nips nips2006 nips2006-82 knowledge-graph by maker-knowledge-mining

82 nips-2006-Gaussian and Wishart Hyperkernels

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Author: Risi Kondor, Tony Jebara

Abstract: We propose a new method for constructing hyperkenels and deﬁne two promising special cases that can be computed in closed form. These we call the Gaussian and Wishart hyperkernels. The former is especially attractive in that it has an interpretable regularization scheme reminiscent of that of the Gaussian RBF kernel. We discuss how kernel learning can be used not just for improving the performance of classiﬁcation and regression methods, but also as a stand-alone algorithm for dimensionality reduction and relational or metric learning. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract We propose a new method for constructing hyperkenels and deﬁne two promising special cases that can be computed in closed form. [sent-6, score-0.017]

2 The former is especially attractive in that it has an interpretable regularization scheme reminiscent of that of the Gaussian RBF kernel. [sent-8, score-0.054]

3 We discuss how kernel learning can be used not just for improving the performance of classiﬁcation and regression methods, but also as a stand-alone algorithm for dimensionality reduction and relational or metric learning. [sent-9, score-0.206]

4 1 Introduction The performance of kernel methods, such as Support Vector Machines, Gaussian Processes, etc. [sent-10, score-0.148]

5 Conceptually, the kernel captures our prior knowledge of the data domain. [sent-12, score-0.148]

6 There is a small number of popular kernels expressible in 2 closed form, such as the Gaussian RBF kernel k(x, x ) = exp(− x − x /(2σ 2 )), which boasts attractive and unique properties from an abstract function approximation point of view. [sent-13, score-0.266]

7 In real world problems, however, and especially when the data is heterogenous or discrete, engineering an appropriate kernel is a major part of the modelling process. [sent-14, score-0.165]

8 It is natural to ask whether instead it might be possible to learn the kernel itself from the data. [sent-15, score-0.148]

9 Recent years have seen the development of several approaches to kernel learning [5][1]. [sent-16, score-0.148]

10 Arguably the most principled method proposed to date is the hyperkernels idea introduced by Ong, Smola and Williamson [8][7][9]. [sent-17, score-0.364]

11 The current paper is a continuation of this work, introducing a new family of hyperkernels with attractive properties. [sent-18, score-0.408]

12 Recently there has been increasing interest in using the kernel in its own right to answer relational questions about the dataset. [sent-20, score-0.167]

13 Instead of predicting individual labels, a kernel characterizes which pairs of labels are likely to be the same, or related. [sent-21, score-0.176]

14 A diﬀerent application is to use the learnt kernel to produce a low dimensional embedding via kernel PCA. [sent-23, score-0.318]

15 In this sense, kernel learning can be also be regarded as a dimensionality reduction or metric learning algorithm. [sent-24, score-0.214]

16 2 Hyperkernels We begin with a brief review of the kernel and hyperkernel formalism. [sent-25, score-0.934]

17 By kernel we mean a symmetric function k : X × X → R that is positive deﬁnite on X . [sent-30, score-0.188]

18 Whenever we refer to a function being positive deﬁnite, we assume that it is also symmetric. [sent-31, score-0.04]

19 Positive deﬁniteness guarantees that k induces a Reproducing Kernel Hilbert Space (RKHS) F, which is a vector space of functions spanned by { kx (·) = k(x, ·) | x ∈ X } and endowed with an inner product satisfying kx , kx = k(x, x ). [sent-32, score-0.265]

20 By the Representer Theorem [2], f is expressible ˆ = m αi k(xi , x) for some α1 , α2 , . [sent-34, score-0.04]

21 in the form (x) i=1 The idea expounded in [8] is to set up an analogous optimization problem for ﬁnding k itself in the RKHS of a hyperkernel K : X × X → R, where X = X 2 . [sent-38, score-0.786]

