo-boolean function (i.e. each monomial depending on at most d variables)
where the sparsity is bounded by s = O(r2d?1 ) [9].
On the graph sketching problem, [18] uses O(sn4 ) random queries to sketch the sparse temporal changes of a hypergraph in polynomial time poly(nd ). However, [9] shows that it becomes computationally infeasible for small graphs (e.g. n = 200 nodes, r = 3 edges with
d = 4), while the LearnGraph algorithm [9] runs in time O(2rd M + n2 d log n) using M =
O(2rd d log n + 22d+1 d2 (log n + rd)) queries. Although this significantly reduces the runtime compared to [14, 18], the algorithm only tackles very sparse graphs due to the scaling 2r and n2 . This
implies that the sketching needs to be done on relatively small graphs (i.e. n = 1000 nodes) over fine
sketching intervals (i.e. minutes) to suppress the sparsity (i.e. r = 10 within the sketching interval).
In this work, we adapt and apply our learning framework to derive an efficient sketching algorithm,
whose runtime scales as O(ds log s(log n + log s)) by using O(ds(log n + log s)) queries. We use
our adapted algorithm on real datasets and find that we can handle much coarser sketching intervals
(e.g. half an hour) and much larger hypergraphs (e.g. n = 105 nodes).
2
Learning Framework
Our learning framework consists of a query generator and a reconstruction engine. Given the sparsity s and the number of variables n, the query generator strategically constructs queries (randomly)
and the reconstruction engine recovers the s-sparse vector ?. For notation convenience, we replace
each boolean variable xi = (?1)m[i] with a binary variable m[i] ? F2 for all i ? [n]. Using the
notation m = [m[1], ? ? ? , m[n]]T in the Fourier expansion (1), we have
X
u[m] =
?[k](?1)hm,ki + ?[m],
(2)
k?Fn
2
3
where hm, ki = ?i?[n] m[i]k[i] over F2 . Now the coefficients ?[k] can be interpreted as the WalshHadamard Transform (WHT) coefficients of the polynomial f (x) for x ? {?1, 1}n .
2.1
Membership Query: A Coding-Theoretic Design
The building block of our query generator is the basic query set by subsampling and tiny WHTs:
? Subsampling: we choose B = 2b samples u[m] indexed selectively by m = M` + d for
` ? Fb2 , where M ? Fn?b
is the subsampling matrix and d ? Fn2 is the subsampling offset.
2
? WHT: a very small B-point WHT is performed over the samples u[M` + d] for ` ? Fb2 ,
where each output coefficient can be obtained according to the aliasing property of WHT:
X
?[k](?1)hd,ki + W [j], j ? Fb2 ,
(3)
U [j] =
k:MT k=j
where W [j] =
?1
B
P
`?Fb2
?[M` + d](?1)h`,ji is the observation noise with variance ? 2 .
The B-point basic query set (3) implies that each coefficient U [j] is the weighted hash output of ?[k]
under the hash function MT k = j. From a coding-theoretic perspective, the coefficient U [j] for
constitutes a parity constraint of the coefficients ?[k], where ?[k] enters the j-th parity if MT k = j.
If we can induce a set of parity constraints that mimic good error-correcting codes with respect to
the unknown coefficients ?[k], the coefficients can be recovered iteratively in the spirit of peeling
decoding, similar to that in LDPC codes. Now it boils down to the following questions:
? How to choose the subsampling matrix M and how to choose the query set size B?
? How to recover the coefficients ?[k] from their aliased observations U [j]?
In the following, we illustrate the principle of our learning framework through a simple example
with n = 4 boolean variables and sparsity s = 4.
2.2
Main Idea: A Simple Example
Suppose that the s = 4 non-zero coefficients are ?[0100], ?[0110], ?[1010] and ?[1111]. We choose
B = s = 4 and use two patterns M1 = [0T2?2 , IT2?2 ]T and M2 = [IT2?2 , 0T2?2 ]T for subsampling,
where all queries made using the same pattern Mi are called a query group.
In this example, by enforcing a zero subsampling offset d = 0, we generate only one set of queries
{Uc [j]}j?Fb2 under each pattern Mc according to (3). For example, under pattern M1 , the chosen
samples are u[0000], u[0001], u[0010], u[0011]. Then, the observations are obtained by a B-point
WHT coefficients of these chosen samples.
For illustration we assume the queries are noiseless:
U1 [00] = ?[0000] + ?[0100] + ?[1000] + ?[1100],
U1 [01] = ?[0001] + ?[0101] + ?[1001] + ?[1101],
U1 [10] = ?[0010] + ?[0110] + ?[1010] + ?[1110],
U1 [11] = ?[0011] + ?[0111] + ?[1011] + ?[1111].
Generally speaking, it is impossible to reconstruct the coefficients from these queries. However, since the coefficients
are sparse, then the observations are reduced to
U1 [00] = ?[0100],
U1 [01] = 0,
U1 [10] = ?[0110] + ?[1010],
U1 [11] = ?[1111],
U2 [00] = 0
U2 [01] = ?[0100] + ?[0110]
U2 [10] = ?[1010]
U2 [11] = ?[1111].
The observations are captured by a bipartite graph, which
consists of s = 4 left nodes and 8 right nodes (see Fig. 2).
4
00
?
?[0100]
?
?[0110]
?
?[0100]
?
01
?
10
?
?[0110]+?[1010]
?
11
?
?[1111]
?
Query Stage 1
?[1010]
?
?[1111]
?
00
?
01
?
?[0100]+?[0110]
?
10
?
?[1010]
?
11
?
?[1111]
?
Query Stage 2
Figure 2:
Example of a bipartite
graph for the observations.
2.2.1
Oracle-based Decoding
We illustrate how to decode the unknown ?[k] from the bipartite graph in Fig. 2 with the help of an
?oracle?, and then introduce how to get rid of this oracle. The right nodes can be categorized as:
? Zero-ton: a right node is a zero-ton if it is not connected to any left node.
? Single-ton: a right node is a single-ton if it is connected to only one left node. We refer to
the index k and its associated value ?[k] as the index-value pair (k, ?[k]).
? Multi-ton: a right node is a multi-ton if it is connected to more than one left node.
The oracle informs the decoder exactly which right nodes are single-tons as well as the corresponding index-value pair (k, ?[k]). Then, we can learn the coefficients iteratively as follows:
Step (1) select all edges in the bipartite graph with right degree 1 (i.e. detect presence of single-tons
and the index-value pairs informed by the oracle);
Step (2) remove (peel off) these edges and the left and right end nodes of these single-ton edges;
Step (3) remove (peel off) other edges connected to the left nodes that are removed in Step (2);
Step (4) remove contributions of the left nodes removed in Step (3) from the remaining right nodes.
Finally, decoding is successful if all edges are removed. Clearly, this simple example is only an illustration. In general, if there are C query groups associated with the subsampling patterns {Mc }C
c=1
and query set size B, we define the bipartite graph ensemble below and derive the guidelines for
choosing them to guarantee successful peeling-based recovery.
Definition 2 (Sparse Graph Ensemble). The bipartite graph ensemble G(s, ?, C, {Mc }c?[C] ) is a
collection of C-regular bipartite graphs where
? there are s left nodes, each associated with a distinct non-zero coefficient ?[k];
? there are C groups of right nodes and B = 2b = ?s right nodes per group, and each right
node is characterized by the observation Uc [j] indexed by j ? Fb2 in each group;
? there exists an edge between left node ?[k] and right node Uc [j] in group c if MTc k = j,
and thus each left node has a regular degree C.
Using the construction of {Mc }C
c=1 given in the supplemental material, the decoding is successful
over the ensemble G(s, ?, C, {Mc }c?[C] ) if C and B are chosen appropriately. The key idea is to
avoid excessive aliasing by exploiting a sufficiently large but finite number of groups C for diversity
and maintaining the query set size B on par with the sparsity O(s).
Lemma 1. If we construct our query generator using C query groups with B = ?s = 2b for some
redundancy parameter ? > 0 satisfying:
C
?
2
1.0000
3
0.4073
4
0.3237
5
0.2850
6
0.2616
???
???
Table 1: Minimum value for ? given the number of groups C
then the oracle-based decoder learns f in O(s) peeling iterations with probability 1 ? O(1/s).
2.2.2
Getting Rid of the Oracle
Now we explain how to detect single-tons and obtain the index-value pair without an oracle. We ex?n
ploit the diversity of subsampling offsets d from (3). Let Dc ? FP
be the offset matrix containing
2
P subsampling offsets, where each row is a chosen offset. Denote by U c [j] := [? ? ? , Uc,p [j], ? ? ? ]T
the vector of observations (called observation bin) associated with the P offsets at the j-th right
node, we have the general observation model for each right node in the bipartite graph as follows.
Proposition 1. Given the offset matrix D ? F2P ?n , we have
X
U c [j] =
?[k](?1)Dc k + wc [j],
(4)
k : MT
c k=j
where wc [j] , [? ? ? , Wc,p [j], ? ? ? ]T contains noise samples with variance ? 2 , (?1)(?) is an elementwise exponentiation operator and (?1)Dc k is the offset signature associated with ?[k].
5
In the same simple example, we keep the subsampling matrix M1 and use the set of offsets d0 =
[0, 0, 0, 0]T , d1 = [1, 0, 0, 0]T , d2 = [0, 1, 0, 0]T , d3 = [0, 0, 1, 0]T and d4 = [0, 0, 0, 1]T such that
D1 = [01?4 ; I4 ]. The observation bin associated with the subsampling pattern M1 is:
U 1 [j] = [U1,0 [j], U1,1 [j], U1,2 [j], U1,3 [j], U1,4 [j]]T .
(5)
For example, observations U 1 [01] and U 1 [10] are given as
?
?
?
?
?
?
1
1
1
0
0
1
?(?1) ?
?(?1) ?
?(?1) ?
?
?
?
?
?
?
U 1 [01] = ?[0100] ? ?(?1)1 ? , U 1 [10] = ?[0110] ? ?(?1)1 ? + ?[1010] ? ?(?1)0 ? .
?(?1)0 ?
?(?1)1 ?
?(?1)1 ?
(?1)0
(?1)0
(?1)0
With these bin observations, one can effectively determine if a check node is a zero-ton, a singleton or a multi-ton. For example, a single-ton, say U 1 [01], satisfies |U1,0 [01]| = |U1,1 [01]| =
|U1,2 [01]| = |U1,3 [01]| = |U1,4 [01]|. Then, the index k = [k[1], k[2], k[3], k[4]]T and the value
of a single-ton can be obtained by a simple ratio test
?
?
b
b
U1,1 [01]
k[1]
0
k[1] = 0
?
(?1)
=
=
(?1)
?
?
?
?
U1,0 [01]
?
?
b
?
?
b
U
[01]
1
?(?1)k[2] = 1,2
?k[2] = 1
U1,0 [01] = (?1)
=? b
k[3] = 0
b
U1,3 [01]
k[3]
0
?
?
(?1)
=
=
(?1)
?
?
U
[01]
1,0
?
?b
k[4] = 0
?
?
?
?
b
U [01]
0
? b
(?1)k[4] = U1,4
=
(?1)
[01]
?
b[k] = U1,0 [01]
1,0
The above tests are easy to verify for all observations such that the index-value pair is obtained
for peeling. In fact, this detection scheme for obtaining the oracle information is mentioned in the
noiseless scenario [16] by using P = n + 1 offsets. However, this procedure fails in the presence of
noise. In the following, we propose the general detection scheme for the noisy scenario while using
P = O(n) offsets.
3
Learning in the Presence of Noise
In this section, we propose a robust bin detection scheme that identifies the type of each observation
bin and estimate the index-value pair (k, ?[k]) of a single-ton in the presence of noise. For convenience, we drop the group index c and the node index j without loss of clarity, because the detection
scheme is identical for all nodes from all groups. The bin detection scheme consists of the single-ton
detection scheme and the zero-ton/multi-ton detection scheme, as described next.
3.1
Single-ton Detection
Proposition 2. Given a single-ton with (k, ?[k]) observed in the presence of noise N (0, ? 2 ), then
by collecting the signs of the observations, we have
c = Dk ? sgn [?[k]] ? z
2
2
where z contains P independent Bernoulli variables with probability at most Pe = e??B?min /2? ,
and the sign function is defined as sgn [x] = 1 if x < 0 and sgn [x] = 0 if x > 0.
Note that the P -bit vector c is a received codeword of the n-bit message k over a binary symmetric
channel (BSC) under an unknown flip sgn [?[k]]. Therefore, we can design the offset matrix D
according to linear block codes. The codes should include 1 as a valid codeword such that both Dk
and Dk ? 1 can be decoded correctly and then obtain the correct codeword Dk and hence k.
?n
Definition 3. Let the offset matrix D ? FP
constitute a P ? n generator matrix of some linear
2
code, which satisfies a minimum distance ?P with a code rate R(?) > 0 and ? > Pe .
Since there are n information bits in the index k, there exists some linear code (i.e. D) with block
length P = n/R(?) that achieves a minimum distance of ?P , where R(?) is the rate of the code
[15]. As long as ? > Pe , it is obvious that the unknown k can be decoded with exponentially
decaying probability of error. Excellent examples include the class of expander codes or LDPC
codes, which admits a linear time decoding algorithm. Therefore, the single-ton detection can be
performed in time O(n), same as the noiseless case.
6
3.2
Zero-ton and Multi-ton Detection
The single-ton detection scheme works when the underlying bin is indeed a single-ton. However,
it does not work on isolating single-tons from zero-tons and multi-tons. We address this issue by
further introducing P extra random offsets.
e ? FP ?n constitute a P ? n random matrix consisting of
Definition 4. Let the offset matrix D
2
independent identically distributed (i.i.d.) Bernoulli entries with probability 1/2.
e = [U
e we perform the following:
e1 , ? ? ? , U
eP ]T the observations associated with D,
Denote by U
e k2 /P ? (1+?)? 2 /B for some ? ? (0, 1).
? zero-ton verification: the bin is a zero-ton if kU
b 2
ek
D
b
e ??
? multi-ton verification: the bin is a multi-ton if kU
b[k](?1)
k ? (1 + ?)? 2 /B,
b
b
where (k, ?
b[k]) are the single-ton detection estimates.
It is shown in the supplemental material that this bin detection scheme works with probability at
least 1 ? O(1/s). Together with Lemma 1, the learning framework in the presence of noise succeeds
with probability at least 1 ? O(1/s). As detailed in the supplemental material, this leads to a overall
sample complexity of O(sn) and runtime of O(ns log s).
4
Application in Hypergraph Sketching
Consider a d-rank hypergraph G = (V, E) with |E| = r edges, where V = {1, ? ? ? , n}. A cut
S ? V is a set of selected vertices, denoted by the boolean cube x = [x1 , ? ? ? , xn ] over {?1}n ,
where xi = ?1 if i ? S and xi = 1 if i ?
/ S. The value of a specific cut x can be written as
"
!#
X
Y (1 + xi ) Y (1 ? xi )
f (x) =
1?
+
.
(6)
2
2
i?e
i?e
e?E
P
Letting xi = (?1)m[i] , we have f (x) = u[m] = k?Fn c[k](?1)hk,mi with xi = (?1)m[i] for all
2
i ? [n], where the coefficient c[k] is a scaled WHT coefficient. Clearly, if the number of hyperedges
is small r 2n and the maximum size of each hyperedge is small d n, the coefficients c[k]?s
are sparse and the sparsity can be well upper bounded by s ? r2d?1 . Now, we can use our learning
framework to compute the sparse coefficients c[k] from only a few cut queries. Note that in the
graph sketching problem, the weight of k is bounded by d due to the special structure of cut function.
Therefore, in the noiseless setting, we can leverage the sparsity d and use much fewer offsets P n
in the spirit of compressed sensing. In the supplemental material, we adapt our framework to derive
the GraphSketch bin detection scheme with even lower query costs and runtime.
Proposition 3. The GraphSketch bin detection scheme uses P = O(d(log n + log s)) offsets and
successfully detects single-tons and their index-value pairs with probability at least 1 ? O(1/s).
Next, we provide numerical experiments of our learning algorithm for sketching large random hypergraphs as well as actual hypergraphs formed by real datasets2 . In Fig. 3, we compare the probability
of success in sketching hypergraphs with n = 1000 nodes over 100 trials against the LearnGraph
procedure3 in [9], by randomly generating r = 1 to 10 hyperedges with rank d = 5. The performance is plotted against the number of edges r and the query complexity of learning. As seen from
Fig. 3, the query complexity of our framework is significantly lower (? 1%) than that of [9].
4.1
Sketching the Yahoo! Messenger User Communication Pattern Dataset
We sketch the hypergraphs extracted from Yahoo! Messenger User Communication Pattern Dataset
[19], which records communications for 28 days. The dataset is recorded entry-wise as (day, time,
transmitter, origin-zipcode, receiver, flag), where day and time represent the time stamp of each
message, the transmitter and receiver represent the IDs of the sender and the recipient, the zipcode is
a spatial stamp of each message, and the flag indicates if the recipient is in the contact list. There are
105 unique users and 5649 unique zipcodes. A hidden hypergraph structure is captured as follows.
2
3
We used MATLAB on a Macbook Pro with an Intel Core i5 processor at 2.4 GHz and 8 GB RAM.
We would like to acknowledge and thank the authors [9] for providing their source codes.
7
Run?time
Prob. of Success
1
1
2
2
0.8
Run?time (secs)
# of Edges
3
4
0.6
5
6
0.4
7
1.5
1
0.5
0
10
8
0.2
3
9
2.5
5
10
1
1.5
2
# of Queries
2.5
0
3
# of Edges
2
0
4
x 10
1.5
1
# of Queries
4
x 10
(a) Our Framework
(b) Our Framework
Run?time
Prob. of Success
1
1
2
40
0.8
Run?time (secs)
# of Edges
3
4
0.6
5
6
0.4
7
30
20
10
0
10
8
0.2
3
9
2.5
5
10
1
1.5
2
# of Queries
2.5
0
3
# of Edges
2
0
6
6
x 10
1.5
1
# of Queries
x 10
(c) LearnGraph
(d) LearnGraph
Figure 3: Sketching performance of random hypergraphs with n = 1000 nodes.
Over an interval ?t, each sender with a unique zipcode forms a hyperedge, and the recipients are
the members of the hyperedge. By considering T consecutive intervals ?t over a set of ?z 5649
zipcodes, the communication pattern gives rise to a hypergraph with only few hyperedges in each
interval and each hyperedge contains only few d nodes. The complete set of nodes in the hypergraph
n is the set of recipients who are active during the T intervals. In Table 2, we choose the sketching
interval ?t = 0.5hr and consider T = 5 intervals. For each interval, we extract the communication
hypergraph from the dataset by sketching the communications originating from a set of ?z = 20
zipcodes4 by posing queries constructed at random in our framework. We average our performance
over 100 trial runs and obtain the success probability.
Temporal Graph
(9:00 a.m. ? 9:30 a.m.)
(9:30 a.m. ? 10:00 a.m.)
(10:00 a.m. ? 10:30 a.m.)
(10:30 a.m. ? 11:00 a.m.)
(11:00 a.m. ? 11:00 a.m.)
n
12648
12648
12648
12648
12648
# of edges (E)
87
102
109
84
89
degree (d)
9
8
7
9
10
1 ? PF
0.97
0.99
0.99
0.93
0.93
Run-time (sec)
422.3
310.1
291.4
571.3
295.1
Table 2: Sketching performance with C = 8 groups and P = 421 query sets of size B = 128.
We maintain C = 8 groups of queries with P = 421 query sets of size B = 256 per group
throughout all the experiments (i.e., 8.6 ? 105 queries ? 60n). It is also seen that we can sketch the
temporal communication hypergraphs from the real dataset over much larger intervals (0.5 hr) than
that by LearnGraph (around 30 sec to 5 min), also more reliably in terms of success probability.
5
Conclusions
In this paper, we introduce a coding-theoretic active learning framework for sparse polynomials under a much more challenging sparsity regime. The proposed framework effectively lowers the query
complexity and especially the computational complexity. Our framework is useful in sketching large
hypergraphs, where the queries are obtained by specific graph cuts. We further show via experiments
that our learning algorithm performs very well over real datasets compared with existing approaches.
4
We did now show the performance of LearnGraph because it fails to work on hypergraphs with the number
of hyperedges at this scale with a reasonable number of queries (i.e., ? 1000n), as mentioned in [9].
8
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[19] Yahoo. Yahoo! webscope dataset ydata-ymessenger-user-communication-pattern-v1 0.
9
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5,189 | 5,698 | Discrete R?enyi Classifiers
Meisam Razaviyayn?
[email protected]
Farzan Farnia?
[email protected]
David Tse?
[email protected]
Abstract
Consider the binary classification problem of predicting a target variable Y from
a discrete feature vector X = (X1 , . . . , Xd ). When the probability distribution
P(X, Y ) is known, the optimal classifier, leading to the minimum misclassification rate, is given by the Maximum A-posteriori Probability (MAP) decision
rule. However, in practice, estimating the complete joint distribution P(X, Y ) is
computationally and statistically impossible for large values of d. Therefore, an
alternative approach is to first estimate some low order marginals of the joint probability distribution P(X, Y ) and then design the classifier based on the estimated
low order marginals. This approach is also helpful when the complete training
data instances are not available due to privacy concerns.
In this work, we consider the problem of finding the optimum classifier based on
some estimated low order marginals of (X, Y ). We prove that for a given set of
marginals, the minimum Hirschfeld-Gebelein-R?enyi (HGR) correlation principle
introduced in [1] leads to a randomized classification rule which is shown to have
a misclassification rate no larger than twice the misclassification rate of the optimal classifier. Then, under a separability condition, it is shown that the proposed
algorithm is equivalent to a randomized linear regression approach. In addition,
this method naturally results in a robust feature selection method selecting a subset of features having the maximum worst case HGR correlation with the target
variable. Our theoretical upper-bound is similar to the recent Discrete Chebyshev
Classifier (DCC) approach [2], while the proposed algorithm has significant computational advantages since it only requires solving a least square optimization
problem. Finally, we numerically compare our proposed algorithm with the DCC
classifier and show that the proposed algorithm results in better misclassification
rate over various UCI data repository datasets.
1
Introduction
Statistical classification, a core task in many modern data processing and prediction problems, is
the problem of predicting labels for a given feature vector based on a set of training data instances
containing feature vectors and their corresponding labels. From a probabilistic point of view, this
problem can be formulated as follows: given data samples (X1 , Y 1 ), . . . , (Xn , Y n ) from a probability distribution P(X, Y ), predict the target label y test for a given test point X = xtest .
Many modern classification problems are on high dimensional categorical features. For example, in the genome-wide association studies (GWAS), the classification task is to predict a trait
of interest based on observations of the SNPs in the genome. In this problem, the feature vector
X = (X1 , . . . , Xd ) is categorical with Xi ? {0, 1, 2}.
What is the optimal classifier leading to the minimum misclassification rate for such a classification
problem with high dimensional categorical feature vectors? When the joint probability distribution
of the random vector (X, Y ) is known, the MAP decision rule defined by ? MAP , argmaxy P(Y =
?
Department of Electrical Engineering, Stanford University, Stanford, CA 94305.
1
y|X = x) achieves the minimum misclassification rate. However, in practice the joint probability
distribution P(X, Y ) is not known. Moreover, estimating the complete joint probability distribution
is not possible due to the curse of dimensionality. For example, in the above GWAS problem, the
dimension of the feature vector X is d ? 3, 000, 000 which leads to the alphabet size of 33,000,000
for the feature vector X. Hence, a practical approach is to first estimate some low order marginals of
P(X, Y ), and then use these low order marginals to build a classifier with low misclassification rate.
This approach, which is the sprit of various machine learning and statistical methods [2?6], is also
useful when the complete data instances are not available due to privacy concerns in applications
such as medical informatics.
In this work, we consider the above problem of building a classifier for a given set of low order
marginals. First, we formally state the problem of finding the robust classifier with the minimum
worst case misclassification rate. Our goal is to find a (possibly randomized) decision rule which has
the minimum worst case misclassification rate over all probability distributions satisfying the given
low order marginals. Then a surrogate objective function, which is obtained by the minimum HGR
correlation principle [1], is used to propose a randomized classification rule. The proposed classification method has the worst case misclassification rate no more than twice the misclassification rate
of the optimal classifier. When only pairwise marginals are estimated, it is shown that this classifier
is indeed a randomized linear regression classifier on indicator variables under a separability condition. Then, we formulate a feature selection problem based on the knowledge of pairwise marginals
which leads to the minimum misclassification rate. Our analysis provides a theoretical justification
for using group lasso objective function for feature selection over the discrete set of features. Finally,
we conclude by presenting numerical experiments comparing the proposed classifier with discrete
Chebyshev classifier [2], Tree Augmented Naive Bayes [3], and Minimax Probabilistic Machine [4].
In short, the contributions of this work is as follows.
? Providing a rigorous theoretical justification for using the minimum HGR correlation principle for binary classification problem.
? Proposing a randomized classifier with misclassification rate no larger than twice the misclassification rate of the optimal classifier.
? Introducing a computationally efficient method for calculating the proposed randomized
classifier when pairwise marginals are estimated and a separability condition is satisfied.
? Providing a mathematical justification based on maximal correlation for using group lasso
problem for feature selection in categorical data.
Related Work: The idea of learning structures in data through low order marginals/moments is
popular in machine learning and statistics. For example, the maximum entropy principle [7], which
is the spirit of the variational method in graphical models [5] and tree augmented naive Bayes [3],
is based on the idea of fixing the marginal distributions and fitting a probabilistic model which
maximizes the Shannon entropy. Although these methods fit a probabilistic model satisfying the low
order marginals, they do not directly optimize the misclassification rate of the resulting classifier.
Another related information theoretic approach is the minimum mutual information principle [8]
which finds the probability distribution with the minimum mutual information between the feature
vector and the target variable. This approach is closely related to the framework of this paper;
however, unlike the minimum HGR principle, there is no known computationally efficient approach
for calculating the probability distribution with the minimum mutual information.
In the continuous setting, the idea of minimizing the worst case misclassification rate leads to the
minimax probability machine [4]. This algorithm and its analysis is not easily extendible to the
discrete scenario.
The most related algorithm to this work is the recent Discrete Chebyshev Classifier (DCC) algorithm [2]. The DCC is based on the minimization of the worst case misclassification rate over the
class of probability distributions with the given marginals of the form (Xi , Xj , Y ). Similar to our
framework, the DCC method achieves the misclassification rate no larger than twice the misclassification rate of the optimum classifier. However, computation of the DCC classifier requires solving
a non-separable non-smooth optimization problem which is computationally demanding, while the
proposed algorithm results in a least squares optimization problem with a closed form solution.
Furthermore, in contrast to [2] which only considers deterministic decision rules, in this work we
2
consider the class of randomized decision rules. Finally, it is worth noting that the algorithm in [2]
requires tree structure to be tight, while our proposed algorithm works on non-tree structures as long
as the separability condition is satisfied.
2
Problem Formulation
Consider the binary classification problem with d discrete features X1 , X2 , . . . , Xd ? X and a target
variable Y ? Y , {0, 1}. Without loss of generality, let us assume that X , {1, 2, . . . , m} and the
? X,Y (x, y). If the joint
data points (X, Y ) are coming from an underlying probability distribution P
?
probability distribution P(x, y) is known, the optimal classifier is given by the maximum a posteriori
?
= y | X = x). However, the
probability (MAP) estimator, i.e., yb MAP (x) , argmaxy?{0,1} P(Y
? y) is often not known in practice. Therefore, in order to utilize
joint probability distribution P(x,
? y) using the training data instances. Unfortunately,
the MAP rule, one should first estimate P(x,
?
estimating the joint probability distribution requires estimating the value of P(X
= x, Y = y) for
all (x, y) ? X d ? Y which is intractable for large values of d. Therefore, as mentioned earlier, our
?
approach is to first estimate some low order marginals of the joint probability distribution P(?);
and
then utilize the minimax criterion for classification.
Let C be the class of probability distributions satisfying the estimated marginals. For example, when
? is estimated, the set C is the class of
only pairwise marginals of the ground-truth distribution P
distributions satisfying the given pairwise marginals, i.e.,
? X ,X (xi , xj ), PX ,Y (xi , y) = P
? X ,Y (xi , y),
Cpairwise , PX,Y (?, ?) PXi ,Xj (xi , xj ) = P
i
j
i
i
(1)
?xi , xj ? X , ?y ? Y, ?i, j .
In general, C could be any class of probability distributions satisfying a set of estimated low order
marginals.
Let us also define ? to be a randomized classification rule with
(
0 with probability q?x
?(x) =
1 with probability 1 ? q?x ,
for some q?x ? [0, 1], ?x ? X d . Given a randomized decision rule ? and a joint probability distribution PX,Y (x, y), we can extend P(?) to include our randomized decision rule. Then the misclassification rate of the decision rule ?, under the probability distribution P(?), is given by P(?(X) 6= Y ).
Hence, under minimax criterion, we are looking for a decision rule ? ? which minimizes the worst
case misclassification rate. In other words, the robust decision rule is given by
? ? ? argmin max P (?(X) 6= Y ) ,
??D
(2)
P?C
where D is the set of all randomized decision rules. Notice that the optimal decision rule ? ? may not
be unique in general.
3
Worst Case Error Minimization
In this section, we propose a surrogate objective for (2) which leads to a decision rule with misclassification rate no larger than twice of the optimal decision rule ? ? . Later we show that the proposed
surrogate objective is connected to the minimum HGR principle [1].
Let us start by rewriting (2) as an optimization problem over real valued variables. Notice that each
probability distribution PX,Y (?, ?) can be represented by a probability vector p = [px,y | (x, y) ?
P
d
X d ? Y] ? R2m with px,y = P(X = x, Y = y) and x,y px,y = 1. Similarly, every randomized
d
rule ? can be represented by a vector q? = [q?x | x ? X d ] ? Rm . Adopting these notations, the
set C can be rewritten in terms of the probability vector p as
C , p Ap = b, 1T p = 1, p ? 0 ,
3
where the system of linear equations Ap = b represents all the low order marginal constraints in
B; and the notation 1 denotes the vector of all ones. Therefore, problem (2) can be reformulated as
X
q?? ? argmin max
(q?x px,1 + (1 ? q?x )px,0 ) ,
(3)
p?C
0?q? ?1
x
where px,0 and px,1 denote the elements of the vector p corresponding to the probability values
P(X = x, Y = 0) and P(X = x, Y = 1), respectively. The simple application of the minimax
theorem [9] implies that the saddle point of the above optimization problem exists and moreover,
the optimal decision rule is a MAP rule for a certain probability distribution P? ? C. In other words,
there exists a pair (? ? , P? ) for which
P(? ? (X) 6= Y ) ? P? (? ? (X) 6= Y ), ? P ? C and P? (?(X) 6= Y ) ? P? (? ? (X) 6= Y ), ?? ? D.
Although the above observation characterizes the optimal decision rule to some extent, it does not
provide a computationally efficient approach for finding the optimal decision rule. Notice that it
is NP-hard to verify the existence of a probability distribution satisfying a given set of low order
marginals [10]. Based on this observation and the result in [11], we conjecture that in general,
solving (2) is NP-hard in the number variables and the alphabet size even when the set C is nonempty. Hence, here we focus on developing a framework to find an approximate solution of (2).
Let us continue by utilizing the minimax theorem
P [9] and obtain the worst case probability distribution in (3) by p? ? argmaxp?C min0?q? ?1 x (qx? px,1 + (1 ? qx? )px,0 ) , or equivalently,
X
p? ? argmax
min {px,0 , px,1 } .
(4)
p?C
x
Despite convexity of the above problem, there are two sources of hardness which make the problem
intractable for moderate and large values of d. Firstly, the objective function is non-smooth. Secondly, the number of optimization variables is 2md and grows exponentially with the alphabet size.
To deal with the first issue, notice that the function inside the summation is the max-min fairness objective between the two quantities px,1 and px,0 . Replacing this objective with the harmonic average
leads to the following smooth convex optimization problem:
X px,1 px,0
e ? argmax
p
.
(5)
px,1 + px,0
p?C
x
It is worth noting that the harmonic mean of the two quantities is intuitively a reasonable surrogate
for the original objective function since
px,1 px,0
2px,1 px,0
? min {px,0 , px,1 } ?
.
(6)
px,1 + px,0
px,1 + px,0
Although this inequality suggests that the objective functions in (5) and (4) are close to each other,
e leads to any classification rule having low misclassification
it is not clear whether the distribution p
e , the first naive approach
rate for all distributions in C. In order to obtain a classification rule from p
e . However, the following result shows that this decision rule
is to use MAP decision rule based on p
does not achieve the factor two misclassification rate obtained in [2].
e x,y with the worst case error probability
Theorem 1 Let us define ?emap (x) , argmaxy?Y p
map
map
?
e
ee
, maxP?C P ?
(X) 6= Y . Then, e ? ee map ? 4e? , where e? is the worst case
misclassification rate of the optimal decision rule ? ? , that is, e? , maxP?C P (? ? (X) 6= Y ) .
Proof The proof is similar to the proof of next theorem and hence omitted here.
Next we show that, surprisingly, one can obtain a randomized decision rule based on the solution of
(5) which has a misclassification rate no larger than twice of the optimal decision rule ? ? .
e as the optimal solution of (5), define the random decision rule ?e as
Given p
e
?(x)
=
?
? 0
with probability
2
p
ex,0
2 +e
2
p
ex,0
px,1
? 1
with probability
2
p
ex,1
2 +e
2
p
ex,0
px,1
4
(7)
e i.e.,
Let e? be the worst case classification error of the decision rule ?,
e
ee , max P ?(X)
6= Y .
P?C
?
Clearly, e ? ee according to the definition of the optimal decision rule e? . The following theorem
shows that ee is also upper-bounded by twice of the optimal misclassification rate e? .
Theorem 2 Define
? , max
p?C
X px,1 px,0
px,1 + px,0
x
(8)
Then, ? ? ee ? 2? ? 2e? . In other words, the worst case misclassification rate of the decision rule
?e is at most twice the optimal decision rule ? ? .
Proof The proof is relegated to the supplementary materials.
So far, we have resolved the non-smoothness issue in solving (4) by using a surrogate objective
function. In the next section, we resolve the second issue by establishing the connection between
problem (5) and the minimum HGR correlation principle [1]. Then, we use the existing result in [1]
? for Cpairwise .
to develop a computationally efficient approach for calculating the decision rule ?(?)
4
Connection to Hirschfeld-Gebelein-R?enyi Correlation
A commonplace approach to infer models from data is to employ the maximum entropy principle [7].
This principle states that, given a set of constraints on the ground-truth distribution, the distribution
with the maximum (Shannon) entropy under those constraints is a proper representer of the class.
To extend this rule to the classification problem, the authors in [8] suggest to pick the distribution
maximizing the target entropy conditioned to features, or equivalently minimizing mutual information between target and features. Unfortunately, this approach does not lead to a computationally
efficient approach for model fitting and there is no guarantee on the misclassification rate of the resulting classifier. Here we study an alternative approach of minimum HGR correlation principle [1].
This principle suggests to pick the distribution in C minimizing HGR correlation between the target
variable and features. The HGR correlation coefficient between the two random objects X and Y ,
which was first introduced by Hirschfeld and Gebelein [12, 13] and then studied by R?enyi [14], is
defined as ?(X, Y ) , supf,g E [f (X)g(Y )] , where the maximization is taken over the class of all
measurable functions f (?) and g(?) with E[f (X)] = E[g(Y )] = 0 and E[f 2 (X)] = E[g 2 (Y )] = 1.
The HGR correlation coefficient has many desirable properties. For example, it is normalized to be
between 0 and 1. Furthermore, this coefficient is zero if and only if the two random variables are
independent; and it is one if there is a strict dependence between X and Y . For other properties of
the HGR correlation coefficient see [14, 15] and the references therein.
Lemma 1 Assume the random variable Y is binary and define q , P(Y = 0). Then,
s
X PX,Y (x, 0)PX,Y (x, 1)
1
?(X, Y ) = 1 ?
,
q(1 ? q) x PX,Y (x, 0) + PX,Y (x, 1)
Proof The proof is relegated to the supplementary material.
This lemma leads to the following observation.
Observation: Assume the marginal distribution P(Y = 0) and P(Y = 1) is fixed for any
distribution P ? C. Then, the distribution in C with the minimum HGR correlation between X and
e obtained by solving (5). In other words, ?(X, Y ; P)
e ? ?(X, Y ; P), ? P ? C,
Y is the distribution P
where ?(X, Y ; P) denotes the HGR correlation coefficient under the probability distribution P.
e in (7) as the ?R?enyi clasBased on the above observation, from now on, we call the classifier ?(?)
sifier?. In the next section, we use the result of the recent work [1] to compute the R?enyi classifier
e for a special class of marginals C = Cpairwise .
?(?)
5
5
Computing R?enyi Classifier Based on Pairwise Marginals
In many practical problems, the number of features d is large and therefore, it is only computationally tractable to estimate marginals of order at most two. Hence, hereafter, we restrict ourselves
? is estimated, i.e.,
to the case where only the first and second order marginals of the distribution P
C = Cpairwise . In this scenario, in order to predict the output of the R?enyi classifier for a given data
e x,0 and p
e x,1 . Next, we state a result from [1] which sheds
point x, one needs to find the value of p
e x,0 and p
e x,1 . To state the theorem, we need the following definitions:
light on the computation of p
Let the matrix Q ? Rdm?dm and the vector d ? Rdm?1 be defined through their entries as
? i+1 = k, Xj+1 = `), dmi+k = P(X
? i+1 = k, Y = 1) ? P(X
? i+1 = k, Y = 0),
Qmi+k,mj+` = P(X
for every i, j = 0, . . . , d ? 1 and k, ` = 1, . . . , m. Also define the function h(z) : Rmd?1 7? R as
Pd
h(z) , i=1 max{zmi?m+1 , zmi?m+2 , . . . , zmi }. Then, we have the following theorem.
Theorem 3 (Rephrased from [1]) Assume Cpairwise 6= ?. Let
?,
Then,
q
1?
?
q(1?q)
min
z?Rmd?1
1
zT Qz ? dT z + .
4
(9)
? minP?Cpairwise ?(X, Y ; P), where the inequality holds with equality if and
only if there exists a solution z? to (9) such that h(z? ) ? 21 and h(?z? ) ? 12 ; or equivalently, if and
only if the following separability condition is satisfied for some P ? Cpairwise .
