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applied to various fields, ranging from collaborative filtering [15], to computer vision [17], and medical imaging [9], among others. In this paper, we propose a new method to tensor completion, which is based on a convex regularizer which encourages low rank tensors and develop an algorithm for solving the associated regularization problem.Arguably the most widely used convex approach to tensor completion is based upon the extension of trace norm regularization [24] to that context. This involves computing the average of the trace norm of each matricization of the tensor [16]. A key insight behind using trace norm regularization for matrix completion is that this norm provides a tight convex relaxation of the rank of a matrix defined on the spectral unit ball [8]. Unfortunately, the extension of this methodology to the more general tensor setting presents some difficulties. In particular, we shall prove in this paper that the tensor trace norm is not a tight convex relaxation of the tensor rank.The above negative result stems from the fact that the spectral norm, used to compute the convex relaxation for the trace norm, is not an invariant property of the matricization of a tensor. This observation leads us to take a different route and study afresh the convex relaxation of tensor rank on the Euclidean ball. We show that this relaxation is tighter than the tensor trace norm, and we describe a technique to solve the associated regularization problem. This method builds upon the alternating direction method of multipliers and a subgradient method to compute the proximity operator of the proposed regularizer. Furthermore, we present numerical experiments on one synthetic dataset and two real-life datasets, which indicate that the proposed method improves significantly over tensor trace norm regularization in terms of estimation error, while remaining computationally tractable.The paper is organized in the following manner. In Section 2, we describe the tensor completion framework. In Section 3, we highlight some limitations of the tensor trace norm regularizer and present an alternative convex relaxation for the tensor rank. In Section 4, we describe a method to solve the associated regularization problem. In Section 5, we report on our numerical experience with the proposed method. Finally, in Section 6, we summarize the main contributions of this paper and discuss future directions of research.2 Preliminaries In this section, we begin by introducing some notation and then proceed to describe the learning problem. We denote by N the set of natural numbers and, for every k ? N, we define [k] = {1, . . . , k}. Let N ? N and let1 p1, . . . , pN ? 2. An N -order tensor W ? Rp1×···×pN , is a collection of real numbers (Wi1,...,iN : in ? [pn], n ? [N ]). Boldface Euler scripts, e.g. W , will be used to denote tensors of order higher than two. Vectors are 1-order tensors and will be denoted by lower case letters, e.g. x or a; matrices are 2-order tensors and will be denoted by upper case letters, e.g. W . If x ? Rd then for every r ? s ? d, we define xr:s := (xi : r ? i ? s). We also use the notation pmin = min{p1, . . . , pN} and pmax = max{p1, . . . , pN}.A mode-n fiber of a tensor W is a vector composed of the elements of W obtained by fixing all indices but one, corresponding to the n-th mode. This notion is a higher order analogue of columns (mode-1 fibers) and rows (mode-2 fibers) for matrices. The mode-n matricization (or unfolding) of W , denoted by W(n), is a matrix obtained by arranging the mode-n fibers of W so that each of them is a column of W(n) ? Rpn×Jn , where Jn := ? k 6=n pk. Note that the ordering of the columns is not important as long as it is used consistently.We are now ready to describe the learning problem. We choose a linear operator I : Rp1×···×pN ? R m, representing a set of linear measurements obtained from a target tensor W0 as y = I(W0)+?, where ? is some disturbance noise. Tensor completion is an important example of this setting, in this case the operator I returns the known elements of the tensor. That is, we have I(W0) = (W0i1(j),...,iN (j) : j ? [m]), where, for every j ? [m] and n ? [N ], the index in(j) is a prescribed integer in the set [pn]. Our aim is to recover the tensor W 0 from the data (I, y). To this end, we solve the regularization problem min {?y ? I(W)?22 + ?R(W) : W ? Rp1×···×pN} (1) where ? is a positive parameter which may be chosen by cross validation. The role of the regularizer R is to encourage solutions W which have a simple structure in the sense that they involve a small number of “degrees of freedom”. A natural choice is to consider the average of the rank of the tensor’s matricizations. Specifically, we consider the combinatorial regularizer R(W) = N N? n=1 rank(W(n)). (2) Finding a convex relaxation of this regularizer has been the subject of recent works [9, 17, 23]. They all agree to use the sum of nuclear norms as a convex proxy of R. This is defined as the average of the trace norm of each matricization of W , that is, ?W?tr = 1 N N? n=1 ?W(n)?tr (3) where ?W(n)?tr is the trace (or nuclear) norm of matrix W(n), namely the ?1-norm of the vector of singular values of matrix W(n) (see, e.g. [14]). Note that in the particular case of 2-order tensors, functions (2) and (3) coincide with the usual notion of rank and trace norm of a matrix, respectively.A rational behind the regularizer (3) is that the trace norm is the tightest convex lower bound to the rank of a matrix on the spectral unit ball, see [8, Thm. 1]. This lower bound is given by the convex envelope of the function ?(W ) = { rank(W ), if ?W?? ? 1 +?, otherwise (4) 1For simplicity we assume that pn ? 2 for every n ? [N ], otherwise we simply reduce the order of the tensor without loss of information.where ? · ?? is the spectral norm, namely the largest singular value of W . The convex envelope can be derived by computing the double conjugate of ?. This is defined as ???(W ) = sup {?W,S? ???(W ) : S ? Rp1×p2} (5) where ?? is the conjugate of ?, namely ??(S) = sup {?W,S? ??(W ) :W ? Rp1×p2}.Note that ? is a spectral function, that is, ?(W ) = ?(?(W )) where ? : Rd+ ? R denotes the associated symmetric gauge function. Using von Neumann’s trace theorem (see e.g. [14]) it is easily seen that ??(S) is also a spectral function. That is, ??(S) = ??(?(S)), where ??(?) = sup {??,w? ? ?