22 To induce an RKHS K must be positive deﬁnite in the latter sense. [sent-40, score-0.04]

23 Additionaly, we have to ensure that the solution of our regularized risk minimization problem is itself a kernel. [sent-41, score-0.038]

24 To this end, we require that the functions Kx1 ,x1 (x2 , x2 ) that we get by ﬁxing the ﬁrst two arguments of K((x1 , x1 ), (x2 , x2 )) be symmetric and positive deﬁnite kernel in the remaining two arguments. [sent-42, score-0.213]

25 Then K is called a hyperkernel on X if and only if 1. [sent-45, score-0.786]

26 Denoting the RKHS of K by K, potential kernels lie in the cone K pd = { k ∈ K | k is pos. [sent-48, score-0.054]

27 Unfortunately, there is no simple way of restricting kernel learning algorithms to Kpd . [sent-51, score-0.148]

28 Instead, we will restrict ourselves to the positive quadrant K + = k ∈ K | k, Kx ≥ 0 ∀ x ∈ X , which is a subcone of Kpd . [sent-52, score-0.057]

29 If K∗ has the property that for any S ⊂ X the orthogonal projection of any k ∈ K ∗ to the subspace spanned by Kx | x ∈ X remains in K∗ , then k is expressible as m k(x, x ) = m αij K(xi ,xj ) (x, x ) = i,j=1 αij K((xi , xj ), (x, x )) (2) i,j=1 for some real coeﬃcients (αij )i. [sent-55, score-0.098]

30 Finding functions that satisfy Deﬁnition 1 and also make sense in terms of regularization theory or practical problem domains in not trivial. [sent-60, score-0.046]

31 Let {gz : X → R} be a family of functions indexed by z ∈ Z and let h : Z×Z → R be a kernel. [sent-65, score-0.016]

32 Then k(x, x ) = gz (x) h(z, z ) gz (x ) dz dz (3) is a kernel on X . [sent-66, score-0.488]

33 3 Convolution hyperkernels One interpreation of a kernel k(x, x ) is that it quantiﬁes some notion of similarity between points x and x . [sent-68, score-0.538]

34 For the Gaussian RBF kernel, and heat kernels in general, this similarity can be regarded as induced by a diﬀusion process in the ambient space [4]. [sent-69, score-0.111]

35 Just as physical substances diﬀuse in space, the similarity between x and x is mediated by intermediate points, in the sense that by virtue of x being similar to some x0 and x0 being similar to x , x and x themselves become similar to each other. [sent-70, score-0.063]

36 Speciﬁcally, the normalized Gaussian kernel on Rn of variance 2t = σ 2 , kt (x, x ) = 1 (4πt) n/2 e− x−x 2 /(4t) , satisﬁes the well known convolution property kt (x, x ) = kt/2 (x, x0 ) kt/2 (x0 , x) dx0 . [sent-72, score-0.248]

37 (5) Such kernels are by deﬁnition homogenous and isotropic in the ambient space. [sent-73, score-0.08]

38 What we hope for from the hyperkernels formalism is to be able to adapt to the inhomogeneous and anisotropic nature of training data, while retaining the transitivity idea in some form. [sent-74, score-0.494]

39 Hyperkernels achieve this by weighting the integrand of (5) in relation to what is “on the other side” of the hyperkernel. [sent-75, score-0.028]

40 Speciﬁcally, we deﬁne convolution hyperkernels by setting gz (x, x ) = r(x, z) r(x , z) in (4) for some r : X × X → R. [sent-76, score-0.496]

41 By (3), the resulting hyperkernel always satisﬁes the conditions of Deﬁnition 1. [sent-77, score-0.786]

42 Given functions r : X ×X → R and h : X ×X → R where h is positive deﬁnite, the convolution hyperkernel induced by r and h is K ((x1 , x1 ) , (x2 , x2 )) = r(x1 , z1 ) r(x1 , z1 ) h(z1 , z2 ) r(x2 , z2 ) r(x2 , z2 ) dz1 dz2 . [sent-79, score-0.882]