EP [Y |X = x] =
d
X
?i (xi ), ?x ? X d ,
(10)
i=1
for some functions ?1 , . . . , ?d . Moreover, if the separability condition holds with equality, then
d
X
?
e = y X = (x1 , . . . , xd )) = 1 ? (?1)y
P(Y
z(i?1)m+x
.
i
2
i=1
(11)
Combining the above theorem with the equality
e2 (Y = 0, X = x)
e2 (Y = 0 | X = x)
P
P
=
e2 (Y = 0, X = x) + P
e2 (Y = 1, X = x)
e2 (Y = 0 | X = x) + P
e2 (Y = 1 | X = x)
P
P
implies that the decision rule ?e and ?e map can be computed in a computationally efficient manner
under the separability condition. Notice that when the separability condition is not satisfied, the
approach proposed in this section would provide a classification rule whose error rate is still bounded
by 2?. However, this error rate does no longer provide a 2-factor approximation gap. It is also worth
mentioning that the separability condition is a property of the class of distribution Cpairwise and is
independent of the classifier at hand. Moreover, this condition is satisfied with a positive measure
over the simplex of the all probability distributions, as discussed in [1]. Two remarks are in order:
Inexact knowledge of marginal distribution: The optimization problem (9) is equivalent to solving the stochastic optimization problem
h
2 i
z? = argmin E WT z ? C
,
z
where W ? {0, 1}md?1 is a random vector with Wm(i?1)+k = 1 if Xi = k in the and
Wm(i?1)+k = 0, otherwise. Also define the random variable C ? {? 21 , 12 } with C = 12 if the
random variable Y = 1 and C = ? 12 , otherwise. Here the expectation could be calculated with respect to any distribution in C. Hence, in practice, the above optimization problem can be estimated
using Sample Average Approximation (SAA) method [16, 17] through the optimization problem
n
b
z = argmin
z
2
1X
(wi )T z ? ci ,
n i=1
6
where (wi , ci ) corresponds to the i-th training data point (xi , y i ). Clearly, this is a least square
problem with a closed form solution. Notice that in order to bound the SAA error and avoid overfitting, one could restrict the search space for b
z [18]. This could also be done using regularizers such
as ridge regression by solving
n
2
1X
b
z ridge = argmin
(wi )T z ? ci + ?ridge kzk22 .
n i=1
z
Beyond pairwise marginals: When d is small, one might be interested in estimating higher order
marginals for predicting Y . In this scenario,
a simple modification
n
o for the algorithm is to define
e
the new set of feature random variables Xij = (Xi , Xj ) | i 6= j ; and apply the algorithm to the
new set of feature variables. It is not hard to see that this approach utilizes the marginal information
P(Xi , Xj , Xk , X` ) and P(Xi , Xj , Y ).
6
Robust R?enyi Feature Selection
The task of feature selection for classification purposes is to preselect a subset of features for use in
model fitting in prediction. Shannon mutual information, which is a measure of dependence between
two random variables, is used in many recent works as an objective for feature selection [19, 20].
In these works, the idea is to select a small subset of features with maximum dependence with the
target variable Y . In other words, the task is to find a subset of variables S ? {1, . . . , d} with
|S| ? k based on the following optimization problem
S MI , argmax I(XS ; Y ),
(12)
S?{1,...,d}
where XS , (Xi )i?S and I (XS ; Y ) denotes the mutual information between the random variable
XS and Y . Almost all of the existing approaches for solving (12) are based on heuristic approaches
and of greedy nature which aim to find a sub-optimal solution of (12). Here, we suggest to replace
mutual information with the maximal correlation. Furthermore, since estimating the joint distribution of X and Y is computationally and statistically impossible for large number of features d, we
?
suggest to estimate some low order marginals of the groundtruth distribution P(X,
Y ) and then solve
the following robust R?enyi feature selection problem:
S RFS , argmax min ?(XS , Y ; P).
(13)
S?{1,...,d} P?C
When only pairwise
q marginals are estimated from the training data, i.e., C = Cpairwise , maximizing
?
the lower-bound 1 ? q(1?q)
instead of (13) leads to the following optimization problem
s
1
1
RFS
b
S
, argmax 1 ?
min zT Qz ? dT z + ,
q(1 ? q) z?ZS
4
|S|?k
or equivalently,
Sb RFS , argmin min zT Qz ? dT z,
|S|?k
z?ZS
Pm
where ZS , z ? Rmd
/ S . This problem is of combinatorial nak=1 |zmi?m+k | = 0, ?i ?
ture. Howevre, using the standard group Lasso regularizer leads to the feature selection procedure
in Algorithm 1.
Algorithm 1 Robust R?enyi Feature Selection
Choose a regularization parameter ? > 0 and define h(z) ,
Let b
zRFS ? argminz zT Qz ? dT z + ?h(|z|).
Pm
RFS
Set S = {i | k=1 |zmi?m+k
| > 0}.
Pd
i=1
max{zmi?m+1 , . . . , zmi }.
Notice that, when the pairwise marginals are estimated from a set of training data points, the above
feature selection procedure is equivalent to applying the group Lasso regularizer to the standard
linear regression problem over the domain of indicator variables. Our framework provides a justification for this approach based on the robust maximal correlation feature selection problem (13).
7
Remark 1 Another natural approach to define the feature selection procedure is to select a subset of
features S by minimizing the worst case classification error, i.e., solving the following optimization
problem
min min max P(?(X) 6= Y ),
(14)
|S|?k ??DS P?C
where DS is the set of randomized decision rules which only uses the feature variables in
S. Define F(S) , min??DS maxP?C P(?(X) 6= Y ). It can be shown that F(S) ?
min|S|?k minz?ZS zT Qz ? dT z + 41 . Therefore, another justification for Algorithm 1 is to
minimize an upper-bound of F(S) instead of itself.
Remark 2 Alternating Direction Method of Multipliers (ADMM) algorithm [21] can be used for
solving the optimization problem in Algorithm 1; see the supplementary material for more details.
7
Numerical Results
We evaluated the performance of the R?enyi classifiers ?e and ?e map on five different binary classification datasets from the UCI machine learning data repository. The results are compared with five
different benchmarks used in [2]: Discrete Chebyshev Classifier [2], greedy DCC [2], Tree Augmented Naive Bayes [3], Minimax Probabilistic Machine [4], and support vector machines (SVM).
In addition to the classifiers ?e and ?e map which only use pairwise marginals, we also use higher
order marginals in ?e2 and ?e2map . These classifiers are obtained by defining the new feature variables
eij = (Xi , Xj )} as discussed in section 5. Since in this scenario, the number of features is large,
{X
we combine our R?enyi classifier with the proposed group lasso feature selection. In other words,
eij } and then find the maximum correlation classifier for the selected
we first select a subset of {X
ridge
features. The value of ?
and ? is determined through cross validation. The results are averaged
over 100 Monte Carlo runs each using 70% of the data for training and the rest for testing. The
results are summarized in the table below where each number shows the percentage of the error of
each method. The boldface numbers denote the best performance on each dataset.
As can be seen in this table, in four of the tested datasets, at least one of the proposed methods
outperforms the other benchmarks. Furthermore, it can be seen that the classifier ?emap on average
? This fact could be due to the specific properties of the underlying probability
performs better than ?.
distribution in each dataset.
Datasets
adult
credit
kr-vs-kp
promoters
votes
?emap
17
13
5
6
3
?e
21
16
10
16
4
?e2map
16
16
5
3
3
?e2
20
17
14
4
4
map
?eFS,2
16
16
5
3
2
?eFS,2
20
17
14
4
4
DCC
18
14
10
5
3
gDCC
18
13
10
3
3
MPM
22
13
5
6
4
TAN
18
17
7
44
8
SVM
22
16
3
9
5
In order to evaluate the computational efficiency of the R?enyi classifier, we compare its running
time with SVM over the synthetic data set with d = 10, 000 features and n = 200 data points.
Each feature Xi is generated by i.i.d. Bernoulli distribution with P(Xi = 1) = 0.7. The target
variable y is generated by y = sign(?T X + n) with n ? N (0, 1); and ? ? Rd is generated with
30% nonzero elements each drawn from standard Gaussian distribution N (0, 1). The results are
averaged over 1000 Monte-Carlo runs of generating the data set and use 85% of the data points
for training and 15% for test. The R?enyi classifier is obtained by gradient descent method with
regularizer ?ridge = 104 . The numerical experiment shows 19.7% average misclassification rate for
SVM and 19.9% for R?enyi classifier. However, the average training time of the R?enyi classifier is
0.2 seconds while the training time of SVM (with Matlab SVM command) is 1.25 seconds.
Acknowledgments: The authors are grateful to Stanford University supporting a Stanford Graduate
Fellowship, and the Center for Science of Information (CSoI), an NSF Science and Technology
Center under grant agreement CCF-0939370 , for the support during this research.
8
References
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9
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5,190 | 5,699 | GAP Safe screening rules for sparse multi-task and
multi-class models
Eugene Ndiaye Olivier Fercoq Alexandre Gramfort Joseph Salmon
LTCI, CNRS, T?el?ecom ParisTech, Universit?e Paris-Saclay
Paris, 75013, France
[email protected]
Abstract
High dimensional regression benefits from sparsity promoting regularizations.
Screening rules leverage the known sparsity of the solution by ignoring some
variables in the optimization, hence speeding up solvers. When the procedure
is proven not to discard features wrongly the rules are said to be safe. In this paper
we derive new safe rules for generalized linear models regularized with `1 and
`1 {`2 norms. The rules are based on duality gap computations and spherical safe
regions whose diameters converge to zero. This allows to discard safely more variables, in particular for low regularization parameters. The GAP Safe rule can cope
with any iterative solver and we illustrate its performance on coordinate descent
for multi-task Lasso, binary and multinomial logistic regression, demonstrating
significant speed ups on all tested datasets with respect to previous safe rules.
1
Introduction
The computational burden of solving high dimensional regularized regression problem has lead to a
vast literature in the last couple of decades to accelerate the algorithmic solvers. With the increasing
popularity of `1 -type regularization ranging from the Lasso [18] or group-Lasso [24] to regularized
logistic regression and multi-task learning, many algorithmic methods have emerged to solve the
associated optimization problems. Although for the simple `1 regularized least square a specific
algorithm (e.g., the LARS [8]) can be considered, for more general formulations, penalties, and
possibly larger dimension, coordinate descent has proved to be a surprisingly efficient strategy [12].
Our main objective in this work is to propose a technique that can speed-up any solver for such
learning problems, and that is particularly well suited for coordinate descent method, thanks to
active set strategies.
The safe rules introduced by [9] for generalized `1 regularized problems, is a set of rules that allows
to eliminate features whose associated coefficients are proved to be zero at the optimum. Relaxing
the safe rule, one can obtain some more speed-up at the price of possible mistakes. Such heuristic
strategies, called strong rules [19] reduce the computational cost using an active set strategy, but
require difficult post-precessing to check for features possibly wrongly discarded. Another road to
speed-up screening method has been the introduction of sequential safe rules [21, 23, 22]. The idea
is to improve the screening thanks to the computations done for a previous regularization parameter.
This scenario is particularly relevant in machine learning, where one computes solutions over a
grid of regularization parameters, so as to select the best one (e.g., to perform cross-validation).
Nevertheless, such strategies suffer from the same problem as strong rules, since relevant features
can be wrongly disregarded: sequential rules usually rely on theoretical quantities that are not known
by the solver, but only approximated. Especially, for such rules to work one needs the exact dual
optimal solution from the previous regularization parameter.
1
Recently, the introduction of safe dynamic rules [6, 5] has opened a promising venue by letting
the screening to be done not only at the beginning of the algorithm, but all along the iterations.
Following a method introduced for the Lasso [11], we generalize this dynamical safe rule, called
GAP Safe rules (because it relies on duality gap computation) to a large class of learning problems
with the following benefits:
?
?
?
?
?
a unified and flexible framework for a wider family of problems,
easy to insert in existing solvers,
proved to be safe,
more efficient that previous safe rules,
achieves fast true active set identification.
We introduce our general GAP Safe framework in Section 2. We then specialize it to important
machine learning use cases in Section 3. In Section 4 we apply our GAP Safe rules to a multitask Lasso problem, relevant for brain imaging with magnetoencephalography data, as well as to
multinomial logistic regression regularized with `1 {`2 norm for joint feature selection.
2
GAP Safe rules
2.1
Model and notations
We denote by rds the set t1, . . . , du for any integer d P N, and by QJ the transpose of a matrix
Q. Our observation matrix is Y P Rn?q where n represents the number of samples, and q the
number of tasks or classes. The design matrix X ? rxp1q , . . . , xppq s ? rx1 , . . . , xn sJ P Rn?p has
p explanatory variables (or features) column-wise, and n observations row-wise. The standard `2
norm is written } ? }2 , the `1 norm } ? }1 , the `8 norm } ? }8 . The `2 unit ball is denoted by B2 (or
simply B) and we ?
write Bpc,
`2 ball with center c and radius r. For a matrix B P Rp?q , we
?q rq the
p
2
2
denote by }B}2 ? j?1 k?1 Bj,k the Frobenius norm, and by x?, ?y the associated inner product.
We consider the general optimization problem of minimizing a separable function with a groupLasso regularization. The parameter to recover is a matrix B P Rp?q , and for any j in Rp , Bj,: is the
j-th row of B, while for any k in Rq , B:,k is the k-th column. We would like to find
n
?
p p?q P arg min
B
fi pxJ
(1)
i Bq ` ??pBq ,
BPRp?q i?1
loooooooooooomoooooooooooon
P? pBq
?n
where fi : R
?? R is a convex function with 1{?-Lipschitz gradient. So F : B ? i?1 fi pxJ
i Bq
p?q
is
?? R` is the `1 {`2 norm ?pBq ?
?palso convex with Lipschitz gradient. The function ? : R
j?1 }Bj,: }2 promoting a few lines of B to be non-zero at a time. The ? parameter is a non-negative
constant controlling the trade-off between data fitting and regularization.
1?q
Some elements of convex analysis used in the following are introduced here. For a convex function
f : Rd ? r?8, `8s the Fenchel-Legendre transform1 of f , is the function f ? : Rd ? r?8, `8s
defined by f ? puq ? supzPRd xz, uy ? f pzq. The sub-differential of a function f at a point x is
denoted by Bf pxq. The dual norm of ? is the `8 {`2 norm and reads ?? pBq ? maxjPrps }Bj,: }2 .
Remark 1. For the ease of reading, all groups are weighted with equal strength, but extension
of our results to non-equal weights as proposed in the original group-Lasso [24] paper would be
straightforward.
2.2
Basic properties
First we recall the associated Fermat?s condition and a dual formulation of the optimization problem:
Theorem 1. Fermat?s condition (see [3, Proposition 26.1] for a more general result)
For any convex function f : Rn ? R:
x P arg min f pxq ? 0 P Bf pxq.
xPRn
1
this is also often referred to as the (convex) conjugate of a function
2
(2)
Theorem 2 ([9]). A dual formulation of (1) is given by
n
?
p p?q ? arg max ?
?
fi? p???i,: q .
?P?X
i?1
looooooooomooooooooon
(3)
D? p?q
where ?X ? t? P Rn?q : @j P rps, }x
and dual solutions are linked by
pjq J
@i P rns,
?}2 ? 1u ? t? P Rn?q : ?? pX J ?q ? 1u. The primal
p p?q ? ??fi pxJ B
p p?q q{?.
?
i
i,:
(4)
Furthermore, Fermat?s condition reads:
@j P rps,
pjq J p p?q
x
?
$"
&
P
??
B
j,;
? ? }2
}B
j,;
%
B2 ,
*
p p?q ? 0,
, if B
j,:
(5)
p p?q ? 0.
if B
j,:
Remark 2. Contrarily to the primal, the dual problem has a unique solution under our assumption
on fi . Indeed, the dual function is strongly concave, hence strictly concave.
Remark 3. For any ? P Rn?q let us introduce Gp?q ? r?f1 p?1,: qJ , . . . , ?fn p?n,: qJ s P Rn?q .
p p?q ? ?GpX B
p p?q q{? .
Then the primal/dual link can be written ?
2.3
Critical parameter: ?max
For ? large enough the solution of the primal problem is simply 0. Thanks to the Fermat?s rule (2),
0 is optimal if and only if ??F p0q{? P B?p0q. Thanks to the property of the dual norm ?? , this is
equivalent to ?? p?F p0q{?q ? 1 where ?? is the dual norm of ?. Since ?F p0q ? X J Gp0q, 0 is a
J
primal solution of P? if and only if ? ? ?max :? maxjPrps }xpjq Gp0q}2 ? ?? pX J Gp0qq.
This development shows that for ? ? ?max , Problem (1) is trivial. So from now on, we will only
focus on the case where ? ? ?max .
2.4
Screening rules description
Safe screening rules rely on a simple consequence of the Fermat?s condition:
J p?q
p }2 ? 1 ? B
p p?q ? 0 .
}xpjq ?
j,:
(6)
p p?q is unknown (unless ? ? ?max ). However,
Stated in such a way, this relation is useless because ?
it is often possible to construct a set R ? Rn?q , called a safe region, containing it. Then, note that
J
p p?q ? 0 .
max }xpjq ?}2 ? 1 ? B
j,:
?PR
(7)
The so called safe screening rules consist in removing the variable j from the problem whenever the
p p?q is then guaranteed to be zero. This property leads to considerable
previous test is satisfied, since B
j,:
speed-up in practice especially with active sets strategies, see for instance [11] for the Lasso case. A
natural goal is to find safe regions as narrow as possible: smaller safe regions can only increase the
number of screened out variables. However, complex regions could lead to a computational burden
limiting the benefit of screening. Hence, we focus on constructing R satisfying the trade-off:
p p?q .
? R is as small as possible and contains ?
pjq J
? Computing max?PR }x
?}2 is cheap.
2.5
Spheres as safe regions
Various shapes have been considered in practice for the set R such as balls (referred to as spheres)
[9], domes [11] or more refined sets (see [23] for a survey). Here we consider the so-called
?sphere regions? choosing a ball R ? Bpc, rq as a safe region. One can easily obtain a control
3
J
on max?PBpc,rq }xpjq ?}2 by extending the computation of the support function of a ball [11, Eq.
J
J
(9)] to the matrix case: max }xpjq ?}2 ? }xpjq c}2 ` r}xpjq }2 .
?PBpc,rq
Note that here the center c is a matrix in Rp?q . We can now state the safe sphere test:
Sphere test:
2.6
If
J
}xpjq c}2 ` r}xpjq }2 ? 1,
p p?q ? 0.
then B
j,:
(8)
GAP Safe rule description
In this section we derive a GAP Safe screening rule extending the one introduced in [11]. For this,
we rely on the strong convexity of the dual objective function and on weak duality.
Finding a radius: Remember that @i P rns, fi is differentiable with a 1{?-Lipschitz gradient.
As a consequence, @i P rns, fi? is ?-strongly convex [14, Theorem 4.2.2, p. 83] and so D? is
??2 -strongly concave:
@p?1 , ?2 q P Rn?q ? Rn?q ,
D? p?2 q ? D? p?1 q ` x?D? p?1 q, ?2 ? ?1 y ?
??2
}?1 ? ?2 }2 .
2
p p?q , ?2 ? ? P ?X , one has
Specifying the previous inequality for ?1 ? ?
2
p p?q q ` x?D? p?
p p?q q, ? ? ?
p p?q y ? ?? }?
p p?q ? ?}2 .
D? p?q ? D? p?
2
p p?q maximizes D? on ?X , so we have: x?D? p?
p p?q q, ? ? ?
p p?q y ? 0. This implies
By definition, ?
2
p p?q q ? ?? }?
p p?q ? ?}2 .
D? p?q ? D? p?
2
p p?q q ? P? pBq, so : @B P Rp?q , @? P ?X , D? p?q ? P? pBq ?
By weak duality @B P Rp?q , D? p?
2
?? p p?q
? ?}2 , and we deduce the following theorem:
2 }?
d
Theorem 3.
p p?q ? ?
? 2pP? pBq ? D? p?qq ?: r?? pB, ?q.
(9)
@B P Rp?q , @? P ?X ,
?
??2
2
Provided one knows a dual feasible point ? P ?X and a B P Rp?q , it is possible to construct a safe
sphere with radius r?? pB, ?q centered on ?. We now only need to build a (relevant) dual point to
center such a ball. Results from Section 2.3, ensure that ?Gp0q{?max P ?X , but it leads to a static
rule, a introduced in [9]. We need a dynamic center to improve the screening as the solver proceeds.
p p?q ? ?GpX B
p p?q q{?. Now assume that one has a converging
Finding a center: Remember that ?
p p?q . Hence, a natural choice for creating a dual
algorithm for the primal problem, i.e., Bk ? B
feasible point ?k is to choose it proportional to ?GpXBk q, for instance by setting:
#
Rk
,
if ?? pX J Rk q ? ?,
?k ? ? R k
where Rk ? ?GpXBk q .
(10)
otherwise.
?? pX J Rk q ,
A refined method consists in solving the one dimensional problem: arg max?P?X XSpanpRk q D? p?q.
In the Lasso and group-Lasso case [5, 6, 11] such a step is simply a projection on the intersection of
a line and the (polytope) dual set and can be computed efficiently. However for logistic regression
the computation is more involved, so we have opted for the simpler solution in Equation (10). This
still provides converging safe rules (see Proposition 1).
Dynamic GAP Safe rule summarized
We can now state our dynamical GAP Safe rule at the k-th step of an iterative solver:
1. Compute Bk , and then obtain ?k and r?? pBk , ?k q using (10).
4
J
p p?q ? 0 and remove xpjq from X.
2. If }xpjq ?k }2 ` r?? pBk , ?k q}xpjq }2 ? 1, then set B
j,:
Dynamic safe screening rules are more efficient than existing methods in practice because they can
increase the ability of screening as the algorithm proceeds. Since one has sharper and sharper dual
regions available along the iterations, support identification is improved. Provided one relies on a
primal converging algorithm, one can show that the dual sequence we propose is converging too.
The convergence of the primal is unaltered by our GAP Safe rule: screening out unnecessary coefficients of Bk can only decrease its distance with its original limits. Moreover, a practical consequence
is that one can observe surprising situations where lowering the tolerance of the solver can reduce
the computation time. This can happen for sequential setups.
p p?q and ?k defined in Eq. (10) be the current
Proposition 1. Let Bk be the current estimate of B
p p?q . Then limk?`8 Bk ? B
p p?q implies limk?`8 ?k ? ?
p p?q .
estimate of ?
Note that if the primal sequence is converging to the optimal, our dual sequence is also converging.
But we know that the radius of our safe sphere is p2pP? pBk q ? D? p?k qq{p??2 qq1{2 . By strong
duality, this radius converges to 0, hence we have certified that our GAP Safe regions sequence
Bp?k , r?? pBk , ?k qq is a converging safe rules (in the sense introduced in [11, Definition 1]).
Remark 4. The active set obtained by our GAP Safe rule (i.e., the indexes of non screened-out
J p?q
p }2 ? 1u, allowing us
variables) converges to the equicorrelation set [20] E? :? tj P p : }xpjq ?
to early identify relevant features (see Proposition 2 in the supplementary material for more details).
3
Special cases of interest
We now specialize our results to relevant supervised learning problems, see also Table 1.
3.1
Lasso
In
B ? ? P Rp , F p?q ? 1{2}y ? X?}22 ?
?nthe Lasso Jcase2 q ? 1, the parameter is a vector:
2
i?1 pyi ? xi ?q , meaning that fi pzq ? pyi ? zq {2 and ?p?q ? }?}1 .
`1 {`2 multi-task regression
3.2
In the multi-task Lasso, which is a special
we assume that the observation is
?n case of group-Lasso,
1
1
n?q
2
J
2
2
Y
P
R
,
F
pBq
?
}Y
?XB}
?
}Y
?x
B}
(i.e.,
f
i,:
i pzq ? }Yi,: ?z} {2) and ?pBq ?
2
2
i
i?1
2
2
?p
j?1 }Bj,: }2 . In signal processing, this model is also referred to as Multiple Measurement Vector
(MMV) problem. It allows to jointly select the same features for multiple regression tasks [1, 2].
Remark 5. Our framework could encompass easily the case of non-overlapping groups with various size and weights presented in [6]. Since our aim is mostly for multi-task and multinomial
applications, we have rather presented a matrix formulation.
`1 regularized logistic regression
3.3
Here, we consider the formulation given in [7, Chapter 3] for the two classes logistic regression. In
such a context, one observes for each i P rns a class label ci P t1, 2u. This information can be recast
as yi ? 1tci ?1u , and it is then customary to minimize (1) where
n
?
`
`
` J ???
F p?q ?
?yi xJ
,
(11)
i ? ` log 1 ` exp xi ?
i?1
with B ? ? P Rp (i.e., q ? 1), fi pzq ? ?yi z ` logp1 ` exppzqq and the penalty is simply the `1
norm: ?p?q ? }?}1 . Let us introduce Nh, the (binary) negative entropy function defined by 2 :
"
x logpxq ` p1 ? xq logp1 ? xq, if x P r0, 1s ,
Nhpxq ?
(12)
`8,
otherwise .
Then, one can easily check that fi? pzi q ? Nhpzi ` yi q and ? ? 4.
2
with the convention 0 logp0q ? 0
5
Lasso
Multi-task regr.
Logistic regr.
fi pzq
pyi ?zq2
2
}Yi,: ?z}2
2
logp1 ` ez q ? yi z
fi? puq
pyi ?uq2 ?yi2
2
?pBq
}Yi,: ?u}2 ?}Yi,: }22
2
p
?
}?}1
Multinomial regr.
q
q
? ?
`?
Yi,k zk
log
ezk ?
k?1
k?1
Nhpu ` yi q
}?}1
}Bj,: }2
j?1
NHpu ` Yi,: q
p
?
}Bj,: }2
j?1
?max
}X J y}8
?? pX J Y q
Gp?q
??y
??Y
?
1
1
}X J p1n {2 ? yq}8
ez
1`ez
?y
4
?? pX J p1n?q {q ? Y qq
RowNormpe? q ? Y
1
Table 1: Useful ingredients for computing GAP Safe rules. We have used lower case to indicate
when the parameters are vectorial (i.e., q ? 1). The function RowNorm consists in normalizing a
(non-negative) matrix row-wise, such that each row sums to one.
3.4
`1 {`2 multinomial logistic regression
We adapt the formulation given in [7, Chapter 3] for the multinomial regression. In such a context,
one observes for each i P rns a class label ci P t1, . . . , qu. This information can be recast into a
matrix Y P Rn?q filled by 0?s and 1?s: Yi,k ? 1tci ?ku . In the same spirit as the multi-task Lasso, a
matrix B P Rp?q is formed by q vectors encoding the hyperplanes for the linear classification. The
multinomial `1 {`2 regularized regression reads:
? q
? q
??
n
?
?
?
` J
?
J
F pBq ?
?Yi,k xi B:,k ` log
exp xi B:,k
,
(13)
i?1
k?1
k?1
?q
?q
with fi pzq ? k?1 ?Yi,k zk ` log p k?1 exp pzk qq to recover the formulation as in (1). Let us
introduce NH, the negative entropy function defined by (still with the convention 0 logp0q ? 0)
"?q
?q
xi logpxi q, if x P ?q ? tx P Rq` : i?1 xi ? 1u,
i?1
NHpxq ?
(14)
`8,
otherwise.
Again, one can easily check that fi? pzq ? NHpz ` Yi,: q and ? ? 1.
Remark 6. For multinomial logistic regression, D? implicitly encodes the additional constraint
? P dom D? ? t?1 : @i P rns, ???1i,: ` Yi,: P ?q u where ?q is the q dimensional simplex, see
(14). As 0 and Rk {? both belong to this set, any convex combination of them, such as ?k defined
in (10), satisfies this additional constraint.
Remark 7. The intercept has been neglected in our models for simplicity. Our GAP Safe framework
can also handle such a feature at the cost of more technical details (by adapting the results from [15]
for instance). However, in practice, the intercept can be handled in the present formulation by adding
a constant column to the design matrix X. The intercept is then regularized. However, if the constant
is set high enough, regularization is small and experiments show that it has little to no impact for
high-dimensional problems. This is the strategy used by the Liblinear package [10].
4
Experiments
In this section we present results obtained with the GAP Safe rule. Results are on high dimensional
data, both dense and sparse. Implementation have been done in Python and Cython for low critical
parts. They are based on the multi-task Lasso implementation of Scikit-Learn [17] and coordinate
descent logistic regression solver in the Lightning software [4]. In all experiments, the coordinate
descent algorithm used follows the pseudo code from [11] with a screening step every 10 iterations.
6
Figure 1: Experiments on MEG/EEG brain imaging dataset (dense data with n ? 360, p ? 22494
and q ? 20). On the left: fraction of active variables as a function of ? and the number of iterations
K. The GAP Safe strategy has a much longer range of ? with (red) small active sets. On the right:
Computation time to reach convergence using different screening strategies.
Note that we have not performed comparison with the sequential screening rule commonly acknowledge as the state-of-the-art ?safe? screening rule (such as th EDDP+ [21]), since we can show that
this kind of rule is not safe. Indeed, the stopping criterion is based on dual gap accuracy, and comparisons would be unfair since such methods sometimes do not converge to the prescribed accuracy.
This is backed-up by a counter example given in the supplementary material. Nevertheless, modifications of such rules, inspired by our GAP Safe rules, can make them safe. However the obtained
sequential rules are still outperformed by our dynamic strategies (see Figure 2 for an illustration).
4.1
`1 {`2 multi-task regression
To demonstrate the benefit of the GAP Safe screening rule for a multi-task Lasso problem we used
neuroimaging data. Electroencephalography (EEG) and magnetoencephalography (MEG) are brain
imaging modalities that allow to identify active brain regions. The problem to solve is a multi-task
regression problem with squared loss where every task corresponds to a time instant. Using a multitask Lasso one can constrain the recovered sources to be identical during a short time interval [13].
This corresponds to a temporal stationary assumption. In this experiment we used a joint MEG/EEG
data with 301 MEG and 59 EEG sensors leading to n ? 360. The number of possible sources is
p ? 22, 494 and the number of time instants q ? 20. With a 1 kHz sampling rate it is equivalent to
say that the sources stay the same for 20 ms.
Results are presented in Figure 1. The GAP Safe rule is compared with the dynamic safe rule
from [6]. The experimental setup consists in estimating the solutions of the multi-task Lasso problem
for 100 values of ? on a logarithmic grid from ?max to ?max {103 . For the experiments on the left
a fixed number of iterations from 2 to 211 is allowed for each ?. The fraction of active variables
is reported. Figure 1 illustrates that the GAP Safe rule screens out much more variables than the
compared method, as well as the converging nature of our safe regions. Indeed, the more iterations
performed the more the rule allows to screen variables. On the right, computation time confirms the
effective speed-up. Our rule significantly improves the computation time for all duality gap tolerance
from 10?2 to 10?8 , especially when accurate estimates are required, e.g., for feature selection.
4.2
`1 binary logistic regression
Results on the Leukemia dataset are reported in Figure 2. We compare the dynamic strategy of GAP
Safe to a sequential and non dynamic rule such as Slores [22]. We do not compare to the actual
Slores rule as it requires the previous dual optimal solution, which is not available. Slores is indeed
not a safe method (see Section B in the supplementary materials). Nevertheless one can observe that
dynamic strategies outperform pure sequential one, see Section C in the supplementary material).
7
No screening
GAP Safe (sequential)
GAP Safe (dynamic)
No screening
GAP Safe (sequential)
GAP Safe (dynamic)
Figure 2: `1 regularized binary logistic regression on the Leukemia dataset (n = 72 ; m = 7,129 ;
q = 1). Simple sequential and full dynamic screening GAP Safe rules are compared. On the left:
fraction of the variables that are active. Each line corresponds to a fixed number of iterations for
which the algorithm is run. On the right: computation times needed to solve the logistic regression
path to desired accuracy with 100 values of ?.
4.3
`1 {`2 multinomial logistic regression
We also applied GAP Safe to an `1 {`2 multinomial logistic regression problem on a sparse dataset.
Data are bag of words features extracted from the News20 dataset (TF-IDF removing English stop
words and words occurring only once or more than 95% of the time). One can observe on Figure 3
the dynamic screening and its benefit as more iterations are performed. GAP Safe leads to a significant speedup: to get a duality gap smaller than 10?2 on the 100 values of ?, we needed 1,353 s
without screening and only 485 s when GAP Safe was activated.
Figure 3: Fraction of the variables that are active for `1 {`2 regularized multinomial logistic
regression on 3 classes of the News20 dataset
(sparse data with n = 2,757 ; m = 13,010 ;
q = 3). Computation was run on the best 10%
of the features using ?2 univariate feature selection [16]. Each line corresponds to a fixed
number of iterations for which the algorithm is
run.
5
Conclusion
This contribution detailed new safe rules for accelerating algorithms solving generalized linear models regularized with `1 and `1 {`2 norms. The rules proposed are safe, easy to implement, dynamic
and converging, allowing to discard significantly more variables than alternative safe rules. The
positive impact in terms of computation time was observed on all tested datasets and demonstrated
here on a high dimensional regression task using brain imaging data as well as binary and multiclass
classification problems on dense and sparse data. Extensions to other generalized linear model,
e.g., Poisson regression, are expected to reach the same conclusion. Future work could investigate
optimal screening frequency, determining when the screening has correctly detected the support.
Acknowledgment
We acknowledge the support from Chair Machine Learning for Big Data at T?el?ecom ParisTech and
from the Orange/T?el?ecom ParisTech think tank phi-TAB. This work benefited from the support of
the ?FMJH Program Gaspard Monge in optimization and operation research?, and from the support
to this program from EDF.
8
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9
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5,191 | 570 | A Self-Organizing Integrated Segmentation And
Recognition Neural Net
Jim Keeler *
MCC
3500 West Balcones Center Drive
Austin, TX 78729
David E. Rumelhart
Psychology Department
Stanford University
Stanford, CA 94305
Abstract
We present a neural network algorithm that simultaneously performs segmentation and recognition of input patterns that self-organizes to detect
input pattern locations and pattern boundaries. We demonstrate this neural network architecture on character recognition using the NIST database
and report on results herein. The resulting system simultaneously segments and recognizes touching or overlapping characters, broken characters, and noisy images with high accuracy.
1
INTRODUCTION
Standard pattern recognition systems usually involve a segmentation step prior to
the recognition step. For example, it is very common in character recognition to
segment characters in a pre-processing step then normalize the individual characters
and pass them to a recognition engine such as a neural network, as in the work of
LeCun et al. 1988, Martin and Pittman (1988).
This separation between segmentation and recognition becomes unreliable if the
characters are touching each other, touching bounding boxes, broken, or noisy.
Other applications such as scene analysis or continuous speech recognition pose
similar and more severe segmentation problems. The difficulties encountered in
these applications present an apparent dilemma: one cannot recognize the patterns
*[email protected] Reprint requests: [email protected] or at the above address.
496
A Self-Organizing Integrated Segmentation and Recognition Neural Net
Sz Outputs: pz = - I + Sz
I
I
I
ItfIM.............
I
I
I
5
Summing Units:
Sz =
LXxyz
xy
I~~~
I
I
I
I
I
,
2.AFI
1ABY/
?AW
Grey-scale
Input image
I(.X,y)
Figure 1: The ISR network architecture. The input image may contain several
characters and is presented to the network in a two-dimensional grey-scale image.
The units in the first block, hij", have linked-local receptive field connections to the
input image. Block 2, Hr'JI'z" has a three-dimensional linked-local receptive field
to block 1, and the exponential unit block, block 3, has three-dimensional linkedlocal receptive field connections to block 2. These linked fields insure translational
invariance (except for edge-effects at the boundary). The exponential unit block
has one layer for each output category. These units are the output units in the test
mode, but hidden units during training: the exponential unit activity is summed
over (sz) to project out the positional information, then converted to a probability
Pz. Once trained, the exponential unit layers serve as "smart histograms" giving
sharp peaks of activity directly above the corresponding characters in the input
image, as shown to the left.
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Keeler and Rumelhart
until they are segmented, yet in many cases one cannot segment the patterns until
they are recognized.
A solution to this apparent dilemm is to simultaneously segment and recognize
the patterns. Integration of the segmentation and recognition steps is essential for
further progress in these difficult pattern recognition tasks, and much effort has been
devoted to this topic in speech recognition. For example, Hidden Markov models
integrate the task of segmentation and recognition as a part of the word-recognition
module. Nevertheless, little neural network research in pattern recognition has
focused on the integrated segmentation and recognition (ISR) problem.
There are several ways to achieve ISR in a neural network. The first use of backpropagation ISR neural networks for character recognition was reported by Keeler,
Rumelhart and Leow (1991a). The ISR neural network architecture is similar to
the time-delayed neural network architecture for speech recognition used by Lang,
Hinton, and Waibel (1990).
The following section outlines the neural network algorithm and architecture. Details and rationale for the exact structure and assumptions of the network can be
found in Keeler et al. (1991a,b).
2
NETWORK ARCHITECTURE AND ALGORITHM
The basic organization of the network is illustrated in Figure 2. The input consists
of a twcrdimensional grey-scale image representing the pattern to be processed. We
designate this input pattern by the twcrdimensional field lex, y). In general, we
assume that any pattern can be presented at any location and that the characters
may touch, overlap or be broken or noisy. The input then projects to a linked-Iocalreceptive-field block of sigmoidal hidden units (to enforce translational invariance).
We designate the activation of the sigmoidal units in this block by h ij It.
The second block of hidden units, H 1:'J/' z', is a linked-local receptive field block of
sigmoidal units that receives input from a three-dimensional receptive field in the
hiilt block. In a standard neural network architecture we would normally connect
block H to the output units. However we connect block H to a block of exponential
units X1:J/z, The X block serves as the outputs after the network has been trained;
there is a sheet of exponential units for each output category. These units are
e"''''"?,
connected to block H via a linked-local receptive field structure. X1:J/z
where the net input to the unit is
=
TJ1:J/Z =
L W:,~~z,H1:'J/'z' + /3z,
(1)
1:'J/'
and W:,~~z' is the weight from hidden unit H1:'J/'z' to the exponential unit X1:J/z,
Since we use linked weights in each block, the entire structure is translationally
invariant. We make use of this property in our training algorithm and project out
the positional information by summing over the entire layer, Sz = L1:Y X1:J/z, This
allows us to give non-specific target information in the form of "the input contains
a 5 and a 3, but I will not say where." We do this by converting the summed
information in"to an output probability, pz = 1!5?.