(w) : w ? Rd+} , with d := min(p1, p2).We refer to [8] for a detailed discussion of these ideas. We will use this equivalence between spectral and gauge functions repeatedly in the paper.3 Alternative Convex Relaxation In this section, we show that the tensor trace norm is not a tight convex relaxation of the tensor rank R in equation (2). We then propose an alternative convex relaxation for this function.Note that due to the composite nature of the function R, computing its convex envelope is a chal- lenging task and one needs to resort to approximations. In [22], the authors note that the tensor trace norm ? · ?tr in equation (3) is a convex lower bound to R on the set G? := { W ? Rp1×···×pN : ??W(n)??? ? 1, ?n ? [N ]} .The key insight behind this observation is summarized in Lemma 4, which we report in Appendix A.However, the authors of [22] leave open the question of whether the tensor trace norm is the convex envelope of R on the set G?. In the following, we will prove that this question has a negative answer by showing that there exists a convex function ? 6= ? · ?tr which underestimates the function R on G? and such that for some tensor W ? G? it holds that ?(W) > ?W?tr.To describe our observation we introduce the set G2 := { W ? Rp1×...×pN : ?W?2 ? 1 } where ? · ?2 is the Euclidean norm for tensors, that is, ?W?22 := p1? i1=1 · · · pN? iN=1 (Wi1,...,iN )2.We will choose ?(W) = ??(W) := N N? n=1 ???? ( ? ( W(n) )) (6) where ???? is the convex envelope of the cardinality of a vector on the ?2-ball of radius ? and we will choose ? = ? pmin. Note, by Lemma 4 stated in Appendix A, that for every ? > 0, function ?? is a convex lower bound of function R on the set ?G2.Below, for every vector s ? Rd we denote by s? the vector obtained by reordering the components of s so that they are non increasing in absolute value, that is, |s?1| ? · · · ? |s?d|.Lemma 1. Let ???? be the convex envelope of the cardinality on the ?2-ball of radius ?. Then, for every x ? Rd such that ?x?2 = ?, it holds that ???? (x) = card (x).This lemma is proved in Appendix B. The function ???? resembles the norm developed in [1], which corresponds to the convex envelope of the indicator function of the cardinality of a vector in the ?2 ball. The extension of its application to tensors is not straighforward though, as it is required to specify beforehand the rank of each matricization.The next lemma provides, together with Lemma 1, a sufficient condition for the existence of a tensor W ? G? at which the regularizer in equation (6) is strictly larger than the tensor trace norm.Lemma 2. If N ? 3 and p1, . . . , pN are not all equal to each other, then there exists W ? Rp1×···×pN such that: (a) ?W?2 = ?pmin, (b) W ? G?, (c) min n?[N ] rank(W(n)) < max n?[N ] rank(W(n)).The proof of this lemma is presented in Appendix C. We are now ready to formulate the main result of this section.Proposition 3. Let p1, . . . , pN ? N, let ? · ?tr be the tensor trace norm in equation (3) and let ?? be the function in equation (6) for ? = ? pmin. If pmin < pmax, then there are infinitely many tensors W ? G? such that ??(W) > ?W?tr. Moreover, for every W ? G2, it holds that ?1(W) ? ?W?tr.Proof. By construction ??(W) ? R(W) for every W ? ?G2. Since G? ? ?G2 then ?? is a convex lower bound for the tensor rank R on the set G? as well. The first claim now follows by Lemmas 1 and 2. Indeed, all tensors obtained following the process described in the proof of Lemma 2 (in Appendix C) have the property that ?W?tr = 1 N N? n=1 ??(W(n))?1 = 1 N ( pmin(N ? 1) + ? p2min + pmin ) < N (pmin(N ? 1) + pmin + 1) = ?(W) = R(W).Furthermore there are infinitely many such tensors which satisfy this claim (see Appendix C).With respect to the second claim, given that ???1 is the convex envelope of the cardinality card on the Euclidean unit ball, then ???1 (?) ? ???1 for every vector ? such that ???2 ? 1. Consequently, ?1(W) = N ?N n=1 ? ?? ( ? ( W(n) )) ? 1 N ?N n=1 ??(W(n))?1 = ?W?tr.The above result stems from the fact that the spectral norm is not an invariant property of the matri- cization of a tensor, whereas the Euclidean (Frobenius) norm is. This observation leads us to further study the function ??.4 Optimization Method In this section, we explain how to solve the regularization problem associated with the regularizer (6). For this purpose, we first recall the alternating direction method of multipliers (ADMM) [4], which was conveniently applied to tensor trace norm regularization in [9, 22].4.1 Alternating Direction Method of Multipliers (ADMM) To explain ADMM we consider a more general problem comprising both tensor trace norm regular- ization and the regularizer we propose, min W { E (W) + ? N? n=1 ? ( W(n) )} (7) where E(W) is an error term such as ?y ? I(W)?22 and ? is a convex spectral function. It is defined, for every matrix A, as ?(A) = ?(?(A)) where ? is a gauge function, namely a function which is symmetric and invariant under permuta- tions. In particular, if ? is the ?1 norm then problem (7) corresponds to tensor trace norm regular- ization, whereas if ? = ???? it implements the proposed regularizer.Problem (7) poses some difficulties because the terms under the summation are interdependent, due to the different matricizations of W having the same elements rearranged in a different way. In order to overcome this difficulty, the authors of [9, 22] proposed to use ADMM as a natural way to decouple the regularization term appearing in problem (7). This strategy is based on the introduction of N auxiliary tensors, B1, . . . ,BN ? Rp1×···×pN , so that problem (7) can be reformulated as2 min W,B1,...,BN { ? E (W) + N? n=1 ? ( Bn(n) ) : Bn = W , n ? [N ] } (8) The corresponding augmented Lagrangian (see e.g. [4, 5]) is given by L (W ,B,A) = 1 ? E (W) + N? n=1 ( ? ( Bn(n) )? ?An,W ?Bn?+ ? ?W ?Bn?22 ) , (9) where ?·, ·? denotes the scalar product between tensors, ? is a positive parameter and A1, . . .AN ? R p1×···×pN are the set of Lagrange multipliers associated with the constraints in problem (8).ADMM is based on the following iterative scheme W 
[i+1] ? argmin W L ( W ,B[i],A[i] ) (10) B 
[i+1] n ? argmin Bn L ( W 
[i+1],B,A[i] ) (11) A 