43 (6) A good way to visualize the structure of convolution hyperkernels is to note that (6) is proportional to the likelihood of the graphical model in the ﬁgure to the right. [sent-80, score-0.42]

44 The only requirements on the graphical model are to have the same potential function ψ1 at each of the extremities and to have a positive deﬁnite potential function ψ2 at the core. [sent-81, score-0.082]

45 1 The Gaussian hyperkernel To make the foregoing more concrete we now investigate the case where r(x, x ) and h(z, z ) are Gaussians. [sent-83, score-0.803]

46 The Gaussian hyperkernel on X = Rn is then deﬁned as K((x1 , x1 ), (x2 , x2 )) = x1 , z X X σ2 z, x1 σ2 z, z 2 σh x2 , z σ2 z , x2 σ2 dz dz . [sent-85, score-0.974]

47 (7) Fixing x and completing the square we have x1 , z σ2 z, x1 σ2 = 1 1 exp − 2 2 )n 2σ (2πσ 1 n exp (2πσ 2 ) − 1 σ2 z− z−x1 x1 +x1 2 2 2 + z−x1 x1 −x1 4σ 2 − 2 = 2 = x 1 , x1 z, x1 2σ 2 σ 2/2 , where xi = (xi +xi )/2. [sent-86, score-0.098]

48 By the convolution property of Gaussians it follows that K((x1 , x1 ), (x2 , x2 )) = x1 , x1 2σ 2 x2 , x2 2σ 2 x1 , z X X z, z σ 2 /2 2 σh x1 , x1 2σ 2 z, x2 σ 2/2 x2 , x2 dz dz = 2σ 2 x1 , x 2 2 σ 2 +σh . [sent-87, score-0.244]

49 (8) It is an important property of the Gaussian hyperkernel that it can be evaluated in closed 2 form. [sent-88, score-0.803]

50 At 2 the opposite extreme, in the limit σh → ∞, the hyperkernel decouples into the product of two RBF kernels. [sent-90, score-0.803]

51 Since the hyperkernel expansion (2) is a sum over hyperkernel evaluations with one pair of arguments ﬁxed, it is worth examining what these functions look like: 2 Kx1 ,x1 (x2 , x2 ) ∝ exp − x1 − x2 2 2 (σ 2 + σh ) exp − x2 − x2 2σ 2 2 (9) √ with σ = 2σ. [sent-91, score-1.661]

52 This is really a conventional Gaussian kernel between x2 and x2 multiplied by a spatially varying Gaussian intensity factor depending on how close the mean of x 2 and x2 is to the mean of the training pair. [sent-92, score-0.199]

53 This can be regarded as a localized Gaussian, and the full kernel (2) will be a sum of such terms with positive weights. [sent-93, score-0.232]

54 As x 2 and x2 move around in X , whichever localized Gaussians are centered close to their mean will dominate the sum. [sent-94, score-0.034]

55 By changing the (αij ) weights, the kernel learning algorithm can choose k from a highly ﬂexible class of potential kernels. [sent-95, score-0.169]

56 ˆ ˜ ˆ Omitting details for brevity, the consequences of this include that K = K × K, where K ˜ is the RKHS of a Gaussian kernel over X , while K is the one-dimensional space generˆ x ˜ ated by x, 0 σ2 : each k ∈ K can be written as k(ˆ, x) = k(ˆ) x, 0 σ2 . [sent-97, score-0.148]

57 In summary, K behaves in (ˆ1 , x2 ) like a Gaussian kernel with variance x ˆ 2 2(σ 2 + σh ), but in x it just eﬀects a one-dimensional feature mapping. [sent-99, score-0.148]

58 ˜ 4 Anisotropic hyperkernels With the hyperkernels so far far we can only learn kernels that are a sum of rotationally invariant terms. [sent-100, score-0.761]