A Self-Organizing Integrated Segmentation and Recognition Neural Net
2.1
The learning Rule
There are two objective functions that we have used to train ISR networks: cross
entropy and total-sum-square-error. I Ez tzlnpz + (1 - t z )ln(l - Pz), where t z
equals 1 if pattern z is presented and 0 otherwise. Computing the gradient with
respect to the net input to a particular exponential unit yields the following term
in our learning rule:
=
~ -- (t z _ pz )
8TJ~yz
X~yz
(2)
E~y X~yz
It should be noted that this is a kind of competitive rule in which the learning is
proportional to the relative strength of the activation at the unit at a particular
location in the X layer to the strength of activation in the entire layer. For example,
suppose that X2,3,5
1000 and X5,3,5= 100. Given the above rules, X2,3,5 would
receive about 10 times more of the output error than the unit X5,3,5. Thus the units
compete with each other for the credit or blame of the output, and the "rich get
richer" until the proper target is achieved. This favors self-organization of highly
localized spikes of activity in the exponential layers directly above the particular
character that the exponential layer detects ("smart histograms" as shown in Figure 1). Note that we never give positional information in the network but that the
network self-organizes the exponential unit activity to discern the positional information. The second function is the total-sum-square error, E = Ez(tz - pz)2. For
the total-sum-square error measure, the gradient term becomes
=
8E
uTJ~yz
-~-
=
(
tz -
pz
)
X~yz
~
(1 + L.~y X~yz
)2 .
(3)
Again this has a competitive term, but the competition is only important for X~yz
large, otherwise the denominator is dominated by 1 for small E~y X~yz. We used
the quadratic error function for the networks reported in the next section.
3
3.1
NIST DATABASE RECOGNITION
Data
We tested this neural network algorithm on the problem of segmenting and recognizing handwritten numerals from the NIST database. This database contains
approximately 273,000 samples of handwritten numerals collected from the Bureau
of Census field staff. There were 50 different forms used in the study, each with
33 fields, 28 of which contain handwritten numerals ranging in length from 2 to 10
digits per field. We only used fields of length 2 to 6 (field numbers 6 to 30). We
used two test sets: a small test set, Test Set A of approximately 4,000 digits, 1,000
fields, from forms labeled f1800 to f1840 and a larger test set, Test Set B, containing
20,000 numerals 5,000 fields and 200 forms from f1800 to f1899 and f2000 to f2199.
We used two different training sets: a hand-segmented training set containing approximately 33,000 digits from forms mooo to m636 (the Segmented Training Set)
and another training set that was never hand-segmented from forms mooo to f1800
(the Unsegmented Training Set. We pre-processed the fields with a simple boxremoval and size-normalization program before they were input to the ISR net.
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Keeler and Rumelhart
The hand segmentation was conventional in the sense that boxes were drawn around
each of the characters, but we the boxes included any other portions of characters
that may be nearby or touching in the natural context. Note that precise labeling of
the characters is not essential at all. We have trained systems where only the center
information the characters was used and found no degradation in performance. This
is due to the fact that the system self-organizes the positional information, so it is
only required that we know whether a character is in a field, not precisely where.
3.2
TRAINING
We trained several nets on the NIST database. The best training procedure was
as follows: Step 1): train the network to an intermediate level of accuracy (96%
or so on single characters, about 12 epochs of training set 1). Note that when we
train on single characters, we do not need isolated characters - there are often
portions of other nearby characters within the input field. Indeed, it helps the ISR
performance to use this natural context. There are two reasons for this step: the
first is speed - training goes much faster with single characters because we can use a
small network. We also found a slight generalization accuracy benefit by including
this training step. Step 2): copy the weights of this small network into a larger
network and start training on 2 and 3 digit fields from the database without hand
segmentation. These are fields numbered 6,7,11,15,19,20,23,24,27, and 28. The
reason that we use these fields is that we do not have to hand-segment them - we
present the fields to the net with the answer that the person was supposed to write
in the field. (There were several cases where the person wrote the wrong numbers
or didn't write anything. These cases were NOT screened from the training set.)
Taking these fields from forms mooo to f1800 gives us another 45,000 characters to
train on without ever segmenting them.
There were several reasons that we use fields of length 2 and 3 and not fields of
4,5,or 6 for training (even though we used these in testing). First, 3 characters
covers the most general case: a character either has no characters on either side,
one to the left, one to the right or one on both sides (3 characters total). If we train
on 3 characters and duplicate the weights, we have covered the most general case for
any number of characters, and it is clearly faster to train on shorter fields. Second,
training with more characters confuses the net. As pointed out in our previous
work (keeler 1991a), the learning algorithm that we use is only valid for one or no
characters of a given type presented in the input field. Thus, the field '39541' is ok
to train on, but the field '288' violates one of the assumptions of the training rule.
In this case the two 8 's would be competing with each other for the answer and
the rule favors only one winner. Even though this problem occurs 1/lth of the
time for two digit fields, it is not serious enough to prevent the net from learning.
(Clearly it would not learn fields of length 10 where all of the target units are
turned on and there would be no chance for discrimination.) This problem could
be avoided by incorporating order information into training and we have proposed
several mechanisms for incorporating order information in training, but do not use
them in the present system. Note that this biases the training toward the a-priori
distribution of characters in the 2 and 3 digit fields, which is a different distribution
from that of the testing set.
The two networks that we used had the following architectures: Net1: Input: 28x24
A Self-Organizing Integrated Segmentation and Recognition Neural Net
receptive fields 6x6 shift 2x2. hidden 1: 12xllx12 receptive fields 4x4x12 shift
2x2x12. hidden 2: 5x4x18 receptive fields 3x3x18 shift lxlxl8. exponentials (block
3): 3x2xlO 10 summing, 10 outputs.
Net2: Input: 28x26 receptive fields 6x6 shift 2x4. hidden 1: 12x6x12 receptive
fields 5x4x12 shift lx2xl2. hidden 2: 8x2x18 receptive fields 5x2x18 shift lxlxl8.
exponentials (block 3): 4xlxlO 10 summing, 10 outputs.
100
0/0
c
99
0
98
r
r
97
e
c
t
A
n1&2
B
96
99. 5 t--+-+--t---:lhr-t:~rI-',
n2
95
94
93
92
91 . . . .______. ._ .....
o
5
10
15
20
25
3C
98 ----'--I-
,6
0/0 Rejected
97.5 .........._ ....~............_ ..............
5
10 15 20 25 30 35 40 45 50 55 60
Figure 2: Average combined network performance on the NIST database. Figure
2A shows the generalization performance of two neural networks on the NIST Test
Set A. The individual nets Netl and Net2 (nl, n2 respectively) and the combined
performance of nets 1 and 2 are shown where fields are rejected when the nets differ.
The curves show results for fields ranging length 2 to 6 averaged over all fields for
1,000 total fields, 4,000 characters. Note that Net2 is not nearly as accurate as Netl
on fields, but that the combination of the two is significantly better than either.
For this test set the rejection rate is 17% (83% acceptance) with an accuracy rate of
99.3% (error rate 0.7%) overall on fields of average length 4. Figure 2B shows the
per-field performance for test-set B (5,000 fields, 20,000 digits) Again both nets are
used for the rejection criterion. For comparison, 99% accuracy on fields of length 4
is achieved at 23% rejection.
Figure 2 shows the generalization performance on the NIST database for Netl, Net2
and their combination. For the combination, we accepted the answer only when the
networks agreed and rejected further based on a simple confidence measure (the
difference of the two highest activations) of each individual net.
501
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Keeler and Rumelhart
../f.
.!
~./.~;,
I .I
1"'"
I
Figure 3: Examples of correctly recognized fields in the NIST database. This figure
shows examples of fields that were correctly recognized by the ISR network. Note
the cases of touching characters, multiple touching characters, characters touching
in multiple places, fields with extrinsic noise, broken characters and touching, broken
characters with noise. Because of the integrated nature of the segmentation and
recognition, the same system is able to handle all of these cases.
4
DISCUSSION AND CONCLUSIONS
This investigation has demonstrated that the ISR algorithm can be used for integrated segmentation and recognition and achieve high-accuracy results on a large
database of hand-printed numerals. The overall accuracy rates of 83% acceptance
with 99.3% accuracy on fields of average length 4 is competitive with accuracy reported in commercial products. One should be careful making such comparisons.
We found a variance of 7% or more in rejection performance on different test sets
with more than 1,000 fields (a good statistical sample). Perhaps more important
than the high accuracy, we have demonstrated that the ISR system is able to deal
with touching, broken and noisy characters. In other investigations we have demonstrated the ISR system on alphabetic characters with good results, and on speech
recognition (Keeler, Rumelhart, Zand-Biglari, 1991) where the results are slightly
better than Hidden Markov Model results.
There are several attractive aspects about the ISR algorithm: 1) Labeling can be
"sloppy" in the sense that the borders of the characters do not have to be defined.
This reduces the labor burden of getting a system running. 2) The final weights can
be duplicated so that the system can all run in parallel. Even with both networks
running, the number of weights and activations needed to be stored in memory is
quite small - about 30,000 floating point numbers, and the system is quite fast
in the feed-forward mode: peak performance is about 2.5 characters/sec on a Dec
5000 (including everything: both networks running, input pre-processing, parsing
the answers, printing results, etc.). This structure is ideal for VLSI implementation
since it contains a very small number of weights (about 5,000). This is one possible
way around the computational bottleneck facing encountered in processing complex
scenes - the ISR net can do very-fast first-cut scene analysis with good discrimi-
A Self-Organizing Integrated Segmentation and Recognition Neural Net
nation of similar objects - an extremely difficult task. 3) The ISR algorithm and
architecture presents a new and powerful approach of using forward models to convert position-independent training information into position-specific error signals.
4) There is no restriction to one-dimension; The same ISR structure has been used
for two-dimensional parsing.
Nevertheless, there are several aspects of the ISR net that require improvement for
future progress. First, the algorithmic assumption of having one pattern of a given
type in the input field is too restrictive and can cause confusion in some training
examples. Second, we are throwing some information away when we project out
all of the positional information order information could be incorporated into the
training information. This extra information should improve training performance
due to the more-specific error signals. Finally, normalization is still a problem.
We do a crude normalization, and the networks are able to segment and recognize
characters as long as the difference in size is not too large. A factor of two in
size difference is easily handled with the ISR system, but a factor of four decreases
recognition accuracy by about 3-5% on the character recognition rates. This requires a tighter coupling between the segmentation/recognition and normalization.
Just as one must segment and recognize simultaneously, in many cases one can't
properly normalize until segmentation/recognition has occurred. Fortunately, in
most document processing applications, crude normalization to within a factor of
two is simple to achieve, allowing high accuracy networks.
Acknowledgements
We thank Wee-Kheng Leow, Steve O'Hara, John Canfield, for useful discussions
and coding.
References
(1] J.D. Keeler, D.E. Rumelhart, and W.K. Leow (1991a) "Integrated Segmentation and Recognition of Hand-printed Numerals". In: Lippmann, Moody and
Touretzky, Editors, Neural Information Processing Systems 3, 557-563.
[2] J.D. Keeler, D.E. Rumelhart, and S. Zand-Biglari (1991b) "A Neural Network
For Integrated Segmentation and Recognition of Continuous Speech". MCC
Technical Report ACT-NN-359-91.
[3] K. Lang, A. Waibel, G. Hinton. (1990) A time delay Neural Network Architecture for Isolated Word Recognition. Neural Networks, 3 23-44.
[4] Y. Le Cun, B. Boser, J .S. Denker, S. Solla, R. Howard, and L. Jackel.
(1990) "Back-Propagation applied to Handwritten Zipcode Recognition." Neural Computation 1(4):541-551.
[5] G. Martin, J. Pittman (1990) "Recognizing hand-printed letters and digits."
In D. Touretzky (Ed.). Neural Information Processing Systems 2, 405-414,
Morgan Kauffman Publishers, San Mateo, CA.
[6] The NIST database can be obtained by writing to: Standard Reference Data
National Institute of Standards and Technology 221/ A323 Gaithersburg, MD
20899 USA and asking for NIST special database 1 (HWDB).
503
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5,192 | 5,700 | Decomposition Bounds for Marginal MAP
?
Wei Ping?
Qiang Liu?
Alexander Ihler?
?
Computer Science, UC Irvine
Computer Science, Dartmouth College
{wping,ihler}@ics.uci.edu [email protected]
Abstract
Marginal MAP inference involves making MAP predictions in systems defined
with latent variables or missing information. It is significantly more difficult than
pure marginalization and MAP tasks, for which a large class of efficient and convergent variational algorithms, such as dual decomposition, exist. In this work, we
generalize dual decomposition to a generic power sum inference task, which includes marginal MAP, along with pure marginalization and MAP, as special cases.
Our method is based on a block coordinate descent algorithm on a new convex
decomposition bound, that is guaranteed to converge monotonically, and can be
parallelized efficiently. We demonstrate our approach on marginal MAP queries
defined on real-world problems from the UAI approximate inference challenge,
showing that our framework is faster and more reliable than previous methods.
1
Introduction
Probabilistic graphical models such as Bayesian networks and Markov random fields provide a useful framework and powerful tools for machine learning. Given a graphical model, inference refers to
answering probabilistic queries about the model. There are three common types of inference tasks.
The first are max-inference or maximum a posteriori (MAP) tasks, which aim to find the most probable state of the joint probability; exact and approximate MAP inference is widely used in structured
prediction [26]. Sum-inference tasks include calculating marginal probabilities and the normalization constant of the distribution, and play a central role in many learning tasks (e.g., maximum
likelihood). Finally, marginal MAP tasks are ?mixed? inference problems, which generalize the first
two types by marginalizing a subset of variables (e.g., hidden variables) before optimizing over the
remainder.1 These tasks arise in latent variable models [e.g., 29, 25] and many decision-making
problems [e.g., 13]. All three inference types are generally intractable; as a result, approximate
inference, particularly convex relaxations or upper bounding methods, are of great interest.
Decomposition methods provide a useful and computationally efficient class of bounds on inference
problems. For example, dual decomposition methods for MAP [e.g., 31] give a class of easy-toevaluate upper bounds which can be directly optimized using coordinate descent [36, 6], subgradient
updates [14], or other methods [e.g., 22]. It is easy to ensure both convergence, and that the objective is monotonically decreasing (so that more computation always provides a better bound). The
resulting bounds can be used either as stand-alone approximation methods [6, 14], or as a component of search [11]. In summation problems, a notable decomposition bound is tree-reweighted BP
(TRW), which bounds the partition function with a combination of trees [e.g., 34, 21, 12, 3]. These
bounds are useful in joint inference and learning (or ?inferning?) frameworks, allowing learning
with approximate inference to be framed as a joint optimization over the model parameters and decomposition bound, often leading to more efficient learning [e.g., 23]. However, far fewer methods
have been developed for marginal MAP problems.
1
In some literature [e.g., 28], marginal MAP is simply called MAP, and the joint MAP task is called MPE.
1
In this work, we deveop a decomposition bound that has a number of desirable properties: (1) Generality: our bound is sufficiently general to be applied easily to marginal MAP. (2) Any-time: it yields
a bound at any point during the optimization (not just at convergence), so it can be used in an anytime way. (3) Monotonic and convergent: more computational effort gives strictly tighter bounds;
note that (2) and (3) are particularly important for high-width approximations, which are expensive
to represent and update. (4) Allows optimization over all parameters, including the ?weights?, or
fractional counting numbers, of the approximation; these parameters often have a significant effect
on the tightness of the resulting bound. (5) Compact representation: within a given class of bounds,
using fewer parameters to express the bound reduces memory and typically speeds up optimization.
We organize the rest of the paper as follows. Section 2 gives some background and notation, followed by connections to related work in Section 3. We derive our decomposed bound in Section 4,
and present a (block) coordinate descent algorithm for monotonically tightening it in Section 5. We
report experimental results in Section 6 and conclude the paper in Section 7.
2
Background
Here, we review some background on graphical models and inference tasks. A Markov random field
(MRF) on discrete random variables x = [x1 , . . . , xn ] ? X n is a probability distribution,
hX
i
hX
i
X
p(x; ?) = exp
?? (x? ) ? ?(?) ; ?(?) = log
exp
?? (x? ) ,
(1)
x?X n
??F
??F
where F is a set of subsets of the variables, each associated with a factor ?? , and ?(?) is the log
partition function. We associate an undirected graph G = (V, E) with p(x) by mapping each xi to
a node i ? V , and adding an edge ij ? E iff there exists ? ? F such that {i, j} ? ?. We say node
i and j are neighbors if ij ? E. Then, F is the subset of cliques (fully connected subgraphs) of G.
The use and evaluation of a given MRF often involves different types of inference tasks. Marginalization, or sum-inference tasks perform a sum over the configurations to calculate the log partition
function ? in (1), marginal probabilities, or the probability of some observed evidence. On the other
hand, the maximum a posteriori (MAP), or max-inference tasks perform
joint maximization to find
P
configurations with the highest probability, that is, ?0 (?) = maxx ??F ?? (x? ).
A generalization of max- and sum- inference is marginal MAP, or mixed-inference, in which we are
interested in first marginalizing a subset A of variables (e.g., hidden variables), and then maximizing
the remaining variables B (whose values are of direct interest), that is,
hX
i
X
?AB (?) = max Q(xB ) = max log
exp
?? (x? ) ,
(2)
xB
xB
xA
??F
where A ? B = V (all the variables) and A ? B = ?. Obviously, both sum- and max- inference are
special cases of marginal MAP when A = V and B = V , respectively.
It will be useful to define an even more general inference task, based on a power sum operator:
?i
X
X
1/? ?i
f (xi ) =
f (xi ) i ,
xi
xi
where f (xi ) is any non-negative function and ?i is a temperature or weight parameter. The power
sum reduces to a standard sum when ?i = 1, and approaches maxx f (x) when ?i ? 0+ , so that we
define the power sum with ?i = 0 to equal the max operator.
The power sum is helpful for unifying max- and sum- inference [e.g., 35], as well as marginal
MAP [15]. Specifically, we can apply power sums with different weights ?i to each variable xi along
a predefined elimination order (e.g., [x1 , . . . , xn ]), to define the weighted log partition function:
?n
?1
?
X
X
X
exp(?(x)) = log
...
exp(?(x)),
(3)
?? (?) = log
x
xn
x1
where we note that the value of (3) depends on the elimination order unless all the weights are equal.
Obviously, (3) includes marginal MAP (2) as a special case by setting weights ?A = 1 and ?B = 0.
This representation provides a useful tool for understanding and deriving new algorithms for general
inference tasks, especially marginal MAP, for which relatively few efficient algorithms exist.
2
3
Related Work
Variational upper bounds on MAP and the partition function, along with algorithms for providing
fast, convergent optimization, have been widely studied in the last decade. In MAP, dual decomposition and linear programming methods have become a dominating approach, with numerous optimization techniques [36, 6, 32, 14, 37, 30, 22], and methods to tighten the approximations [33, 14].
For summation problems, most upper bounds are derived from the tree-reweighted (TRW) family
of convex bounds [34], or more generally conditional entropy decompositions [5]. TRW bounds
can be framed as optimizing over a convex combination of tree-structured models, or in a dual
representation as a message-passing, TRW belief propagation algorithm. This illustrates a basic
tension in the resulting bounds: in its primal form 2 (combination of trees), TRW is inefficient:
it maintains a weight and O(|V |) parameters for each tree, and a large number of trees may be
required to obtain a tight bound; this uses memory and makes optimization slow. On the other hand,
the dual, or free energy, form uses only O(|E|) parameters (the TRW messages) to optimize over the
set of all possible spanning trees ? but, the resulting optimization is only guaranteed to be a bound
at convergence, 3 making it difficult to use in an anytime fashion. Similarly, the gradient of the
weights is only correct at convergence, making it difficult to optimize over these parameters; most
implementations [e.g., 24] simply adopt fixed weights.
Thus, most algorithms do not satisfy all the desirable properties listed in the introduction. For example, many works have developed convergent message-passing algorithms for convex free energies
[e.g., 9, 10]. However, by optimizing the dual they do not provide a bound until convergence, and the
representation and constraints on the counting numbers do not facilitate optimizing the bound over
these parameters. To optimize counting numbers, [8] adopt a more restrictive free energy form requiring positive counting numbers on the entropies; but this cannot represent marginal MAP, whose
free energy involves conditional entropies (equivalent to the difference between two entropy terms).
On the other hand, working in the primal domain ensures a bound, but usually at the cost of enumerating a large number of trees. [12] heuristically select a small number of trees to avoid being
too inefficient, while [21] focus on trying to speed up the updates on a given collection of trees.
Another primal bound is weighted mini-bucket (WMB, [16]), which can represent a large collection
of trees compactly and is easily applied to marginal MAP using the weighted log partition function
viewpoint [15, 18]; however, existing optimization algorithms for WMB are non-monotonic, and
often fail to converge, especially on marginal MAP tasks.
While our focus is on variational bounds [16, 17], there are many non-variational approaches for
marginal MAP as well. [27, 38] provide upper bounds on marginal MAP by reordering the order
in which variables are eliminated, and using exact inference in the reordered join-tree; however,
this is exponential in the size of the (unconstrained) treewidth, and can easily become intractable.
[20] give an approximation closely related to mini-bucket [2] to bound the marginal MAP; however,
unlike (weighted) mini-bucket, these bounds cannot be improved iteratively. The same is true for
the algorithm of [19], which also has a strong dependence on treewidth. Other examples of marginal
MAP algorithms include local search [e.g., 28] and Markov chain Monte Carlo methods [e.g., 4, 39].
4
Fully Decomposed Upper Bound
In this section, we develop a new general form of upper bound and provide an efficient, monotonically convergent optimization algorithm. Our new bound is based on fully decomposing the graph
into disconnected cliques, allowing very efficient local computation, but can still be as tight as WMB
or the TRW bound with a large collection of spanning trees once the weights and shifting variables
are chosen or optimized properly. Our bound reduces to dual decomposition for MAP inference, but
is applicable to more general mixed-inference settings.
Our main result is based on the following generalization of the classical H?older?s inequality [7]:
2
Despite the term ?dual decomposition? used in MAP tasks, in this work we refer to decomposition bounds
as ?primal? bounds, since they can be viewed as directly bounding the result of variable elimination. This
is in contrast to, for example, the linear programming relaxation of MAP, which bounds the result only after
optimization.
3
See an example on Ising model in Supplement A.
3
Theorem 4.1. For a given graphical model p(x; ?) in (1) with cliques F = {?} and a set of nonnegative weights ? = {?i ? 0, i ? V }, we definePa set of ?split weights? w? = {wi? ? 0, i ? ?}
on each variable-clique pair (i, ?), that satisfies ?|?3i wi? = ?i . Then we have
? Y
w?
X
YX
exp ?? (x? ) ?
exp ?? (x? ) ,
(4)
x ??F
??F x?
where the left-hand side is the powered-sum along order [x1 , . . . , xn ] as defined in (3), and the
right-hand side is the product of the powered-sums on subvector x? with weights w? along
Pw?
Pwk?
Pwk?1
c ???
the same elimination order; that is, x? exp ?? (x? ) =
xk1 exp ?? (x? ) , where
xk c
x? = [xk1 , . . . , xkc ] should be ranked with increasing index, consisting with the elimination order
[x1 , . . . , xn ] as used in the left-hand side.
Proof details can be found in Section E of the supplement. A key advantage of the bound (4) is
that it decomposes the joint power sum on x into a product of independent power sums over smaller
cliques x? , which significantly reduces computational complexity and enables parallel computation.
4.1
Including Cost-shifting Variables
In order to increase the flexibility of the upper bound, we introduce a set of cost-shifting or reparameterization variables ? = {?i? (xi ) | ?(i, ?), i ? ?} on each variable-factor pair (i, ?), which can
be optimized to provide a much tighter upper bound. Note that ?? (?) can be rewritten as,
?? (?) = log
?
X
exp
hX X
x
?i? (xi ) +
i?V ??Ni
X
??F
?? (x? ) ?
X
i
?i? (xi ) ,
i??
where Ni = {? | ? 3 i} is the set of cliques incident to i. Applying inequality (4), we have that
?? (?) ?
X
i?V
log
wi
X
xi
exp
h X
w?
h
i
i X
X
X ?
def
exp ?? (x? ) ?
?i (xi ) == L(?, w),
?i? (xi ) +
log
??Ni
x?
??F
(5)
i??
where the nodes i ? V are also treated as cliques within inequality (4), and a new weight wi is
introduced on each variable i; the new weights w = {wi , wi? | ?(i, ?), i ? ?} should satisfy
X
wi +
wi? = ?i , wi ? 0, wi? ? 0, ?(i, ?).
(6)
??Ni
The bound L(?, w) is convex w.r.t. the cost-shifting variables ? and weights w, enabling an efficient
optimization algorithm that we present in Section 5. As we will discuss in Section 5.1, these shifting
variables correspond to Lagrange multipliers that enforce a moment matching condition.
4.2
Dual Form and Connection With Existing Bounds
It is straightforward to see that our bound in (5) reduces to dual decomposition [31] when applied
on MAP inference with all ?i = 0, and hence wi = wi? = 0. On the other hand, its connection with
sum-inference bounds such as WMB and TRW is seen more clearly via a dual representation of (5):
Theorem 4.2. The tightest upper bound obtainable by (5), that is,
X
XX
min min L(?, w) = min max h?, bi +
wi H(xi ; bi ) +
wi? H(xi |xpa?i ; b? ) , (7)
w
?
w b?L(G)
??F i??
i?V
where b = {bi (xi ), b? (x? ) | ?(i, ?), i ? ?} is a set of pseudo-marginals (or beliefs) defined on the
singleton variables and theP
cliques, and L isPthe corresponding local consistency polytope defined
by L(G) = {b | bi (xi ) = x?\i b? (x? ),
xi bi (xi ) = 1}. Here, H(?) are their corresponding
?
marginal or conditional entropies, and pai is the set of variables in ? that rank later than i, that is,
for the global elimination order [x1 , . . . , xn ], pa?
i = {j ? ? | j i}.
The proof details can be found in Section F of the supplement. It is useful to compare Theorem 4.2 with other dual representations. As the sum of non-negatively weighted conditional entropies, the bound is clearly convex and within the general class of conditional entropy decompositions (CED) [5], but unlike generic CED it has a simple and efficient primal form (5). 4 Comparing
4
The primal form derived in [5] (a geometric program) is computationally infeasible.
4
(a) 3 ? 3 grid
(b) WMB: covering tree
(c) Full decomposition
(d) TRW
Figure 1: Illustrating WMB, TRW and our bound on (a) 3 ? 3 grid. (b) WMB uses a covering tree
with a minimal number of splits and cost-shifting. (c) Our decomposition (5) further splits the graph
into small cliques (here, edges), introducing additional cost-shifting variables but allowing for easier,
monotonic optimization. (d) Primal TRW splits the graph into many spanning trees, requiring even
more cost-shifting variables. Note that all three bounds attain the same tightness after optimization.
to the dual form of WMB in Theorem 4.2 of [16], our bound is as tight as WMB, and hence the
class of TRW / CED bounds attainable by WMB
P [16]. Most duality-based forms [e.g., 9, 10] are
expressed in terms of joint entropies, h?, bi + ? c? H(b? ), rather than conditional entropies; while
the two can be converted, the resulting counting numbers c? will be differences of weights {wi? }, 5
which obfuscates its convexity, makes it harder to maintain the relative constraints on the counting
numbers during optimization, and makes some counting numbers negative (rendering some methods inapplicable [8]). Finally, like most variational bounds in dual form, the RHS of (7) has a inner
maximization and hence guaranteed to bound ?? (?) only at its optimum.
In contrast, our Eq. (5) is a primal bound (hence, a bound for any ?). It is similar to the primal form
of TRW, except that (1) the individual regions are single cliques, rather than spanning trees of the
graph, 6 and (2) the fraction weights w? associated with each region are vectors, rather than a single
scalar. The representation?s efficiency can be seen with an example in Figure 1, which shows a 3 ? 3
grid model and three relaxations that achieve the same bound. Assuming d states per variable and
ignoring the equality constraints, our decomposition in Figure 1(c) uses 24d cost-shifting parameters
(?), and 24 weights. WMB (Figure 1(b)) is slightly more efficient, with only 8d parameters for ? and
and 8 weights, but its lack of decomposition makes parallel and monotonic updates difficult. On the
other hand, the equivalent primal TRW uses 16 spanning trees, shown in Figure 1(d), for 16 ? 8 ? d2
parameters, and 16 weights. The increased dimensionality of the optimization slows convergence,
and updates are non-local, requiring full message-passing sweeps on the involved trees (although
this cost can be amortized in some cases [21]).
5
Monotonically Tightening the Bound
In this section, we propose a block coordinate descent algorithm (Algorithm 1) to minimize the
upper bound L(?, w) in (5) w.r.t. the shifting variables ? and weights w. Our algorithm has a
monotonic convergence property, and allows efficient, distributable local computation due to the
full decomposition of our bound. Our framework allows generic powered-sum inference, including
max-, sum-, or mixed-inference as special cases by setting different weights.
5.1
Moment Matching and Entropy Matching
We start with deriving the gradient of L(?, w) w.r.t. ? and w. We show that the zero-gradient
equation w.r.t. ? has a simple form of moment matching that enforces a consistency between the
singleton beliefs with their related clique beliefs, and that of weights w enforces a consistency of
marginal and conditional entropies.
Theorem 5.1. (1) For L(?, w) in (5), its zero-gradient w.r.t. ?i? (xi ) is
X
?L
= ?i (xi ) ?
?? (x? ) = 0,
?
??i (xi )
x
(8)
?\i
5
See more details of this connection in Section F.3 of the supplement.
While non-spanning subgraphs can be used in the primal TRW form, doing so leads to loose bounds; in
contrast, our decomposition?s terms consist of individual cliques.
6
5
Algorithm 1 Generalized Dual-decomposition (GDD)
Input: weights {?i | i ? V }, elimination order o.
Output: the optimal ? ? , w? giving tightest upper bound L(? ? , w? ) for ?? (?) in (5).
initialize ? = 0 and weights w = {wi , wi? }.
repeat
for node i (in parallel with node j, (i, j) 6? E) do
if ?i = 0 then
update ? Ni = {?i? |?? ? Ni } with the closed-form update (11);
else if ?i 6= 0 then
update ? Ni and wNi with gradient descent (8) and(12), combined with line search;
end if
end for
until convergence
? ? ? ?, w? ? w, and evaluate L(? ? , w? ) by (5);
Remark. GDD solves max-, sum- and mixed-inference by setting different values of weights {?i }.
P
where ?i (xi ) ? exp w1i ??Ni ?i? (xi ) can be interpreted as a singleton belief on xi , and ?? (x? )
can be viewed
Qc as clique belief on x? , defined with a chain rule (assuming x?? = [x1 , . . . , xc ]),
?? (x? ) = i=1 ?? (xi |xi+1:c ); ?? (xi |xi+1:c ) = (Zi?1 (xi:c )/Zi (xi+1:c ))1/wi , where Zi is the
partial powered-sum up to x1:i on the clique, that is,
w1?
wi?
h
i
h
i
X
X
X
X
???
exp ?? (x? ) ?
?i? (xi ) , Z0 (x? ) = exp ?? (x? ) ?
?i? (xi ) ,
Zi (xi+1:c ) =
xi
x1
i??
i??
where the summation order should be consistent with the global elimination order o = [x1 , . . . , xn ].
(2) The gradients of L(?, w) w.r.t. the weights {wi , wi? } are marginal and conditional entropies
defined on the beliefs {?i , ?? }, respectively,
X
?L
?L
= H(xi |xi+1:c ; ?? ) = ?
?? (x? ) log ?? (xi |xi+1:c ).
(9)
= H(xi ; ?i ),
?
?wi
?wi
x
?
Therefore, the optimal weights should
satisfy the following KKT condition
wi H(xi ; ?i ) ? H?i = 0, wi? H(xi |xi+1:c ; ?? ) ? H?i = 0, ?(i, ?)
(10)
P ?
?
where Hi = wi H(xi ; ?i ) +
w H(xi |xi+1:c ; ?? ) is the (weighted) average entropy on node i.
?
i
The proof details can be found in Section G of the supplement. The matching condition (8) enforces
that ? = {?i , ?? | ?(i, ?)} belong to the local consistency polytope L as defined in Theorem 4.2;
similar moment matching results appear commonly in variational inference algorithms [e.g., 34].
[34] also derive a gradient of the weights, but it is based on the free energy form and is correct only
after optimization; our form holds at any point, enabling efficient joint optimization of ? and w.
5.2
Block Coordinate Descent
We derive a block coordinate descent method in Algorithm 1 to minimize our bound, in which
we sweep through all the nodes i and update each block ? Ni = {?i? (xi ) | ?? ? Ni } and
wNi = {wi , wi? | ?? ? Ni } with the neighborhood parameters fixed. Our algorithm applies two
update types, depending on whether the variables have zero weight: (1) For nodes with ?i = 0 (corresponding to max nodes i ? B in marginal MAP), we derive a closed-form coordinate descent rule
for the associated shifting variables ? Ni ; these nodes do not require to optimize wNi since it is fixed
to be zero. (2) For nodes with ?i 6= 0 (e.g., sum nodes i ? A in marginal MAP), we lack a closed
form update for ? Ni and wNi , and optimize by local gradient descent combined with line search.
The lack of a closed form coordinate update for nodes ?i 6= 0 is mainly because the order of power
sums with different weights cannot be exchanged. However, the gradient descent inner loop is still
efficient, because each gradient evaluation only involves the local variables in clique ?.
Closed-form Update. For any node i with ?i = 0 (i.e., max nodes i ? B in marginal MAP), and
its associated ? Ni = {?i? (xi ) | ?? ? Ni }, the following update gives a closed form solution for the
zero (sub-)gradient equation in (8) (keeping the other {?j? |j 6= i, ?? ? Ni } fixed):
6
?i? (xi ) ?
X ?
1
|Ni |
?i? (xi ) ?
?i (xi ),
|Ni | + 1
|Ni | + 1
(11)
??Ni \?
?
Pw\i
where |Ni | is the
number of neighborhood cliques, and ?i? (xi ) = log x?\i
exp ?? (x? ) ?
P
?
j??\i ?j (xj ) . Note that the update in (11) works regardless of the weights of nodes {?j | ?j ?
?, ?? ? Ni } in the neighborhood cliques; when all the neighboring nodes also have zero weight
(?j = 0 for ?j ? ?, ?? ? Ni ), it is analogous to the ?star? update of dual decomposition for
MAP [31]. The detailed derivation is shown in Proposition H.2 in the supplement.
The update in (11) can be calculated with a cost of only O(|Ni | ? d|?| ), where d is the number of
states of xi , and |?| is the clique size, by computing and saving all the shared {?i? (xi )} before
updating ? Ni . Furthermore, the updates of ? Ni for different nodes i are independent if they are not
directly connected by some clique ?; this makes it easy to parallelize the coordinate descent process
by partitioning the graph into independent sets, and parallelizing the updates within each set.
Local Gradient Descent. For nodes with ?i 6= 0 (or i ? A in marginal MAP), there is no closedform solution for {?i? (xi )} and {wi , wi? } to minimize the upper bound. However, because of the
fully decomposed form, the gradient w.r.t. ? Ni and wNi , (8)?(9), can be evaluated efficiently via
local computation with O(|Ni | ? d|?| ), and again can be parallelized between nonadjacent nodes.
To handle the normalization
P constraint (6) on wNi , we use an exponential
P gradient descent: let
wi = exp(vi )/ exp(vi ) + ? exp(vi? ) and wi? = exp(vi? )/ exp(vi ) + ? exp(vi? ) ; taking the
gradient w.r.t.vi and vi? and transforming
update
back? gives?the following
?
?
wi ? wi exp ? ?wi H(xi ; ?i ) ? Hi , wi ? wi exp ? ?wi H(xi |xpa?i ; ?? ) ? H?i , (12)
where ? is the step size and pa?
i = {j ? ? | j i}. In our implementation, we find that a few gradient
steps (e.g., 5) with a backtracking line search using the Armijo rule works well in practice. Other
more advanced optimization methods, such as L-BFGS and Newton?s method are also applicable.
6
Experiments
In this section, we demonstrate our algorithm on a set of real-world graphical models from recent
UAI inference challenges, including two diagnostic Bayesian networks with 203 and 359 variables
and max domain sizes 7 and 6, respectively, and several MRFs for pedigree analysis with up to 1289
variables, max domain size of 7 and clique size 5.7 We construct marginal MAP problems on these
models by randomly selecting half of the variables to be max nodes, and the rest as sum nodes.
We implement several algorithms that optimize the same primal marginal MAP bound, including
our GDD (Algorithm 1), the WMB algorithm in [16] with ibound = 1, which uses the same cliques
and a fixed point heuristic for optimization, and an off-the-shelf L-BFGS implementation that directly optimizes our decomposed bound. For comparison, we also computed several related primal
bounds, including standard mini-bucket [2] and elimination reordering [27, 38], limited to the same
computational limits (ibound = 1). We also tried MAS [20] but found its bounds extremely loose.8
Decoding (finding a configuration x
?B ) is more difficult in marginal MAP than in joint MAP. We use
the same local decoding procedure that is standard in dual decomposition [31]. However, evaluating
the objective Q(?
xB ) involves a potentially difficult sum over xA , making it hard to score each
decoding. For this reason, we evaluate the score of each decoding, but show the most recent decoding
rather than the best (as is standard in MAP) to simulate behavior in practice.
Figure 2 and Figure 3 compare the convergence of the different algorithms, where we define the
iteration of each algorithm to correspond to a full sweep over the graph, with the same order of
time complexity: one iteration for GDD is defined in Algorithm 1; for WMB is a full forward and
backward message pass, as in Algorithm 2 of [16]; and for L-BFGS is a joint quasi-Newton step
on all variables. The elimination order that we use is obtained by a weighted-min-fill heuristic [1]
constrained to eliminate the sum nodes first.
Diagnostic Bayesian Networks.
Figure 2(a)-(b) shows that our GDD converges quickly and
monotonically on both the networks, while WMB does not converge without proper damping; we
7
See http://graphmod.ics.uci.edu/uai08/Evaluation/Report/Benchmarks.
The instances tested have many zero probabilities, which make finding lower bounds difficult; since MAS?
bounds are symmetrized, this likely contributes to its upper bounds being loose.
8
7
10
?10
?20
?30
WMB?0.015
WMB?0.020
WMB?0.025
GDD
L?BFGS
MBE
Elimination reordering
Decoded value (WMB)
Decoded value (GDD)
0
?10
Bound
0
Bound
10
WMB?0.025
WMB?0.035
WMB?0.045
GDD
L?BFGS
MBE
Elimination reordering
Decoded value (WMB)
Decoded value (GDD)
?20
?30
?40
?40
?50
?50
0
5
10
Iterations
15
?60
0
20
(a) BN-1 (203 nodes)
5
10
Iterations
15
20
(b) BN-2 (359 nodes)
Figure 2: Marginal MAP results on BN-1 and BN-2 with 50% randomly selected max-nodes (additional plots are in the supplement B). We plot the upper bounds of different algorithms across
iterations; the objective function Q(xB ) (2) of the decoded solutions xB are also shown (dashed
lines). At the beginning, Q(xB ) may equal to ?? because of zero probabiliy.
