Sparse Prediction with the k-overlap Norm

Transcription

1 Sparse Prediction with the -Overlap Norm Andreas Argyriou Toyota Technological Institute at Chicago Rina Foygel University of Chicago Nathan Srebro Toyota Technological Institute at Chicago Abstract We derive a novel norm that corresponds to the tightest convex relaxation of sparsity combined with an l penalty and can also be interpreted as a group Lasso norm with overlaps. We show that this new norm provides a tighter relaxation than the elastic net and suggest using it as a replacement for the Lasso or the elastic net in sparse prediction problems. Introduction Regularizing with the l norm, when we expect a sparse solution to a regression problem, is often justified by w being the convex envelope of w 0 (the numberofnon-zerocoordinatesofavectorw R d ). Thatis, w isthetightest convex lower bound on w 0. But we must be careful with this statement for sparsevectorswith largeentries, w 0 can be small while w is large. In order to discuss convex lower bounds on w 0, we must impose some scale constraint. A more accurate statement is that w w w 0, and so, when the magnitudes of entries in w are bounded by, then w w 0, and indeed it is the largest such convex lower bound. Viewed as a convex outer relaxation, S ( ) := { w w 0, w } { w w }. Intersectingthe right-hand-sidewith the l unit ball, weget the tightest convex outer bound (convex hull) of S ( ) : { w w, w } = conv(s ( ) ).

2 However, in our view, this relationship between w and w 0 yields disappointing learning guarantees, and does not appropriately capture the success of the l norm as a surrogate for sparsity. In particular, the sample complexity of learning a linear predictor with non-zero entries by empirical ris minimization inside this class (an NP-hard optimization problem) scales as O(logd), but relaxing to the constraint w yields a sample complexity which scales as O( logd) (since the sample complexity of l -regularized learning scales quadratically with the l norm [0, 0]). Perhaps a better reason for the l norm being a good surrogate for sparsity is because, not only do we expect the magnitude of each entry of w to be bounded, but we further expect w to be small. In a regression setting, with a vector of features x, this can be justified when E[(x w) ] is bounded (a reasonable assumption) and the features are not too correlated see, e.g. [6]. More broadly, especially in the presence of correlations, we might require this as a modeling assumption to aid in robustness and generalization. In any case, we then have w w w 0, and so if we are interested in predictors with bounded l norm, we can motivate the l norm through the following relaxation of sparsity, where the scale is now set by the l norm: { w w 0, w B } { w w B }. The sample complexity when using the relaxation now scales as O(logd). Sparse + l constraint. Ourstartingpointisthenthatofcombiningsparsity and l regularization, and learning a sparse predictor with small l norm. We are thus interested in classes of the form S () := { w w 0, w }. As discussed above, the class { w } (corresponding to the standard Lasso) provides a convex relaxation of S (). But it is clear that we can get a tighter convex relaxation by eeping the l constraint as well: { S () w w }, w (.) { w w }. This is based on bounding the VC-subgraph dimension of this class, which is essentially the effective number of parameters. The sample complexity is O(B logd), where the dependence on B is to be expected. Note that if feature vectors are l -bounded (i.e. individual features are bounded), the sample complexity when using only w B (without a sparsity or l constraint) scales as O(B d). That is, even after identifying the correct support, we still need a sample complexity that scales with B.