[i+1] n ? A[i]n ? ( ?W [i+1] ?B[i+1]n ) . (12) Step (12) is straightforward, whereas step (10) is described in [9]. Here we focus on the step (11) since this is the only problem which involves function ?. We restate it with more explanatory notations as argmin Bn(n) { ? ( Bn(n) )? ?An(n),W(n) ?Bn(n)?+ ? ??W(n) ?Bn(n)??22 } .By completing the square in the right hand side, the solution of this problem is given by Bˆn(n) = prox 1 ? ? (X) := argmin Bn(n) { ? ? ( Bn(n) ) + ??Bn(n) ?X??22 } , where X =W(n)? 1?An(n). By using properties of proximity operators (see e.g. [2, Prop. 3.1]) we know that if ? is a gauge function then prox 1 ? ? (X) = UXdiag ( prox 1 ? ? (?(X)) ) V ?X , where UX and VX are the orthogonal matrices formed by the left and right singular vectors of X , respectively. If we choose ? = ?·?1 the associated proximity operator is the well-known soft thresholding operator, that is, prox 1 ? ?·?1 (?) = v, where the vector v has components vi = sign (?i) ( |?i| ? 1 ? ) .On the other hand, if we choose ? = ???? , we need to compute prox 1 ? ???? . In the next section, we describe a method to accomplish this task.4.2 Computation of the Proximity Operator To compute the proximity operator of the function 1 ? ???? we will use several properties of proximity calculus. First, we use the formula (see e.g. [7]) proxg? (x) = x? proxg (x) for g? = 1????? . Next we use a property of conjugate functions from [21, 13], which states that g(·) = 1 ? ???(?·). Finally, by the scaling property of proximity operators [7], we have that proxg (x) = ? prox???? (?x).2The somewhat cumbersome notation B n(n) denotes the mode-n matricization of tensor Bn, that is, B n(n) = (Bn)(n).Algorithm 1 Computation of prox????(y) Input: y ? Rd, ?, ? > 0.Output: wˆ ? Rd.Initialization: initial step ?0 = 2 , initial and best found solution w 0 = wˆ = PS(y) ? Rd.for t = 1, 2, . . . do ? ? ?0? t Find k such that k ? argmax{??wt?11:r ?2 ? r : 0 ? r ? d} w˜1:k ? wt?11:k ? ? ( wt?11:k ( 1 + ???wt?11:k ?2 ) ? y1:k ) w˜k+1:d ? wt?1k+1:d ? ? ( wt?1k+1:d ? yk+1:d ) wt ? P˜S (w˜) If h(wt) < h(wˆ) then wˆ ? wt If “Stopping Condition = True” then terminate.end for It remains to compute the proximity operator of a multiple of the function ??? in equation (13), that is, for any ? > 0, y ? S , we wish to compute prox???? (y) = argmin w {h (w) : w ? S} where we have defined S := {w ? Rd : w1 ? · · · ? wd ? 0} and h (w) = ?w ? y?22 + ? d max r=0 {? ?w1:r?2 ? r} .In order to solve this problem we employ the projected subgradient method, see e.g. [6]. It consists in applying two steps at each iteration. First, it advances along a negative subgradient of the current solution; second, it projects the resultant point onto the feasible set S . In fact, according to [6], it is sufficient to compute an approximate projection, a step which we describe in Appendix D. To compute a subgradient of h at w, we first find any integer k such that k ? dargmax r=0 {? ?w1:r?2 ? r}.Then, we calculate a subgradient g of the function h at w by the formula gi = { ( 1 + ???w1:k?2 ) wi ? yi, if i ? k, wi ? yi, otherwise.Now we have all the ingredients to apply the projected subgradient method, which is summarized in Algorithm 1. In our implementation we stop the algorithm when an update of wˆ is not made for more than 102 iterations.5 Experiments We have conducted a set of experiments to assess whether there is any advantage of using the pro- posed regularizer over the tensor trace norm for tensor completion3. First, we have designed a synthetic experiment to evaluate the performance of both approaches under controlled conditions.Then, we have tried both methods on two tensor completion real data problems. In all cases, we have used a validation procedure to tune the hyper-parameter ?, present in both approaches, among the values { 10j : j = ?7,?6, . . . , 1}. In our proposed approach there is one further hyper-parameter, ?, to be specified. It should take the value of the Euclidean norm of the underlying tensor. Since this is unknown, we propose to use the estimate ?ˆ = ?????w?22 + (mean(w)2 + var(w)) ( N? i=1 pi ?m ) , where m is the number of known entries and w ? Rm contains their values. This estimator assumes that each value in the tensor is sampled from N (mean(w), var(w)), where mean(w) and var(w) are the average and the variance of the elements in w.3The code is available at http://romera-paredes.com/code/tensor-completion ?5 ?4 ?3 ?2 ?1 0.0085 0.009 0.0095 0.01 0.0105 0.011 0.0115 log ?2 R M S E Tensor Trace Norm Proposed Regularizer 50 100 150 200 p S e c o n d s Tensor Trace Norm Proposed Regularizer Figure 1: Synthetic dataset: (Left) Root Mean Squared Error (RMSE) of tensor trace norm and the proposed regularizer. (Right) Running time execution for different sizes of the tensor.5.1 Synthetic Dataset We have generated a 3-order tensor W0 ? R40×20×10 by the following procedure. First we gener- ated a tensor W with ranks (12, 6, 3) using Tucker decomposition (see e.g. [16]) Wi1,i2,i3 = 12? j1=1 6? j2=1 3? j3=1 Cj1,j2,j3M (1)i1,j1M (2) i2,j2 M (3) i3,j3 , (i1, i2, i3) ? [40]× [20]× [10] where each entry of the Tucker decomposition components is sampled from the standard Gaussian distribution N (0, 1). We then created the ground truth tensor W0 by the equation W0i1,i2,i3 = Wi1,i2,i3 ?mean(W)? Nstd(W) + ?i1,i2,i3 where mean(W) and std(W) are the mean and standard deviation of the elements of W , N is the total number of elements of W , and the ?i1,i2,i3 are i.i.d. Gaussian random variables with zero mean and variance ?2. We have randomly sampled 10% of the elements of the tensor to compose the training set, 45% for the validation set, and the remaining 45% for the test set. After repeating this process 20 times, we report the average results in Figure 1 (Left). Having conducted a paired t-test for each value of ?2, we conclude that the visible differences in the performances are highly significant, obtaining always p-values less than 0.01 for ?2 ? 10?2.Furthermore, we have conducted an experiment to test the running time of both approaches. We have generated tensors W0 ? Rp×p×p for different values of p ? {20, 40, . . . , 200}, following the same procedure as outlined above. The results are reported in Figure 1 (Right). For low values of p, the ratio between the running time of our approach and that of the trace norm regularization method is quite high. For example in the lowest value tried for p in this experiment, p = 20, this ratio is 22.661. However, as the volume of the tensor increases, the ratio quickly decreases. For example, for p = 200, the running time ratio is 1.9113. These outcomes are expected because when p is low, the most demanding routine in our method is the one described in Algorithm 1, where each iteration is of order O (p) and O ( p2 ) in the best and worst case, respectively. However, as p increases the singular value decomposition routine, which is common to both methods, becomes the most demanding because it has a time complexity O ( p3 ) 