?40
?45
?50
WMB?0.01
WMB?0.02
WMB?0.03
WMB?0.04
WMB?0.05
GDD
MBE
Elim reordering
?80
?100
?120
?140
?55
0
?60
5
10
15
Iterations
(a) pedigree1 (334 nodes)
20
0
?20
WMB?0.01
WMB?0.02
WMB?0.03
WMB?0.04
WMB?0.05
GDD
MBE
Elim reordering
?40
Upper Bound
WMB?0.01
WMB?0.02
WMB?0.03
WMB?0.04
WMB?0.05
GDD
MBE
Elim reordering
Upper Bound
Upper Bound
?35
?60
?80
?100
?120
?140
5
10
15
Iterations
(b) pedigree7 (1068 nodes)
20
0
5
10
15
Iterations
20
(c) pedigree9 (1118 nodes)
Figure 3: Marginal MAP inference on three pedigree models (additional plots are in the supplement
C). We randomly select half the nodes as max-nodes in these models. We tune the damping rate of
WMB from 0.01 to 0.05.
experimented different damping ratios for WMB, and found that it is slower than GDD even with the
best damping ratio found (e.g., in Figure 2(a), WMB works best with damping ratio 0.035 (WMB0.035), but is still significantly slower than GDD). Our GDD also gives better decoded marginal
MAP solution xB (obtained by rounding the singleton beliefs). Both WMB and our GDD provide a much tighter bound than the non-iterative mini-bucket elimination (MBE) [2] or reordered
elimination [27, 38] methods.
Genetic Pedigree Instances. Figure 3 shows similar results on a set of pedigree instances. Again,
GDD outperforms WMB even with the best possible damping, and out-performs the non-iterative
bounds after only one iteration (pass through the graph).
7
Conclusion
In this work, we propose a new class of decomposition bounds for general powered-sum inference,
which is capable of representing a large class of primal variational bounds but is much more computationally efficient. Unlike previous primal sum bounds, our bound decomposes into computations
on small, local cliques, increasing efficiency and enabling parallel and monotonic optimization. We
derive a block coordinate descent algorithm for optimizing our bound over both the cost-shifting parameters (reparameterization) and weights (fractional counting numbers), which generalizes dual
decomposition and enjoy similar monotonic convergence property. Taking the advantage of its
monotonic convergence, our new algorithm can be widely applied as a building block for improved
heuristic construction in search, or more efficient learning algorithms.
Acknowledgments
This work is sponsored in part by NSF grants IIS-1065618 and IIS-1254071. Alexander Ihler is
also funded in part by the United States Air Force under Contract No. FA8750-14-C-0011 under the
DARPA PPAML program.
8
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9
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5,193 | 5,701 | Anytime Influence Bounds and the Explosive
Behavior of Continuous-Time Diffusion Networks
1
Kevin Scaman1
R?emi Lemonnier1,2
Nicolas Vayatis1
CMLA, ENS Cachan, CNRS, Universit?e Paris- Saclay, France, 2 1000mercis, Paris, France
{scaman, lemonnier, vayatis}@cmla.ens-cachan.fr
Abstract
The paper studies transition phenomena in information cascades observed along
a diffusion process over some graph. We introduce the Laplace Hazard matrix
and show that its spectral radius fully characterizes the dynamics of the contagion both in terms of influence and of explosion time. Using this concept, we
prove tight non-asymptotic bounds for the influence of a set of nodes, and we
also provide an in-depth analysis of the critical time after which the contagion becomes super-critical. Our contributions include formal definitions and tight lower
bounds of critical explosion time. We illustrate the relevance of our theoretical results through several examples of information cascades used in epidemiology and
viral marketing models. Finally, we provide a series of numerical experiments for
various types of networks which confirm the tightness of the theoretical bounds.
1
Introduction
Diffusion networks capture the underlying mechanism of how events propagate throughout a complex network. In marketing, social graph dynamics have caused large transformations in business
models, forcing companies to re-imagine their customers not as a mass of isolated economic agents,
but as customer networks [1]. In epidemiology, a precise understanding of spreading phenomena
is heavily needed when trying to break the chain of infection in populations during outbreaks of
viral diseases. But whether the subject is a virus spreading across a computer network, an innovative product among early adopters, or a rumor propagating on a network of people, the questions
of interest are the same: how many people will it infect? How fast will it spread? And, even more
critically for decision makers: how can we modify its course in order to meet specific goals? Several papers tackled these issues by studying the influence maximization problem. Given a known
diffusion process on a graph, it consists in finding the top-k subset of initial seeds with the highest expected number of infected nodes at a certain time distance T . This problem being NP-hard
[2], various heuristics have been proposed in order to obtain scalable suboptimal approximations.
While the first algorithms focused on discrete-time models and the special case T = +? [3, 4],
subsequent papers [5, 6] brought empirical evidences of the key role played by temporal behavior.
Existing models of continuous-time stochastic processes include multivariate Hawkes processes [7]
where recent progress in inference methods [8, 9] made available the tools for the study of activity
shaping [10], which is closely related to influence maximization. However, in the most studied case
in which each node of the network can only be infected once, the most widely used model remains
the Continuous-Time Information Cascade (CTIC) model [5]. Under this framework, successful
inference [5] as well as influence maximization algorithms have been developed [11, 12].
However, if recent works [13, 14] provided theoretical foundations for the inference problem, assessing the quality of influence maximization remains a challenging task, as few theoretical results
exist for general graphs. In the infinite-time setting, studies of the SIR diffusion process in epidemiology [15] or percolation for specific graphs [16] provided a more accurate understanding of these
processes. More recently, it was shown in [17] that the spectral radius of a given Hazard matrix
1
played a key role in influence of information cascades. This allowed the authors to derive closedform tight bounds for the influence in general graphs
? and characterize epidemic thresholds under
which the influence of any set of nodes is at most O( n).
In this paper, we extend their approach in order to deal with the problem of anytime influence bounds
for continuous-time information cascades. More specifically, we define the Laplace Hazard matrices
and show that the influence at time T of any set of nodes heavily depends on their spectral radii.
Moreover, we reveal the existence and characterize the behavior of critical times at which supercritical processes explode. We show that before these times, super-critical processes will behave
sub-critically and infect at most o(n) nodes. These results can be used in various ways. First, they
provide a way to evaluate influence maximization algorithms without having to test all possible set
of influencers, which is intractable for large graphs. Secondly, critical times allow decision makers
to know how long a contagion will remain in its early phase before becoming a large-scale event,
in fields where knowing when to act is nearly as important as knowing where to act. Finally, they
can be seen as the first closed-form formula for anytime influence estimation for continuous-time
information cascades. Indeed, we provide empirical evidence that our bounds are tight for a large
family of graphs at the beginning and the end of the infection process.
The rest of the paper is organized as follows. In Section 2, we recall the definition of Information
Cascades Model and introduce useful notations. In Section 3, we derive theoretical bounds for the
influence. In Section 4, we illustrate our results by applying them on specific cascade models. In
Section 5, we perform experiments in order to show that our bounds are sharp for a family of graphs
and sets of initial nodes. All proof details are provided in the supplementary material.
2
2.1
Continuous-Time Information Cascades
Information propagation and influence in diffusion networks
We describe here the propagation dynamics introduced in [5]. Let G = (V, E) be a directed network
of n nodes. We equip each directed edge (i, j) ? E with a time-varying probability distribution
pij (t) over R+ ? {+?} (pij is thus a sub-probability measure on R+ ) and define the cascade
behavior as follows. At time t = 0, only a subset A ? V of influencers is infected. Each node i
infected at time ?i may transmit the infection at time ?i + ?ij along its outgoing edge (i, j) ? E with
probability density pij (?ij ), and independently of other transmission events. The process ends for a
given T > 0.
For each node v ? V, we will denote as ?v the (possibly infinite) time at which it is reached by the
infection. The influence of A at time T , denoted as ?A (T ), is defined as the expected number of
nodes reached by the contagion at time T originating from A, i.e.
X
?A (T ) = E[
1{?v ?T } ],
(1)
v?V
where the expectation is taken over cascades originating from A (i.e. ?v = 0 ? 1{v?A} ).
Following the percolation literature, we will differentiate between sub-critical cascades whose size
is o(n) and super-critical cascades whose size is proportional to n, where n denotes the size of
the network. This work focuses on upper bounding the influence ?A (T ) for any given time T and
characterizing the critical times at which phase transitions occur between sub-critical and supercritical behaviors.
2.2
The Laplace Hazard Matrix
We extend here the concept of hazard matrix first introduced in [17] (different from the homonym
notion of [13]), which plays a key role in the influence of the information cascade.
Definition 1. Let G = (V, E) be a directed graph, and pij be integrable edge transmission probR +?
abilities such that 0 pij (t)dt < 1. For s ? 0, let LH(s) be the n ? n matrix, denoted as the
Laplace hazard matrix, whose coefficients are
(
R
?1
R +?
+?
??
p
(s)
p
(t)dt
ln
1
?
p
(t)dt
if (i, j) ? E .
ij
ij
ij
0
0
LHij (s) =
(2)
0
otherwise
2
where p?ij (s) denotes the Laplace transform of pij defined for every s ? 0 by p?ij (s) =
R +?
pij (t)e?st dt. Note that the long term behavior of the cascade is retrieved when s = 0 and
0
coincides with the concept of hazard matrix used in [17].
We recall that for any square matrix M of size n, its spectral radius ?(M ) is the maximum of the
>
absolute values of its eigenvalues. If M is moreover real and positive, we also have ?( M +M
)=
2
>
x Mx
supx?Rn x> x .
2.3
Existence of a critical time of a contagion
In the following, we will derive critical times before which the contagion is sub-critical, and above
which the contagion is super-critical. We now formalize this notion of critical time via limits of
contagions on networks.
Theorem 1. Let (Gn )n?N be a sequence of networks of size n, and (pnij )n?N be transmission probability functions along the edges of Gn . Let also ?n (t) be the maximum influence in Gn at time t from
a single influencer. Then there exists a critical time T c ? R+ ? {+?} such that, for every sequence
of times (Tn )n?N :
? If lim supn?+? Tn < T c , then ?n (Tn ) = o(n),
? If ?n (Tn ) = o(n), then lim inf n?+? Tn ? T c .
Moreover, such a critical time is unique.
In other words, the critical time is a time before which the regime is sub-critical and after which no
contagion can be sub-critical. The next proposition shows that, after the critical time, the contagion
is super-critical.
n)
Proposition 1. If (Tn )n?N is such that lim inf n?+? Tn > T c , then lim inf n?+? ?n (T
> 0 and
n
?n (Tn )
the contagion is super-critical. Conversely, if (Tn )n?N is such that lim inf n?+? n > 0, then
lim supn?+? Tn ? T c .
In order to simplify notations, we will omit in the following the dependence in n of all the variables
whenever stating results holding in the limit n ? +?.
3
Theoretical bounds for the influence of a set of nodes
We now present our upper bounds on the influence at time T and derive a lower bound on the critical
time of a contagion.
3.1
Upper bounds on the maximum influence at time T
The next proposition provides an upper bound on the influence at time T for any set of influencers A
such that |A| = n0 . This result may be valuable for assessing the quality of influence maximization
algorithms in a given network.
>
Proposition 2. Define ?(s) = ?( LH(s)+LH(s)
). Then, for any A such that |A| = n0 < n, denoting
2
by ?A (T ) the expected number of nodes reached by the cascade starting from A at time T :
?A (T ) ? n0 + (n ? n0 ) min ?(s)esT .
s?0
where ?(s) is the smallest solution in [0, 1] of the following equation:
?(s)n0
= 0.
?(s) ? 1 + exp ??(s)?(s) ?
?(s)(n ? n0 )
3
(3)
(4)
Corollary 1. Under the same assumptions:
s
?A (T ) ? n0 +
p
n0 (n ? n0 )
min
{s?0|?(s)<1}
?(s) sT
e
1 ? ?(s)
!
,
(5)
Note that the long-term upper bound in [17] is a corollary of Proposition 2 using s = 0. When
?(0) < 1, Corollary 1 with s = 0 implies that the regime is sub-critical for all T ? 0. When
?(0) ? 1, the long-term behavior may be super-critical and the influence may reach linear values in
n. However, at a cost growing
? exponentially with T , it is always possible to choose a s such that
?(s) < 1 and retrieve a O( n) behavior. While the exact optimal parameter s is in general not
explicit, two choices of s derive relevant results: either simplifying esT by choosing s = 1/T , or
keeping ?(s) sub-critical by choosing s s.t. ?(s) < 1. In particular, the following corollary shows
?1
that the contagion explodes at most as e? (1?)T for any ? [0, 1].
Corollary 2. Let ? [0, 1] and ?(0) ? 1. Under the same assumptions:
r
n0 (n ? n0 ) ??1 (1?)T
?A (T ) ? n0 +
e
.
(6)
Remark. Since this section focuses on bounding ?A (T ) for a given T ? 0, all the aforementioned
results also hold for pTij (t) = pij (t)1{t?T } . This is equivalent to integrating everything on [0, T ]
RT
RT
RT
instead of R+ , i.e. LHij (s) = ? ln(1 ? 0 pij (t)dt)( 0 pij (t)dt)?1 0 pij (t)e?st dt. This choice
of LH is particularly useful when some edges are transmitting the contagion with probability 1, see
for instance the SI epidemic model in Section 4.3).
3.2
Lower bound on the critical time of a contagion
The previous section presents results about how explosive a contagion is. These findings suggest
that the speed at which a contagion explodes is bounded by a certain quantity, and thus that the
process needs a certain amount of time to become super-critical. This intuition is made formal in
the following corollary:
??1 (1?
1
)
ln n
= 1. If the sequence (Tn )n?N
Corollary 3. Assume ?n ? 0, ?n (0) ? 1 and limn?+? n ??1 (1)
n
is such that
2??1 (1)Tn
< 1.
(7)
lim sup n
ln n
n?+?
Then,
?A (Tn ) = o(n).
(8)
In other words, the regime of the contagion is sub-critical before
T c ? lim inf
n?+?
The technical condition limn?+?
ln n
2??1
n (1)
and
ln n
.
2??1
n (1)
(9)
1
??1
??1
n (1? ln n )
n (1?)
=
1
imposes
that,
for
large
n,
lim
con?1
?0
?n (1)
??1
n (1)
1
?1
?1
?n (1 ? ln n ) has the same behavior than ?n (1). This condition
verges sufficiently fast to 1 so that
is not very restrictive, and is met for the different case studies considered in Section 4.
This result may be valuable for decision makers since it provides a safe time region in which the
contagion has not reached a macroscopic scale. It thus provides insights into how long do decision
makers have to prepare control measures. After T c , the process can explode and immediate action
is required.
4
Application to particular contagion models
In this section, we provide several examples of cascade models that show that our theoretical bounds
are applicable in a wide range of scenarios and provide the first results of this type in many areas,
including two widely used epidemic models.
4
4.1
Fixed transmission pattern
When the transmission probabilities are of the form pij (t) = ?ij p(t) s.t.
R +?
0
p(t) = 1 and ?ij < 1,
LHij (s) = ? ln(1 ? ?ij )?
p(s),
(10)
?(s) = ?? p?(s),
(11)
and
ln(1??ij )+ln(1??ji )
?(?
)
2
where ?? = ?(0) =
is the long-term hazard matrix defined in [17]. In
these networks, the temporal and structural behaviors are clearly separated. While ?? summarizes
the structure of the network and how connected the nodes are to one another, p?(s) captures how fast
the transmission probabilities are fading through time.
When ?? ? 1, the long-term behavior is super-critical and the bound on the critical times is given
by inverting p?(s)
ln n
T c ? lim inf ?1
,
(12)
n?+? 2?
p (1/?? )
where p??1 (1/?? ) exists and is unique since p?(s) is decreasing from 1 to 0. In general, it is not
possible to give a more explicit version of the critical time of Corollary 3, or of the anytime influence
bound of Proposition 2. However, we investigate in the rest of this section specific p(t) which lead
to explicit results.
4.2
Exponential transmission probabilities
A notable example of fixed transmission pattern is the case of exponential probabilities pij (t) =
?ij ?e??t for ? > 0 and ?ij ? [0, 1[. Influence maximization algorithms under this specific choice
of transmission functions have been for instance developed in [11]. In such a case, we can calculate
the spectral radii explicitly:
?
?? ,
(13)
?(s) =
s+?
ln(1?? )+ln(1?? )
ij
ji
where ?? = ?(?
) is again the long-term hazard matrix. When ?? > 1, this
2
leads to a critical time lower bounded by
T c ? lim inf
n?+?
ln n
.
2?(?? ? 1)
(14)
The influence bound of Corollary 1 can also be reformulated in the following way:
Corollary 4. Assume ?? ? 1, or else ?T (1 ? ?? ) < 21 . Then the minimum in Eq. 5 is met for
1
s = 2T
+ ?(?? ? 1) and Corollary 1 rewrites:
p
p
(15)
?A (T ) ? n0 + n0 (n ? n0 ) 2eT ??? e?T (?? ?1) .
If ?? < 1 and ?T (1 ? ?? ) ? 12 , the minimum in Eq. 5 is met for s = 0 and Corollary 1 rewrites:
r
p
??
?A (T ) ? n0 + n0 (n ? n0 )
.
(16)
1 ? ??
Note that, in particular, the condition of
? Corollary 4 is always met in the super-critical case where
?? > 1. Moreover, we retrieve the O( n) behavior when T < ?(??1?1) . Concerning the behavior
in T , the bound matches exactly the infinite-time bound when T is very large in the sub-critical case.
However,
? for sufficiently small T , we obtain a greatly improved result with a very instructive growth
in O( T ).
4.3
SI and SIR epidemic models
Both epidemic models SI and SIR are particular cases of exponential transmission probabilities.
SIR model ([18]) is a widely used epidemic model that uses three states to describe the spread of an
infection. Each node of the network can be either : susceptible (S), infected (I), or removed (R). At
5
t = 0, a subset A of n0 nodes is infected. Then, each node i infected at time ?i is removed at an
exponentially-distributed time ?i of parameter ?. Transmission along its outgoing edge (i, j) ? E
occurs at time ?i + ?ij with conditional probability density ? exp(???ij ), given that node i has not
been removed at that time. When the removing events are not observed, SIR is equivalent to CT IC,
except that transmission along outgoing edges of one node are positively correlated. However, our
results still hold in case of such a correlation, as shown in the following result.
Proposition 3. Assume the propagation follow a SIR model of transmission parameter ? and removal parameter ?. Define pij (t) = ? exp(?(? + ?)t) for (i, j) ? E. Let A = 1{(i,j)?E} ij
be the adjacency matrix of the underlying undirected network. Then, results of Proposition 2 and
subsequent corollaries still hold with ?(s) given by:
LH(s) + LH(s)>
?
?+?
?(s) = ?
= ln 1 +
?(A)
(17)
2
? s+?+?
From this proposition, the same analysis than in the independent transmission events case can be
derived, and the critical time for the SIR model is
T c ? lim inf
n?+?
ln n
.
2(? + ?)(ln(1 + ?? )?(A) ? 1)
(18)
Proposition 4. Consider the SIR model with transmission rate ?, recovery rate ? and adjacency
matrix An . Assume lim inf n?+? ln(1 + ?? )?(An ) > 1, and the sequence (Tn )n?N is such that
lim sup
n?+?
2(? + ?)(ln(1 + ?? )?(An ) ? 1)Tn
< 1.
ln n
(19)
Then,
?A (Tn ) = o(n).
(20)
This is a direct corollary of Corollary 3 with ??1 (1) = (? + ?)(ln(1 + ?? )?(An ) ? 1).
The SI model is a simpler model in which individuals of the network remain infected and contagious
through time (i.e. ? = 0). Thus, the network is totally infected at the end of the contagion and
limn?+? ?A (T ) = n. For this reason, the previous critical time for the more general SIR model is
of no use here, and a more precise analysis is required. Following the remark of Section 3.1, we can
integrate pij on [0, T ] instead of R+ , which leads to the following result:
Proposition 5. Consider the SI model with transmission rate ? and adjacency matrix An . Assume
lim inf n?+? ?(An ) > 0 and the sequence (Tn )n?N is such that
lim sup q
n?+?
?Tn
ln n
2?(An ) (1
?e
?
q
ln n
2?(An )
< 1.
(21)
)
Then,
?A (Tn ) = o(n).
In other words, the critical time for the SI model is lower bounded by
s
q
ln n
1
ln n
? 2?(A
c
n ) ).
T ? lim inf
(1 ? e
n?+? ?
2?(An )
(22)
(23)
If ?(An ) = o(ln n) (e.g. for sparse
q networks with a maximum degree in O(1)), the critical time
1
ln n
resumes to Tc ? lim inf n?+? ? 2?(A
. However, when the graph is denser and ?(An )/ ln n ?
n)
+?, then Tc ? lim inf n?+?
4.4
ln n
2??(An ) .
Discrete-time Information Cascade
A final example is the discrete-time contagion in which a node infected at time t makes a unique
attempt to infect its neighbors at a time t + T0 . This defines the Information Cascade model, the
6
1000
totally connected
erdos renyi
preferential attachment
small world
contact network
upper bound
60
800
influence (?A(T))
influence (?A(T))
80
40
20
600
400
200
0
2
4
6
spectral radius (??)
8
0
0
10
2
4
6
spectral radius (??)
1000
800
800
600
400
200
0
0
10
8
10
(b) T = 1
1000
influence (?A(T))
influence (?A(T))
(a) T = 0.1
8
600
400
200
2
4
6
spectral radius (??)
8
0
0
10
(c) T = 5
2
4
6
spectral radius (??)
(d) T = 100
Figure 1: Empirical maximum influence w.r.t. the spectral radius ?? defined in Section 4.2 for various network types. Simulation parameters: n = 1000, n0 = 1 and ? = 1.
discrete-time diffusion model studied by the first works on influence maximization [2, 19, 3, 4]. In
this setting, pij (t) = ?ij ?T0 (t) where ?T0 is the Dirac distribution centered at T0 . The spectral radii
are given by
?(s) = ?? e?sT0 ,
(24)
and the influence bound of Corollary 1 simplifies to:
Corollary 5. Let ?? ? 1, or else T ?
T0
2(1??? ) .
If T < T0 , then ?A (T ) = n0 . Otherwise,
r
p
2eT TT0
?A (T ) ? n0 + n0 (n ? n0 )
?? .
T0
(25)
Moreover, the critical time is lower bounded by
T c ? lim inf
n?+?
ln n
T0 .
2 ln ??
(26)
A notable difference from the exponential transmission probabilities is that T c is here inversely
proportional to ln ?? , instead of ?? in Eq. 4.2, which implies that, for the same long-term influence, a discrete-time contagion will explode much slower than one with a constant infection rate.
This is probably due to the existence of very small infection times for contagions with exponential
transmission probabilities.
5
Experimental results
This section provides an experimental validation of our bounds, by comparing them to the empirical
influence simulated on several network types. In all our experiments, we simulate a contagion
with exponential transmission probabilities (see Section 4.2) on networks of size n = 1000 and
generated random networks of 5 different types (for more information on the respective random
generators, see e.g [20]): Erd?os-R?enyi networks, preferential attachment networks, small-world
networks, geometric random networks ([21]) and totally connected networks with fixed weight b ?
[0, 1] except for the ingoing and outgoing edges of a single node having, respectively, weight 0
and a > b. The reason for simulating on such totally connected networks is that the influence
over these networks tend to match our upper bounds more closely, and plays the role of a best case
7
influence (?A(T))
50
40
influence (?A(T))
totally connected
erdos renyi
preferential attachment
small world
contact network
upper bound
60
30
20
1000
1000
800
800
influence (?A(T))
70
600
400
200
600
400
200
10
0
0
200
400
600
number of nodes (n)
(a) T = 0.2T c?
800
1000
0
0
200
400
600
number of nodes (n)
(b) T = 2T c?
800
1000
0
0
200
400
600
number of nodes (n)
800
1000
(c) T = 5T c?
Figure 2: Empirical maximum influence w.r.t. the network size for various network types. Simulan
tion parameters: n0 = 1, ? = 1 and ?? = 4. In such a setting, T c ? = 2(?ln
= 1.15. Note the
? ?1)?
sub-linear (a) versus linear behavior (b and c).
scenario. More precisely, the transmission probabilities are of the form pij (t) = ?e?t for each edge
(i, j) ? E, where ? ? [0, 1[ (and ? = 1 in the formulas of Section 4.2).
We first investigate the tightness of the upper bound on the maximum influence given in Proposition 2. Figure 1 presents the empirical influence w.r.t. ?? = ? ln(1 ? ?)?(A) (where A is the
adjacency matrix of the network) for a large set of network types, as well as the upper bound in
Proposition 2. Each point in the figure corresponds to the maximum influence on one network. The
influence was averaged over 100 cascade simulations, and the best influencer (i.e. whose influence
was maximal) was found by performing an exhaustive search. Our bounds are tight for all values
of T ? {0.1, 1, 5, 100} for totally connected networks in the sub-critical regime (?? < 1). For the
super-critical regime (?? > 1), the behavior in T is very instructive. For T ? {0.1, 5, 100}, we are
tight for most network types when ?? is high. For T = 1 (the average transmission time for the
(?ij )(i,j)?E ), the maximum influence varies a lot across different graphs. This follows the intuition
that this is one of the times where, for a given final number of infected node, the local structure of
the networks will play the largest role through precise temporal evolution of the infection. Because
?? explains quite well the final size of the infection, this discrepancy appears on our graphs at ??
fixed. While our bound does not seem tight for this particular time, the order of magnitude of the
explosion time is retrieved and our bounds are close to optimal values as soon as T = 5.
In order to further validate that our bounds give meaningful insights on the critical time of explosion
for super-critical graphs, Figure 2 presents the empirical influence with respect to the size of the
network n for different network types and values of T , with ?? fixed to ?? = 4. In this setting, the
n
critical time of Corollary 3 is given by T c ? = 2(?ln
= 1.15. We see that our bounds are tight
? ?1)?
for totally connected networks for all values of T ? {0.2, 2, 5}. Moreover, the accuracy of critical
time estimation is proved by the drastic change of behavior around T = T c ? , with phase transitions
having occurred for most network types as soon as T = 5T c ? .
6
Conclusion
In this paper, we characterize the phase transition in continuous-time information cascades between
their sub-critical and super-critical behavior. We provide for the first time general influence bounds
that apply for any time horizon, graph and set of influencers. We show that the key quantities
governing this phenomenon are the spectral radii of given Laplace Hazard matrices. We prove the
pertinence of our bounds by deriving the first results of this type in several application fields. Finally,
we provide experimental evidence that our bounds are tight for a large family of networks.
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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5,194 | 5,702 | Estimating Mixture Models via Mixtures of
Polynomials
Sida I. Wang Arun Tejasvi Chaganty Percy Liang
Computer Science Department, Stanford University, Stanford, CA, 94305
{sidaw,chaganty,pliang}@cs.stanford.edu
Abstract
Mixture modeling is a general technique for making any simple model more expressive through weighted combination. This generality and simplicity in part
explains the success of the Expectation Maximization (EM) algorithm, in which
updates are easy to derive for a wide class of mixture models. However, the likelihood of a mixture model is non-convex, so EM has no known global convergence
guarantees. Recently, method of moments approaches offer global guarantees for
some mixture models, but they do not extend easily to the range of mixture models that exist. In this work, we present Polymom, an unifying framework based on
method of moments in which estimation procedures are easily derivable, just as
in EM. Polymom is applicable when the moments of a single mixture component
are polynomials of the parameters. Our key observation is that the moments of
the mixture model are a mixture of these polynomials, which allows us to cast
estimation as a Generalized Moment Problem. We solve its relaxations using
semidefinite optimization, and then extract parameters using ideas from computer
algebra. This framework allows us to draw insights and apply tools from convex
optimization, computer algebra and the theory of moments to study problems in
statistical estimation. Simulations show good empirical performance on several
models.
1
Introduction
Mixture models play a central role in machine learning and statistics, with diverse applications
including bioinformatics, speech, natural language, and computer vision. The idea of mixture
modeling is to explain data through a weighted combination of simple parametrized distributions
[1, 2]. In practice, maximum likelihood estimation via Expectation Maximization (EM) has been
the workhorse for these models, as the parameter updates are often easily derivable. However, EM
is well-known to suffer from local optima. The method of moments, dating back to Pearson [3] in
1894, is enjoying a recent revival [4, 5, 6, 7, 8, 9, 10, 11, 12, 13] due to its strong global theoretical guarantees. However, current methods depend strongly on the specific distributions and are not
easily extensible to new ones.
In this paper, we present a method of moments approach, which we call Polymom, for estimating
a wider class of mixture models in which the moment equations are polynomial equations (Section
2). Solving general polynomial equations is NP-hard, but our key insight is that for mixture models,
the moments equations are mixtures of polynomials equations and we can hope to solve them if the
moment equations for each mixture component are simple polynomials equations that we can solve.
Polymom proceeds as follows: First, we recover mixtures of monomials of the parameters from the
data moments by solving an instance of the Generalized Moment Problem (GMP) [14, 15] (Section
3). We show that for many mixture models, the GMP can be solved with basic linear algebra and
in the general case, can be approximated by an SDP in which the moment equations are linear
constraints. Second, we extend multiplication matrix ideas from the computer algebra literature [16,
1
mixture model
xt
data point (RD )
zt
latent mixture component ([K])
?k
parameters of component k (RP )
?k
mixing proportion of p(z = k)
[? k ]K
all model parameters
k=1
moments of data
observation function
n (x)
fn (?)
observation function
moments of parameters
Ly
the Riesz linear functional
y?
y? = Ly (? ? ), ?th moment
?
probability measure for y
y
(y? )? the moment sequence
Mr (y) moment matrix of degree r
sizes
D
data dimensions
K
mixture components
P
parameters of mixture components
T
data points
N
constraints
[N ]
{1, . . . , N }
r
degree of the moment matrix
s(r)
size of the degree r moment matrix
polynomials
R[?]
polynomial ring in variables ?
N
set of non-negative integers
?, ,
vector of exponents (in NP or ND )
QP
?
?
?
monomial p=1 ?p p
?
an?
coefficient of ? in fn (?)
Table 1: Notation: We use lowercase letters (e.g., d) for indexing, and the corresponding uppercase
letter to denote the upper limit (e.g., D, in ?sizes?). We use lowercase letters (e.g., ?k,p ) for scalars,
lowercase bold letters (e.g., ?) for vectors, and bold capital letters (e.g., M) for matrices.
1. Write down a mixture model
z
4. Recover parameter moments (y)
z ? Multinomial(?1 , ?2 )
x | z ? N (?z ,
x
? k = (?k ,
z)
k)
2R
1
2 R2
Mr (y) =
?
?
2
2
2
1
y0,0
6 y1,0
6
4 y2,0
y0,1
?
?2
y1,0
y2,0
y3,0
y1,1
y2,0
y3,0
y4,0
y2,1
minimize tr(Mr (y))
2. Derive single mixture
moment equations
(x)
x
x2
x3
..
.
E
!
f (?)
?
?2 + 2
? 3 + 3?
..
.
3. Add data
2
3
y0,1
y1,1 7
7
y2,1 5
y0,2
Mr (y) = VPV>
# sim. diag.
P = diag([?1 , ?2 ])
Mr (y) ? 0, y0,0 = 1
PT
y1,0 = T1 t xt
PT
y2,0 + y0,1 = T1 t x2t
PT
y3,0 + 3y1,1 = T1 t x3t
...
s.t.
xT ? p(x; ? ? )
2
1
y
x1 ? p(x; ? ? )
...
2
5. Solve for parameters
?
2
V=
?2
?
2
4
?2
2
6
6
6
6
6
6
6
6
4
v(? 1 )
v(? 2 )
1
2
3
4
6
9
12
1
2
5
4
10
25
20
3
7
7
7
7
7
7
7
7
5
user specified framework specified
Figure 1: An overview of applying the Polymom framework.
17, 18, 19] to extract the parameters by solving a certain generalized eigenvalue problem (Section
4).
Polymom improves on previous method of moments approaches in both generality and flexibility.
First, while tensor factorization has been the main driver for many of the method of moments approaches for many types of mixture models, [6, 20, 9, 8, 21, 12], each model required specific adaptations which are non-trivial even for experts. In contrast, Polymom provides a unified principle for
tackling new models that is as turnkey as computing gradients or EM updates. To use Polymom
(Figure 1), one only needs to provide a list of observation functions ( n ) and derive their expected
values expressed symbolically as polynomials in the parameters of the specified model (fn ). Polymom then estimates expectations of n and outputs parameter estimates of the specified model.
Since Polymom works in an optimization framework, we can easily incorporate constraints such
as non-negativity and parameter tying which is difficult to do in the tensor factorization paradigm.
In simulations, we compared Polymom with EM and tensor factorization and found that Polymom
performs similarly or better (Section 5). This paper assumes identifiability and infinite data. With
the exception of a few specific models in Section 5, we defer issues of general identifiability and
sample complexity to future work.
2
2
2.1
Problem formulation
The method of moments estimator
In a mixture model, each data point x 2 RD is associated with a latent component z 2 [K]:
z ? Multinomial(?),
x | z ? p(x; ? ?z ),
? ?k
(1)
where ? = (?1 , . . . , ?K ) are the mixing coefficients,
2 R are the true model parameters for
the k th mixture component, and x 2 RD is the random variable representing data. We restrict our
attention to mixtures where each component distribution comes from the same parameterized family.
For example, for a mixture of Gaussians, ? ?k = (?k? 2 RD , ??k 2 RD?D ) consists of the mean and
covariance of component k.
P
We define N observation functions n : RD ! R for n 2 [N ] and define fn (?) to be the expectation
of n over a single component with parameters ?, which we assume is a simple polynomial:
X
fn (?) := Ex?p(x;?) [ n (x)] =
an? ? ? ,
(2)
?
where ?
?
QP
?p
p=1 ?p .
The expectation of each observation function E[ n (x)] can then be exPK
pressed as a mixture of polynomials of the true parameters E[ n (x)] = k=1 ?k E[ n (x)|z = k] =
PK
?
k=1 ?k fn (? k ).
=
The method of moments for mixture models seeks parameters [? k ]K
k=1 that satisfy the moment conditions
E[
n (x)]
=
K
X
?k fn (? k ).
(3)
k=1
PT
p
where E[ n (x)] can be estimated from the data: T1 t=1 n (xt ) ! E[ n (x)]. The goal of this work
is to find parameters satisfying moment conditions that can be written in the mixture of polynomial
form (3). We assume that the N observations functions 1 , . . . , N uniquely identify the model
parameters (up to permutation of the components).
Example 2.1 (1-dimensional Gaussian mixture). Consider a K-mixture of 1D Gaussians with parameters ? k = [?k , k2 ] corresponding to the mean and variance, respectively, of the k-th component
(Figure 1: steps 1 and 2). We choose the observation functions, (x) = [x1 , . . . , x6 ], which have
corresponding moment polynomials, f (?) = [?, ? 2 + 2 , ? 3 + 3? 2 , . . . ]. For example, instantiating
PK
(3), E[x2 ] = k=1 ?k (?k2 + k2 ). Given (x) and f (? ? ), and data, the Polymom framework can
recover the parameters. Note that the 6 moments we use have been shown by [3] to be sufficient for
a mixture of two Gaussians.
Example 2.2 (Mixture of linear regressions). Consider a mixture of linear regressions [22, 9],
where each data point x = [x, y] is drawn from component k by sampling x from an unknown
distribution independent of k and setting y = wk x + ?, where ? ? N (0, k2 ). The parameters
? k = (wk , k2 ) are the slope and noise variance for each component k. Let us take our observation
functions to be (x) = [x, xy, xy 2 , x2 , . . . , x3 y 2 ], for which the moment polynomials are f (?) =
[E[x], E[x2 ]w, E[x3 ]w2 + E[x] 2 , E[x2 ], . . .].
In Example 2.1, the coefficients an? in the polynomial fn (?) are just constants determined by integration. For the conditional model in Example 2.2, the coefficients depends on the data. However,
we cannot handle arbitrary data dependence, see Section D for sufficient conditions and counterexamples.
2.2
Solving the moment conditions
Our goal is to recover model parameters ? ?1 , . . . , ? ?K 2 RP for each of the K components of the
mixture model that generated the data as well as their respective mixing proportions ?1 , . . . , ?K 2
R. To start, let?s ignore sampling noise and identifiability issues and suppose that we are given exact
moment conditions as defined in (3). Each condition fn 2 R[?] is a polynomial of the parameters
?, for n = 1, . . . , N .
3
Equation 3 is a polynomial system of N equations in the K + K ? P variables [?1 , . . . , ?K ] and
[? 1 , . . . , ? K ] 2 RP ?K . It is natural to ask if standard polynomial solving methods can solve (3) in
the case where each fn (?) is simple. Unfortunately, the complexity of general polynomial equation
solving is lower bounded by the number of solutions, and each of the K! permutations of the mixture
components corresponds to a distinct solution of (3) under this polynomial system representation.
While several methods can take advantage of symmetries in polynomial systems [23, 24], they still
cannot be adapted to tractably solve (3) to the best of our knowledge.
The key idea of Polymom is to exploit the mixture representation of the moment equations (3).
Specifically, let ?? be a particular ?mixture? over the component parameters ? ?1 , . . . , ? ?k (i.e. ?? is a
probability measure). Then we can express the moment conditions (3) in terms of ?? :
E[
n (x)]
=
Z
fn (?) ?? (d?), where ?? (?) =
K
X
?k (?
? ?k ).
(4)
k=1
As a result, solving the original moment conditions (3) is equivalent to solving the following feasibility problem over ?, but where we deliberately ?forget? the permutation of the components by
using ? to represent the problem:
find
s.t.
P
P
R? 2 M+ (R ), the set of probability measures over R
fn (?) ?(d?) = E[ n (x)], n = 1, . . . , N
? is K-atomic (i.e. sum of K deltas).
(5)
If the true model parameters [? ?k ]K
k=1 can be identified by the N observed moments up to permutaP
K
?
tion, then the measure ? (?) = k=1 ?k (? ? ?k ) solving Problem 5 is also unique.
Polymom solves Problem 5 in two steps:
1. Moment completion (Section 3): We show that Problem 5 over the measure ? can be
relaxed to an SDP over a certain (parameter) moment matrix Mr (y) whose optimal solution
PK
is Mr (y? ) = k=1 ?k vr (? ?k )vr (? ?k )> , where vr (? ?k ) is the vector of all monomials of
degree at most r.