3 Constraining (or equivalently, penalizing) both the l and l norms, as in (.), is nown as the elastic net [] and has indeed been advocated as a better alternative to the Lasso [7]. In this paper, we as whether the elastic net is the tightest convex relaxation to sparsity plus l (that is, to S () ) or whether a tighter, and better, convex relaxation is possible. A new norm. We consider the convex hull (tightest convex outer bound) of S (), C := conv(s () ) = conv{ w w 0, w }. (.) We study the gauge function associated with this convex set, that is, the norm whose unit ball is given by (.), which we call the -overlap norm. We show that, for >, this is indeed a tighter convex relaxation than the elastic net (that is, both inequalities in (.) are in fact strict inequalities), and is therefore a better convex constraint than the elastic net when seeing a sparse, low l - norm linear predictor. We thus advocate using it as a replacement for the elastic net. However,wealsoshowthat the gapbetween the elasticnet andthe -overlap norm is at most a factor of, corresponding to a factor of two difference in the sample complexity. Thus, our wor can also be interpreted as justifying the use of the elastic net, viewing it as a fairly good approximation to the tightest possible convex relaxation of sparsity intersected with an l constraint. Still, even a factor of two should not necessarily be ignored and, as we show in our experiments, using the tighter -overlap norm can indeed be beneficial. To better understand the -overlap norm, we show in Section that it can also ( be described as the group lasso with overlap norm [9] corresponding to all d ) subsets of features, hence its name. Despite the exponential number of groups in this description, we show that the -overlap norm can be calculated efficiently in time O(dlogd) and that its dual is given simply by the l norm of the largest entries. We also provide efficient first-order optimization algorithms for learning with the -overlapnorm in Section 4. In Section 5, we demonstrate how it can indeed outperform the Lasso and the elastic net.. Related Wor In many learning problems of interest, Lasso has been observed to shrin too many of the variables of w to zero. In particular, in many applications, when a group of variables is highly correlated, the Lasso may prefer a sparse solution, but we might gain more predictive accuracy by including all the correlated variables in our model. These drawbacs have recently motivated the use of various other regularization methods, such as the elastic net[], which penalizes the regression coefficients w with a combination of l and l norms: { } min Xw y +λ w +λ w : w R d, (.3) 3

4 where for a sample of size n, y R n is the vector of response values, and X R n d is a matrix with column j containing the values of feature j. The elastic net can be viewed as a trade-off between l regularization (the Lasso) and l regularization (Ridge regression [8]), depending on the relative values of λ and λ. In particular, when λ = 0, (.3) is equivalent to the Lasso. This method, and the other methods discussed below, have been observed to significantly outperform Lasso in many real applications. The pairwise elastic net (PEN), proposed by [], has a penalty function that accounts for similarity among features: w PEN R = w + w w R w, where R [0,] p p is a matrix with R j measuring similarity between features X j and X. The trace Lasso [5] is a second method proposed to handle correlations within X, defined by w trace X = Xdiag(w), where denotes the matrix trace-norm (the sum of the singular values) and promotes a low-ran solution. If the features are orthogonal, then both the PEN and the Trace Lasso are equivalent to the Lasso. If the features are all identical, then both penalties are equivalent to Ridge regression (penalizing w ). Another existing penalty is OSCAR [3], given by w OSCAR c = w +c j<max{ w j, w }. Lie the elastic net, each one of these three methods also prefers averaging similar features over selecting a single feature. The -Overlap Norm One argument for the elastic net has been the flexibility of tuning the cardinality of the regression vector w. Thus, when groups of correlated variables are present, a larger may be learned, which corresponds to a higher λ in (.3). A more natural way to obtain such an effect of tuning the cardinality is to consider the convex hull of cardinality vectors, C = conv(s () ) = conv{w Rd w 0, w }. Clearly the sets C are nested, and C and C d are the unit balls for the l and l norms, respectively. Consequently we define the -overlap norm as the norm whose unit ball equals C (the gauge function associated with C ). An equivalent definition is the following variational formula: Definition.. Let {,...,d}. The -overlap norm ov is defined, for every w R d, as { } w ov := min v I : supp(v I ) I, v I = w, I G I G 4

5 Figure : Unit ball of the -overlap norm (left) and of the elastic net (right) on R 3. where G denotes the set of all subsets of {,...,d} of cardinality at most. The equivalence is immediate by rewriting v I = µ I z I in the above definition, where µ I 0,z I C, I G, µ I =. In addition, this immediately I G implies that ov is indeed a norm. In fact, the -overlapnorm is equivalent to the norm used by the group lasso with overlaps [9], when the set of overlapping groups is chosen to be G (although the group lasso has traditionally been used for applications with nown grouped structure, unlie the case considered here). Although the variational definition. is not amenable to computation because of the exponential growth of the set of groups G, the -overlap norm is computationally very tractable, with an O(d log d) algorithm described in Section.. As already mentioned, ov = and ov d =. The unit ball of this new norm in R 3 for = is depicted in Figure. We immediately notice severaldifferences between this unit ball and the l, l and elastic net unit balls. For example, at points with cardinality and l norm equal to, the -overlap norm is not differentiable, but unlie the l or elastic norm, it is differentiable at points with cardinality less than. Thus, the -overlap norm is less biased towards sparse vectors than the elastic net and the l norm.. The Dual Norm It is interesting and useful to compute the dual of the -overlap norm. We follow the notation of [] for ordered vectors: for any w R d, w is the vector of absolute values, and w i is the i-th largest element of w. We have u ov = max{ w,u : w ov } = ( ) ( ) max u i : I G = ( u i ) i I i= =: u () (). This is the l -norm of the largest entries in u, and is nown as the - symmetric gauge norm []. Not surprisingly, this dual norm interpolates between the l norm (when = d and all entries are taen) and the l norm (when = and only the 5