[10]. Therefore, we can conclude that even though our approach is slower than the trace norm based method, this difference becomes much smaller as the size of the tensor increases.5.2 School Dataset The first real dataset we have tried is the Inner London Education Authority (ILEA) dataset. It is composed of examination marks ranging from 0 to 70, of 15362 students who are described by a set of attributes such as school and ethnic group. Most of these attributes are categorical, thereby we can think of exam mark prediction as a tensor completion problem where each of the modes corresponds to a categorical attribute. In particular, we have used the following attributes: school (139), gender (2), VR-band (3), ethnic (11), and year (3), leading to a 5-order tensor W ? R139×2×3×11×3.4000 6000 8000 10000 12000 10.4 10.6 10.8 11.2 11.4 11.6 m (Training Set Size) R M S E Tensor Trace Norm Proposed Regularizer 2 4 6 8 10 12 14 16 x 10 m (Training Set Size) R M S E Tensor Trace Norm Proposed Regularizer Figure 2: Root Mean Squared Error (RMSE) of tensor trace norm and the proposed regularizer for ILEA dataset (Left) and Ocean video (Right).We have selected randomly 5% of the instances to make the test set and another 5% of the instances for the validation set. From the remaining instances, we have randomly chosenm of them for several values of m. This procedure has been repeated 20 times and the average performance is presented in Figure 2 (Left). There is a distinguishable improvement of our approach with respect to tensor trace norm regularization for values of m > 7000. To check whether this gap is significant, we have conducted a set of paired t-tests in this regime. In all these cases we obtained a p-value below 0.01.5.3 Video Completion In the second real-data experiment we have performed a video completion test. Any video can be treated as a 4-order tensor: “width” × “height” × “RGB” × “video length”, so we can use tensor completion algorithms to rebuild a video from a few inputs, a procedure that can be useful for compression purposes. In our case, we have used the Ocean video, available at [17]. This video sequence can be treated as a tensor W ? R160×112×3×32. We have randomly sampled m tensors elements as training data, 5% of them as validation data, and the remaining ones composed the test set. After repeating this procedure 10 times, we present the average results in Figure 2 (Right). The proposed approach is noticeably better than the tensor trace norm in this experiment. This apparent outcome is strongly supported by the paired t-tests which we run for each value of m, obtaining always p-values below 0.01, and for the cases m > 5× 104, we obtained p-values below 10?6.6 Conclusion In this paper, we proposed a convex relaxation for the average of the rank of the matricizations of a tensor. We compared this relaxation to a commonly used convex relaxation used in the context of tensor completion, which is based on the trace norm. We proved that this second relaxation is not tight and argued that the proposed convex regularizer may be advantageous. Our numerical experience indicates that our method consistently improves in terms of estimation error over tensor trace norm regularization, while being computationally comparable on the range of problems we considered. In the future it would be interesting to study methods to speed up the computation of the proximity operator of our regularizer and investigate its utility in tensor learning problems beyond tensor completion such as multilinear multitask learning [20].Acknowledgements We wish to thank Andreas Argyriou, Raphael Hauser, Charles Micchelli and Marco Signoretto for useful comments. A valuable contribution was made by one of the anonymous referees. Part of this work was supported by EPSRC Grant EP/H017178/1, EP/H027203/1 and Royal Society Interna- tional Joint Project 2012/R2.
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Over the past decade, there has been a fever of activity to address the drawbacks of PCA with a class of techniques called sparse PCA that combine the essence of PCA with the assumption that the phe- nomena of interest depend mostly on a few variables. Examples include algorithmic [e.g., 1, 3–10] and theoretical [e.g., 11–14] developments. However, much of this work has focused on the first principal component. One rationale behind this focus is by analogy with ordinary PCA: additional components can be found by iteratively deflating the input matrix to account for variation uncovered by previous components. However, the use of deflation with sparse PCA entails complications of non-orthogonality, sub-optimality, and multiple tuning parameters [15]. Identifiability and consis- tency present more subtle issues. The principal directions of variation correspond to eigenvectors of some population matrix ?. There is no reason to assume a priori that the d largest eigenvalues of ? are distinct. Even if the eigenvalues are distinct, estimates of individual eigenvectors can be unreliable if the gap between their eigenvalues is small. So it seems reasonable, if not necessary, to de-emphasize eigenvectors and to instead focus on their span, i.e. the principal subspace of variation.There has been relatively little work on the problem of estimating the principal subspace or even multiple eigenvectors simultaneously. Most works that do are limited to iterative deflation schemes or optimization problems whose global solution is intractable to compute. Sole exceptions are the diagonal thresholding method [2], which is just ordinary PCA applied to the subset of variables with largest marginal sample variance, or refinements such as iterative thresholding [16], which use diagonal thresholding as an initial estimate. These works are limited, because they cannot be used when the variables have equal variances (e.g., correlation matrices). Theoretical results are equally limited in their applicability. Although the optimal minimax rates for the sparse principal subspace problem are known in both the spiked [17] and general [18] covariance models, existing statistical guarantees only hold under the restrictive spiked covariance model, which essentially guarantees that diagonal thresholding has good properties, or for estimators that are computationally intractable.In this paper, we propose a novel convex optimization problem to estimate the d-dimensional princi- pal subspace of a population matrix? based on a noisy input matrix S. We show that if S is a sample covariance matrix and the projection ? of the d-dimensional principal subspace of ? depends only on s variables, then with a suitable choice of regularization parameter, the Frobenius norm of the error of our estimator X? is bounded with high probability |||X? ??|||2 = O ( (?1/?)s ? log p/n ) where ?1 is the largest eigenvalue of ? and ? the gap between the dth and (d + 1)th largest eigen- values of ?. This rate turns out to be nearly minimax optimal (Corollary 3.3), and under additional assumptions on signal strength, it also allows us to recover the support of the principal subspace (Theorem 3.2). Moreover, we provide easy to verify conditions (Theorem 3.3) that yield near- optimal statistical guarantees for other choices of input matrix, such as Pearson’s correlation and Kendall’s tau correlation matrices (Corollary 3.4).Our estimator turns out to be a semidefinite program (SDP) that generalizes the DSPCA approach of [1] to d ? 