2. Solution extraction (Section 4): We then take Mr (y) and construct a series of generalized
eigendecomposition problems, whose eigenvalues yield [? ?k ]K
k=1 .
Remark. From this point on, distributions and moments refer to ?? which is over parameters, not
over the data. All the structure about the data is captured in the moment conditions (3).
3
Moment completion
The first step is to reformulate Problem 5 as an instance of the Generalized Moment Problem (GMP)
introduced by [15]. A reference on the GMP, algorithms for solving GMPs, and its various extensions is [14]. We start by observing that Problem 5 really only depends on the integrals of monomials
under the measure ?: for example, if fn (?)
2?13 ?12 ?2 , then we
R =
R only need to know the integrals
3
over the constituent monomials (y3,0 := ?1 ?(d?) and y2,1 := ?12 ?2 ?(d?)) in order to evaluate
the integral over fn . This suggests that we can optimize over the (parameter) moment sequence
y = (y? )?2NP , rather than Rthe measure ? itself. We say that the moment sequence y has a representing measure ? if y? = ? ? ?(d?) for all ?, but we do not assume that such a ? exists. The
Riesz linear functional Ly : R[?] ! R is defined to be the linear map such that Ly (? ? ) := y? and
Ly (1) = 1. For example, Ly (2?13 ?12 ?2 + 3) = 2y3,0 y2,1 + 3. If y has a representing measure
?, then Ly simply maps polynomials f to integrals of f against ?.
The key idea of the GMP approach is to convexify the problem by treating y as free variables
and then introduce constraints to guarantee that y has a representing measure. First, let vr (?) :=
[? ? : |?| ? r] 2 R[?]s(r) be the vector of all s(r) monomials of degree no greater than r. Then,
define the truncated moment matrix as Mr (y) := Ly (vr (?)vr (?)T ), where the linear functional
Ly is applied elementwise (see Example 3.1 below). If y has a representing measure ?, then Mr (y)
is simply a (positive) integral over rank 1 matrices vr (?)vr (?)T with respect to ?, so necessarily
4
Mr (y) ? 0 holds. Furthermore, by Theorem 1 [25], for y to have a K-atomic representing measure,
it is sufficient that rank(Mr (y)) = rank(Mr 1 (y)) = K. So Problem 5 is equivalent to
find
s.t.
y 2 RN (or equivalently, find M(y))
P
? an? y? = E[ n (x)], n = 1, . . . , N
Mr (y) ? 0, y0 = 1
rank(Mr (y)) = K and rank(Mr 1 (y)) = K.
(6)
Unfortunately, the rank constraints in Problem 6 are not tractable. We use the following relaxation
to obtain our final (convex) optimization problem
minimize tr(CMr (y))
y
P
s.t.
? an? y? = E[ n (x)], n = 1, . . . , N
Mr (y) ? 0, y0 = 1
(7)
where C
0 is a chosen scaling matrix. A common choice is C = Is(r) corresponding to minimizing the nuclear norm of the moment matrix, the usual convex relaxation for rank. Section A
discusses some other choices of C.
Example 3.1 (moment matrix for a 1-dimensional Gaussian mixture). Recall that the parameters
? = [?, 2 ] are the mean and variance of a one dimensional Gaussian. Let us choose the monomials
v2 (?) = [1, ?, ? 2 , 2 ]. Step 4 for Figure 1 shows the moment matrix when using r = 2. Each row
and column of the moment matrix is labeled with a monomial and entry (i, j) is subscripted by the
product of the monomials in row i and column j. For 2 (x) := x2 , we have f2 (?) = ? 2 + c, which
leads to the linear constraint y2,0 + y0,1 E[x2 ] = 0. For 3 (x) = x3 , f3 (?) = ? 3 + 3?c, leading
to the constraint y3,0 + 3y1,1 E[x3 ] = 0.
Related work. Readers familiar with the sum of squares and polynomial optimization literature [26, 27, 28, 29] will note that Problem 7 is similar to the SDP relaxation of a polynomial
optimization problem. However, in typical polynomial optimization, we are only interested in solutions ? ? that actually satisfy the given constraints, whereas here we are interested in K solutions
[? ?k ]K
k=1 , whose mixture satisfies constraints corresponding to the moment conditions (3). Within
machine learning, generalized PCA has been formulated as a moment problem [30] and the Hankel
matrix (basically the moment matrix) has been used to learn weighted automata [13]. While similar
tools are used, the conceptual approach and the problems considered are different. For example,
the moment matrix of this paper consists of unknown moments of the model parameters, whereas
exisiting works considered moments of the data that are always directly observable.
Constraints. Constraints such as non-negativity (for parameters which represent probabilities or
variances) and parameter tying [31] are quite common in graphical models and are not easily addressed with existing method of moments approaches. The GMP framework allows us to incorporate
some constraints using localizing matrices [32]. Thus, we can handle constraints during the estimation procedure rather than projecting back onto the constraint set as a post-processing step. This is
necessary for models that only become identifiable by the observed moments after constraints are
taken into account. We describe this method and its learning implications in Section C.1.
Guarantees and statistical efficiency. In some circumstances, e.g. in three-view mixture models
or the mixture of linear regressions, the constraints fully determine the moment matrix ? we consider
these cases in Section 5 and Appendix B. While there are no general guarantee on Problem 7, the
flat extension theorem tells us when the moment matrix corresponds to a unique solution (more
discussions in Appendix A):
Theorem 1 (Flat extension theorem [25]). Let y be the solution to Problem 7 for a particular r. If
Mr (y) ? 0 and rank(Mr 1 (y)) = rank(Mr (y)) then y is the optimal solution to Problem 6 for
K = rank(Mr (y)) and there exists a unique K-atomic supporting measure ? of Mr (y).
Recovering Mr (y) is linearly dependent on small perturbations of the input [33], suggesting that
the method has polynomial sample complexity for most models where the moments concentrate at
a polynomially rate. Finally, in Appendix C, we discuss a few other important considerations like
noise robustness, making Problem 7 more statistical efficient, along with some technical results on
the moment completion problem and some open problems.
5
4
Solution extraction
Having completed the (parameter) moment matrix Mr (y) (Section 3), we now turn to the problem
of extracting the model parameters [? ?k ]K
k=1 . The solution extraction method we present is based on
ideas from solving multivariate polynomial systems where the solutions are eigenvalues of certain
multiplication matrices [16, 17, 34, 35].1 The main advantage of the solution extraction view is
that higher-order moments and structure in parameters are handled in the framework without modelspecific effort.
PK
?
? T
Recall that the true moment matrix is Mr (y? ) =
k=1 ?k v(? k )v(? k ) , where v(?) :=
s(r)
?1
?s(r)
[? , . . . , ?
] 2 R[?]
contains all the monomials up to degree r. We use ? = [?1 , . . . , ?P ]
for variables and [? ?k ]K
for
the
true solutions to these variables (note the boldface). For example,
k=1
?
?k,p
:= (? ? k )p denotes the pth value of the k th component, which corresponds to a solution for the
variable ?p . Typically, s(r)
K, P and the elements of v(?) are arranged in a degree ordering so
that ||?i ||1 ? ||?j ||1 for i ? j. We can also write Mr (y? ) as Mr (y? ) = VPV> , where the canonical basis V := [v(? ?1 ), . . . , v(? ?K )] 2 Rs(r)?K and P := diag(?1 , . . . , ?K ). At the high level, we
want to factorize Mr (y ? ) to get V, however we cannot simply eigen-decompose Mr (y? ) since V is
not orthogonal. To overcome this challenge, we will exploit the internal structure of V to construct
several other matrices that share the same factors and perform simultaneous diagonalization.
Specifically, let V[ 1 ; . . . ; K ] 2 RK?K be a sub-matrix of V with only the rows corresponding to
monomials with exponents 1 , . . . , K 2 NP . Typically, 1 , . . . , K are just the first K monomials
in v. Now consider the exponent p 2 NP which is 1 in position p and 0 elsewhere, corresponding
to the monomial ? p = ?p . The key property of the canonical basis is that multiplying each column
?
k by a monomial ?k,p
just performs a ?shift? to another set of rows:
?
?
?
?
V[ 1 ; . . . ; K ] Dp = V 1 + p ; . . . ; K + p , where Dp := diag(?1,p
, . . . , ?K,p
). (8)
Note that Dp contains the pth parameter for all K mixture components.
Example 4.1 (Shifting the canonical basis). Let ? = [?1 , ?2 ] and the true solutions be ? ?1 = [2, 3]
?
?
and ? ?2 = [ 2, 5]. To extract the solution for ?1 (which are (?1,1
, ?2,1
)), let 1 = (1, 0), 2 = (1, 1),
and 1 = (1, 0).
1
?1
?2
V=
?12
?1 ?2
?22
?12 ?2
2
6
6
6
6
6
6
4
v(? 1 )
v(? 2 )
1
2
3
4
6
9
12
1
2
5
4
10
25
20
3
7
7
7
7
7
7
5
?1
?1 ?2
|
?
V[
v1
2
6
{z
1;
v2
2
10
2]
?
} |
2
0
{z
0
2
=
}
diag(?1,1 ,?2,1 )
?12
?12 ?2
|
V[
?
1+
v1
v2
4
12
{z
4
20
1;
2+
1]
(9)
}
While the above reveals the structure of V, we don?t know V. However, we recover its column space
U 2 Rs(r)?K from the moment matrix Mr (y ? ), for example with an SVD. Thus, we can relate
U and V by a linear transformation: V = UQ, where Q 2 RK?K is some unknown invertible
matrix.
Equation 8 can now be rewritten as:
U[
1; . . . ;
K ]Q Dp
=U
?
1
+
p; . . . ;
K
+
p
?
Q,
p = 1, . . . , P,
(10)
which is a generalized eigenvalue problem where Dp are the eigenvalues and Q are the eigenvectors.
?
?
Crucially, the eigenvalues, Dp = diag(?1,p
, . . . , ?K,p
) give us solutions to our parameters. Note
that for any choice of 1 , . . . , K and p 2 [P ], we have generalized eigenvalue problems that
share eigenvectors Q, though their eigenvectors Dp may differ. Corresponding eigenvalues (and
hence solutions) can be obtained by solving a simultaneous generalized eigenvalue problem, e.g., by
using random projections like Algorithm B of [4] or more robust [37] simutaneous diagonalization
algorithms [38, 39, 40].
1
[36] is a short overview and [35] is a comprehensive treatment including numerical issues.
6
Table 2: Applications of the Polymom framework. See Appendix B.2 for more details.
Mixture of linear regressions
Model
Observation functions
x = [x, ] is observed where x 2 RD is drawn
from an unspecified distribution and
? N (w ? x, 2 I), and is known. The
parameters are ? ?k = (wk ) 2 RD .
?,b (x)
= x?
b
for 0 ? |?| ? 3, b 2 [2].
Moment polynomials
P
?+ p
f?,1 (?) = P
]wp
p=1 E[x
PP
? 2
f?,2 (?) = E[x ] + p,q=1 E[x? xp xq ]wp wq ,
where the p 2 NP is 1 in position p and 0 elsewhere.
Mixture of Gaussians
Model
Observation functions
x 2 RD is observed where x is drawn from a
Gaussian with diagonal covariance:
x ? N (?, diag(c)). The parameters are
? ?k = (?k , ck ) 2 RD+D .
? (x)
= x? for 0 ? |?| ? 4.
MomentQ
polynomials
f? (?) =
D
d=1
h?d (?d , cd ).
2
Multiview mixtures
Model
Observation functions
With 3 views, x = [x(1) , x(2) , x(3) ] is observed
where x(1) , x(2) , x(3) 2 RD and x(`) is drawn
from an unspecified distribution with mean ?(`)
for ` 2 [3]. The parameters are
(1)
(2)
(3)
? ?k = (?k , ?k , ?k ) 2 RD+D+D .
ijk (x)
(1) (2) (3)
= xi xj xk where 1 ? i, j, k ? D.
Moment polynomials
(1) (2) (3)
fijk (?) = ?i ?j ?k .
We describe one approach to solve (10), which is similar to Algorithm B of [4]. The idea is to take P
random weighted combinations of the equations (10) and solve the resulting (generalized) eigendecomposition problems. Let R 2 RP ?P be a random?matrix whose entries are drawn from N?(0, 1).
?
?
1
PP
Then for each q = 1, . . . Q, solve U[ 1 ; . . . ; K ]
Q=
1 + p; . . . ; K + p
p=1 Rq,p U
QDq . The resulting eigenvalues can be collected in ? 2 RP ?K , where ?q,k = Dq,k,k . Note that
PP
?
?
?
1
by definition ?q,k =
?.
p=1 Rq,p ? k,p , so we can simply invert to obtain [? 1 , . . . , ? K ] = R
Although this simple approach does not have great numerical properties, these eigenvalue problems
are solvable if the eigenvalues [ q,1 , . . . , q,K ] are distinct for all q, which happens with probability
1 as long as the parameters ? ?k are different from each other.
In Appendix B.1, we show how a prior tensor decomposition algorithm from [4] can be seen as
solving Equation 10 for a particular instantiation of 1 , . . . K .
5
Applications
Let us now look at some applications of Polymom. Table 2 presents several models with corresponding observation functions and moment polynomials. It is fairly straightforward to write down
observation functions for a given model. The moment polynomials can then be derived by computing expectations under the model? this step can be compared to deriving gradients for EM.
We implemented Polymom for several mixture models in Python (code: https://github.
com/sidaw/polymom). We used CVXOPT to handle the SDP and the random projections algorithm from to extract solutions. In Table 3, we show the relative error maxk ||? k ? ?k ||2 /||? ?k ||2
averaged over 10 random models of each class.
In the rest of this section, we will discuss guarantees on parameter recovery for each of these models.
P
h? (?, c) = b?/2c
a?,? 2i ? ? 2i ci and a?,i be the absolute value of the coefficient of the degree i term
i=0
of the ?th (univariate) Hermite polynomial. For example, the first few are h1 (?, c) = ?, h2 (?, c) = ? 2 + c,
h3 (?, c) = ? 3 + 3?c, h4 (?, c) = ? 4 + 6? 2 c + 3c2 .
2
7
Gaussians
spherical
diagonal
constrained
Others
3-view
lin. reg.
Methd.
K, D
2, 2
2, 2
2, 2
K, D
3, 3
2, 2
EM
0.37
0.44
0.49
0.38
-
TF
T = 103
2.05
2.15
7.52
T = 104
0.51
-
Poly
EM
0.58
0.48
0.38
0.24
0.48
0.47
0.57
3.51
0.31
-
TF
T = 104
0.73
4.03
2.56
T = 105
0.33
-
Poly
EM
0.29
0.40
0.30
0.19
0.38
0.34
0.26
2.60
0.36
-
TF
T = 105
0.36
2.46
3.02
T = 106
0.16
-
Poly
0.14
0.35
0.29
0.12
2.52
Table 3: T is the number of samples, and the error metric is defined above. Methods: EM: sklearn
initialized with k-means using 5 random restarts; TF: tensor power method implemented in Python;
Poly: Polymom by solving Problem 7. Models: for mixture of Gaussians, we have ? 2||?1
?2 ||2 . spherical and diagonal describes the type of covariance matrix. The mean parameters of
constrained Gaussians satisfies ?1 + ?2 = 1. The best result is bolded. TF only handles spherical
variance, but it was of interest to see what TF does if the data is drawn from mixture of Gaussians
with diagonal covariance, these results are in strikeout.
Mixture of Linear Regressions. We can guarantee that Polymom can recover parameters for this
model when K ? D by showing that Problem 6 can be solved exactly: observe that while no entry
of the moment matrix M3 (y) is directly observed, each observation gives us a linear constraint on
the entries of the moment matrix and when K ? D, there are enough equations that this system
admits an unique solution for y.
Chaganty et al. [9] were also able to recover parameters for this model under the same conditions
(K ? D) by solving a series of low-rank tensor recovery problems, which ultimately requires the
computation of the same moments described above. In contrast, the Polymom framework makes the
dependence on moments upfront and takes care of the heavy-lifting in a problem-agnostic manner.
Lastly, the model can be extended to handle per component noise by including as a parameter, an
extension that is not possible using the method in [9].
Multiview Mixtures. We can guarantee parameter recovery when K ? D by proving that Problem 7 can be solved exactly (see Section B.2).
Mixture of Gaussians. In this case however, the moment conditions are non-trivial and we cannot
guarantee recovery of the true parameters. However, Polymom is guaranteed to recover a mixture of
Gaussians that match the moments. We can also apply constraints to the model: consider the case
of 2d mixture where the mean parameters for all components lies on a parabola ?1 ?22 = 0. In this
case, we just need to add constraints to Problem 7: y(1,0)+
y(0,2)+ = 0 for all 2 N2 up to
degree | | ? 2r 2. By incorporating these constraints at estimation time, we can possibly identify
the model parameters with less moments. See Section C for more details.
6
Conclusion
We presented an unifying framework for learning many types of mixture models via the method
of moments. For example, for the mixture of Gaussians, we can apply the same algorithm to both
mixtures in 1D needing higher-order moments [3, 11] and mixtures in high dimensions where lowerorder moments suffice [6]. The Generalized Moment Problem [15, 14] and its semidefinite relaxation hierarchies is what gives us the generality, although we rely heavily on the ability of nuclear
norm minimization to recover the underlying rank. As a result, while we always obtain parameters satisfying the moment conditions, there are no formal guarantees on consistent estimation. The
second main tool is solution extraction, which characterizes a more general structure of mixture
models compared the tensor structure observed by [6, 4]. This view draws connections to the literature on solving polynomial systems, where many techniques might be useful [35, 18, 19]. Finally,
through the connections we?ve drawn, it is our hope that Polymom can make the method of moments as turnkey as EM on more latent-variable models, as well as improve the statistical efficiency
of method of moments procedures.
Acknowledgments. This work was supported by a Microsoft Faculty Research Fellowship to the
third author and a NSERC PGS-D fellowship for the first author.
8
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9
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5,195 | 5,703 | Robust Gaussian Graphical Modeling with the
Trimmed Graphical Lasso
Aur?elie C. Lozano
IBM T.J. Watson Research Center
[email protected]
Eunho Yang
IBM T.J. Watson Research Center
[email protected]
Abstract
Gaussian Graphical Models (GGMs) are popular tools for studying network structures. However, many modern applications such as gene network discovery and
social interactions analysis often involve high-dimensional noisy data with outliers or heavier tails than the Gaussian distribution. In this paper, we propose the
Trimmed Graphical Lasso for robust estimation of sparse GGMs. Our method
guards against outliers by an implicit trimming mechanism akin to the popular
Least Trimmed Squares method used for linear regression. We provide a rigorous
statistical analysis of our estimator in the high-dimensional setting. In contrast,
existing approaches for robust sparse GGMs estimation lack statistical guarantees. Our theoretical results are complemented by experiments on simulated and
real gene expression data which further demonstrate the value of our approach.
1
Introduction
Gaussian graphical models (GGMs) form a powerful class of statistical models for representing distributions over a set of variables [1]. These models employ undirected graphs to encode conditional
independence assumptions among the variables, which is particularly convenient for exploring network structures. GGMs are widely used in variety of domains, including computational biology [2],
natural language processing [3], image processing [4, 5, 6], statistical physics [7], and spatial statistics [8]. In many modern applications, the number of variables p can exceed the number of observations n. For instance, the number of genes in microarray data is typically larger than the sample
size. In such high-dimensional settings, sparsity constraints are particularly pertinent for estimating GGMs, as they encourage only a few parameters to be non-zero and induce graphs with few
edges. The most widely used estimator among others (see e.g. [9]) minimizes the Gaussian negative
log-likelihood regularized by the `1 norm of the entries (or the off-diagonal entries) of the precision
matrix (see [10, 11, 12]). This estimator enjoys strong statistical guarantees (see e.g. [13]). The
corresponding optimization problem is a log-determinant program that can be solved with interior
point methods [14] or by co-ordinate descent algorithms [11, 12]. Alternatively neighborhood selection [15, 16] can be employed to estimate conditional independence relationships separately for
each node in the graph, via Lasso linear regression, [17]. Under certain assumptions, the sparse
GGM structure can still be recovered even under high-dimensional settings.
The aforementioned approaches rest on a fundamental assumption: the multivariate normality of the
observations. However, outliers and corruption are frequently encountered in high-dimensional data
(see e.g. [18] for gene expression data). Contamination of a few observations can drastically affect
the quality of model estimation. It is therefore imperative to devise procedures that can cope with
observations deviating from the model assumption. Despite this fact, little attention has been paid to
robust estimation of high-dimensional graphical models. Relevant work includes [19], which leverages multivariate t-distributions for robustified inference and the EM algorithm. They also propose
an alternative t-model which adds flexibility to the classical t but requires the use of Monte Carlo
EM or variational approximation as the likelihood function is not available explicitly. Another per1
tinent work is that of [20] which introduces a robustified likelihood function. A two-stage procedure
is proposed for model estimation, where the graphical structure is first obtained via coordinate gradient descent and the concentration matrix coefficients are subsequently re-estimated using iterative
proportional fitting so as to guarantee positive definiteness of the final estimate.
In this paper, we propose the Trimmed Graphical Lasso method for robust Gaussian graphical modeling in the sparse high-dimensional setting. Our approach is inspired by the classical Least Trimmed
Squares method used for robust linear regression [21], in the sense that it disregards the observations
that are judged less reliable. More specifically the Trimmed Graphical Lasso seeks to minimize a
weighted version of the negative log-likelihood regularized by the `1 penalty on the concentration
matrix for the GGM and under some simple constraints on the weights. These weights implicitly
induce the trimming of certain observations. Our key contributions can be summarized as follows.
? We introduce the Trimmed Graphical Lasso formulation, along with two strategies for solving the objective. One involves solving a series of graphical lasso problems; the other is
more efficient and leverages composite gradient descent in conjunction with partial optimization.
? As our key theoretical contribution, we provide statistical guarantees on the consistency of
our estimator. To the best of our knowledge, this is in stark contrast with prior work on
robust sparse GGM estimation (e.g. [19, 20]) which do not provide any statistical analysis.
? Experimental results under various data corruption scenarios further demonstrate the value
of our approach.
2
Problem Setup and Robust Gaussian Graphical Models
Notation. For matrices U ? Rp?p and V ? Rp?p , hhU, V ii denotes the trace inner product
tr(A B T ). For a matrix U ? Rp?p and parameter a ? [1, ?], kU ka denotes the element-wise
`a norm, andPkU ka,off does the element-wise `a norm only for off-diagonal entries. For example,
kU k1,off := i6=j |Uij |. Finally, we use kU kF and |||U |||2 to denote the Frobenius and spectral norms,
respectively.
Setup. Let X = (X1 , X2 , . . . , Xp ) be a zero-mean Gaussian random field parameterized by p ? p
concentration matrix ?? :
1
P(X; ?? ) = exp ? hh?? , XX > ii ? A(?? )
(1)
2
where A(?? ) is the log-partition function of Gaussian random field. Here, the probability density
function in (1) is associated with p-variate Gaussian distribution, N (0, ?? ) where ?? = (?? )?1 .
Given n i.i.d. samples X (1) , . . . , X (n) from high-dimensional Gaussian random field (1), the
standard way to estimate the inverse covariance matrix is to solve the `1 regularized maximum likelihood estimator (MLE) that can be written as the following regularized log-determinant program:
n
DD
1 X (i) (i) > EE
minimize ?,
X (X )
? log det(?) + ?k?k1,off
(2)
???
n i=1
where ? is the space of the symmetric positive definite matrices, and ? is a regularization parameter
that encourages a sparse graph model structure.
In this paper, we consider the case where the number of random variables p may be substantially
larger than the number of sample size n, however, the concentration parameter of the underlying
distribution is sparse:
(C-1) The number of non-zero off-diagonal entries of ?? is at most k, that is |{??ij : ??ij 6=
0 for i 6= j}| ? k.
Now, suppose that n samples are drawn from this underlying distribution (1) with true parameter
?? . We further allow some samples are corrupted and not drawn from (1). Specifically, the set of
sample indices {1, 2, . . . , n} is separated into two disjoint subsets: if i-th sample is in the set of
?good? samples, which we name G, then it is a genuine sample from (1) with the parameter ?? . On
2
Algorithm 1 Trimmed Graphical Lasso in (3)
Initialize ?(0) (e.g. ?(0) = (S + ?I)?1 )
repeat
Compute w(t) given ?(t?1) , by assigning a weight of one to the h observations with lowest negative
log-likelihood and a weight of zero to the remaining ones.
P
(t) (i)
?L(t) ? h1 n
(X (i) )> ? (?(t?1) )?1
i=1 wi X
(t)
Line search. Choose ? (See Nesterov (2007) for a discussion of how the stepsize may be chosen),
checking that the following update maintains positive definiteness. This can be verified via Cholesky
factorization (as in [23]).
Update. ?(t) ? S?(t) ? (?(t?1) ? ? (t) ?L(t) ), where S is the soft-thresholding operator: [S? (U )]i,j =
sign(Ui,j ) max(|Ui,j | ? ?, 0) and is only applied to the off-diagonal elements of matrix U.
Compute (?(t) )?1 reusing the Cholesky factor.
until stopping criterion is satisfied
the other hand, if the i-th sample is in the set of ?bad? samples, B, the sample is corrupted. The
identifications of G and B are hidden to us. However, we naturally assume that only a small number
of samples are corrupted:
(C-2) Let h be the number of good samples: h := |G| and hence |B| = n ? h. Then, we assume
that larger portion of samples are genuine and uncorrupted so that |G|?|B|
? ? where 0 < ? ? 1.
|G|
0.6n?0.4n
1
If we assume that 40% of samples are corrupted, then ? = 0.6n = 3 .
In later sections, we will derive a robust estimator for corrupted samples of sparse Gaussian graphical
models and provide statistical guarantees of our estimator under the conditions (C-1) and (C-2).
2.1
Trimmed Graphical Lasso
We now propose a Trimmed Graphical Lasso for robust estimation of sparse GGMs:
n
DD
EE
1X
minimize ?,
wi X (i) (X (i) )> ? log det(?) + ?k?k1,off
???,w
h i=1
n
(3)
>
s. t. w ? [0, 1] , 1 w = h , k?k1 ? R
where ? is a regularization parameter to decide the sparsity of our estimation, and h is another
parameter, which decides the number of samples (or sum of weights) used in the training. h is
ideally set as the number of uncorrupted samples in G, but practically we can tune the parameter h by
cross-validation. Here, the constraint k?k1 ? R is required to analyze this non-convex optimization
problem as discussed in [22]. For another tuning parameter R, any positive real value would be
sufficient for R as long as k?? k1 ? R. Finally note that when h is fixed as n (and R is set as
infinity), the optimization problem (3) will be simply reduced to the vanilla `1 regularized MLE for
sparse GGM without concerning outliers.
The optimization problem (3) is convex in w as well as in ?, however this is not the case jointly.
Nevertheless, we will show later that any local optimum of (3) is guaranteed to be strongly consistent
under some fairly mild conditions.
Optimization As we briefly discussed above, the problem (3) is not jointly convex but biconvex.
One possible approach to solve the objective of (3) thus is to alternate between solving for ? with
fixed w and solving for w with fixed ?. Given ?, solving for w is straightforward and boils down to
assigning a weight of one to the h observations with lowest negative log-likelihood and a weight of
zero to the remaining ones. Given w, solving for ? can be accomplished by any algorithm solving
the ?vanilla? graphical Lasso program, e.g. [11, 12]. Each step solves a convex problem hence the
objective is guaranteed to decrease at each iteration and will converge to a local minima.
A more efficient optimization approach can be obtained by adopting a partial minimization strategy
for ?: rather than solving to completion for ? each time w is updated, one performs a single
step update. This approach stems from considering the following equivalent reformulation of our
3
objective:
minimize
???
n
DD
EE
1X
?,
wi (?)X (i) (X (i) )> ? log det(?) + ?k?k1,off
h i=1
n
DD
EE
1X
s. t. wi (?) =
argmin
?,
wi X (i) (X (i) )> , k?k1 ? R ,
h i=1
w?[0,1]n , 1> w=h
(4)
On can then leverage standard first-order methods such as projected and composite gradient descent
[24] that will converge to local optima. The overall procedure is depicted in Algorithm 1. Therein we
assume that we pick R sufficiently large, so one does not need to enforce the constraint k?k1 ? R
explicitly. If needed the constraint can be enforced by an additional projection step [22].
3
Statistical Guarantees of Trimmed Graphical Lasso
One of the main contributions of this paper is to provide the statistical guarantees of our Trimmed
Graphical Lasso estimator for GGMs. The optimization problem (3) is non-convex, and therefore
the gradient-type methods solving (3) will find estimators by local minima. Hence, our theory in this
section provides the statistical error bounds on any local minimum measured by k ? kF and k ? k1,off
norms simultaneously.
e w)
Suppose that we have some local optimum (?,
e of (3) by arbitrary gradient-based method. While
?? is fixed unconditionally, we define w? as follows: for a sample index i ? G, wi? is simply set
to w
ei so that wi? ? w
ei = 0. Otherwise for a sample index i ? B, we set wi? = 0. Hence, w? is
dependent on w.
e
In order to derive the upper bound on the Frobenius norm error, we first need to assume the standard
restricted strong convexity condition of (3) with respective to the parameter ?:
(C-3) (Restricted strong convexity condition) Let ? be an arbitrary error of parameter ?. That
is, ? := ? ? ?? . Then, for any possible error ? such that k?kF ? 1,
DD
?1
?1 EE
??
? ?? + ?
, ? ? ?l k?k2F
(5)
where ?l is a curvature parameter.
Note that in order to guarantee the Frobenius norm-based error bounds, (C-3) is required even for the
vanilla Gaussian graphical models without outliers, which has been well studied by several works
such as the following lemma:
Lemma 1 (Section B.4 of [22]). For any ? ? Rp?p such that k?kF ? 1,
DD
?1
?1 EE
?2
??
? ?? + ?
, ? ? |||?? |||2 + 1
k?k2F ,
thus (C-3) holds with ?l = |||?? |||2 + 1
?2
.
While (C-3) is a standard condition that is also imposed for the conventional estimators under clean
set of of samples, we additionally require the following condition for a successful estimation of (3)
on corrupted samples:
e w).
e := ?
e ? ?? and ?
e := w
(C-4) Consider arbitrary local optimum (?,
e Let ?
e ? w? . Then,
n
DD 1 X
EE
e i X (i) (X (i) )> , ?
e ? ?1 (n, p)k?k
e F + ?2 (n, p)k?k
e 1
?
h i=1
with some positive quantities ?1 (n, p) and ?2 (n, p) on n and p. These will be specified below for
some concrete examples.
(C-4) can be understood as a structural incoherence condition between the model parameter ? and
the weight parameter w. Such a condition is usually imposed when analyzing estimators with multiple parameters (for example, see [25] for a robust linear regression estimator). Since w? is defined
4
depending on w,
e each local optimum has its own (C-4) condition. We will see in the sequel that
under some reasonable cases, this condition for any local optimum holds with high probability. Also
e i = 0 for
note that for the case with clean samples, the condition (C-4) is trivially satisfied since ?
all i ? {1, . . . , n} and hence the LHS becomes 0.
Armed with these conditions, we now state our main theorem on the error bounds of our estimator
(3):
e w)
Theorem 1. Consider corrupted Gaussian graphical models. Let (?,
e be an any local optie
mum of M -estimator (3). Suppose that (?, w)
e satisfies the condition (C-4). Suppose also that the
regularization parameter ? in (3) is set such that
(
)
n
1 X
?l ? ?1 (n, p)
? (i)
(i) >
? ?1
4 max
w X (X ) ? (? )
, ?2 (n, p) ? ? ?
.
(6)
h i=1 i
3R
?
e w)
Then, this local optimum (?,
e is guaranteed to be consistent as follows:
?
e ? ?? kF ? 1 3? k + p + ?1 (n, p)
and
k?
?l
2
p
2
e ? ?? k1,off ? 2 3? k + p + ?1 (n, p) .
k?
? ?l
(7)
The statement in Theorem 1 holds deterministically, and the probabilistic statement comes where
e w)
we show (C-4) and (6) for a given (?,
e are satisfied. Note that, defining L(?, w :=
Pn
?, h1 i=1 wi X (i) (X (i) )> ? log det(?), it is a standard way of choosing ? based on
?
k?? L ?? , w? k? (see [26], for details). Also it is important to note that the term k + p captures
the relation between element-wise `1 norm and the error norm k ? kF including diagonal entries. Due
to the space limit, the proof of Theorem 1 (and all other proofs) are provided in the Supplements
[27].
Now, it is natural to ask how easily we can satisfy the conditions in Theorem 1. Intuitively it is
impossible to recover true parameter by weighting approach as in (3) when the amount of corruptions
exceeds that of normal observation errors.
To this end, suppose that we have some upper bound on the corruptions:
(C-B1) For some function f (?), we have |||X B |||2
2
?
? f (X B ) h log p
where X B denotes the sub-design matrix in R|B|?p corresponding to outliers. Under this assumption, we can properly choose the regularization parameter ? satisfying (6) as follows:
Corollary 1. Consider corrupted Gaussian graphical models with conditions (C-2) and (C-B1).
Suppose that we choose the regularization parameter
q
s
)
(
r
|B| log p
B
?
?
f
(X
)
l
10? log p |B| ?
log p
h
B
?
? = 4 max 8(max ?ii )
+
k? k? , f (X )
?
.
i
h ? |B|
h
h
3R
Then, any local optimum of (3) is guaranteed to satisfy (C-4) and have the error bounds in (7) with
probability at least 1 ? c1 exp(?c01 h?2 ) for some universal positive constants c1 and c01 .
?
If we further assume the number of corrupted samples scales with n at most :
?
(C-B2) |B| ? a n for some constant a ? 0,
then we can derive the following result as another corollary of Theorem 1:
Corollary 2. Consider corrupted Gaussian graphical models. Suppose that the conditions
q (C2), (C-B1) and (C-B2) hold. Also suppose that the regularization parameter ? is set as c logn p
?
?
k? ?
?
where c := 4 max 16(maxi ??ii ) 5? + 2ak?
, 2f (X B ) . Then, if the sample size n is lower
log p
5
bounded as
2
p
4
n ? max 16a2 , |||?? |||2 + 1
3Rc + f (X B ) 2|B| (log p) ,
then any local optimum of (3) is guaranteed to satisfy (C-4) and have the following error bound:
!
r
r
2|B| log p
1 3c (k + p) log p
B
?
e
(8)
+ f (X )
k? ? ? kF ?
?l 2
n
n
with probability at least 1 ? c1 exp(?c01 h?2 ) for some universal positive constants c1 and c01 .
Note that the k ? k1,off norm based error bound also can be easily
? derived using the selection of ?
from (7). Corollary 2 reveals an interesting result: even when O( n) samples out of total n samples
are corrupted, our estimator (3) can successfully
recover the true parameter with guaranteed error
q
(k+p) log p
which exactly recovers the Frobenius error
in (8). The first term in this bound is O
n
bound for the case without outliers (see [13, 22] for example).
Due to the outliers, we have the
q
|B| log p
performance degrade with the second term, which is O
. To the best of our knowledge,
n
this is the first statistical error bounds on the parameter estimation for Gaussian graphical models
with outliers. Also note that Corollary 1 only concerns on any local optimal point derived by an
arbitrary optimization algorithm. For the guarantees of multiple local optima simultaneously, we
may use a union bound from the corollary.
When Outliers Follow a Gaussian Graphical Model Now let us provide a concrete example and
show how f (X B ) in (C-B1) is precisely specified in this case:
(C-B3) Outliers in the set B are drawn from another Gaussian graphical model (1) with a parameter (?B )?1 .
This can be understood as the Gaussian mixture model where the most of the samples are drawn
from (?? )?1 that we want to estimate, and small portion of samples are drawn from ?B . In this
case, Corollary 2 can be further shaped as follows:
Corollary 3. Suppose that the conditions (C-2), (C-B2) and (C-B3) hold. Then the statement in
2
?
?
4 2a 1+ log p |||?B |||2
?
.
Corollary 2 holds with f (X B ) :=
log p
4
Experiments
In this section we corroborate the performance of our Trimmed Graphical Lasso (trim-glasso) algorithm on simulated data. We compare against glasso: the vanilla Graphical Lasso [11]; the t-Lasso
and t*-lasso methods [19], and robust-LL: the robustified-likelihood approach of [20].
4.1
Simulated data
Our simulation setup is similar to [20] and is a akin to gene regulatory networks. Namely we consider four different scenarios where the outliers are generated from models with different graphical
structures. Specifically, each sample is generated from the following mixture distribution:
p0
p0
yk ? (1 ? p0 )Np (0, ??1 ) + Np (??, ??1
Np (?, ??1
o )+
o ), k = 1, . . . , n,
2
2
where po = 0.1, n = 100, and p = 150. Four different outlier distributions are considered:
?
M1: ? = (1, . . . , 1)T , ?o = ?,
?
M2: ? = (1.5, . . . , 1.5)T , ?o = ?,
M3: ? = (1, . . . , 1)T , ?o = Ip ,
M4: ? = (1.5, . . . , 1.5)T , ?o = Ip .
We also consider the scenario where the outliers are not symmetric about the mean and simulate
data from the following model
6
0.6
0.2
0.3
0.4
0.0
0.1
0.2
0.3
1-specificity
(a) M1
(b) M2
0.4
0.6
0.5
0.4
sensitivity
0.1
0.2
0.3
0.6
0.5
0.4
0.3
0.1
0.2
sensitivity
0.4
0.2
0.1
0.5
1-specificity
0.7
0.1
0.7
0.0
0.3
sensitivity
0.5
0.6
0.5
0.4
0.3
sensitivity
0.1
0.2
glasso
t-lasso
t*-lasso
robust-LL (best)
trim-glasso (best)
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
1-specificity
1-specificity
(c) M3
(d) M4
0.4
0.5
Figure 1: Average ROC curves for the comparison methods for contamination scenarios M1-M4.
M5: yk ? (1 ? po )Np (0, ??1 ) + po Np (2, Ip ), k = 1, . . . , n.
For each simulation run, ? is a randomly generated precision matrix corresponding to a network
with 9 hub nodes simulated as follows. Let A be the adjacency of the network. For all i < j we set
Aij = 1 with probability 0.03, and zero otherwise. We set Aji = Aij . We then randomly select 9
hub nodes and set the elements of the corresponding rows and columns of A to one with probability
0.4 and zero otherwise. Using A, the simulated nonzero coefficients of the precision matrix are
sampled as follows. First we create a matrix E so that Ei,j = 0 if Ai,j = 0, and Ei,j is sampled
T
uniformly from [?0.75, ?0.23] ? [0.25, 0.75] if Ai,j 6= 0. Then we set E = E+E
. Finally we set
2
?