6 largest entry is taen). This parallels the interpolation of the -overlap norm between the l and l norms. Lie the l p norms and elastic net, the -overlap norm and its dual are symmetric gauge functions, that is, sign- and permutation-invariant norms. For properties of such norms, see [].. Computation of the Norm In this section, we derive an alternative formula for the -overlap norm, which leads to computation of the value of the norm in O(dlogd) steps. Proposition.. For every w R d, w ov = r i= ( w i ) + r+ ( d i= r where r is the unique integer in {0,..., } satisfying w i ), w r > r+ (Here we use the notation w 0 = +.) d i= r w i w r. (.) This result shows that ov trades off between the l and l norms in a way that favors sparse vectors but allows for cardinality larger than. It combines the uniform shrinage of an l penalty for the largest components, with the sparse shrinage of an l penalty for the smallest components. Proof of Proposition.. We will use the inequality w,u w,u [6]. We have { ( w ov ) = max u,w } ( u () () ) : u R d { d } = max α i w i α i : α α d 0 i= i= { } d = max α i w i +α w i α i : α α 0. i= i= i= Let A r := d i= r w i for r {0,..., }. If A 0 < w then the solution α is given by α i = w i for i =,...,( ), α i = A 0 for i =,...,d. If 6

7 A 0 w then the optimal α, α lie between w and A 0, and have to be equal. So, the maximization becomes max { } α i w i α i + A α α : α α 0. i= i= If A 0 w and w > A then the solution is α i = w i for i =,...,( ), α i = A for i = ( ),...,d. Otherwise we proceed as before and continue this process. At stage r the process terminates if A 0 w Ar,..., r w r, A r r+ < w r and all but the last two inequalities are redundant. Hence the condition can be rewritten as (.). One optimal solution is α i = w i for i =,..., r, α i = Ar r+ for i = r,...,d. This proves the claim..3 Learning with the -overlap norm We thus propose using learning rules with -overlap norm regularization. These are appropriate when we would lie to learn a sparse predictor that also has low l norm, and are especially relevant when features might be correlated (that is, in almost all learning tass) but the correlation structure is not nown in advance. For regression problems with squared error loss, the resulting learning rule is of the form { min Xw y + λ } ( w ov ) : w R d (.) with λ > 0 a regularization parameter and {,...,d} also a parameter to be tuned. As typical in regularization-based methods, both λ and can be selected by cross validation [7]. Although we have motivated this norm by considering S (), the set of -sparse unit vectors, the parameter does not necessarily correspond to the sparsity level of the fitted vector of coefficients, and should be chosen via cross-validation independently of the desired sparsity level. 3 Relation to the Elastic Net Recall that the elastic net with penalty parameters λ and λ selects a vector of coefficients given by { } arg min Xw y +λ w +λ w. (3.) Foreaseofcomparisonwiththe-overlapnorm, wefirstshowthatthe elastic net is equivalent to regularizing with the norm { w el := max w, w }, 7

8 for some [,d], corresponding to the unit ball in (.) (note that is not necessarily an integer). To see this, let ŵ be a solution to (3.), and let ( ) ŵ := [,d]. ŵ Then for any w ŵ, if w el ŵ el, then w p ŵ p for p =,. Since ŵ is a solution to (3.), therefore, Xw y Xŵ y. This proves that, for some constraint parameter B, { } ŵ = argmin n Xw y : w el B. Lie the -overlap norm, the elastic net interpolates between the l and l norms. In fact, when is an integer, any -sparse unit vector w R d must lie in the unit ball of el. Since the -overlap norm gives the convex hull of all -sparse unit vectors, this immediately implies that w el w ov w Rd. The two norms are not equal, however. The difference between the two is illustrated in Figure, where we see that the -overlap norm is more rounded. To see an example where the two norms are not equal, we set d = + for some large, and let w = (.5,,,...,) R d. Then { 3 w el = max +,.5 + } ( =.5 + ). Taing u = (,,,..., ), we have u () () <, and therefore w ov > w,u =.5 + =.5. In this example, we see that the two norms can differ by as much as a factor of. We now show that this is actually the most by which they can differ. Proposition 3.. el ov < el. Proof. We show that these bounds hold in the duals of the two norms. First, since el is a maximum over the l and l norms, its dual is given by u (el) { := inf a + } u a a R d Now tae any u R d. First we show u () () u (el) generality, we tae u u d 0. For any a R d,. Without loss of u () () = u : a : + u : a : a + u a. 8