1 dimensions. It is based on a convex body, called the Fantope, that provides a tight relaxation for simultaneous rank and orthogonality constraints on the positive semidefinite cone.Solving the SDP is non-trivial. We find that an alternating direction method of multipliers (ADMM) algorithm [e.g., 19] can efficiently compute its global optimum (Section 4).In summary, the main contributions of this paper are as follows.1. We formulate the sparse principal subspace problem as a novel semidefinite program with a Fantope constraint (Section 2).2. We show that the proposed estimator achieves a near optimal rate of convergence in sub- space estimation without assumptions on the rank of the solution or restrictive spiked co- variance models. This is a first for both d = 1 and d > 1 (Section 3).3. We provide a general theoretical framework that accommodates other matrices, in addition to sample covariance, such as Pearson’s correlation and Kendall’s tau.4. We develop an efficient ADMM algorithm to solve the SDP (Section 4), and provide nu- merical examples that demonstrate the superiority of our approach over deflation methods in both computational and statistical efficiency (Section 5).The remainder of the paper explains each of these contributions in detail, but we defer all proofs to Appendix A.Related work Existing work most closely related to ours is the DSPCA approach for single com- ponent sparse PCA that was first proposed in [1]. Subsequently, there has been theoretical analysis under a spiked covariance model and restrictions on the entries of the eigenvectors [11], and algo- rithmic developments including block coordinate ascent [9] and ADMM [20]. The crucial difference with our work is that this previous work only considered d = 1. The d > 1 case requires invention and novel techniques to deal with a convex body, the Fantope, that has never before been used in sparse PCA.Notation For matrices A,B of compatible dimension ?A,B? := tr(ATB) is the Frobenius inner product, and |||A|||22 := ?A,A? is the squared Frobenius norm. ?x?q is the usual ?q norm with ?x?0 defined as the number of nonzero entries of x. ?A?a,b is the (a, b)-norm defined to be the ?b norm of the vector of rowwise ?a norms of A, e.g. ?A?1,? is the maximum absolute row sum. For a symmetric matrix A, we define ?1(A) ? ?2(A) ? · · · to be the eigenvalues of A with multiplicity.When the context is obvious we write ?j := ?j(A) as shorthand. For two subspacesM1 andM2, sin?(M1,M2) is defined to be the matrix whose diagonals are the sines of the canonical angles between the two subspaces [see 21, §VII].2 Sparse subspace estimation Given a symmetric input matrix S, we propose a sparse principal subspace estimator X? that is defined to be a solution of the semidefinite program maximize ?S,X? ? ??X?1,1 subject to X ? Fd, (1) in the variable X , where Fd := {X : 0 ? X ? I and tr(X) = d} is a convex body called the Fantope [22, §2.3.2], and ? ? 0 is a regularization parameter that encourages sparsity. When d = 1, the spectral norm bound in Fd is redundant and (1) reduces to the DSPCA approach of [1]. The motivation behind (1) is based on two key insights.The first insight is a variational characterization of the principal subspace of a symmetric matrix.The sum of the d largest eigenvalues of a symmetric matrix A can be expressed as d? i=1 ?i(A) (a) = max V TV=Id ?A, V V T ? (b)= max X?Fd ?A,X? . (2) Identity (a) is an extremal property known as Ky Fan’s maximum principle [23]; (b) is based on the less well known observation that Fd = conv({V V T : V TV = Id}) , i.e. the extremal points of Fd are the rank-d projection matrices. See [24] for proofs of both. Thus, the constraint in (1) is a convex relaxation of an orthogonality constraint.The second insight is a connection between the (1, 1)-norm and a notion of subspace sparsity intro- duced by [18]. Any X ? 0 can be factorized (non-uniquely) as X = V V T .Lemma 2.1. If X = V V T , then ?X?1,1 ? ?V ?22,1 ? ?V ?22,0 tr(X).Consequently, any X ? Fd that has at most s non-zero rows necessarily has ?X?1,1 ? s2d. Thus, ?X?1,1 is a convex relaxation of what [18] call row sparsity for subspaces.These two insights reveal that (1) is a semidefinite relaxation of the non-convex problem maximize ?S, V V T ? ? ??V ?22,0d subject to V TV = Id .
[18] proposed solving an equivalent form of the above optimization problem and showed that the estimator corresponding to its global solution is minimax rate optimal under a general statistical model for S. Their estimator requires solving an NP-hard problem. The advantage of (1) is that it is computationally tractable.Subspace estimation The constraint X? ? Fd guarantees that its rank is? d. However X? need not be an extremal point ofFd, i.e. a rank-d projection matrix. In order to obtain a proper d-dimensional subspace estimate, we can extract the d leading eigenvectors of X? , say V? , and form the projection matrix ?? = V? V? T . The projection is unique, but the choice of basis is arbitrary. We can follow the convention of standard PCA by choosing an orthogonal matrixO so that (V? O)TS(V? O) is diagonal, and take V? O as the orthonormal basis for the subspace estimate.3 Theory In this section we describe our theoretical framework for studying the statistical properties of X? given by (1) with arbitrary input matrices that satisfy the following assumptions.Assumption 1 (Symmetry). S and ? are p× p symmetric matrices.Assumption 2 (Identifiability). ? = ?(?) = ?d(?)? ?d+1(?) > 0.Assumption 3 (Sparsity). The projection ? onto the subspace spanned by the eigenvectors of ? corresponding to its d largest eigenvalues satisfies ???2,0 ? s, or equivalently, ?diag(?)?0 ? s.The key result (Theorem 3.1 below) implies that the statistical properties of the error of the estimator ? := X? ?? , can be derived, in many cases, by routine analysis of the entrywise errors of the input matrix W := S ? ? .There are two main ideas in our analysis of X? . The first is relating the difference in the values of the objective function in (1) at ? and X? to ?. The second is exploiting the decomposability of the regularizer. Conceptually, this is the same approach taken by [25] in analyzing the statistical properties of regularized M -estimators. It is worth noting that the curvature result in our problem comes from the geometry of the constraint set in (1). It is different from the “restricted strong convexity” in [25], a notion of curvature tailored for regularization in the form of penalizing an unconstrained convex objective.3.1 Variational analysis on the Fantope The first step of our analysis is to establish a bound on the curvature of the objective function along the Fantope and away from the truth.Lemma 3.1 (Curvature). Let A be a symmetric matrix and E be the projection onto the subspace spanned by the eigenvectors of A corresponding to its d largest eigenvalues ?1 ? ?2 ? · · · . If ?A = ?d ? ?d+1 > 0, then ?A |||E ? F |||22 ? ?A,E ? F ? for all F satisfying 0 ? F ? I and tr(F ) = d.A version of Lemma 3.1 first appeared in [18] with the additional restriction that F is a projection matrix. Our proof of the above extension is a minor modification of their proof.The following is an immediate corollary of Lemma 3.1 and the Ky Fan maximal principle.Corollary 3.1 (A sin? theorem [18]). Let A,B be symmetric matrices and MA, MB be their respective d-dimensional principal subspaces. If ?A,B = [?d+1(A)??d(A)]? [?d+1(B)??d(B)], then |||sin?