? = E + (0.1 ? ?min (E))Ip , where ?min (E) is the smallest eigenvalue of E. ? is a randomly
generated precision matrix in the same way ? is generated.
For the robustness parameter ? of the robust-LL method, we consider ? ? {0.005, 0.01, 0.02, 0.03}
as recommended in [20]. For the trim-glasso method we consider 100h
n ? {90, 85, 80}. Since all the
robust comparison methods converge to a stationary point, we tested various initialization strategies
for the concentration matrix, including Ip , (S + ?Ip )?1 and the estimate from glasso. We did not
observe any noticeable impact on the results.
Figure 1 presents the average ROC curves of the comparison methods over 100 simulation data sets
for scenarios M1-M4 as the tuning parameter ? varies. In the figure, for robust-LL and trim-glasso
methods, we depict the best curves with respect to parameter ? and h respectively. Due to space
constraints, the detailed results for all the values of ? and h considered, as well as the results for
model M5 are provided in the Supplements [27].
From the ROC curves we can see that our proposed approach is competitive compared the alternative
robust approaches t-lasso, t*-lasso and robust-LL. The edge over glasso is even more pronounced for
7
8
6
4
Frequency
2
0
-3
-2
-1
0
1
2
rescaled ORC3 gene expression
Figure 2: (a) Histogram of standardized gene expression levels for gene ORC3. (b) Network estimated by trim-glasso
scenarios M2, M4 and M5. Surprisingly, trim-glasso with h/n = 80% achieves superior sensitivity
for nearly any specificity.
Computationally the trim-glasso method is also competitive compared to alternatives. The average
run-time over the path of tuning parameters ? is 45.78s for t-lasso, 22.14s for t*-lasso, 11.06s
for robust-LL, 1.58s for trimmed lasso, 1.04s for glasso. Experiments were run on R in a single
computing node with a Intel Core i5 2.5GHz CPU and 8G memory. For t-lasso, t*-lasso and robustLL we used the R implementations provided by the methods? authors. For glasso we used the
glassopath package.
4.2
Application to the analysis of Yeast Gene Expression Data
We analyze a yeast microarray dataset generated by [28]. The dataset concerns n = 112 yeast
segregants (instances). We focused on p = 126 genes (variables) belonging to cell-cycle pathway as
provided by the KEGG database [29]. For each of these genes we standardize the gene expression
data to zero-mean and unit standard deviation. We observed that the expression levels of some genes
are clearly not symmetric about their means and might include outliers. For example the histogram
of gene ORC3 is presented in Figure 2(a). For the robust-LL method we set ? = 0.05 and for trimglasso we use h/n = 80%. We use 5-fold-CV to choose the tuning parameters for each method.
After ? is chosen for each method, we rerun the methods using the full dataset to obtain the final
precision matrix estimates.
Figure 2(b) shows the cell-cycle pathway estimated by our proposed method. For comparison the
cell-cycle pathway from the KEGG [29] is provided in the Supplements [27]. It is important to note
that the KEGG graph corresponds to what is currently known about the pathway. It should not be
treated as the ground truth. Certain discrepancies between KEGG and estimated graphs may also
be caused by inherent limitations in the dataset used for modeling. For instance, some edges in
cell-cycle pathway may not be observable from gene expression data. Additionally, the perturbation
of cellular systems might not be strong enough to enable accurate inference of some of the links.
glasso tends to estimate more links than the robust methods. We postulate that the lack of robustness
might result in inaccurate network reconstruction and the identification of spurious links. Robust
methods tend to estimate networks that are more consistent with that from the KEGG (F1 -score
of 0.23 for glasso, 0.37 for t*-lasso, 0.39 for robust-NLL and 0.41 for trim-glasso, where the F1
score is the harmonic mean between precision and recall). For instance our approach recovers
several characteristics of the KEGG pathway. For instance, genes CDC6 (a key regulator of DNA
replication playing important roles in the activation and maintenance of the checkpoint mechanisms
coordinating S phase and mitosis) and PDS1 (essential gene for meiotic progression and mitotic cell
cycle arrest) are identified as a hub genes, while genes CLB3,BRN1,YCG1 are unconnected to any
other genes.
8
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5,196 | 5,704 | Matrix Completion from Fewer Entries:
Spectral Detectability and Rank Estimation
Alaa Saade1 and Florent Krzakala1,2
Laboratoire de Physique Statistique, CNRS & ?cole Normale Sup?rieure, Paris, France.
2
Sorbonne Universit?s, Universit? Pierre et Marie Curie Paris 06, F-75005, Paris, France
1
Lenka Zdeborov?
Institut de Physique Th?orique, CEA Saclay and CNRS UMR 3681, 91191 Gif-sur-Yvette, France
Abstract
The completion of low rank matrices from few entries is a task with many practical
applications. We consider here two aspects of this problem: detectability, i.e. the
ability to estimate the rank r reliably from the fewest possible random entries,
and performance in achieving small reconstruction error. We propose a spectral
algorithm for these two tasks called MaCBetH (for Matrix Completion with the
Bethe Hessian). The rank is estimated as the number of negative eigenvalues of
the Bethe Hessian matrix, and the corresponding eigenvectors are used as initial
condition for the minimization of the discrepancy between the estimated matrix
and the revealed entries. We analyze the performance in a random matrix setting
using results from the statistical mechanics of the Hopfield neural network, and
show in particular that?MaCBetH efficiently detects the rank r of a large n ?
m matrix from C(r)r nm entries, where C(r) is a constant close to 1. We
also evaluate the corresponding root-mean-square error empirically and show that
MaCBetH compares favorably to other existing approaches.
Matrix completion is the task of inferring the missing entries of a matrix given a subset of known
entries. Typically, this is possible because the matrix to be completed has (at least approximately)
low rank r. This problem has witnessed a burst of activity, see e.g. [1, 2, 3], motivated by many
applications such as collaborative filtering [1], quantum tomography [4] in physics, or the analysis
of a covariance matrix [1]. A commonly studied model for matrix completion assumes the matrix
to be exactly low rank, with the known entries chosen uniformly at random and observed without
noise. The most widely considered question in this setting is how many entries need to be revealed
such that the matrix can be completed exactly in a computationally efficient way [1, 3]. While our
present paper assumes the same model, the main questions we investigate are different.
The first question we address is detectability: how many random entries do we need to reveal in order
to be able to estimate the rank r reliably. This is motivated by the more generic problem of detecting
structure (in our case, low rank) hidden in partially observed data. It is reasonable to expect the
existence of a region where exact completion is hard or even impossible yet the rank estimation is
tractable. A second question we address is what is the minimum achievable root-mean-square error
(RMSE) in estimating the unknown elements of the matrix. In practice, even if exact reconstruction
is not possible, having a procedure that provides a very small RMSE might be quite sufficient.
In this paper we propose an algorithm called MaCBetH that gives the best known empirical performance for the two tasks above when the rank r is small. The rank in our algorithm is estimated as the
number of negative eigenvalues of an associated Bethe Hessian matrix [5, 6], and the corresponding
eigenvectors are used as an initial condition for the local optimization of a cost function commonly
considered in matrix completion (see e.g. [3]). In particular, in the random matrix setting, we show
1
?
that MaCBetH detects the rank of a large n ? m matrix from C(r)r nm entries, where C(r) is a
small constant, see Fig.?2, and C(r) ? 1 as r ? ?. The RMSE is evaluated empirically and, in the
regime close to C(r)r nm, compares very favorably to existing approache such as OptSpace [3].
This paper is organized as follows. We define the problem and present generally our approach in the
context of existing work in Sec. 1. In Sec. 2 we describe our algorithm and motivate its construction
via a spectral relaxation of the Hopfield model of neural network. Next, in Sec. 3 we show how the
performance of the proposed spectral method can be analyzed using, in parts, results from spin glass
theory and phase transitions, and rigorous results on the spectral density of large random matrices.
Finally, in Sec. 4 we present numerical simulations that demonstrate the efficiency of MaCBetH.
Implementations of our algorithms in the Julia and Matlab programming languages are available at
the SPHINX webpage http://www.lps.ens.fr/~krzakala/WASP.html.
1
Problem definition and relation to other work
Let Mtrue be a rank-r matrix such that
Mtrue = XY T ,
(1)
where X ? Rn?r and Y ? Rm?r are two (unknown) tall matrices. We observe only a small
fraction of the elements of Mtrue , chosen uniformly at random. We call E the subset of observed
entries, and M the (sparse) matrix supported on E whose nonzero elements are the revealed entries
T
of Mtrue . The aim is to reconstruct the rank r matrix Mtrue = XY
given M. An important
?
parameter which controls the difficulty of the problem is = |E|/ nm. In the case of a square
matrix M, this is the average number of revealed entries per line or column.
In our numerical examples and theoretical justifications we shall generate the low rank matrix
Mtrue = XY T , using tall matrices X and Y with iid Gaussian elements, we call this the random matrix setting. The MaCBetH algorithm is, however, non-parametric and does not use any
prior knowledge about X and Y . The analysis we perform applies to the limit n ? ? while
m/n = ? = O(1) and r = O(1).
The matrix completion problem was popularized in [1] who proposed nuclear norm minimization
as a convex relaxation of the problem. The algorithmic complexity of the associated semidefinite
programming is, however, O(n2 m2 ). A low complexity procedure to solve the problem was later
proposed by [7] and is based on singular value decomposition (SVD). A considerable step towards
theoretical understanding of matrix completion from few entries was made in [3] who proved that
with the use of trimming
pthe performance of SVD-based matrix completion can be improved and a
RMSE proportional to nr/|E| can be achieved. The algorithm of [3] is referred to as OptSpace,
and empirically it achieves state-of-the-art RMSE in the regime of very few revealed entries.
OptSpace proceeds in three steps [3]. First, one trims the observed matrix M by setting to zero all
rows (resp. columns) with more revealed entries than twice the average number of revealed entries
per row (resp. per column). Second, a singular value decompositions is performed on the matrix
and only the first r components are kept. When the rank r is unknown it is estimated as the index for
which the ratio between two consecutive singular values has a minimum. Third, a local minimization
of the discrepancy between the observed entries and the estimate is performed. The initial condition
for this minimization is given by the first r left and right singular vectors from the second step.
In this work we improve upon OptSpace by replacing the first two steps by a different spectral
procedure that detects the rank and provides a better initial condition for the discrepancy minimization. Our method leverages on recent progress made in the task of detecting communities in the
stochastic block model [8, 5] with spectral methods. Both in community detection and matrix completion, traditional spectral methods fail in the very sparse regime due to the existence of spurious
large eigenvalues (or singular values) corresponding to localized eigenvectors [8, 3]. The authors
of [8, 5, 9] showed that using the non-backtracking matrix or the closely related Bethe Hessian as
a basis for the spectral method in community detection provides reliable rank estimation and better
inference performance. The present paper provides an analogous improvement for the matrix completion problem. In particular, we shall analyze the algorithm using tools from spin glass theory in
statistical mechanics, and show that there exists a phase transition between a phase where it is able
to detect the rank, and a phase where it is unable to do so.
2
2
2.1
Algorithm and motivation
The MaCBetH algorithm
A standard approach to the completion problem (see e.g. [3]) is to minimize the cost function
X
min
[Mij ? (XY T )ij ]2
X,Y
(2)
(ij)?E
over X ? Rn?r and Y ? Rm?r . This function is non-convex, and global optimization is hard.
One therefore resorts to a local optimization technique with a careful choice of the initial conditions
X0 , Y0 . In our method, given the matrix M, we consider a weighted bipartite undirected graph with
adjacency matrix A ? R(n+m)?(n+m)
0
M
A=
.
(3)
MT
0
We will refer to the graph thus defined as G. We now define the Bethe Hessian matrix H(?) ?
R(n+m)?(n+m) to be the matrix with elements
!
X
1
2
Hij (?) = 1 +
sinh ?Aik ?ij ? sinh(2?Aij ) ,
(4)
2
k??i
where ? is a parameter that we will fix to a well-defined value ?SG depending on the data, and ?i
stands for the neighbors of i in the graph G. Expression (4) corresponds to the matrix introduced in
[5], applied to the case of graphical model (6). The MaCBetH algorithm that is the main subject of
this paper is then, given the matrix A, which we assume to be centered:
Algorithm (MaCBetH)
1. Numerically solve for the value of ??SG such that F (??SG ) = 1, where
X
1
F (?) := ?
tanh2 (?Mij ) .
nm
(5)
(i,j)?E
2. Build the Bethe Hessian H(??SG ) following eq. (4).
3. Compute all its negative eigenvalues ?1 , ? ? ? , ?r? and corresponding eigenvectors
v1 , ? ? ? , vr?. r? is our estimate for the rank r. Set X0 (resp. Y0 ) to be the first n lines
(resp. the last m lines) of the matrix [v1 v2 ? ? ? vr?].
4. Perform local optimization of the cost function (2) with rank r? and initial condition X0 , Y0 .
In step 1, ??SG is an approximation of the optimal value of ?, for which H(?) has a maximum number
of negative eigenvalues (see section 3). Instead of this approximation, ? can be chosen in such a
way as to maximize the number of negative eigenvalues. We however observed numerically that
the algorithm is robust to some imprecision on the value of ??SG . In step 2 we could also use the
non-backtracking matrix weighted by tanh ?Mij , it was shown in [5] that the spectrum of the Bethe
Hessian and the non-backtracking matrix are closely related. In the next section, we will motivate
and analyze this algorithm (in the setting where Mtrue was generated from element-wise random
X and Y ) and show that in this case MaCBetH is able to infer the rank whenever > c . Fig. 1
illustrates the spectral properties of the Bethe Hessian that justify this algorithm: the spectrum is
composed of a few informative negative eigenvalues, well separated from the bulk (which remains
positive). In particular, as observed in [8, 5], it avoids the spurious eigenvalues with localized
eigenvectors that make trimming necessary in the case of [3]. This algorithm is computationally
efficient as it is based on the eigenvalue decomposition of a sparse, symmetric matrix.
2.2
Motivation from a Hopfield model
We shall now motivate the construction of the MaCBetH algorithm from a graphical model perspective and a spectral relaxation. Given the observed matrix M from the previous section, we consider
3
the following graphical model
?
?
X
1
P ({s}, {t}) = exp ??
Mij si tj ? ,
Z
(6)
(i,j)?E
where the {si }1?i?n and {tj }1?j?m are binary variables, and ? is a parameter controlling the
strength of the interactions. This model is a (generalized) Hebbian Hopfield model on a bipartite
sparse graph, and is therefore known to have r modes (up to symmetries) correlated with the lines of
X and Y [10]. To study it, we can use the standard Bethe approximation which is widely believed
to be exact for such problems on large random graphs [11, 12]. In this approximation the means
E(si ), E(tj ) and moments E(si tj ) of each variable are approximated by the parameters bi , cj and
?ij that minimize the so-called Bethe free energy FBethe ({bi }, {cj }, {?ij }) that reads
X
X X 1 + bi si + cj tj + ?ij si tj
FBethe ({bi }, {cj }, {?ij }) = ?
Mij ?ij +
?
4
s ,t
(i,j)?E
n
X
(i,j)?E
i
j
m
X
X 1 + bi si
X 1 + cj t j
+
+
,
(1 ? di ) ?
(1 ? dj ) ?
2
2
s
t
i=1
j=1
i
(7)
j
where ?(x) := x ln x, and di , dj are the degrees of nodes i and j in the graph G. Neural network
models such as eq. (6) have been extensively studied over the last decades (see e.g. [12, 13, 14, 15,
16] and references therein) and the phenomenology, that we shall review briefly here, is well known.
In particular, for ? small enough, the global minimum of the Bethe free energy corresponds to the
so-called paramagnetic state
?i, j,
bi = cj = 0, ?ij = tanh (?Mij ).
(8)
As we increase ?, above a certain value ?R , the model enters a retrieval phase, where the free energy
has local minima correlated with the factors X and Y . There are r local minima, called retrieval
l
states ({bli }, {clj }, {?ij
}) indexed by l = 1, ? ? ? , r such that, in the large n, m limit,
n
?l = 1 ? ? ? r,
1X
Xi,l bli > 0,
n i=1
m
1X
Yj,l clj > 0 .
m j=1
(9)
These retrieval states are therefore convenient initial conditions for the local optimization of eq. (2),
and we expect their number to tell us the correct rank. Increasing ? above a critical value ?SG the
system eventually enters a spin glass phase, marked by the appearance of many spurious minima.
It would be tempting to continue the Bethe approach leading to belief propagation, but we shall
instead consider a simpler spectral relaxation of the problem, following the same strategy as used
in [5, 6] for graph clustering. First, we use the fact that the paramagnetic state (8) is always a
stationary point of the Bethe free energy, for any value of ? [17, 18]. In order to detect the retrieval
states, we thus study its stability by looking for negative eigenvalues of the Hessian of the Bethe
free energy evaluated at the paramagnetic state (8). At this point, the elements of the Hessian
involving one derivative with respect to ?ij vanish, while the block involving two such derivatives
is a diagonal positive definite matrix [5, 17]. The remaining part is the matrix called Bethe Hessian
in [5] (which however considers a different graphical model than (6)). Eigenvectors corresponding
to its negative eigenvalues are thus expected to give an approximation of the retrieval states (9). The
picture exposed in this section is summarized in Figure 1 and motivates the MaCBetH algorithm.
Note that a similar approach was used in [16] to detect the retrieval states of a Hopfield model using
the weighted non-backtracking matrix [8], which linearizes the belief propagation equations rather
than the Bethe free energy, resulting in a larger, non-symmetric matrix. The Bethe Hessian, while
mathematically closely related, is also simpler to handle in practice.
3
Analysis of performance in detection
We now show how the performance of MaCBetH can be analyzed, and the spectral properties of the
matrix characterized using both tools from statistical mechanics and rigorous arguments.
4
7
6
4
1.2
0.9
3
0
0.7
0
0.2
0.8
0.2
0.9
0.4
0.6
0.5
?
1
0
1.2
? = 0.12824
0.09
0.2
0
-0.5
0
1
2
3
?
4
0
0.25
-1
0
5
6
0.5
7
0.5
-0.5
0
0.5
1
1.5
2
2.5
? = 0.25
0.12
0.03
0.08
0
0.04
0
8
?
Direct diag
BP
0.06
0.16
0.3
0.1
0
-1.5
0.2
?(?)
?(?)
0.8
Direct diag
0.18
BP
0.4
0
0.15
0.6
0.4
2
1
0.8
? = 0.05
Direct diag
BP
0.3
1
?(?)
?(?)
5
1.4
? = 0.01
Direct diag
BP
1.8
0
5
10
0
0.6
?
15
1.2
20
25
Figure 1: Spectral density of the Bethe Hessian for various values of the parameter ?. Red dots
are the result of the direct diagonalisation of the Bethe Hessian for a rank r = 5 and n = m = 104
matrix, with = 15 revealed entries per row on average. The black curves are the solutions of (18)
computed with belief propagation on a graph of size 105 . We isolated the 5 smallest eigenvalues,
represented as small bars for convenience, and the inset is a zoom around these smallest eigenvalues.
For ? small enough (top plots), the Bethe Hessian is positive definite, signaling that the paramagnetic
state (8) is a local minimum of the Bethe free energy. As ? increases, the spectrum is shifted towards
the negative region and has 5 negative eigenvalues at the approximate value of ??SG = 0.12824 (to
be compared to ?R = 0.0832 for this case) evaluated by our algorithm (lower left plot). These
eigenvalues, corresponding to the retrieval states (9), become positive and eventually merge in the
bulk as ? is further increased (lower right plot), while the bulk of uninformative eigenvalues remains
at all values of ? in the positive region.
3.1
Analysis of the phase transition
We start by investigating the phase transition above which our spectral method will detect the correct
rank. Let xp = (xlp )1?l?r , yp = (ypl )1?l?r be random vectors with the same empirical distribution
as the lines of X and Y respectively. Using the statistical mechanics correspondence between the
negative eigenvalues of the Bethe Hessian and the appearance of phase transitions in model (6), we
can compute the values ?R and ?SG where instabilities towards, respectively, the retrieval states and
the spurious glassy states, arise. We have repeated the computations of [13, 14, 15, 16] in the case
of model (6), using the cavity method [12]. We refer the reader interested in the technical details of
the statistical mechanics approach to neural networks to [14, 15, 16].
Following a standard computation for locating phase transitions in the Bethe approximation (see e.g.
[12, 19]), the stability of the paramagnetic state (8) towards these two phases can be monitored in
terms of the two following parameters:
s
r
r
1
hY
X
X
i 2s
,
(10)
?(?) = lim E
tanh2 ? xlp ypl tanh2 ? xlp+1 ypl
s??
p=1
l=1
l=1
s
r
r
1
hY
i 2s
X
X
?(?) = lim E
tanh ?|x1p yp1 | + ? xlp ypl tanh ?|x1p+1 yp1 | + ? xlp+1 ypl
,
s??
p=1
l=2
(11)
l=2
where the expectation is over the distribution of the vectors xp , yp . The parameter ?(?) controls the
sensitivity of the paramagnetic solution to random noise, while ?(?) measures its sensitivity to a
perturbation in the direction of a retrieval state. ?SG and ?R are defined implicitly as ?(?SG ) = 1
and ?(?R ) = 1, i.e. the value beyond which the perturbation diverges. The existence of a retrieval
phase is equivalent to the condition ?SG > ?R , so that there exists a range of values of ? where the
retrieval states exist, but not the spurious ones. If this condition is met, by setting ? = ?SG in our
algorithm, we ensure the presence of meaningful negative eigenvalues of the Bethe Hessian.
5
We define the critical value of = c such that ?SG > ?R if and only if > c . In general, there is
no closed-form formula for this critical value, which is defined implicitly in terms of the functions ?
and ?. We thus computed c numerically using a population dynamics algorithm [12] and the results
?
for C(r) = c /r are presented on Figure 2. Quite remarkably, with the definition = |E|/ nm,
the critical value c does not depend on the ratio m/n, only on the rank r.
1.5
C(r)
C(r ? ?)
1 + 0.812 r?3/4
1.4
1.3
C(r)
In the limit of large and r it is possible to
obtain a simple closed-form formula. In this
case the observed entries of the matrix become
jointly Gaussian distributed, and uncorrelated,
and therefore independent. Expression (10)
then simplifies to
r
h
X
i
?(?) =r?? E tanh2 ? xl y l . (12)
1.2
1.1
1
l=1
0.9
Note that the MaCBetH algorithm uses an empirical estimator F (?) ' ?(?) (5) of this
quantity to compute an approximation ??SG of
?SG purely from the revealed entries. In the
large r, regime, both ?SG , ?R decay to 0, so
that we can further approximate
5
10
15
20
25
r
Figure 2: Location of the critical value as a function of the rank r. MaCBetH is able
? to estimate
the correct rank from |E| > C(r)r nm known
entries. We used a population dynamics algorithm
6
2
1 = ?(?SG ) ?r?? r?SG
E[x2 ]E[y 2 ] , (13) with a population of size 10 to compute the funcp
tions ? and ? from (10,11). The dotted line is a fit
1 = ?(?R ) ?r?? ?R E[x2 ]E[y 2 ] , (14) suggesting that C(r) ? 1 = O(r?3/4 ).
so that we reach the simple asymptotic expression, in the large , r limit, that c = r, or equivalently C(r) = 1. Interestingly, this result was
obtained as the detectability threshold in completion of rank r = O(n) matrices from O(n2 ) entries
in the Bayes optimal setting in [20].
? Notice, however, that exact completion in the setting of [20] is
only possible for > r(m+n)/ nm: clearly detection and exact completion are different phenomena. The previous analysis can be extended beyond the random setting assumption, as long as the
empirical distribution of the entries is well defined, and the lines of X (resp. Y ) are approximately
orthogonal and centered. This condition is related to the standard incoherence property [1, 3].
3.2
Computation of the spectral density
In this section, we show how the spectral density of the Bethe Hessian can be computed analytically
on tree-like graphs such as those generated by picking uniformly at random the observed entries of
the matrix XY T . This further motivates our algorithm and in particular our choice of ? = ??SG ,
independently of section 3. The spectral density is defined as
?(?) =
n+m
1 X
?(? ? ?i ) ,
n,m?? n + m
i=1
lim
(15)
where the ?i ?s are the eigenvalues of the Bethe Hessian. Using again the cavity method, it can be
shown [21] that the spectral density (in which potential delta peaks have been removed) is given by
?(?) =
n+m
X
1
Im?i (?) ,
n,m?? ?(n + m)
i=1
lim
(16)
where the ?i are complex variables living on the vertices of the graph G, which are given by:
?1
X
X1
sinh2 (2?Ail )?l?i
,
(17)
?i = ? ? + 1 +
sinh2 ?Aik ?
4
k??i
l??i
where ?i is the set of neighbors of i. The ?i?j are the (linearly stable) solution of the following
belief propagation recursion:
?1
X 1
X
?i?j = ? ? + 1 +
sinh2 ?Aik ?
sinh2 (2?Ail )?l?i
.
(18)
4
k??i
l??i\j
6
Rank 10
Rank 3
10
3
Mean inferred rank
9
8
2.5
7
2
6
5
1.5
n = m = 500
n = m = 2000
n = m = 8000
n = m = 16000
Transition ?c
1
0.5
0
2
3
4
5
6
7
8
9
4
3
2
1
10
?
0
9
10
11
12
13
14
15
16
17
18
19
?
Figure 3: Mean inferred rank as a function of , for different sizes, averaged over 100 samples of
n ? m XY T matrices. The entries of X, Y are drawn from a Gaussian distribution of mean 0 and
variance 1. The theoretical transition is computed with a population dynamics algorithm (see section
3.1). The finite size effects are considerable but consistent with the asymptotic prediction.
This formula can be derived by turning the computation of the spectral density into a marginalization
problem for a graphical model on the graph G and then solving it using loopy belief propagation.
Quite remarkably, this approach leads to an asymptotically exact (and rigorous [22]) description of
the spectral density on Erd?os-R?nyi random graphs. Solving equation (18) numerically we obtain
the results shown on Fig. 1: the bulk of the spectrum, in particular, is always positive.
We now demonstrate that for any value of ? < ?SG , there exists an open set around ? = 0 where
the spectral density vanishes. This justifies independently or choice for the parameter ?. The proof
follows [5] and begins by noticing that ?i?j = cosh?2 (?Aij ) is a fixed point of the recursion (18)
for ? = 0. Since this fixed point is real, the corresponding spectral density is 0. Now consider
a small perturbation ?ij of this solution such that ?i?j = cosh?2 (?Aij )(1 + cosh?2 (?Aij )?ij ).
P
The linearized version of (18) writes ?i?j = l??i\j tanh2 (?Ail )?i?l . The linear operator thus
defined is a weighted version of the non-backtracking matrix of [8]. Its spectral radius is given by
? = ?(?), where ? is defined in 10. In particular, for ? < ?SG , ? < 1, so that a straightforward
application [5] of the implicit function theorem allows to show that there exists a neighborhood U of
0 such that for any ? ? U , there exists a real, linearly stable fixed point of (18), yielding a spectral
density equal to 0. At ? = ??SG , the informative eigenvalues (those outside of the bulk), are therefore
exactly the negative ones, which motivates independently our algorithm.
4
Numerical tests
Figure 3 illustrates the ability of the Bethe Hessian to infer the rank above the critical value c in
the limit of large size n, m (see section 3.1). In Figure 4, we demonstrate the suitability of the
eigenvectors of the Bethe Hessian as starting point for the minimization of the cost function (2). We
compare the final RMSE achieved on the reconstructed matrix XY T with 4 other initializations of
the optimization, including the largest singular vectors of the trimmed matrix M [3]. MaCBetH systematically outperforms all the other choices of initial conditions, providing a better initial condition
for the optimization of (2). Remarkably, the performance achieved by MaCBetH with the inferred
rank is essentially the same as the one achieved with an oracle rank. By contrast, estimating the correct rank from the (trimmed) SVD is more challenging. We note that for the choice of parameters
we consider, trimming had a negligible effect. Along the same lines, OptSpace [3] uses a different
minimization procedure, but from our tests we could not see any difference in performance due to
that. When using Alternating Least Squares [23, 24] as optimization method, we also obtained a
similar improvement in reconstruction by using the eigenvectors of the Bethe Hessian, instead of the
singular vectors of M, as initial condition.
7
P(RMSE < 10?1 )
Rank 3
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
Macbeth OR
Tr-SVD OR
Random OR
Macbeth IR
Tr-SVD IR
0.4
0.3
0.2
0.1
0
1
P(RMSE < 10?8 )
Rank 10
1
10
20
30
40
0.4
0.3
0.2
0.1
0
10
1
50
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
10
20
30
40
50
0
10
20
30
20
30
40
50
60
40
50
60
?
?
Figure 4: RMSE as a function of the number of revealed entries per row : comparison between
different initializations for the optimization of the cost function (2). The top row shows the probability that the achieved RMSE is smaller than 10?1 , while the bottom row shows the probability that
the final RMSE is smaller than 10?8 . The probabilities were estimated as the frequency of success
over 100 samples of matrices XY T of size 10000 ? 10000, with the entries of X, Y drawn from a
Gaussian distribution of mean 0 and variance 1. All methods optimize the cost function (2) using a
low storage BFGS algorithm [25] part of NLopt [26], starting from different initial conditions. The
maximum number of iterations was set to 1000. The initial conditions compared are MaCBetH with
oracle rank (MaCBetH OR) or inferred rank (MaCBetH IR), SVD of the observed matrix M after
trimming, with oracle rank (Tr-SVD OR), or inferred rank (Tr-SVD IR, note that this is equivalent
to OptSpace [3] in this regime), and random initial conditions with oracle rank (Random OR). For
the Tr-SVD IR method, we inferred the rank from the SVD by looking for an index for which the
ratio between two consecutive eigenvalues is minimized, as suggested in [27].
5
Conclusion
In this paper, we have presented MaCBetH, an algorithm for matrix completion that is efficient for
two distinct, complementary, tasks: (i) it has the ability to estimate a finite rank r reliably from
fewer random entries than other existing approaches, and (ii) it gives lower root-mean-square reconstruction errors than its competitors. The algorithm is built around the Bethe Hessian matrix and
leverages both on recent progresses in the construction of efficient spectral methods for clustering
of sparse networks [8, 5, 9], and on the OptSpace approach [3] for matrix completion.
The method presented here offers a number of possible future directions, including replacing the
minimization of the cost function by a message-passing type algorithm, the use of different neural
network models, or a more theoretical direction involving the computation of information theoretically optimal transitions for detectability.
Acknowledgment
Our research has received funding from the European Research Council under the European Union?s
7th Framework Programme (FP/2007-2013/ERC Grant Agreement 307087-SPARCS).
8
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9
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5,197 | 5,705 | Robust PCA with compressed data
Wooseok Ha
University of Chicago
[email protected]
Rina Foygel Barber
University of Chicago
[email protected]
Abstract
The robust principal component analysis (RPCA) problem seeks to separate lowrank trends from sparse outliers within a data matrix, that is, to approximate a n?d
matrix D as the sum of a low-rank matrix L and a sparse matrix S. We examine
the robust principal component analysis (RPCA) problem under data compression,
where the data Y is approximately given by (L+S)?C, that is, a low-rank + sparse
data matrix that has been compressed to size n ? m (with m substantially smaller
than the original dimension d) via multiplication with a compression matrix C.
We give a convex program for recovering the sparse component S along with the
compressed low-rank component L ? C, along with upper bounds on the error of
this reconstruction that scales naturally with the compression dimension m and
coincides with existing results for the uncompressed setting m = d. Our results
can also handle error introduced through additive noise or through missing data.
The scaling of dimension, compression, and signal complexity in our theoretical
results is verified empirically through simulations, and we also apply our method
to a data set measuring chlorine concentration across a network of sensors to test
its performance in practice.
1
Introduction
Principal component analysis (PCA) is a tool for providing a low-rank approximation to a data
matrix D 2 Rn?d , with the aim of reducing dimension or capturing the main directions of variation
in the data. More recently, there has been increased focus on more general forms of PCA, that is
more robust to realistic flaws in the data such as heavy-tailed outliers. The robust PCA (RPCA)
problem formulates a decomposition of the data,
D ?L+S ,
into a low-rank component L (capturing trends across the data matrix) and a sparse component S
(capturing outlier measurements that may obscure the low-rank trends), which we seek to separate
based only on observing the data matrix D [3, 10]. Depending on the application, we may be
primarily interested in one or the other component:
? In some settings, the sparse component S may represent unwanted outliers, e.g. corrupted
measurements?we may wish to clean the data by removing the outliers and recovering the
low-rank component L.
? In other settings, the sparse component S may contain the information of interest?for
instance, in image or video data, S may capture the foreground objects which are of interest,
while L may capture background components which we wish to subtract.
Existing methods to separate the sparse and low-rank components include convex [3, 10] and nonconvex [9] methods, and can handle extensions or additional challenges such as missing data [3],
column-sparse rather than elementwise-sparse structure [11], streaming data [6, 7], and different
types of structures superimposed with a low-rank component [1].
1
In this paper, we examine the possibility of demixing sparse and low rank structure, under the
additional challenge of working with data that has been compressed,
Y = D ? C ? (L + S) ? C 2 Rn?m ,
where L, S 2 Rn?d comprise the (approximately) low-rank and (approximately) sparse components
of the original data matrix D, while C 2 Rd?m is a random or fixed compression matrix. In general,
we think of the compression dimension m as being significantly smaller than d, motivated by several
considerations:
? Communication constraints: if the n ? d data matrix consists of d-dimensional measurements taken at n remote sensors, compression would allow the sensors to transmit information of dimension m ? d;
? Storage constraints: storing a matrix with nm many entries instead of nd many entries;
? Data privacy: if the data is represented as the n ? d matrix, where n-dimensional features
were collected from d individuals, we can preserve privacy by compressing the data by a
random linear transformation and allow the access to database only through the compressed
data. This privacy-preserving method has been called matrix masking in the privacy literature and studied by [12] in the context of high-dimensional linear regression.
Random projection methods have been shown to be highly useful for reducing dimensionality without much loss of accuracy for numerical tasks such as least squares regression [8] or low-rank
matrix computations [5]. Here we use random projections to compress data while preserving the
information about the underlying low-rank and sparse structure. [13] also applied random projection
methods to the robust PCA problem, but their purpose is to accelerate the computational task of
low-rank approximation, which is different from the aim of our work.
In the compressed robust PCA setting, we hope to learn about both the low-rank and sparse components. Unlike compressed sensing problems where sparse structure may be reconstructed perfectly
with undersampling, here we face a different type of challenge:
? The sparse component S is potentially identifiable from the compressed component S ? C,
using the tools of compressed sensing; however,
? The low-rank component L is not identifiable from its compression L ? C. Specifically, if
we let PC 2 Rd?d be the projection operator onto the column span of C, then the two
low-rank matrices L and L0 = L ? PC cannot be distinguished after multiplication by C.
Therefore, our goal will be to recover both the sparse component S, and the compressed low-rank
component L ? C. Note that recovering L ? C is similar to the goal of recovering the column span
of L, which may be a useful interpretation if we think of the columns of the data matrix D as data
points lying in Rn ; the column span of L characterizes a low-rank subspace of Rn that captures the
main trends in the data.
Notation We will use the following notation throughout the paper. We write [n] = {1, . . . , n} for
any n 1. We write kvk0 or kM k0 to denote the number of nonzero entries in a vector v or matrix
M (note that this is not in fact a norm). Mi? denotes the ith row of a matrix M and is treated as
a column vector. We will use the matrix norms kM kF (Frobenius norm), kM k1 (elementwise `1
norm), kM k1 (elementwise `1 norm), kM k (spectral norm, i.e. largest singular value), and kM k?
(nuclear norm, also known as the trace norm, given by the sum of the singular values of M ).
2
Problem and method
We begin by formally defining the problem at hand. The data, which takes the form of a n ? d
matrix, is well-approximated by a sum L? + S ? , where L? is low-rank and S ? is sparse. However,
we can only access this data through a (noisy) compression: our observed data is the n ? m matrix
Y = (L? + S ? ) ? C + Z ,
where C 2 R
is the compression matrix, and Z 2 R
noise?we discuss specific models for Z later on.
d?m
2
n?m
(1)
absorbs all sources of error and
Given this model, our goal will be to learn about both the low-rank and sparse structure. In the
ordinary robust PCA setting, the task of separating the low-rank and sparse components has been
known to be possible when the underlying low-rank component L? satisfies certain conditions, e.g.
incoherence condition in [3] or spikiness condition in [1]. In order to successfully decompose the
low-rank and sparse component in the compressed data, we thus need the similar conditions to hold
for the compressed low-rank component, which we define as the product P ? := L? ? C. As we
will see, if L? satisfies the spikiness condition, i.e. kL? k1 ? ?0 , then the compressed low-rank
component P ? satisfies the similar spikiness condition, i.e. a bound on kP ? Ck1 . This motivates
the possibility to recover both the low-rank and sparse components in the case of compressed data.
As discussed above, while we can aim to recover the sparse component S ? , there is no hope to
recover the original low-rank component L? , since L? is not identifiable in the compressed model.
Therefore, we propose a natural convex program for recovering the underlying compressed lowrank component P ? = L? ? C and the sparse component S ? . Note that as discussed in [5], random
projection preserves the column span of L? , and so we can recover the column span of L? via P ? .
We define our estimators of the sparse component S ? , and the low-rank product P ? , as follows:
?
1
b =
(Pb, S)
arg min
kY P S ? Ck2F + ?kP k? + kSk1 .
(2)
(P,S):kP C > k1 ?? 2
Note that we impose the spikiness condition kP C > k1 ? ? on P , in order to guarantee good
performance for demixing such two superimposed components?in later section, we will see that
the same condition holds for P ? . This method is parametrized by the triple (?, ?, ), and natural
scalings for these tuning parameters are discussed alongside our theoretical results.
2.1
Sources of errors and noise
Next, we give several examples of models and interpretations for the error term Z in (1).
Random noise First, we may consider a model where the signal has an exact low-rank + sparse
decomposition, with well-behaved additive noise added before and/or after the compression step:
Y = (L? + S ? + Zpre ) ? C + Zpost ,
where the entries of the pre- and post-compression noise, Zpre and Zpost , are i.i.d. mean-zero subgaussian random variables. In this case, the noise term Z in (1) is given by Z = Zpre ? C + Zpost .
Misspecified model Next, we may consider a case where the original data can be closely approximated by a low-rank + sparse decomposition, but this decomposition is not exact. In this case, we
could express the original (uncompressed) data as L? + S ? + Zmodel , where Zmodel captures the error
of the low-rank + sparse decomposition. Then this model misspecification can be absorbed into the
noise term Z, i.e. Z = Zmodel ? C.