9 Finally, we show that u (el) < u () (). Taing a = (u u +,...,u u +,0,...,0), we have u (el) a + u a = (u i u + ) + u + i= (u i u + u )+ + i= (u i u + )+u + i= = u () (). Furthermore, this yields a strict inequality, because if u > u +, the nextto-last inequality is strict, while if u = = u +, then the last inequality is strict. 4 Optimization Solving the optimization problem (.) efficiently can be done with a first-order proximal algorithm. Proximal methods see [, 4, 5, 8, 9] and references therein are used to solve composite problems of the form min{f(x) +ω(x) : x R d }, where the loss function f(x) and the regularizer ω(x) are convex functions, and f is smooth with an L-Lipschitz gradient. These methods require fast computation of the gradient f and the proximity operator [3] { } prox ω (x) := argmin u x +ω(u) : u R d. In particular, accelerated first-order methods, proposed by Nesterov [4, 5] require two levels of memory at each iteration and exhibit an optimal O ( ) T convergence rate for the objective after T iterations. To obtain a proximal method for -overlap regularization, it suffices to compute the proximity map of g = L ( ov ), for any L > 0. This can be done in O(d( +logd)) steps with Algorithm. Proof. If g is a symmetric gauge function then prox g (v) has the same ordering and signs as v. This is an easy consequence of the v,u v, u inequality. Hence, without loss of generality, we may assume that v v d 0 and require that q q d 0. The latter follows from inequality (4.) and the fact that z is ordered. 9

10 Algorithm Computation of the proximity operator. Input v R d Output q = prox L ( ov )(v) z v Find r {0,..., }, l {,...,d} such that L+ z r > z l > T r,l l +(L+)r+L+ L+ z r (4.) T r,l l +(L+)r+L+ z l+ (4.) where z 0 := +, z d+ :=, T r,l := l z i i= r L L+ z i if i =,..., r T q i z i r,l l +(L+)r+L+ if i = r,...,l 0 if i = l+,...,d Reorder and change the signs of q so that it conforms with v Algorithm Accelerated -overlap regularization. w = α R d, θ for t=,,... do θ t+ + +4θt ( w t+ prox λ αt L ( ov ) L (Xα X t y) ) using Algorithm α t+ w t+ + θt θ t+ (w t+ w t ) end for Note that q = prox g (v) is equivalent to Lz Lq = Lv Lq ( ov ) (q). It suffices to showthat, for w = q, Lz Lq is an optimal α in the proofof Proposition.. Indeed, A r correspondsto d q i = l i= r i= r LT r,l l +(L+)r+L+ ( z i T r,l l +(L+)r+L+ ) = T r,l (l +r+)t r,l l +(L+)r+L+ = (r+) and (.) is equivalent to condition (4.). For i r, we need Lz i Lq i = q i, which holds. For r i l, we need Lz i Lq i = r+ A r, which also holds. For i l +, since q i = 0, any z i such that Lz i Lq i r+ A r are allowed, which is true by (4.). Combining with standard results on accelerated methods [], we obtain the following rate. Corollary 4.. After T iterations of Algorithm with L X, for every 0