(MA,MB)|||2 ? ? ?A,B |||A?B|||2 .The advantage of Corollary 3.1 over the Davis-Kahan Theorem [see, e.g., 21, §VII.3] is that it does not require a bound on the differences between eigenvalues of A and eigenvalues of B. This means that typical applications of the Davis-Kahan Theorem require the additional invocation of Weyl’s Theorem. Our primary use of this result is to show that even if rank(X?) ?= d, its principal subspace will be close to that of ? if ? is small.Corollary 3.2 (Subspace error bound). IfM is the principal d-dimensional subspace of ? and M? is the principal d-dimensional subspace of X? , then |||sin?(M,M?)|||2 ? ? 2|||?|||2 .3.2 Deterministic error With Lemma 3.1, it is straightforward to prove the following theorem.Theorem 3.1 (Deterministic error bound). If ? ? ?W??,? and s ? ???2,0 then |||?|||2 ? 4s?/? .Theorem 3.1 holds for any global optimizer X? of (1). It does not assume that the solution is rank-d as in [11]. The next theorem gives a sufficient condition for support recovery by diagonal thresholding X? .Theorem 3.2 (Support recovery). For all t > 0??{j : ?jj = 0, X?jj ? t}??+ ??{j : ?jj ? 2t, X?jj < t}?? ? |||?|||22 t2 .As a consequence, the variable selection procedure J?(t) := { j : X?jj ? t } succeeds if minj:?jj ?=0?jj ? 2t > 2|||?|||2.3.3 Statistical properties In this section we use Theorem 3.1 to derive the statistical properties of X? in a generic setting where the entries ofW uniformly obey a restricted sub-Gaussian deviation inequality. This is not the most general result possible, but it allows us to illustrate the statistical properties of X? for two different types of input matrices: sample covariance and Kendall’s tau correlation. The former is the standard input for PCA; the latter has recently been shown to be a useful robust and nonparametric tool for high-dimensional graphical models [26].Theorem 3.3 (General statistical error bound). If there exists ? > 0 and n > 0 such that ? and S satisfy max ij P (|Sij ? ?ij | ? t) ? 2 exp (? 4nt2/?2) (3) for all t ? ? and ? = ? ? log p/n ? ? , (4) then |||X? ??|||2 ? 4? ? s ? log p/n with probability at least 1? 2/p2.Sample covariance Consider the setting where the input matrix is the sample covariance matrix of a random sample of size n > 1 from a sub-Gaussian distribution. A random vector Y with ? = Var(Y ) has sub-Gaussian distribution if there exists a constant L > 0 such that P (|?Y ? EY, u?| ? t) ? exp (? Lt2/??1/2u?22) (5) for all u and t ? 0. Under this condition we have the following corollary of Theorem 3.3.Corollary 3.3. Let S be the sample covariance matrix of an i.i.d. sample of size n > 1 from a sub-Gaussian distribution (5) with population covariance matrix ?. If ? is chosen to satisfy (4) with ? = c?1, then |||X? ??|||2 ? C ?1 ? s ? log p/n with probablity at least 1? 2/p2, where c, C are constants depending only on L.Comparing with the minimax lower bounds derived in [17, 18], we see that the rate in Corollary 3.3 is roughly larger than the optimal minimax rate by a factor of? ?1/?d+1 · ? s/d The first term only becomes important in the near-degenerate case where ?d+1 ? ?1. It is possible with much more technical work to get sharp dependence on the eigenvalues, but we prefer to retain brevity and clarity in our proof of the version here. The second term is likely to be unimprovable without additional conditions on S and ? such as a spiked covariance model. Very recently, [14] showed in a testing framework with similar assumptions as ours when d = 1 that the extra factor? s is necessary for any polynomial time procedure if the planted clique problem cannot be solved in randomized polynomial time.Kendall’s tau Kendall’s tau correlation provides a robust and nonparametric alternative to ordi- nary (Pearson) correlation. Given an n × p matrix whose rows are i.i.d. p-variate random vectors, the theoretical and empirical versions of Kendall’s tau correlation matrix are ?ij := Cor ( sign(Y1i ? Y2i) , sign(Y1j ? Y2j) ) ?ˆij := n(n? 1) ? s<t sign(Ysi ? Yti) sign(Ysj ? Ytj) .A key feature of Kendall’s tau is that it is invariant under strictly monotone transformations, i.e.sign(Ysi ? Yti) sign(Ysj ? Ytj)) = sign(fi(Ysi)? fi(Yti)) sign(fj(Ysj)? fj(Ytj)) , where fi, fj are strictly monotone transformations. When Y is multivariate Gaussian, there is also a one-to-one correspondence between ?ij and ?ij = Cor(Y1i, Y1j) [27] : ?ij = ? arcsin(?ij) . (6) These two observations led [26] to propose using T?ij = { sin ( ? 2 ?ˆij ) if i ?= j 1 if i = j .(7) as an input matrix to Gaussian graphical model estimators in order to extend the applicability of those procedures to the wider class of nonparanormal distributions [28]. This same idea was extended to sparse PCA by [29]; they proposed and analyzed using T? as an input matrix to the non-convex sparse PCA procedure of [13]. A shortcoming of that approach is that their theoretical guarantees only hold for the global solution of an NP-hard optimization problem. The following corollary of Theorem 3.3 rectifies the situation by showing that X? with Kendall’s tau is nearly optimal.Corollary 3.4. Let S = T? as defined in (7) for an i.i.d. sample of size n > 1 and let ? = T be the analogous quantity with ?ij in place of ?ˆij . If ? is chosen to satisfy (4) with ? = ? 8?, then |||X? ??|||2 ? ? 2? ? s ? log p/n with probablity at least 1? 2/p2.Note that Corollary 3.4 only requires that ?ˆ be computed from an i.i.d. sample. It does not specify the marginal distribution of the observations. So ? = T is not necessarily positive semidefinite and may be difficult to interpret. However, under additional conditions, the following lemma gives meaning to T by extending (6) to a wide class of distributions, called transelliptical by [29], that includes the nonparanormal. See [29, 30] for further information.Lemma ([29, 30]). If (Y11, . . . , Y1p) has continuous distribution and there exist monotone transfor- mations f1, . . . , fp such that ( f1(Y11), . . . , fp(Y1p) ) has elliptical distribution with scatter matrix ?