Missing data Given an original data matrix D = L? + S ? , we might have access only to a partial
version of this matrix. We write D? to denote the available data, where ? ? [n] ? [d] indexes the
entries where data is available, and (D? )ij = Dij ? ij2? . Then, a low-rank + sparse model for our
compressed data is given by
?
Y = D? ? C = (L? + S?
) ? C + Zmissing ? C ,
?
?
where Zmissing = L? L . In some settings, we may first want to adjust D? before compressing
the data, for instance, by reweighting the observed entries in D? to ensure a closer approximation
e ? , we have compressed data
to D. Denoting the reweighted matrix of partial observations by D
e ? ? C = (L? + Se? ) ? C + Zmissing ? C ,
Y =D
?
?
?
e?
with Zmissing = L
L? , and where Se?
is the reweighted matrix of S?
. Then the error from the
?
missing data can be absorbed into the Z term, i.e. Z = Zmissing ? C.
Combinations Finally, the observed data Y may differ from the compressed low-rank + sparse
decomposition (L? + S ? ) ? C due to a combination of the factors above, in which case we may write
Z = (Zpre + Zmodel + Zmissing ) ? C + Zpost .
3
2.2
Models for the compression matrix C
Next, we consider several scenarios for the compression matrix C.
Random compression In some settings, the original data naturally lies in Rn?d , but is compressed
by the user for some purpose. For instance, if we have data from d individuals, with each data point
lying in Rn , we may compress this data for the purpose of providing privacy to the individuals in the
data set. Alternately, we may compress data to adhere to constraints on communication bandwidth
or on data storage. In either case, we control the choice of the compression matrix C, and are free
to use a simple random model. Here we consider two models:
iid
Gaussian model: the entries of C are generated as Cij ? N (0, 1/m).
p
Orthogonal model: C = d/m ? U ,
where U 2 Rd?m is an orthonormal matrix chosen uniformly at random.
?
?
Note that in each case, E CC > = Id .
(3)
(4)
Multivariate regression / multitask learning In a multivariate linear regression, we observe a
matrix of data Y that follows a model Y = X ? B + W where X is an observed design matrix,
B is an unknown matrix of coefficients (generally the target parameter), and W is a matrix of
noise terms. Often, the rows of Y are thought of as (independent) samples, where each row is
a multivariate response. In this setting, the accuracy of the regression can often be improved by
leveraging low-rank or sparse structure that arises naturally in the matrix of coefficients B. If B is
approximately low-rank + sparse, the methodology of this paper can be applied: taking the transpose
of the multivariate regression model, we have Y > = B > ? X > + W > . Compare to our initial
model (1), where we replace Y with Y > , and use the compression matrix C = X > . Then, if
B > ? L? + S ? is a low-rank + sparse approximation, the multivariate regression can be formulated
as a problem of the form (1) by setting the error term to equal Z = (B > L? S ? ) ? X > + W > .
3
Theoretical results
In this section, we develop theoretical error bounds for the compressed robust PCA problem under
several of the scenarios described above. We first give a general deterministic result in Section 3.1,
then specialize this result to handle scenarios of pre- and post-compression noise and missing data.
Results for multivariate regression are given in the Supplementary Materials.
3.1
Deterministic result
We begin by stating a version of the Restricted Eigenvalue property found in the compressed sensing
and sparse regression literature [2]:
Definition 1. For a matrix X 2 Rm?d and for c1 , c2
0, X satisfies the restricted eigenvalue
property with constants (c1 , c2 ), denoted by REm,d (c1 , c2 ), if
r
log(d)
kXvk2 c1 kvk2 c2 ?
? kvk1 for all v 2 Rd .
(5)
m
We now give our main result for the accuracy of the convex program (2), a theorem that we will
see can be specialized to many of the settings described earlier. This theorem gives a deterministic
result and does not rely on a random model for the compression matrix C or the error matrix Z.
Theorem 1. Let L? 2 Rn?d be any matrix with rank(L? ) ? r, and let S ? 2 Rn?d be any
?
matrix with at most s nonzero entries per row, that is, maxi kSi?
k0 ? s. Let C 2 Rd?m be any
compression matrix and define the data Y and the error/noise term Z as in
(1). Let P ? = L? ? C
p
as before. Suppose that C > satisfies REm,d (c1 , c2 ), where c0 := c1 c2 ? 16s log(d)/m > 0. If
parameters (?, ?, ) satisfy
kL? CC > k1 , ? 2kZk,
2kZC > k1 + 4?,
b to the convex program (2) satisfies
then deterministically, the solution (Pb, S)
?
kPb
P ? k2F + c20 kSb
S ? k2F ? 18r? 2 + 9c0 2 sn
4
2
.
(6)
We now highlight several applications of this theorem to specific settings: a random compression
model with Gaussian or subgaussian noise, and a random compression model with missing data. (An
application to the multivariate linear regression model is given in the Supplementary Materials.)
3.2
Results for random compression with subgaussian noise
Suppose compression matrix C is random, and that the error term Z in the model (1) comes from
i.i.d. subgaussian noise, e.g. measurement error that takes place before and/or after the compression:
Z = Zpre ? C + Zpost .
Our model for this setting is as follows: for fixed matrices L? and S ? , where rank(L? ) ? r and
?
maxi kSi?
k0 ? s, we observe data
Y = (L? + S ? + Zpre ) ? C + Zpost ,
(7)
where the compression matrix C is generated under either the Gaussian (3) or orthogonal (4) model,
and where the noise matrices Zpre , Zpost are independent from each other and from C, with entries
iid
(Zpre )ij ? N (0,
2
pre )
iid
and (Zpost )ij ? N (0,
2
post )
.
For this section, we assume d m without further comment (that is, the compression should reduce
2
2
2
the dimension of the data). Let max
max{ pre
, post
}. Specializing the result of Theorem 1 to
this setting, we obtain the following probablistic guarantee:
?
Theorem 2. Assume the model (7). Suppose that rank(L? ) ? r, maxi kSi?
k0 ? s, and kL? k1 ?
0 00
?0 . Then there exist universal constants c, c , c > 0 such that if we define
r
r
r
d log(nd)
d(n + m)
d log(nd)
? = 5?0
, ? = 24 max
, = 32 max
+ 4? ,
m
m
m
and if m
kPb
b to the convex program (2) satisfies
c ? s log(nd), then the solution (Pb, S)
P ? k2F + kSb
S ? k2F ? c0 ?
d
m
2
max
? r(n + m) + (
2
max
+ ?02 ) ? sn log(nd)
(8)
00
c
with probability at least 1 nd
.
Remark 1. If the entries of Zpre and Zpost are subgaussian rather than Gaussian, then the same result
holds, except for a change in the constants appearing in the parameters (?, ?, ). (Recall that a
?
?
2 2
random variable X is 2 -subgaussian if E etX ? et /2 for all t 2 R.)
Remark 2. In the case d = m, our result matches Corollary 2 in Agarwal et al [1] exactly, except
that our result involves multiplicative logarithm factor log(nd) in the ?0 term whereas theirs does
not.1 This additional log factor arises when we upper bound kL? CC > k1 , which is unavoidable if
we want the bound to hold with high probability.
Remark 3. Theorem 2 shows the natural scaling: the first term r(n+m) is the degree of freedom for
compressed rank r matrix P whereas the term sn log(nd) is the signal complexity of sparse compod 2
nent S, which has sn many nonzero entries. The multiplicative factor m
max can be interpreted as
the noise variance of the problem amplified by the compression.
3.3
Results for random compression with missing data
Next, we consider a missing data scenario where the original n ? d matrix is only partially observed. The original (complete) data is D = L? + S ? 2 Rn?d , a low-rank + sparse decomposition.2 However, only a subset ? ? [n] ? [d] of entries are observed?we are given access to
Dij for each (i, j) 2 ?. After a reweighting step, we compress this data with a compression matrix
C 2 Rd?m , for instance, in order to reduce communication, storage, or computation requirements.
1
Note that s ? n in our paper is equivalent to s in [1], since their work defines s to be the total number of
nonzero entries in S ? while we count entries per row.
2
For clarity of presentation, we do not include additive noise before or after compression in this section.
However, our theoretical analysis for additive noise (Theorem 2) and for missing data (Theorem 3) can be
combined in a straightforward way to obtain an error bound scaling as a sum of the two respective bounds.
5
First, we specify a model for the missing data. For each (i, j) 2 [n] ? [d], let ?ij 2 [0, 1] be
the probability that this entry is observed. Additionally, we assume that the sampling scheme is
independent across all entries, and that the ?ij ?s are known.3
To proceed, we first define a reweighted version of the partially observed data matrix and then
multiply by the compression matrix C:
e ? ? C where (D
e ? )ij = Dij /?ij ? ij2? .
Y =D
(9)
Define also the reweighted versions of the low rank and sparse components,
e ? )ij = Lij /?ij ? ij2? and (Se? )ij = Sij /?ij ? ij2? ,
(L
?
?
and note that we then have
?
?
?
?
?
?
e ?? + Se?
Y = L
? C = L? + Se?
?C +Z ,
(10)
e ? L? ) ? C. The role of the reweighting step (9) is to ensure that this noise term Z has
where Z = (L
?
mean zero. Note that in the reformulation (10) of the model, Y is approximated with a compression
?
?
of L? + Se?
, where L? is the original low rank component while Se?
is defined above. While the
?
original sparse component S , is not identifiable via the missing data model (since we have no
?
?
information to help us recover entries Sij
for (i, j) 62 ?), this new decomposition L? + Se?
now has
?
a sparse component that is identifiable, since by definition, Se?
preserves the sparsity of S ? but has
?
no nonzero entries in unobserved locations, that is, (Se?
)ij = 0 whenever (i, j) 62 ?.
With this model in place, we obtain the following probabilistic guarantee for this setting, which is
another specialized version of Theorem 1. We note that we again have no assumptions on the values
of the entries in S ? , only on the sparsity level?e.g. there is no bound assumed on kS ? k1 .
?
Theorem 3. Assume the model (9). Suppose that rank(L? ) ? r, maxi kSi?
k0 ? s, and kL? k1 ?
?0 . If the sampling scheme satisfies ?ij
?min for all (i, j) 2 [n] ? [d] for some positive constant
?min > 0, then there exist universal constants c, c0 , c00 > 0 such that if we define
s
r
r
d log(nd)
d(n + m) log(nd)
d log2 (nd)
1
1
, ? = 10?min ?0
, = 12?min ?0
+ 4? ,
? = 5?0
m
m
m
and if m
b to the convex program (2) satisfies
c ? s log(nd), then the solution (Pb, S)
d
? 2
kPb P ? k2F + kSb Se?
kF ? c0 ?
? ? 2 ?2 r(n + m) log(nd) + sn log2 (nd)
m min 0
with probability at least 1
4
c00
nd .
Experiments
In this section, we first use simulated data to study the behavior of the convex program (2) for
different compression dimensions, signal complexities and missing levels, which show the close
agreement with the scaling predicted by our theory. We also apply our method to a data set consisting
of chlorine measurements across a network of sensors. For simplicity, in all experiments, we select
? = 1, which is easier for optimization and generally results in a solution that still has low spikiness
(that is, the solution is the same as if we had imposed a bound with finite ?).
4.1
Simulated data
Here we run a series of simulations on compressed data to examine the performance of the convex program (2). In all cases, we used the compression matrix C generated under the orthogonal
model (4). We solve the convex program (2) via alternating minimization over L and S, selecting
the regularization parameters ? and that minimizes the squared Frobenius error. All results are
averaged over 5 trials.
3
?i
In practice, the assumption that ?ij ?s are known is not prohibitive. For example, we might model ?ij =
(the row
locations of the observed entries are chosen independently, e.g. see [4]), or a logistic
? and column
?
j
model, log
?ij
1 ?ij
= ?i +
j.
In either case, fitting a model using the observed set ? is extremely accurate.
6
Total squared error
2
?10 5
n=d=800
n=d=400
1.5
1
0.5
0
0
2
4
6
8
10
Compression ratio d/m
2
?10 4
Dimension n=d=200
Total squared error
Total squared error
Figure 1: Results for the noisy data experiment. The total squared error, calculated as in Theorem 2,
is plotted against the compression ratio d/m. Note the linear scaling, as predicted by the theory.
m=50
m=100
m=150
m=200
1.5
1
0.5
0
0
10
20
30
40
50
?10 4
5
Dimension n=d=400
m=100
m=200
m=300
m=400
4
3
2
1
0
0
10
20
8
?10 4
Dimension n=d=200
m=50
m=100
m=150
m=200
6
4
2
0
0
0.02
0.04
0.06
30
40
50
Rank
Total squared error
Total squared error
Rank
0.08
0.1
Sparsity proportion
15
?10 4
Dimension n=d=400
m=100
m=200
m=300
m=400
10
5
0
0
0.02
0.04
0.06
0.08
0.1
Sparsity proportion
Figure 2: Results for the varying-rank (top row) and varying-sparsity (bottom row) experiments. The
total squared error, calculated as in Theorem 2, is plotted against the rank r or sparsity proportion
s/d. Note the nearly linear scaling for most values of m.
Simulation 1: compression ratio. First we examine the role of the compression dimension
p m. We
fix the matrix dimension n = d 2 {400, 800}. The low-rank component is given by L? = r?U V > ,
where U and V are n ? r and d ? r matrices with i.i.d. N (0, 1) entries, for rank r = 10. The
sparse component S ? has 1% of its entries generated as 5 ? N (0, 1), that is, s = 0.01d. The data
iid
is D = L? + S ? + Z, where Zij ? N (0, 0.25). Figure 1 shows the squared Frobenius error
kPb P ? k2F + kSb S ? k2F plotted against the compression ratio d/m. We see error scaling linearly
with the compression ratio, which supports our theoretical results.
Simulation 2: rank and sparsity. Next we study the role of rank and sparsity, for a matrix of size
n = d = 200 or n = d = 400. We generate the data D as before, but we either vary the rank
r 2 {5, 10, . . . , 50}, or we vary the sparsity s with s/d 2 {0.01, 0.02, . . . , 0.1}. Figure 2 shows the
squared Frobenius error plotted against either the varying rank or the varying sparsity. We repeat this
experiment for several different compression dimensions m. We see a little deviation from linear
scaling for the smallest m, which can be due to the fact that our theorems give upper bounds rather
than tight matching upper and lower bounds (or perhaps the smallest value of m does not satisfy the
condition stated in the theorems). However, for all but the smallest m, we see error scaling nearly
linearly with rank or with sparsity, which is consistent with our theory.
Simulation 3: missing data. Finally, we perform experiments under the existence of missing
entries in the data matrix D = L? + S ? . We fix dimensions n = d = 400 and generate L? and S ?
as before, with r = 10 and s = 0.01d, but do not add noise. To introduce the missing entries in
the data, we use a uniform sampling scheme, where each entry of D is observed with probability ?,
7
?10 5
7
Total squared error
Total squared error
7
m=100
m=200
m=300
m=400
6
5
4
3
2
1
0
0
0.2
0.4
0.6
0.8
m=100
m=200
m=300
m=400
6
5
4
3
2
1
0
1
?10 5
0
20
40
60
80
100
1/?2
?
Figure 3: Results for the missing data experiment. The total squared error, calculated as in Theorem 3, is plotted against ? (proportion of observed data) or against 1/?2 , for various values of m,
based on one trial. Note the nearly linear scaling with respect to 1/?2 .
Log(relative error)
?1
Low?rank + sparse model
Low?rank model
?2
?3
?4
?5
0
1000
2000
3000
Compression dimension m
4000
Figure 4: Results for the chlorine data (averaged over 2 trials), plotting the log of the relative error
on the test set for a low-rank + sparse model and a low-rank-only model. The low-rank + sparse
model performs better across a range of compression dimensions m (up to 8?9% reduction in error).
? 2
with ? 2 {0.1, 0.2, . . . , 1}. Figure 3 shows the squared Frobenius error kPb P ? k2F + kSb Se?
kF
(see Theorem 3 for details) across a range of probabilities ?. We see that the squared error scales
approximately linearly with 1/?2 , as predicted by our theory.
4.2
Chlorine sensor data
To illustrate the application of our method to a specific application, we consider chlorine concentration data from a network of sensors.4 The data contains a realistic simulation of chlorine concentration measurements from n = 166 sensors in a hydraulic system over d = 4310 time points. We
assume D is well approximated with a low-rank + sparse decomposition. We then compress the
data using the orthogonal model (4) and study the performance of our estimators (2) for varying m.
In order to evaluate performance, we use 80% of the entries to fit the model, 10% as a validation set
for selecting tuning parameters, and the final 10% as a test set. We compare against a low-rank mab (Details
trix reconstruction, equivalent to setting Sb = 0 and fitting only the low-rank component L.
are given in the Supplementary Materials.) The results are displayed in Figure 4, where we see that
the error of the recovery grows smoothly with compression dimension m, and that the low-rank +
sparse decomposition gives better data reconstruction than the low-rank-only model.
5
Discussion
In this paper, we have examined the robust PCA problem under data compression, where we seek to
decompose a data matrix into low-rank + sparse components with access only to a partial projection
of the data. This provides a tool for accurate modeling of data with multiple superimposed structures, while enabling restrictions on communication, privacy, or other considerations that may make
compression necessary. Our theoretical results show an intuitive tradeoff between the compression
ratio and the error of the fitted low-rank + sparse decomposition, which coincides with existing
results in the extreme case of no compression (compression ratio = 1). Future directions for this
problem include adapting the method to the streaming data (online learning) setting.
4
Data obtained from http://www.cs.cmu.edu/afs/cs/project/spirit-1/www/
8
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pages 33?40, 2011.
9
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5,198 | 5,706 | Mixed Robust/Average Submodular Partitioning:
Fast Algorithms, Guarantees, and Applications
Kai Wei1
Rishabh Iyer1
Shengjie Wang2
Wenruo Bai1
Jeff Bilmes1
1
Department of Electrical Engineering, University of Washington
2
Department of Computer Science, University of Washington
{kaiwei, rkiyer, wangsj, wrbai, bilmes}@u.washington.edu
Abstract
We investigate two novel mixed robust/average-case submodular data partitioning
problems that we collectively call Submodular Partitioning. These problems generalize purely robust instances of the problem, namely max-min submodular fair
allocation (SFA) [12] and min-max submodular load balancing (SLB) [25], and
also average-case instances, that is the submodular welfare problem (SWP) [26]
and submodular multiway partition (SMP) [5]. While the robust versions have been
studied in the theory community [11, 12, 16, 25, 26], existing work has focused
on tight approximation guarantees, and the resultant algorithms are not generally
scalable to large real-world applications. This is in contrast to the average case,
where most of the algorithms are scalable. In the present paper, we bridge this gap,
by proposing several new algorithms (including greedy, majorization-minimization,
minorization-maximization, and relaxation algorithms) that not only scale to large
datasets but that also achieve theoretical approximation guarantees comparable
to the state-of-the-art. We moreover provide new scalable algorithms that apply
to additive combinations of the robust and average-case objectives. We show that
these problems have many applications in machine learning (ML), including data
partitioning and load balancing for distributed ML, data clustering, and image segmentation. We empirically demonstrate the efficacy of our algorithms on real-world
problems involving data partitioning for distributed optimization (of convex and
deep neural network objectives), and also purely unsupervised image segmentation.
1
Introduction
The problem of data partitioning is of great importance to many machine learning (ML) and data
science applications as is evidenced by the wealth of clustering procedures that have been and continue
to be developed and used. Most data partitioning problems are based on expected, or average-case,
utility objectives where the goal is to optimize a sum of cluster costs, and this includes the ubiquitous
k-means procedure [1]. Other algorithms are based on robust objective functions [10], where the
goal is to optimize the worst-case cluster cost. Such robust algorithms are particularly important
in mission critical applications, such as parallel and distributed computing, where one single poor
partition block can significantly slow down an entire parallel machine (as all compute nodes might
need to spin while waiting for a slow node to complete a round of computation). Taking a weighted
combination of both robust and average case objective functions allows one to balance between
optimizing worst-case and overall performance. We are unaware, however, of any previous work that
allows for a mixing between worst- and average-case objectives in the context of data partitioning.
This paper studies two new mixed robust/average-case partitioning problems of the following form:
1
m
m
h
i
h
i
X
? X
?
?
? min fi (A? ) + ?
?
Prob. 1: max ?
f
(A
)
,
Prob.
2:
min
?
max
f
(A
)
+
fj (A?j ) ,
j
i
i
j
i
i
i
???
???
m j=1
m j=1
? , 1 ? ?, the set of sets ? = (A? , A? , ? ? ? , A? ) is a partition of a finite
where 0 ? ? ? 1, ?
m
1
2
?
set V (i.e, ?i Ai = V and ?i 6= j, A?i ? A?j = ?), and ? refers to the set of all partitions of
V into m blocks. The parameter ? controls the objective: ? = 1 is the average case, ? = 0 is
the robust case, and 0 < ? < 1 is a mixed case. In general, Problems 1 and 2 are hopelessly
intractable, even to approximate, but we assume that the f1 , f2 , ? ? ? , fm are all monotone nondecreasing (i.e., fi (S) ? fi (T ) whenever S ? T ), normalized (fi (?) = 0), and submodular [9] (i.e.,
?S, T ? V , fi (S) + fi (T ) ? fi (S ? T ) + fi (S ? T )). These assumptions allow us to develop fast,
simple, and scalable algorithms that have approximation guarantees, as is done in this paper. These
assumptions, moreover, allow us to retain the naturalness and applicability of Problems 1 and 2 to
a wide variety of practical problems. Submodularity is a natural property in many real-world ML
applications [20, 15, 18, 27]. When minimizing, submodularity naturally model notions of interacting
costs and complexity, while when maximizing it readily models notions of diversity, summarization
quality, and information. Hence, Problem 1 asks for a partition whose blocks each (and that
collectively) are a good, say, summary of the whole. Problem 2 on the other hand, asks for a partition
whose blocks each (and that collectively) are internally homogeneous (as is typical in clustering).
Taken together, we call Problems 1 and 2 Submodular Partitioning. We further categorize these
problems depending on if the fi ?s are identical to each other (homogeneous) or not (heterogeneous).1
The heterogeneous case clearly generalizes the homogeneous setting, but as we will see, the additional
homogeneous structure can be exploited to provide more efficient and/or tighter algorithms.
Problem 1 (Max-(Min+Avg))
Approximation factor
? = 0, B IN S RCH [16]
1/(2m ? 1)
? = 0, M ATCHING [12]
1/(n ? m + 1)
1
3
?
? = 0, E LLIPSOID [11]
O( nm 4 log n log 2 m)
? = 1, G REEDW ELFARE [8]
1/2
?
)
? = 0, G REED S AT?
(1/2 ? ?, 1/2+?
? = 0, MM AX?
O(min
i
??
1/m
max{ ? ?? , ??}
0 < ? < 1, G ENERAL G REED S AT?
?/2
??+?
?
? = 0, MM IN?
1
? |?m log3 m )
|A?
i
? = 0, G REED M AX
0 < ? < 1, C OMB S FA S WP?
? = 0, Hardness
? = 1, Hardness
Problem 2 (Min-(Max+Avg))
Approximation factor
? = 0, BALANCED? [25]
min{m, n/m}
?
? = 0, S AMPLING [25]
O( n log n)
?
? = 0, E LLIPSOID [11]
O( n log n)
? = 1, G REED S PLIT? [29, 22]
2
? = 1, R ELAX [5]
O(log n)
2max|A?
i |
i
?
? ROUND
? = 0, L OV ASZ
0 < ? < 1, C OMB S LB S MP?
? ROUND?
0 < ? < 1, G ENERAL L OV ASZ
?
1/2 [12]
1 ? 1/e [26]
? = 0, Hardness
? = 1, Hardness
m
? + ?)}
min{ m?
, ?(m?
?
m?+?
m
m
2 ? 2/m [7]
Table 1: Summary of our contributions and existing work on Problems 1 and 2.2 See text for details.
Previous work: Special cases of Problems 1 and 2 have appeared previously. Problem 1 with ? = 0
is called submodular fair allocation (SFA), and Problem 2 with ? = 0 is called submodular load
balancing (SLB), robust optimization problems both of which previously have been studied. When
fi ?s are all modular, SLB is called minimum makespan scheduling. An LP relaxation algorithm
provides a 2-approximation for the heterogeneous setting [19]. When the objectives are submodular,
the problem becomes much harder. Even in the homogeneous
p setting, [25] show that the problem
is information theoretically hard to approximate within o( n/ log n). They provide a balanced
partitioning algorithm yielding a factor of min{m,
p n/m} under the homogeneous setting. They also
give a sampling-based algorithm achieving O( n/ log n) for the homogeneous setting. However,
the sampling-based algorithm is not practical and scalable since it involves solving, in the worst-case,
O(n3 log n) instances of submodular function minimization each of which requires O(n5 ? + n6 )
computation [23], where ? is the cost of a function valuation. Another approach approximates
each submodular function by its ellipsoid approximation (again non-scalable) and reduces SLB
to its
? modular version (minimum makespan scheduling) leading to an approximation factor of
O( n log n) [11]. SFA, on the other hand, has been studied mostly in the heterogeneous setting.
When f?i ?s are all modular, the tightest algorithm, so far, is to iteratively round an LP solution achieving
O(1/( m log3 m)) approximation [2], whereas the problem is NP-hard to 1/2 + approximate for
any > 0 [12]. When fi ?s are submodular, [12] gives a matching-based algorithm with a factor
1/(n ? m + 1) approximation that performs poorly when m n. [16] proposes a binary search
algorithm yielding an improved factor of 1/(2m ? 1). Similar to SLB, [11] applies the same ellipsoid
1
2
Similar sub-categorizations have been called the ?uniform? vs. the ?non-uniform? case in the past [25, 11].
Results obtained in this paper are marked as ?. Methods for only the homogeneous setting are marked as ?.
2
?
approximation techniques leading to a factor of O( nm1/4 log n log3/2 m). These approaches are
theoretically interesting, but they do not scale to large problems. Problems 1 and 2, when ? = 1, have
also been previously studied. Problem 2 becomes the submodular multiway partition (SMP) for which
one can obtain a relaxation based 2-approximation [5] in the homogeneous case. In the heterogeneous
case, the guarantee is O(log n) [6]. Similarly, [29, 22] propose a greedy splitting 2-approximation
algorithm for the homogeneous setting. Problem 1 becomes the submodular welfare [26] for which a
scalable greedy algorithm achieves a 1/2 approximation [8]. Unlike the worst case (? = 0), many of
the algorithms proposed for these problems are scalable. The general case (0 < ? < 1) of Problems 1
and 2 differs from either of these extreme cases since we wish both for a robust (worst-case) and
average case partitioning, and controlling ? allows one to trade off between the two. As we shall see,
the flexibility of a mixture can be more natural in certain applications.
Applications: There are a number of applications of submodular partitioning in ML as outlined below.
Some of these we evaluate in Section 4. Submodular functions naturally capture notions of interacting
cooperative costs and homogeneity and thus are useful for clustering and image segmentation [22,
17]. While the average case instance has been used before, a more worst-case variant (i.e., Problem 2
with ? ? 0) is useful to produce balanced clusterings (i.e., the submodular valuations of all the
blocks should be similar to each other). Problem 2 also addresses a problem in image segmentation,
namely how to use only submodular functions (which are instances of pseudo-Boolean functions) for
multi-label (i.e., non-Boolean) image segmentation. Problem 2 addresses this problem by allowing
each segment j to have its own submodular function fj , and the objective measures the homogeneity
fj (A?j ) of segment j based on the image pixels A?j assigned to it. Moreover, by combining the average
case and the worst case objectives, one can achieve a tradeoff between the two. Empirically, we
evaluate our algorithms on unsupervised image segmentation (Section 4) and find that it outperforms
other clustering methods including k-means, k-medoids, spectral clustering, and graph cuts.
Submodularity also accurately represents computational costs in distributed systems, as shown
in [20]. In fact, [20] considers two separate problems: 1) text data partitioning for balancing memory
demands; and 2) parameter partitioning for balancing communication costs. Both are treated by
solving an instance of SLB (Problem 2, ? = 0) where memory costs are modeled using a set-cover
submodular function and the communication costs are modeled using a modular (additive) function.
Another important ML application, evaluated in Section 4, is distributed training of statistical
models. As data set sizes grow, the need for statistical training procedures tolerant of the distributed
data partitioning becomes more important. Existing schemes are often developed and performed
assuming data samples are distributed in an arbitrary or random fashion. As an alternate strategy,
if the data is intelligently partitioned such that each block of samples can itself lead to a good
approximate solution, a consensus amongst the distributed results could be reached more quickly
than when under a poor partitioning. Submodular functions can in fact express the value of a
subset of training data for certain machine learning risk functions, e.g., [27]. Using these functions
within Problem 1, one can expect a partitioning (by formulating the problem as an instance of
Problem 1, ? ? 0) where each block is a good representative of the entire set, thereby achieving
faster convergence in distributed settings. We demonstrate empirically, in Section 4, that this provides
better results on several machine learning tasks, including the training of deep neural networks.
Our Contributions: In contrast to Problems 1 and 2 in the average case (i.e., ? = 1), existing
algorithms for the worst case (? = 0) are not scalable. This paper closes this gap, by proposing
three new classes of algorithmic frameworks to solve SFA and SLB: (1) greedy algorithms; (2)
semigradient-based algorithms; and (3) a Lov?asz extension based relaxation algorithm. For SFA,
when m = 2, we formulate the problem as non-monotone submodular maximization, which can
be approximated up to a factor of 1/2 with O(n) function evaluations [4]. For general m, we give a
simple and scalable greedy algorithm (G REED M AX), and show a factor of 1/m in the homogeneous
setting, improving the state-of-the-art factor of 1/(2m ? 1) under the heterogeneous setting [16].
For the heterogeneous setting, we propose a ?saturate? greedy algorithm (G REED S AT) that iteratively
solves instances of submodular welfare problems. We show G REED S AT has a bi-criterion guarantee
of (1/2 ? ?, ?/(1/2 + ?)), which ensures at least dm(1/2 ? ?)e blocks receive utility at least
?/(1/2 + ?)OP T for any 0 < ? < 1/2. For SLB, we first generalize thephardness result in [25]
and show that it is hard to approximate better than m for any m = o( n/ log n) even in the
? ROUND)
homogeneous setting. We then give a Lov?asz extension based relaxation algorithm (L OV ASZ
yielding a tight factor of m for the heterogeneous setting. As far as we know, this is the first algorithm
achieving a factor of m for SLB in this setting. For both SFA and SLB, we also obtain more efficient
3
algorithms with bounded approximation factors, which we call majorization-minimization (MM IN)
and minorization-maximization (MM AX).
Next we show algorithms to handle Problems 1 and 2 with general 0 < ? < 1. We first give two simple and generic schemes (C OMB S FA S WP and C OMB S LB S MP), both of which efficiently combines an
algorithm for the worst-case problem (special case with ? = 0), and an algorithm for the average case
(special case with ? = 1) to provide a guarantee interpolating between the two bounds. For Problem 1
we generalize G REED S AT leading to G ENERAL G REED S AT, whose guarantee smoothly interpolates in
terms of ? between the bi-criterion factor by G REED S AT in the case of ? = 0 and the constant factor of
? ROUND to ob1/2 by the greedy algorithm in the case of ? = 1. For Problem 2 we generalize L OV ASZ
? ROUND) that achieves an m-approximation for general
tain a relaxation algorithm (G ENERAL L OV ASZ
?. The theoretical contributions and the existing work for Problems 1 and 2 are summarized in Table 1.
Lastly, we demonstrate the efficacy of Problem 2 on unsupervised image segmentation, and the
success of Problem 1 to distributed machine learning, including ADMM and neural network training.
2
Robust Submodular Partitioning (Problems 1 and 2 when ? = 0)
Notation: we define f (j|S) , f (S ? j) ? f (S) as the gain of j ? V in the context of S ? V . We
assume w.l.o.g. that the ground set is V = {1, 2, ? ? ? , n}.
2.1 Approximation Algorithms for SFA (Problem 1 with ? = 0)
We first study approximation algorithms for SFA. When m = 2, the problem becomes maxA?V g(A)
where g(A) = min{f1 (A), f2 (V \ A)} and is submodular thanks to Theorem 2.1.
Theorem 2.1. If f1 and f2 are monotone submodular, min{f1 (A), f2 (V \ A)} is also submodular.
Proofs for all theorems in this paper are given in [28]. The simple bi-directional randomized greedy
algorithm [4] therefore approximates SFA with m = 2 to a factor of 1/2 matching the problem?s
hardness. For general m, we approach SFA from the perspective of the greedy algorithms. In
this work we introduce two variants of a greedy algorithm ? G REED M AX (Alg. 1) and G REED S AT
(Alg. 2), suited to the homogeneous and heterogeneous settings, respectively.
G REED M AX: The key idea of G REED M AX (see Alg. 1) is to greedily add an item with the
maximum marginal gain to the block whose current solution is minimum. Initializing {Ai }m
i=1 with
the empty sets, the greedy flavor also comes from that it incrementally grows the solution by greedily
improving the overall objective mini=1,...,m fi (Ai ) until {Ai }m
i=1 forms a partition. Besides its
simplicity, Theorem 2.2 offers the optimality guarantee.
Theorem 2.2. G REED M AX achieves a guarantee of 1/m under the homogeneous setting.
By assuming the homogeneity of the fi ?s, we obtain a very simple 1/m-approximation algorithm
improving upon the state-of-the-art factor 1/(2m ? 1) [16]. Thanks to the lazy evaluation trick as
described in [21], Line 5 in Alg. 1 need not to recompute the marginal gain for every item in each
round, leading G REED M AX to scale to large data sets.
G REED S AT: Though simple and effective in the homogeneous setting, G REED M AX performs
arbitrarily poorly under the heterogeneous setting. To this end we provide another algorithm ?
?Saturate? Greedy (G REED S AT, see Alg. 2). The key idea of G REED S AT is to relax SFA to a much
simpler problem ? Submodular Welfare (SWP), i.e., Problem 1 with ? = 0. Similar
Pm in flavor to the
one proposed in [18] G REED S AT defines an intermediate objective F? c (?) = i=1 fic (A?i ), where
1
fic (A) = m
min{fi (A), c} (Line 2). The parameter c controls the saturation in each block. fic satisfies submodularity for each i. Unlike SFA, the combinatorial optimization problem max??? F? c (?)
(Line 6) is much easier and is an instance of SWP. In this work, we solve Line 6 by the efficient
greedy algorithm as described in [8] with a factor 1/2. One can also use a more computationally
expensive multi-linear relaxation algorithm as given in [26] to solve Line 6 with a tight factor
? = (1 ? 1/e). Setting the input argument ? as the approximation factor for Line 6, the essential idea
of G REED S AT is to perform a binary search over the parameter c to find the largest c? such that the
?
?
?
returned solution ?
? c for the instance of SWP satisfies F? c (?
? c ) ? ?c? . G REED S AT terminates after
mini fi (V )
solving O(log(
)) instances of SWP. Theorem 2.3 gives a bi-criterion optimality guarantee.
Theorem 2.3. Given > 0, 0 ? ? ? 1 and any 0 < ? < ?, G REED S AT finds a partition such that
?
at least dm(? ? ?)e blocks receive utility at least 1??+?
(max??? mini fi (A?i ) ? ).
4
Algorithm 4: G REED M IN
Algorithm 1: G REED M AX
1:
2:
3:
4:
5:
6:
7:
8:
1:
2:
3:
4:
5:
6:
7:
8:
Input: f , m, V .
Let A1 =, . . . , = Am = ?; R = V .
while R 6= ? do
j ? ? argminj f (Aj );
a? ? argmaxa?R f (a|Aj ? )
Aj ? ? Aj ? ? {a? }; R ? R \ a?
end while
Output {Ai }m
i=1 .
Algorithm 2: G REED S AT
Algorithm 5: MM IN
1: Input: {fi }m
V , ?.
i=1 , m,
1 Pm
?
2: Let F? c (?) = m
i=1 min{fi (Ai ), c}.
3: Let cmin = 0, cmax = mini fi (V )
4: while cmax ? cmin ? do
5:
c = 21 (cmax + cmin )
6:
?
? c ? argmax??? F? c (?)
7:
if F? c (?
? c ) < ?c then
8:
cmax = c
9:
else
10:
cmin = c; ?
???
?c
11:
end if
12: end while
13: Output: ?? .
0
1: Input: {fi }m
i=1 , m, V , partition ? .
2: Let t = 0
3: repeat
4:
for i = 1, . . . , m do
t
5:
Pick a supergradient mi at A?
i for fi .
6:
end for
7:
? t+1 ? argmin??? maxi mi (A?
i )
8:
t = t + 1;
9: until ? t = ? t?1
10: Output: ? t .
Algorithm 6: MM AX
0
1: Input: {fi }m
i=1 , m, V , partition ? .
2: Let t = 0.
3: repeat
4:
for i = 1, . . . , m do
t
5:
Pick a subgradient hi at A?
i for fi .
6:
end for
7:
? t+1 ? argmax??? mini hi (A?
i )
8:
t = t + 1;
9: until ? t = ? t?1
10: Output: ? t .
? ROUND
Algorithm 3: L OV ASZ
1:
2:
3:
4:
5:
6:
7:
Input: f , m, V ;
Let A1 =, . . . , = Am = ?; R = V .
while R 6= ? do
j ? ? argminj f (Aj )
a? ? mina?R f (a|Aj ? )
Aj ? ? Aj ? ? a? ; R ? R \ a?
end while
Output {Ai }m
i=1 .
? m
Input: {fi }m
i=1 , {fi }i=1 , m, V .
Solve for {x?i }m
i=1 via convex relaxation.
Rounding: Let A1 =, . . . , = Am = ?.
for j = 1, . . . , n do
?i ? argmaxi x? (j); A? = A? ? j
i
i
i
end for
m
Output {Ai }i=1 .
For any 0 < ? < ? Theorem 2.3 ensures that the top dm(? ? ?)e valued blocks in the partition
returned by G REED S AT are (?/(1??+?)?)-optimal. ? controls the trade-off between the number of
top valued blocks to bound and the performance guarantee attained for these blocks. The smaller ? is,
the more top blocks are bounded, but with a weaker guarantee. We set the input argument ? = 1/2 (or
? = 1 ? 1/e) as the worst-case performance guarantee for solving SWP so that the above theoretical
analysis follows. However, the worst-case is often achieved only by very contrived submodular
functions. For the ones used in practice, the greedy algorithm often leads to near-optimal solution
([18] and our own observations). Setting ? as the actual performance guarantee for SWP (often very
close to 1) can improve the empirical bound, and we, in practice, typically set ? = 1 to good effect.