11 Table : Mean squared error for the simulated examples (median over 50 data sets). σ = 5 σ = Method MSE MSE Lasso Elastic net overlap w R d, it holds that ( ) Xw T y + λ ( w T ov ) Xw y + λ ( w ov ) + L w w (T +). 5 Empirical Comparisons 5. Simulation For the first simulation we follow [, Sec. 5, example 4]. In this experimental protocol, the target (oracle) vector equals w = (3,...,3,0...,0). }{{}}{{} The input data X were generated from a normal distribution such that components,...,5 have the same random mean Z N(0,), components 6,...,0 have mean Z N(0,) and components,...,5 have mean Z 3 N(0,). A total of 50 data sets were created in this way, each containing 50 training points, 50 validation points and 350 test points. The goal is to achieve good prediction performance on the test data. We compared the -overlap norm with Lasso and the elastic net. We considered the ranges = {,...,d} for -overlap norm regularization, λ = 0 i, i = { 5,...,5}, for the regularization parameter of Lasso and -overlap regularization and the same range for the λ,λ of the elastic net. For each method, the optimal set of parameters was selected based on mean squared error on the validation set. The error reported in Table is the mean squared error with respect to the oracle w, namely MSE = (ŵ w ) V(ŵ w ), where V is the population covariance matrix of X test. We report results on both a high noise (σ = 5 as in []) and a low noise (σ = ) experiment. In the presence of low noise, -overlap regularization predicts significantly better than elastic net, which is even inferior to Lasso. However, in the presence of high noise, none of the three methods could predict well enough. To further illustrate the effect of the -overlap norm with the low noise data sets, in Figure we show the coefficients learned by each method, in absolute value. For each image, one row corresponds to the w learned for one of the 5 5

12 Figure : Solutions learned by each method for all the simulation data sets (σ = ). Left is -overlap, right is Lasso and bottom is elastic net. 50 data sets. Whereas the elastic net can learn higher values at the relevant features, a better feature pattern with less variability emerges when using Lasso and especially when using the -overlap norm. 5. South African Heart Data This is a classificationtas which has been used in [7]. There are 9 variablesand 46 examples, and the response is presence/absence of coronary heart disease. We normalized the data so that each predictor variable has zero mean and unit variance. We then split the data 50 times randomly into training, validation, and test sets of sizes 400, 30, and 3 respectively. For each method, the parameters were selected using the validation data. In Tables, we report the accuracy of each method on the test data. In Table 3, we report the mean squared errors. We observe that all three methods have identical performance Newsgroups This is a binary classification version of 0 newsgroups created in [] which can be found in the LIBSVM data repository. 3 The positive class consists of the 0 groups with names of form sci.*, comp.*, or misc.forsale and the negative class consists of the other 0 groups. To reduce the number of features, we removed the words which appear in less than 3 documents. We randomly split the data into a training, a validation and a test set of sizes 4000,000and 4996, respectively. We report the accuracy on the test data in Table and the mean squared error in Table 3. We found that -overlap regularization gave improved prediction accuracy than both other methods. 4 6 Summary We introduced the -overlap norm as the tightest convex relaxation of sparsity plus l regularization, and showed that it is tighter than the elastic net by exactly a factor of. In our view, this sheds light on the elastic net as a 3 cjlin/libsvmtools /datasets/ 4 Regarding other sparse prediction methods, we did not manage to compare with OSCAR, due to memory limitations, or to PEN or trace Lasso, which do not have code available online.

13 Table : Accuracies for the SA heart data (median over 50 replications) and for the 0 newsgroups data set. Heart 0 newsgroups Method Accuracy (%) Accuracy (%) Lasso Elastic net overlap Table 3: Mean squared errors for the SA heart data(median over 50 replications) and for the 0 newsgroups data set. Heart 0 newsgroups Method MSE MSE Lasso Elastic net overlap close approximation to this tightest possible convex relaxation, and motivates using the -overlap norm when a tighter relaxation is sought. This is also demonstrated in our empirical results. We note that the -overlap norm has better prediction properties, but not necessarily better sparsity-inducing properties, as evident from its more rounded unit ball. It is well understood that there is often a tradeoff between sparsity and good prediction, and that even if the population optimal predictor is sparse, a denser predictor often yields better predictive performance [3, 9, ]. For example, in the presence of correlated features, it is often beneficial to include several highly correlated features rather than a single representative feature. This is exactly the behavior encouraged by l norm regularization, and the elastic net is already nown to yield less sparse (but more predictive) solutions. The -overlap norm goes a step further in this direction, often yielding solutions that are even less sparse (but more predictive) compared to the elastic net. Nevertheless, it is interesting to consider whether compressed sensing results, where l regularizationis of coursecentral, can be refined by using the -overlap norm, which might be able to handle more correlation structure within the set of features. References [] A. Bec and M. Teboulle. A fast iterative shrinage-thresholding algorithm for linear inverse problems. SIAM Journal of Imaging Sciences, ():83 3

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