˜, then Tij = ?˜ij/ ? ?˜ii?˜jj .Moreover, if fj(Y1j), j = 1, . . . , p have finite variance, then Tij = Cor ( fi(Y1i), fj(Y1j) ) .This lemma together with Corollary 3.4 shows that Kendall’s tau can be used in place of the sample correlation matrix for a wide class of distributions without much loss of efficiency.4 An ADMM algorithm The chief difficulty in directly solving (1) is the interaction between the penalty and the Fantope constraint. Without either of these features, the optimization problemwould be much easier. ADMM can exploit this fact if we first rewrite (1) as the equivalent equality constrained problem minimize ? · 1Fd(X)? ?S,X?+ ??Y ?1,1 subject to X ? Y = 0 , (8) Algorithm 1 Fantope Projection and Selection (FPS) Require: S = ST , d ? 1, ? ? 0, ? > 0, ? > 0 Y (0) ? 0, U (0) ? 0 ? Initialization repeat t = 0, 1, 2, 3, . . .X(t+1) ? PFd ( Y (t) ? U (t) + S/?) ? Fantope projection Y (t+1) ? S?/? ( X(t+1) + U (t) ) ? Elementwise soft thresholding U (t+1) ? U (t) +X(t+1) ? Y (t+1) ? Dual variable update until max(|||X(t) ? Y (t)|||22 , ?2|||Y (t) ? Y (t?1)|||22) ? d?2 ? Stopping criterion return Y (t) in the variablesX and Y , where 1Fd is the 0-1 indicator function forFd and we adopt the convention? · 0 = 0. The augmented Lagrangian associated with (8) has the form L?(X,Y, U) :=? · 1Fd(X)? ?S,X?+ ??Y ?1,1 + ? ( |||X ? Y + U |||22 ? |||U |||22 ) , (9) where U = (1/?)Z is the scaled ADMM dual variable and ? is the ADMM penalty parameter [see 19, §3.1]. ADMM consists of iteratively minimizing L? with respect to X , minimizing L? with respect to Y , and then updating the dual variable. Algorithm 1 summarizes the main steps.In light of the separation of X and Y in (9) and some algebraic manipulation, the X and Y updates reduce to computing the proximal operators PFd ( Y ? U + S/?) := argmin X?Fd |||X ? (Y ? U + S/?)|||22 S?/?(X + U) := argmin Y ? ? ?Y ?1,1 + |||(X + U)? Y |||22 .S?/? is the elementwise soft thresholding operator [e.g., 19, §4.4.3] defined as S?/?(x) = sign(x)max(|x|? ?/?, 0) .PFd is the Euclidean projection onto Fd and is given in closed form in the following lemma.Lemma 4.1 (Fantope projection). If X = ? i ?iuiu T i is a spectral decomposition of X , then PFd(X) = ? i ? + i (?)uiu T i , where ? + i (?) = min(max(?i ? ?, 0), 1) and ? satisfies the equation? i ? + i (?) = d.Thus,PFd(X) involves computing an eigendecomposition of Y , and then modifying the eigenvalues by solving a monotone, piecewise linear equation.Rather than fix the ADMM penalty parameter ? in Algorithm 1 at some constant value, we recom- mend using the varying penalty scheme described in [19, §3.4.1] that dynamically updates ? after each iteration of the ADMM to keep the primal and dual residual norms (the two sum of squares in the stopping criterion of Algorithm 1) within a constant factor of each other. This eliminates an additional tuning parameter, and in our experience, yields faster convergence.5 Simulation results We conducted a simulation study to compare the effectiveness of FPS against three deflation-based methods: DSPCA (which is just FPS with d = 1), GPower?1 [7], and SPC [5, 6]. These methods obtain multiple component estimates by taking the kth component estimate vˆk from input matrix Sk, and then re-running the method with the deflated input matrix: Sk+1 = (I ? vˆkvˆTk )Sk(I ? vˆkvˆTk ).The resulting d-dimensional principal subspace estimate is the span of vˆ1, . . . , vˆd. Tuning parameter selection can be much more complicated for these iterative deflation methods. In our simulations, we simply fixed the regularization parameter to be the same for all d components.We generated input matrices by sampling n = 100, i.i.d. observations from a Np(0,?), p = 200 distribution and taking S to be the usual sample covariance matrix. We considered two different types of sparse ? = V V T of rank d = 5: those with disjoint support for the nonzero entries of the s:10, noise:1 s:10, noise:10 s:25, noise:1 s:25, noise:10 ?3 ?2 ?1 ?1 support:disjoint support:shared 5 10 20 30 5 10 20 30 5 10 20 30 5 10 20 30 (2,1)?norm of estimate log (M SE ) Figure 1: Mean squared error of FPS ( ), DSPCAwith deflation ( ), GPower?1 ( ), and SPC ( ) across 100 replicates each of a variety of simulation designs with n = 100, p = 200, d = 5, s ? {10, 25}, noise ?2 ? {1, 10}.columns of V and those with shared support. We generated V by sampling its nonzero entries from a standard Gaussian distribution and then orthnormalizing V while retaining the desired sparsity pattern. In both cases, the number of nonzero rows of V is equal to s ? {10, 25}. We then embedded ? inside the population covariance matrix ? = ?? + (I ? ?)?0(I ? ?), where ?0 is a Wishart matrix with p degrees of freedom and ? > 0 is chosen so that the effective noise level (in the optimal minimax rate [18]), ?2 = ? ?1?d+1/(?d ? ?d+1) ? {1, 10}.Figure 1 summarizes the resulting mean squared error |||?? ? ?|||22 across 100 replicates for each of the different combinations of simulation parameters. Each method’s regularization parameter varies over a range and the x-axis shows the (2, 1)-norm of the corresponding estimate. At the right extreme, all methods essentially correspond to standard PCA. It is clear that regularization is beneficial, because all the methods have significantly smaller MSE than standard PCA when they are sufficiently sparse. Comparing between methods, we see that FPS dominates in all cases, but the competition is much closer in the disjoint support case. Finally, all methods degrade when the number of active variables or noise level increases.6 Discussion Estimating sparse principal subspaces in high-dimensions poses both computational and statistical challenges. The contribution of this paper—a novel SDP based estimator, an efficient algorithm, and strong statistical guarantees for a wide array of input matrices—is a significant leap forward on both fronts. Yet, there are newly open problems and many possible extensions related to this work.For instance, it would be interesting to investigate the performance of FPS a under weak, rather than exact, sparsity assumption on ? (e.g., ?q , 0 < q ? sparsity). The optimization problem (1) and ADMM algorithm can easily be modified handle other types of penalties. In some cases, extensions of Theorem 3.1 would require minimal modifications to its proof. Finally, the choices of dimension d and regularization parameter ? are of great practical interest. Techniques like cross-validation need to be carefully formulated and studied in the context of principal subspace estimation.Acknowledgments This research was supported in part by NSF grants DMS-0903120, DMS-1309998, BCS-0941518, and NIH grant MH057881.