MM AX: Lastly, we introduce another algorithm for the heterogeneous setting, called minorizationmaximization (MM AX, see Alg. 6). Similar to the one proposed in [14], the idea is to iteratively
maximize tight lower bounds of the submodular functions. Submodular functions have tight modular
lower bounds, which are related to the subdifferential ?f (Y ) of the submodular set function f at a set
Y ? V [9]. Denote a subgradient at Y by hY ? ?f (Y ), the extreme points of ?f (Y ) may be computed via a greedy algorithm: Let ? be a permutation of V that assigns the elements in Y to the first
|Y | positions (?(i) ? Y if and only if i ? |Y |). Each such permutation defines a chain with elements
?
?
S0? = ?, Si? = {?(1), ?(2), . . . , ?(i)}, and S|Y
| = Y . An extreme point hY of ?f (Y ) has each entry
?
?
?
?
as hY (?(i))
P= f (Si ) ? f (Si?1 ). Defined as above, hY forms a lower bound of f , tight at Y ? i.e.,
h?Y (X) = j?X h?Y (j) ? f (X), ?X ? V and h?Y (Y ) = f (Y ). The idea of MM AX is to consider a
modular lower bound tight at the set corresponding to each block of a partition. In other words, at itert
ation t + 1, for each block i, we approximate fi with its modular lower bound tight at A?i and solve a
modular version of Problem 1 (Line 7), which admits efficient approximation algorithms [2]. MM AX
is initialized with a partition ? 0 , which isP
obtained by solving Problem 1, where each fi is replaced
with a simple modular function fi0 (A) = a?A fi (a). The following worst-case bound holds:
?
?
?
1+(|A?
i |?1)(1??fi (Ai ))
?
), where
? | m log3 m
|A?
i
f (v|A\v)
?f (A) = 1 ? minv?V f (v) ?
Theorem 2.4. MM AX achieves a worst-case guarantee of O(mini
?
) is the partition obtained by the algorithm, and
?
? = (A?1? , ? ? ? , A?m
[0, 1] is the curvature of a submodular function f at A ? V .
5
2.2
Approximation Algorithms for SLB (Problem 2 with ? = 0)
p
We next investigate SLB, where existing hardness
p results [25] are o( n/ log n), which is independent
of m and implicitly assumes that m = ?( n/ log n). However, applications for SLB are often
dependent on m with m n. We hence offer hardness analysis in terms of m in the following.
Theorem
p 2.5. For any > 0, SLB cannot be approximated to a factor of (1 ? )m for any
m = o( n/ log n) with polynomial number of queries even under the homogeneous setting.
p
For the rest of the paper, we assume m = o( n/ log n) for SLB, unless stated otherwise.
G REED M IN: Theorem 2.5 implies that SLB is hard to approximate better than m. However, an
arbitrary partition ? ? ? already achieves the best approximation factor of m that one can hope
P
0
0
for under the homogeneous setting, since maxi f (A?i ) ? f (V ) ? i f (A?i ) ? m maxi f (A?i )
0
for any ? ? ?. In practice, one can still implement a greedy style heuristic, which we refer to as
G REED M IN (Alg. 4). Very similar to G REED M AX, G REED M IN only differs in Line 5, where the
item with the smallest marginal gain is added. Since the functions are all monotone, any additions
to a block can (if anything) only increase its value, so we choose to add to the minimum valuation
block in Line 4 to attempt to keep the maximum valuation block from growing further.
? ROUND: Next we consider the heterogeneous setting, for which we propose a tight
L OV ASZ
? ROUND (see Alg. 3). The algorithm proceeds as follows: (1) apply the Lov?asz
algorithm ? L OV ASZ
extension of submodular functions to relax SLB to a convex program, which is exactly solved to
a fractional solution (Line 2); (2) map the fractional solution to a partition using the ?-rounding
technique as proposed in [13] (Line 3 - 6). The Lov?asz extension, which naturally connects a
submodular function f with its convex relaxation f?, is defined as follows: given any x ? [0, 1]n ,
we obtain a permutation ?x by ordering its elements in non-increasing order, and thereby a chain of
sets S0?x ?, . . . , ? Sn?x with Sj?x = {?x (1), . . . , ?x (j)} for j = 1, . . . , n. The Lov?asz extension f?
Pn
?x
for f is the weighted sum of the ordered entries of x: f?(x) = j=1 x(?x (j))(f (Sj?x ) ? f (Sj?1
)).
?
Given the convexity of the fi ?s , SLB is relaxed to the following convex program:
m
X
?
min
max fi (xi ), s.t
xi (j) ? 1, for j = 1, . . . , n
(1)
n
x1 ,...,xm ?[0,1]
i
i=1
Denoting the optimal solution for Eqn 1 as {x?1 , . . . , x?m }, the ?-rounding step simply maps each
? ROUND is as follows:
item j ? V to a block ?i such that ?i ? argmaxi x?i (j) . The bound for L OV ASZ
? ROUND achieves a worst-case approximation factor m.
Theorem 2.6. L OV ASZ
? ROUND is the first algorithm that is tight
We remark that, to the best of our knowledge, L OV ASZ
and that gives an approximation in terms of m for the heterogeneous setting.
MM IN: Similar to MM AX for SFA, we propose Majorization-Minimization (MM IN, see Alg. 5)
for SLB. Here, we iteratively choose modular upper bounds, which are defined via superdifferentials
? f (Y ) of a submodular function [15] at Y . Moreover, there are specific supergradients [14] that
define the following two modular upper bounds (when referring to either one, we use mfX ):
mfX,1 (Y ) , f (X) ?
X
j?X\Y
f (j|X\j) +
X
f (j|?), mfX,2 (Y ) , f (X) ?
j?Y \X
X
j?X\Y
f (j|V \j) +
X
f (j|X).
j?Y \X
Then mfX,1 (Y ) ? f (Y ) and mfX,2 (Y ) ? f (Y ), ?Y ? V and mfX,1 (X) = mfX,2 (X) = f (X). At
iteration t + 1, for each block i, MM IN replaces fi with a choice of its modular upper bound mi tight
t
at A?i and solves a modular version of Problem 2 (Line 7), for which there exists an efficient LP relaxation based algorithm [19]. Similar to MM AXP
, the initial partition ? 0 is obtained by solving Problem
2, where each fi is substituted with fi0 (A) = a?A fi (a). The following worst-case bound holds:
Theorem 2.7. MM IN achieves a worst-case guarantee of (2 maxi
?
? =
?
(A?1 , ? ? ?
?
, A?m )
denotes the optimal partition.
6
?
|A?
i |
?
?
?
1+(|Ai |?1)(1??fi (A?
i ))
), where
5-Partition on 20newsgroup with ADMM
30-Partition on TIMIT
5-Partition on MNIST with Distributed NN
86
50
99
45
83
82
81
Test accuracy (%)
98.9
84
Test accuracy (%)
Test accuracy (%)
85
99.1
98.8
98.7
98.6
40
35
30
25
98.5
20
98.4
80
Submodular partition
Random partition
Submodular partition
Random partition
98.3
Submodular partition
Random partition
15
79
5
10
15
20
25
30
35
5
Number of iterations
10
15
5
20
10
15
99.2
84
25
30
35
40
45
50
55
Number of iterations
Number of iterations
10-Partition on 20newsgroup with ADMM
20
40-Block Partition on TIMIT with Distributed NN
10-Partition on MNIST with Distributed NN
50
99
45
80
78
Test accuracy (%)
98.8
Test accuracy (%)
Test accuracy (%)
82
98.6
98.4
98.2
40
35
30
25
76
98
Submodular partition
Adversarial partition
Random partition
74
20
97.8
15
Submodular partition
Random partition
Submodular partition
Random partition
10
5
10
15
20
25
30
Number of iterations
(a)20Newsgroups
35
5
10
Number of iterations
(b) MNIST
15
20
5
10
15
20
25
30
35
40
45
50
55
Number of iterations
(c) TIMIT
Figure 1: Comparison between submodular and random partitions for distributed ML, including
ADMM (Fig 1a) and distributed neural nets (Fig 1b) and (Fig 1c). For the box plots, the central mark
is the median, the box edges are 25th and 75th percentiles, and the bars denote the best and worst cases.
3
General Submodular Partitioning (Problems 1 and 2 when 0 < ? < 1)
In this section we study Problem 1 and Problem 2, in the most general case, i.e., 0 < ? < 1. We first
propose a simple and general ?extremal combination? scheme that works both for problem 1 and 2.
It naturally combines an algorithm for solving the worst-case problem (? = 0) with an algorithm for
solving the average case (? = 1). We use Problem 1 as an example, but the same scheme easily works
for Problem 2. Denote A LG WC as the algorithm for the worst-case problem (i.e. SFA), and A LG AC
as the algorithm for the average case (i.e., SWP). The scheme is to first obtain a partition ?
?1 by running
A LG WC on the instance of Problem 1 with ? = 0 and a second partition ?
?2 by running A LG AC with
? = 1. Then we output one of ?
?1 and ?
?2 , with which the higher valuation for Problem 1 is achieved.
We call this scheme C OMB S FA S WP. Suppose A LG WC solves the worst-case problem with a factor
? ? 1 and A LG AC for the average case with ? ? 1. When applied to Problem 2 we refer to this
scheme as C OMB S LB S MP (? ? 1 and ? ? 1). The following guarantee holds for both schemes:
??
Theorem 3.1. For any ? ? (0, 1) C OMB S FA S WP solves Problem 1 with a factor max{ ??+?
, ??}
?
??
1
in the heterogeneous case, and max{min{?, m
}, ??+?
, ??} in the homogeneous case. Similarly,
?
? + ?)} in the heterogeneous case,
C OMB S LB S MP solves Problem 2 with a factor min{ mm?
, ?(m?
?
?+?
m?
?
and min{m, m?+?
, ?(m? + ?)} in the homogeneous case.
?
The drawback of C OMB S FA S WP and C OMB S LB S MP is that they do not explicitly exploit the tradeoff between the average-case and worst-case objectives in terms of ?. To obtain more practically
interesting algorithms, we also give G ENERAL G REED S AT that generalizes G REED
ProbPmS AT to solve
1
? i (A? )+
lem 1. Similar to G REED S AT we define an intermediate objective: F??c (?) = m
min{?f
i
i=1
Pm
1
?
f
(A
),
c}
?m
in
G
ENERAL
G
REED
S
AT
.
Following
the
same
algorithmic
design
as
in
G
REED
j
j
j=1
S AT, G ENERAL G REED S AT only differs from G REED S AT in Line 6, where the submodular welfare
problem is defined on the new objective F??c (?). In [28] we show that G ENERAL G REED S AT gives ?/2
approximation, while also yielding a bi-criterion guarantee that generalizes Theorem 2.3. In particular
G ENERAL G REED S AT recovers the bicriterion guarantee as shown in Theorem 2.3 when ? = 0. In
the case of ? = 1, G ENERAL G REED S AT recovers the 1/2-approximation guarantee of the greedy
algorithm for solving the submodular welfare problem, i.e., the average-case objective. Moreover an
improved guarantee is achieved by G ENERAL G REED S AT as ? increases. Details are given in [28].
? ROUND leading to G ENERAL L OV ASZ
? ROUND. Similar
To solve Problem 2 we generalize L OV ASZ
? ROUND we relax each submodular objective as its convex relaxation using the Lov?asz
to L OV ASZ
? ROUND, G ENERAL L OV ASZ
? ROUND only differs in Line 2,
extension. Almost the same as L OV ASZ
? maxi f?i (xi ) +
where Problem 2 is relaxed as the following convex program: minx1 ,...,xm ?[0,1]n ?
7
Pm ?
Pm
1
?m
j=1 fj (xj ), s.t
i=1 xi (j) ? 1, for j = 1, . . . , n. Following the same rounding procedure
?
? ROUND is guaranteed to give an m-approximation for
as L OV ASZ ROUND, G ENERAL L OV ASZ
Problem 2 with general ?. Details are given in [28].
4
Experiments and Conclusions
We conclude in this section by empirically evaluating the algorithms proposed for Problems 1 and 2
on real-world data partitioning applications including distributed ADMM, distributed deep neural
network training, and lastly unsupervised image segmentation tasks.
ADMM: We first consider data partitioning for distributed convex optimization. The evaluation
task is text categorization on the 20 Newsgroup data set, which consists of 18,774 articles divided
almost evenly across 20 classes. We formulate the multi-class classification as an `2 regularized
logistic regression, which is solved by ADMM implemented as [3]. We run 10 instances of random
partitioning on the training data as a baseline. In this case, we use the feature based function (same as
the one used in [27]), in the homogeneous setting of Problem 1 (with ? = 0). We use G REED M AX
as the partitioning algorithm. In Figure 1a, we observe that the resulting partitioning performs much
better than a random partitioning (and significantly better than an adversarial partitioning, formed
by grouping similar items together). More details are given in [28].
Distributed Deep Neural Network (DNN) Training: Next we evaluate our framework on distributed deep neural network (DNN) training. We test on two tasks: 1) handwritten digit recognition
on the MNIST database, which consists of 60,000 training and 10,000 test samples; 2) phone
classification on the TIMIT data, which has 1,124,823 training and 112,487 test samples. A 4-layer
DNN model is applied to the MNIST experiment, and we train a 5-layer DNN for TIMIT. For both
experiments the submodular partitioning is obtained by solving the homogeneous case of Problem 1
(? = 0) using G REED M AX on a form of clustered facility location (as proposed and used in [27]).
We perform distributed training using an averaging stochastic gradient descent scheme, similar to the
one in [24]. We also run 10 instances of random partitioning as a baseline. As shown in Figure 1b
and 1c, the submodular partitioning outperforms the random baseline. An adversarial partitioning,
which is formed by grouping items with the same class, in either case, cannot even be trained.
Unsupervised Image SegF-measure on
Original
mentation: We test the
all of GrabCut
efficacy of Problem 2 on
Ground
1.0
unsupervised image segTruth
mentation over the GrabCut
data set (30 color images
k-means
0.810
and their ground truth foreground/background labels).
k-medoids
0.823
By ?unsupervised?, we
mean that no labeled data
Spectral
0.854
Clustering
at any time in supervised
or semi-supervised training,
Graph
0.853
Cut
nor any kind of interactive
segmentation, was used in
Submodular
0.870
Partitioning
forming or optimizing the
objective. The submodular
partitioning for each image Figure 2: Unsupervised image segmentation (right: some examples).
is obtained by solving the
homogeneous case of Problem 2 (? = 0.8) using a modified variant of G REED M IN on the facility location function. We compare our method against the other unsupervised methods k-means, k-medoids,
spectral clustering, and graph cuts. Given an m-partition of an image and its ground truth labels, we
assign each of the m blocks either to the foreground or background label having the larger intersection.
In Fig. 2 we show example segmentation results after this mapping on several example images as well
as averaged F-measure (relative to ground truth) over the whole data set. More details are given in [28].
Acknowledgments: This material is based upon work supported by the National Science Foundation
under Grant No. IIS-1162606, the National Institutes of Health under award R01GM103544, and by
a Google, a Microsoft, and an Intel research award. R. Iyer acknowledges support from a Microsoft
Research Ph.D Fellowship. This work was supported in part by TerraSwarm, one of six centers of
STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA.
8
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9
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5,199 | 5,707 | Subspace Clustering with Irrelevant Features via
Robust Dantzig Selector
Chao Qu
Department of Mechanical Engineering
National University of Singapore
Huan Xu
Department of Mechanical Engineering
National University of Singapore
[email protected]
[email protected]
Abstract
This paper considers the subspace clustering problem where the data contains
irrelevant or corrupted features. We propose a method termed ?robust Dantzig selector? which can successfully identify the clustering structure even with the presence of irrelevant features. The idea is simple yet powerful: we replace the inner
product by its robust counterpart, which is insensitive to the irrelevant features
given an upper bound of the number of irrelevant features. We establish theoretical guarantees for the algorithm to identify the correct subspace, and demonstrate
the effectiveness of the algorithm via numerical simulations. To the best of our
knowledge, this is the first method developed to tackle subspace clustering with
irrelevant features.
1
Introduction
The last decade has witnessed fast growing attention in research of high-dimensional data: images,
videos, DNA microarray data and data from many other applications all have the property that the
dimensionality can be comparable or even much larger than the number of samples. While this
setup appears ill-posed in the first sight, the inference and recovery is possible by exploiting the fact
that high-dimensional data often possess low dimensional structures [3, 14, 19]. On the other hand,
in this era of big data, huge amounts of data are collected everywhere, and such data is generally
heterogeneous. Clean data and irrelevant or even corrupted information are often mixed together,
which motivates us to consider the high-dimensional, big but dirty data problem. In particular, we
study the subspace clustering problem in this setting.
Subspace clustering is an important subject in analyzing high-dimensional data, inspired by many
real applications[15]. Given data points lying in the union of multiple linear spaces, subspace clustering aims to identify all these linear spaces, and cluster the sample points according to the linear
spaces they belong to. Here, different subspaces may correspond to motion of different objects in
video sequence [11, 17, 20], different rotations, translations and thickness in handwritten digit or
the latent communities for the social graph [15, 5].
A variety of algorithms of subspace clustering have been proposed in the last several years including algebraic algorithms [16], iterative methods [9, 1], statistical methods [11, 10], and spectral
clustering-based methods [6, 7]. Among them, sparse subspace clustering (SSC) not only achieves
state-of-art empirical performance, but also possesses elegant theoretical guarantees. In [12], the
authors provide a geometric analysis of SSC which explains rigorously why SSC is successful even
when the subspaces are overlapping [12]. [18] and [13] extend SSC to the noisy case, where data are
contaminated by additive Gaussian noise. Different from these work, we focus on the case where
some irrelevant features are involved.
1
Mathematically, SSC indeed solves for each sample a sparse linear regression problem with the
dictionary being all other samples. Many properties of sparse linear regression problem are well
understood in the clean data case. However, the performance of most standard algorithms deteriorates (e.g. LASSO and OMP) even only a few entries are corrupted. As such, it is well expected
that standard SSC breaks for subspace clustering with irrelevant or corrupted features (see Section 5
for numerical evidences). Sparse regression under corruption is a hard problem, and few work has
addressed this problem [8][21] [4].
Our contribution: Inspired by [4], we use a simple yet powerful tool called robust inner product
and propose the robust Dantzig selector to solve the subspace clustering problem with irrelevant
features. While our work is based upon the robust inner product developed to solve robust sparse
regression, the analysis is quite different from the regression case since both the data structures and
the tasks are completely different: for example, the RIP condition ? essential for sparse regression ?
is hardly satisfied for subspace clustering [18]. We provide sufficient conditions to ensure that
the Robust Dantzig selector can detect the true subspace clustering. We further demonstrate via
numerical simulation the effectiveness of the proposed method. To the best of our knowledge, this
is the first attempt to perform subspace clustering with irrelevant features.
2
2.1
Problem setup and method
Notations and model
The clean data matrix is denoted by XA ? RD?N , where each column corresponds to a data point,
normalized to a unit vector. The data points are lying on a union of L subspace S = ?L
l=1 Sl .
Each subspace Sl is of dimension dl which is smaller than D and contains Nl data samples with
N1 +N2 +? ? ?+NL = N . We denote the observed dirty data matrix by X ? R(D+D1 )?N . Out of the
T
T T
D + D1 features, up to D1 of them are irrelevant. Without loss of generality, let X = [XO
, XA
] ,
where XO ? RD1 ?N denotes the irrelevant data. The subscript A and O denote the set of row
indices corresponding to true and irrelevant features and the superscript T denotes the transpose.
Notice that we do not know O a priori except its cardinality is D1 . The model is illustrated in Figure
(l)
1. Let XA ? RD?Nl denote the selection of columns in XA that belongs to Sl . Similarly, denote
the corresponding columns in X by X (l) . Without loss of generality, let X = [X (1) , X (2) , ..., X (L) ]
be ordered. Further more, we use the subscript ??i?to describe a matrix that excludes the column
(l)
(l)
(l)
(l)
(l)
i, e.g., (XA )?i = [(xA )1 , ..., (xA )i?1 , (xA )i+1 , ..., (xA )Nl ]. We use the superscript lc to describe
c
(1)
(l?1)
(l+1)
(L)
a matrix that excludes column in subspace l, e.g., (XA )l = [XA , ..., XA , XA , ..., XA ].
For a matrix ?, we use ?s,? to denote the submatrix with row indices in set s and column indices
in set ?. For any matrix Z, P (Z) denotes the symmetrized convex hull of its column, i.e., P (Z) =
(l)
l
conv(?z1 , ?z2 , ...., ?zN ) . We define P?i
:= P ((XA )?i ) for simplification, i.e., the symmetrized
convex hull of clean data in subspace l except data i. Finally we use k ? k2 to denote the l2 norm of
a vector and k ? k? to denote infinity norm of a vector or a matrix. Caligraphic letters such as X , Xl
represent the set containing all columns of the corresponding clean data matrix.
Figure 1: Illustration of the model of irrelevant features in the subspace clustering problem. The
left one is the model addressed in this paper: Among total D + D1 features, up tp D1 of them are
irrelevant. The right one illustrates a more general case, where the value of any D1 element of each
column can be arbitrary (e.g., due to corruptions). It is a harder case and left for future work.
2
Figure 2: Illustration of the Subspace Detection Property. Here, each figure corresponds to a matrix
where each column is ci , and non-zero entries are in white. The left figure satisfies this property.
The right one does not.
2.2
Method
In this secion we present our method as well as the intuition that derives it. When all observed data
are clean, to solve the subspace clustering problem, the celebrated SSC [6] proposes to solve the
following convex programming
min kci k1
ci
s.t.
xi = X?i ci ,
(1)
for each data point xi . When data are corrupted by noise of small magnitude such as Gaussian noise,
a straightforward extension of SSC is the Lasso type method called ?Lasso-SSC? [18, 13]
min kci k1 +
ci
?
kxi ? X?i ci k22 .
2
(2)
Note that while Formulation (2) has the same form as Lasso, it is used to solve the subspace clustering task. In particular, the support recovery analysis of Lasso does not extend to this case, as X?i
typically does not satisfy the RIP condition [18].
This paper considers the case where X contains irrelevant/gross corrupted features. As we discussed above, Lasso is not robust to such corruption. An intuitive idea is to consider the following
formulation first proposed for sparse linear regression [21].
min kci k1 +
ci ,E
?
kxi ? (X?i ? E)ci k22 + ?kEk? ,
2
(3)
where k ? k? is some norm corresponding to the sparse type of E. One major challenge of this
formulation is that it is not convex. As such, it is not clear how to efficiently find the optimal
solution, and how to analyze the property of the solution (typically done via convex analysis) in the
subspace clustering task.
Our method is based on the idea of robust inner product. The robust inner product ha, bik is defined
as follows: For vector a ? RD , b ? RD , we compute qi = ai bi , i = 1, ..., N . Then {|qi |} are
P sorted
and the smallest (D ? k) are selected. Let ? be the set of selected indices, then ha, bik = i?? qi ,
i.e., the largest k terms are truncated. Our main idea is to replace all inner products involved by
robust counterparts ha, biD1 , where D1 is the upper bound of the number of irrelevant features.
The intuition is that the irrelevant features with large magnitude may affect the correct subspace
clustering. This simple truncation process will avoid this. We remark that we do not need to know
the exact number of irrelevant feature, but instead only an upper bound of it.
Extending (2) using the robust inner product leads the following formulation:
min kci k1 +
ci
? T?
c ?ci ? ??
? T ci ,
2 i
(4)
? and ?? are robust counterparts of X T X?i and X T xi . Unfortunately, ?
? may not be a
where ?
?i
?i
positive semidefinite matrix, thus (4) is not a convex program. Unlike the work [4][8] which studies
3
non-convexity in linear regression, the difficulty of non-convexity in the subspace clustering task
appears to be hard to overcome.
Instead we turn to the Dantzig Selector, which is essentially a linear program (and hence no positive
semidefiniteness is required):
T
min kci k1 + ?kX?i
(X?i ci ? xi )k? .
(5)
ci
Replace all inner product by its robust counterpart, we propose the following Robust Dantzig Selector, which can be easily recast as a linear program:
? i ? ?? k? ,
min kci k1 + ?k?c
Robust Dantzig Selector:
(6)
ci
Subspace Detection Property: To measure whether the algorithm is successful, we define the
criterion Subspace Detection Property following [18]. We say that the Subspace Detection Property holds, if and only if for all i, the optimal solution to the robust Dantzig Selector satisfies (1)
Non-triviality: ci is not a zero vector; (2) Self-Expressiveness Property: nonzeros entries of ci
correspond to only columns of X sampled from the same subspace as xi . See Figure 2 for illustrations.
3
Main Results
To avoid repetition and cluttered notations, we denote the following primal convex problem by
P (?, ?)
min kck1 + ?k?c ? ?k? .
c
Its dual problem, denoted by D(?, ?), is
maxh?, ?i subject to k?k1 = ?
?
k??k? ? 1.
(7)
Before we presents our results, we define some quantities.
The dual direction is an important geometric term introdcued in analyzing SSC [12]. Here we define
similarly the dual direction of the robust Dantzig selector: Notice that the dual of robust Dantzig
? ?? ), where ?? and ?
? are robust counterparts of X T xi and X T X?i respectively
problem is D(?,
?i
?i
? into two parts ?
? = (XA )T (XA )?i + ?,
?
(recall that X?i and xi are the dirty data). We decompose ?
?i
where the first term corresponds to the clean data, and the second term is due to the irrelevant features
and truncation from the robust inner product. Thus, the second constraint of the dual problem
? ? ? 1. Let ? be the optimal solution to the above optimization
becomes k((XA )T?i (XA )?i + ?)?k
(l)
problem, we define v(xi , X?i , ?) := (XA )?i ? and the dual direction as v l =
v(xli ,X?i ,?)
(l)
kv(xli ,X?i ,?)k2
.
l
Similarly as SSC [12], we define the subspace incoherence. Let V l = [v1l , v2l , ..., vN
]. The incoherl
(k)
ence of a point set Xl to other clean data points is defined as ?(Xl ) = maxk:k6=l k(XA )T V l k? .
? and ?? as ?
? = (XA )T (XA )?i + ?
? and ?? = (XA )T (xA )i + ?? .
Recall that we decompose ?
?i
?i
?
Intuitively, for robust Dantzig selecter to succeed, we want ? and ?? not too large. Particularly, we
assume k(xA )i k? ? 1 and k(XA )?i k? ? 2 .
l
Theorem 1 (Deterministic Model). Denote ?l := ?(Xl ), rl := mini:xi ?Xl r(P?i
), r :=
minl=1,...,L rl and suppose ?l < rl for all l. If
rl ? ul
1
< min
,
l 2D1 2
r2 ? 4D1 1 2 r ? 2D1 22
2 (ul + rl )
(8)
then the subspace detection property holds for all ? in the range
1
rl ? ul
< ? < min
.
l 2D1 2
r2 ? 4D1 1 2 r ? 2D1 22
2 (ul + rl )
4
(9)
In an ideal case when D1 = 0, the condition of the upper bound of ? reduces to rl > ul , similar to
the condition for SSC in the noiseless case [12].
Based on Condition (8), under a randomized generative model, we can derive how many irrelevant
features can be tolerated.
Theorem 2 (Random model). Suppose there are L subspaces and for simplicity, all subspaces have
same dimension d and are chosen uniformly at random. For each subspace there are ?d + 1 points
chosen independently and uniformly at random. Up to D1 features of data are irrelevant. Each
data point (including true and irrelevant features) is independent from other data points. Then for
some universal constants
C1 , C2 , the subspace detection property holds with probability at least
?
1 ? N4 ? N exp(? ?d) if
d?
Dc2 (?) log(?)
,
12 log N
and
log ?
1 2
2 c (?) d
1
1??
D
q
<?<
,
?
C1 D1 (log D+C2 log N )
1
+
?
C
D
(log
D
+ C2 log N )
log ?
1
1
? ( 2c(?)
+
1)
d
D
q
12d log N
where ? =
Dc2 (?) log ? ; c(?) is a constant only depending on the density of data points on
subspace and satisfies (1) c(?) >
? 0 for all ? > 1, (2) there is a numerical value ?0 , such that for all
? > ?0 , one can take c(?) = 1/ 8.
Simplifying the above conditions, we can determine the number of irrelevant features that can be
tolerated. In particular, if d ? 2c2 (?) log ? and we choose the ? as
?=
4d
,
c2 (?) log ?
then the maximal number of irrelevant feature D1 that can be torelated is
1??
C0 Dc2 (?) log ?
c(?)D log ?
,
},
8C1 d(log(D) + C2 log N ) 1 + ? C1 d(log(D) + C2 log N )
?
with probability at least 1 ? N4 ? N exp(? ?d).
D1 = min{
If d ? 2c2 (?) log ?, and we choose the same ?, then the number of irrelevant feature we can tolerate
is
q
Dc(?) logd ?
1??
C0 Dc2 (?) log ?
D1 = min{ ?
,
},
4 2C1 (log(D) + C2 log N ) 1 + ? C1 d(log(D) + C2 log N )
?
with probability at least 1 ? N4 ? N exp(? ?d).
Remark 1. If D is much larger than D1 , the lower bound of ? is proportional to the subspace
dimension d. When d increases, the upper bound of ? decreases, since 1??
1+? decreases. Thus the
valid range of ? shrinks when d increases.
Remark 2. Ignoring the logarithm terms, when d is large, the tolerable D1 is proportional
to
?
D
D
1?? D
min(C1 1??
,
C
).
When
d
is
small,
D
is
proportional
to
min(C
,
C
D/
d)
.
2
1
1
2
1+? d
d
1+? d
4
Roadmap of the Proof
In this section, we lay out the roadmap of proof. In specific we want to establish the condition with
the number of irrelevant features, and the structure of data (i.e., the incoherence ? and inradius r)
for the algorithm to succeed. Indeed, we provide a lower bound of ? such that the optimal solution
ci is not trivial; and an upper bound of ? so that the Self-Expressiveness Property holds. Combining
them together established the theorems.
5
4.1
Self-Expressiveness Property
The Self-Expressiveness Property is related to the upper bound of ?. The proof technique is inspired
by [18] and [12], we first establish the following lemma, which provides a sufficient condition such
that Self-Expressiveness Property holds of the problem 6.
Lemma 1. Consider a matrix ? ? RN ?N and ? ? RN ?1 , If there exist a pair (?
c, ?) such that c?
has a support S ? T and
sgn(?
cs ) + ?s,? ?? = 0,
k?sc ?T,? ?? k? ? 1,
k?k1 = ?,
k?T c ,? ?? k? < 1,
(10)
where ? is the set of indices of entry i such that |(??
c ? ?)i | = k??
c ? ?k? , then for all optimal
solution c? to the problem P (?, ?), we have c?T c = 0.
The variable ? in Lemma 1 is often termed the ?dual certificate?. We next consider an oracle problem
? to construct such a dual certificate. This
? l,l , ??l ), and use its dual optimal variable denoted by ?,
P (?
candidate satisfies all conditions in the Lemma 1 automatically except to show
? lc ,?? ????k? < 1,
k?
(11)
c
where l denotes the set of indices expect the ones corresponding to subspace l. We can compare
c
this condition with the corresponding one in analyzing SSC, in which one need k(X)(l )T vk? < 1,
c
? lc ,?? = (XA )(l )T (XA )?? + ?
? lc ,?? .
where v is the dual certificate. Recall that we can decompose ?
Thus Condition 11 becomes
k(XA )(l
c
)T
? lc ,?? ????k? < 1.
((XA )??????) + ?
(12)
? lc ,?? ????k? .
To show this holds, we need to bound two terms k(XA )??????k2 and k?
? ? , k?
Bounding k?k
? k?
? ? and k?
The following lemma relates D1 with k?k
? k? .
? and ?? are robust counterparts of X T X?i and X T xi respectively and among
Lemma 2. Suppose ?
?i
?i
? and ?? into following form ?
? =
D + D1 features, up to D1 are irrelevant. We can decompose ?
? and ?? = (XA )T (xA )i + ?? . We define ?1 := k?
?
(XA )T?i (XA )?i + ?
?
k
and
?
:=
k
?k
?
2
? .If
?i
k(xA )i k? ? 1 and k(XA )?i k? ? 2 , then ?2 ? 2D1 22 , ?1 ? 2D1 1 2 .
We then bound 1 and 2 in the random model
cap [2].
? using the upper bound of the spherical
?
Indeed we have 1 ? C1 (log D + C2 log N )/ D and 2 ? C1 (log D + C2 log N )/ D with high
probability.
Bounding kX??????k2
By exploiting the feasible condition in the dual of the oracle problem, we obtain the following
bound:
1 + 2D1 ?22
kX??????k2 ?
.
l )
r(P?i
q
log ?
l
?
Furthermore, r(P?i
) can be lower bound by c(?)
d and 2 can be upper bounded by C1 (log D+
2
?
C2 log N )/ D in the random model with high probability. Thus the RHS can be upper bounded.
Plugging this upper bound into (12), we obtain the upper bound of ?.
4.2
Non-triviality with sufficiently large ?
To ensure that the solution is not trivial (i.e., not all-zero), we need a lower bound on ?.
6
If ? satisfies the following condition, the optimal solution to problem 6 can not be zero
?>
l )
r2 (P?i
?
2D1 22
1
.
l )D
? 4r(P?i
1 1 2
(13)
The proof idea is to show when ? is large enough, the trivial solution c = 0 can not be optimal. In
particular, if c = 0, the corresponding value in the primal problem is ?k?
?l k? . We then establish a
?
lower bound of k?
?l k? and a upper bound of kck1 + ?k?l,l c ? ??l k? so that the following inequality
always holds by some carefully choosen c.
? l,l c ? ??l k? < ?k?
kck1 + ?k?
?l k? .
(14)
l
). Notice
We then further lower bound the RHS of Equation (13) using the bound of 1 , 2 and r(P?i
that condition (14) requires that ? > A and condition (11) requires ? < B, where A and B are some
terms depending on the number of irrelevant features. Thus we require A < B to get the maximal
number of irrelevant features that can be tolerated.
5
Numerical simulations
In this section, we use three numerical experiments to demonstrate the effectiveness of our method
to handle irrelevant/corrupted features. In particular, we test the performance of our method and
effect of number of irrelevant features and dimension subspaces d with respect to different ?. In
all experiments, the ambient dimension D = 200, sample density ? = 5, the subspace are drawn
uniformly at random. Each subspace has ?d+1 points chosen independently and uniformly random.
We measure the success of the algorithms using the relative violation of the subspace detection
property defined as follows,
P
|C|i,j
(i,j)?M
/
RelV iolation(C, M) = P
,
(i,j)?M |C|i,j
where C = [c1 , c2 , ..., cN ], M is the ground truth mask containing all (i, j) such that xi , xj belong
to a same subspace. If RelV iolation(C, M) = 0, then the subspace detection property is satisfied.
We also check whether we obtain a trivial solution, i.e., if any column in C is all-zero.
We first compare the robust Dantzig selector(? = 2) with SSC and LASSO-SSC ( ? = 10). The
results are shown in Figure 3. The X-axis is the number of irrelevant features and the Y-axis is the
Relviolation defined above. The ambient dimension D = 200, L = 3, d = 5, the relative sample
density ? = 5. The values of irrelevant features are independently sampled from a uniform distribution in the region [?2.5, 2.5] in (a) and [?10, 10] in (b). We observe from Figure 3 that both SSC
and Lasso SSC are very sensitive to irrelevant information. (Notice that RelViolation=0.1 is pretty
large and can be considered as clustering failure.) Compared with that, the proposed Robust Dantzig
Selector performs very well. Even when D1 = 20, it still detects the true subspaces perfectly. In
the same setting, we do some further experiments, our method breaks when D1 is about 40. We
also do further experiment for Lasso-SSC with different ? in the supplementary material to show
Lasso-SSC is not robust to irrelevant features.
We also examine the relation of ? to the performance of the algorithm. In Figure 4a, we test the
subspace detection property with different ? and D1 . When ? is too small, the algorithm gives a
trivial solution (the black region in the figure). As we increase the value of ?, the corresponding
solutions satisfy the subspace detection property (represented as the white region in the figure).
When ? is larger than certain upper bound, RelV iolation becomes non-zero, indicating errors in
subspace clustering. In Figure 4b, we test the subspace detection property with different ? and d.
Notice we rescale ? with d, since by Theorem 3, ? should be proportional to d. We observe that the
valid region of ? shrinks with increasing d which matches our theorem.
6
Conclusion and future work
We studied subspace clustering with irrelevant features, and proposed the ?robust Dantzig selector?
based on the idea of robust inner product, essentially a truncated version of inner product to avoid
7
Original SSC
Lasso SSC
Robust Dantzig Selector
1
0.8
RelViolation
RelViolation
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
Original SSC
Lasso SSC
Robust Dantzig Selector
1
0
5
10
Number of irrelevant features
15
0
20
0
5
10
Number of irrelevant features
(a)
15
20
(b)
2.5
10.5
10
9.5
9
8.5
8
7.5
7
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
2.3
2.1
1.9
1.7
1.5
?/d
?
Figure 3: Relviolation with different D1 . Simulated with D = 200, d = 5, L = 3, ? = 5, ? = 2,
and D1 from 1 to 20.
1.3
1.1
0.9
0.7
0.5
0.3
0.1
1
2
3
4
5
6
7
8
9
10
4
Number of irrelevant features D1
(a) Exact recovery with different number of
irrelevant features. Simulated with D =
200, d = 5, L = 3, ? = 5 with an increasing D1 from 1 to 10. Black region:
trivial solution. White region: Non-trivial
solution with RelViolation=0. Gray region:
RelViolation> 0.02.
6
8
10
12
14
16
Subspace dimension d
(b) Exact recovery with different subspace
dimension d. Simulated with D = 200,
L = 3, ? = 5, D1 = 5 and an increasing d from 4 to 16. Black region: trivial solution. White region: Non-trivial solution with RelViolation=0. Gray region:
RelViolation> 0.02.
Figure 4: Subspace detection property with different ?, D1 , d.
any single entry having too large influnce on the result. We established the sufficient conditions
for the algorithm to exactly detect the true subspace under the deterministic model and the random
model. Simulation results demonstrate that the proposed method is robust to irrelevant information
whereas the performance of original SSC and LASSO-SSC significantly deteriorates.
We now outline some directions of future research. An immediate future work is to study theoretical
guarantees of the proposed method under the semi-random model, where each subspace is chosen
deterministically, while samples are randomly distributed on the respective subspace. The challenge
here is to bound the subspace incoherence, previous methods uses the rotation invariance of the data,
which is not possible in our case as the robust inner product is invariant to rotations.
Acknowledgments
This work is partially supported by the Ministry of Education of Singapore AcRF Tier Two grant
R-265-000-443-112, and A*STAR SERC PSF grant R-265-000-540-305.
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