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[ Regret(T, piPS? ) T? ??M? ?M]P (M? ?M) ? P ( Regret(T, piPS? ) T? ??M? ?M)P (M? ?M) Therefore via theorem (1), for any ? > 12 : P ( Regret(T, piPS? ) T? ??M? ?M) ? ( 1 P (M? ?M) ) E[Regret(T, piPS? )] T? ? 0 B Bounding the sum of confidence set widths We are interested in bounding min{??m k=1 ?? i=1 min{?kstk+i, atk+i), 1}, T} which we claim is O(?S ? AT log(SAT ) for ?k(s, a) := ? 14S log(2SAmtk) max{1,Ntk (s,a)} .Proof. In a manner similar to [4] we can say: m? k=1 ?? i=1 ? 14S log(2SAmtk) max{1, Ntk (s, a)} ? m? k=1 ?? i=1 1{Ntk??} + m? k=1 ?? i=1 1{Ntk>?} ? 14S log(2SAmtk) max{1, Ntk (s, a)} Now, the consider the event (st, at) = (s, a) and (Ntk (s, a) ? ?). This can happen fewer than 2? times per state action pair. Therefore, ?m k=1 ?? i=1 1(Ntk (s, a) ? ?) ? 2?SA.Now, suppose Ntk (s, a) > ? . Then for any t ? {tk, .., tk+1 ? 1}, Nt(s, a) + 1 ? Ntk (s, a) + ? ? 2Ntk (s, a).Therefore: m? k=1 tk+1?1? t=tk ? 1(Ntk (st, at) > ?) Ntk (st, at) ? m? k=1 tk+1?1? t=tk ? Nt(st, at) + 1 = ? T? t=1 (Nt(st, at) + 1)?1/2 ? ? ? s,a NT+1(s,a)? j=1 j?1/2 ? ? ? s,a ? NT+1(s,a) x=0 x?1/2 dx ? ? 2SA ? s,a NT+1(s, a) = ? 2SAT Note that since all rewards and transitions are absolutely constrained ? [0, 1] our regret min{? m? k=1 ?? i=1 min{?k(stk+i, atk+i), 1}, T} ? min{2?2SA+ ? ? 28S2AT log(SAT ), T} ? ? 2?2SAT + ? ? 28S2AT log(SAT ) ? ?S ? 30AT log(SAT ) Which is our required result.
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convergence can be established with enough softness [18, 14]. The natural actor-critic algorithm, which is convergent and optimal, is placed to the lower left corner by Theorem 1, while the in- evitable non-optimality of soft-greedy variants toward right follows from Theorems 2 and 3. The exact (problem-dependent) place and shape of the line separating non-convergent and convergent soft-greedy variants (dashed line on the map) remains an open problem.The main value of Theorem 1 is in bringing the greedy value function and policy gradient method- ologies closer to each other. In our context, the unifying NAC(?) formulation in (8) permits interpo- lation between the methodologies using the ? parameter. As discussed at the end of Section 4, the policy-forgetting term requires a Markovian problem for being justified: a greedy update implicitly 0? 0 ? ? Non-optimistic hard-greedy 7 Oscillation (Bertsekas, . . . ) 7 Non-optimality Optimistic hard-greedy 7 Chattering (Bertsekas, . . . ) 7 Non-optimality Natural actor-critic 3 Convergence (Theorem 1) 3 Optimality Non-optimistic soft-greedy (small ? ) 7 Non-convergence (Perkins & Precup) 7 Non-optimality (Theorems 2–3) Non-optimistic soft-greedy (large ? ) 3 Convergence (Perkins & Precup) 7 Non-optimality (Theorems 2–3) Optimistic soft-greedy (large ? ) 3 Convergence (Melo et al.) 7 Non-optimality (Theorems 2–3) cf. Fig. 2b cf. F ig. 2 c Figure 1: The hyperparameter space of the general form of (approximate) optimistic policy iteration in (4), with known convergence and optimality properties (see text for assumptions).y1 s1 s2 0 1 1/4 al al ar ar (a) A non-Markovian problem (adapted from 
[24]). The incoming arrow indicates the start state.Arrows leading out indicate termination with the shown reward.0 5 10 15 20 0.5 iteration ? ( le ft ) ? ? ( ri gh t) ? = 0.2, ? = 1 ? = 0.2, ? = 0.2 ? = 0.2, ? = 0.05 (b) Non-optimality or oscillation with ? 6? 0. The variants are marked with in Fig. 1 (schematic).0 5 10 15 20 0.5 iteration ? ( le ft ) ? ? ( ri gh t) ? = ? = 0.2 ? = ? = 0.05 ? = ? = 0.01 NAC (? = 1) (c) Interpolation toward NAC with ?? 0 and ? ? 0. The variants are marked with in Fig. 1 (schematic).Figure 2: Empirical illustration of the behavior of optimistic policy iteration ((1), (2), (4) and (5), with tabular ?) in the proximity of a stochastic optimum. The problem is shown in Fig. 2a. In Figures 2b and 2c, the optimum at ?(left) ? ?(right) = log(2) is denoted by a solid green line. The uniformly stochastic policy is denoted by a dashed red line.stands on aMarkov assumption and the ? parameter in (8) can be interpreted as adjusting the strength of this assumption. In this respect, the policy improvement parameter ? in NAC(?) can be seen (in- versely) as a dual in spirit to the policy evaluation parameter ? in TD(?)-style algorithms. On the policy evaluation side, having ? = 0 obtains variance reduction by assuming and exploiting Marko- vianity of the problem, while ? = 1 obtains unbiased estimates also for non-Markovian problems.On the policy improvement side, with ? = 1, we have strictly greedy updates that gain in speed as the policy can respond instantly to new opportunities appearing in the value function (for empirical observations of such a speed gain, see [11, 25]), and in representational flexibility due to the lack of continuity constraints between successive policies (for a canonical example, consider fitted Q itera- tion). This comes at the price of either oscillation or non-optimality if the Markov assumption fails to hold, which is illustrated in Figure 2b for the problem in 2a. With ? ? 0, we approach natural gradient updates that remain sound also in non-Markovian settings, which is illustrated in Figure 2c.The possibility to interpolate between the approaches might turn out useful in problems with partial Markovianity: a large ? in the NAC(?) formulation can be used to quickly find the rough direction of the strongest attractors, after which gradually decreasing ? allows a convergent final ascent toward an optimum.Acknowledgments This work has been financially supported by the Academy of Finland through project no. 254104, and by the Foundation of Nokia Corporation.
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