59 Consequently, the learnt kernel will have a locally isotropic character. [sent-101, score-0.192]

60 Yet, rescaling of the axes and anisotropic dilations are one of the most common forms of variation in naturally occurring data that we would hope to accomodate by learning the kernel. [sent-102, score-0.108]

61 1 The Wishart hyperkernel We deﬁne the Wishart hyperkernel as K((x1 , x1 ), (x2 , x2 )) = x1 , z Σ 0 where x, x Σ z, x1 Σ X 1 = n/2 x2 , z Σ Σ IW(Σ; C, r) dz dΣ. [sent-104, score-1.666]

62 Σ−1 (x−x )/2 e−(x−x ) 1/2 z, x2 Σ (2π) |Σ| and IW(Σ; C, r) is the inverse Wishart distribution |C | Zr,n | Σ | (10) , r/2 (n+r+1)/2 exp −tr Σ−1 C /2 over positive deﬁnite matrices (denoted Σ 0) [6]. [sent-105, score-0.062]

63 Here r is an integer parameter, C is an n n × n positive deﬁnite parameter matrix and Zr,n = 2 rn/2 π n(n−1)/4 i=1 Γ((r+1−i)/2) is a normalizing factor. [sent-106, score-0.04]

64 The Wishart hyperkernel can be seen as the anisotropic analog of (7) 2 in the limit σh → 0, z, z σ2 → δ(z, z ). [sent-107, score-0.837]

65 (11) By using the identity v A v = tr(A(vv )), x, x Σ IW(Σ; C, r) = |C | r/2 (2π)n/2 Zr,n | Σ | (n+r+2)/2 exp −tr Σ−1 (C +S) /2 = r/2 Zr+1,n |C | IW( Σ ; C +S, r+1 ) , (r+1)/2 n/2 Z (2π) r,n | C + S | where S = (x−x )(x−x ) . [sent-110, score-0.022]

66 We 2 can read oﬀ that for given x1 − x1 , x2 − x2 , and x − x , the hyperkernel will favor quadruples where x1 − x1 , x2 − x2 , and x − x are close to parallel to each other and to the largest eigenvector of C. [sent-112, score-0.786]

67 2 Figure 1: The ﬁrst two panes show the separation of ’3’s and ’8’s in the training and testing sets respectively achieved by the Gaussian hyperkernel (the plots show the data plotted by its ﬁrst two eigenvectors according to the learned kernel k). [sent-157, score-0.97]

68 5 Experiments We conducted preliminary experiments with the hyperkernels in relation learning between pairs of datapoints. [sent-159, score-0.408]

69 The idea here is that the learned kernel k naturally induces a distance metric d(x, x ) = k(x, x) − 2k(x, x ) + k(x , x ), and in this sense kernel learning is equivalent to learning d. [sent-160, score-0.373]

70 Given a labeled dataset, we can learn a kernel which eﬀectively remaps the data in such a way that data points with the same label are close to each other, while those with diﬀerent labels are far apart. [sent-161, score-0.148]

72 As an illustrative example we learned a kernel (and hence, a metric) between a subset of the NIST handwritten digits1 . [sent-168, score-0.166]

73 The training data consisted of 20 ’3’s and 20 ’8’s randomly rotated by ±45 degrees to make the problem slightly harder. [sent-169, score-0.018]

74 Figure 1 shows that a kernel learned by the above strategy with a Gaussian hyperkernel with parameters set by cross validation is extremely good at separating the two classes in training as well as testing. [sent-170, score-0.97]

75 In comparison, in a similar plot for a ﬁxed RBF kernel the ’3’s and ’8’s are totally intermixed. [sent-171, score-0.148]

76 Interpreting this as an information retrieval problem, we can imagine inﬂating a ball around each data point in the test set and asking how many other data points in this ball are of the same class. [sent-172, score-0.038]

77 We ran a similar experiment but with multiple classes on the Olivetti faces dataset, which consists of 92 × 112 pixel normalized gray-scale images of 30 individuals in 10 diﬀerent poses. [sent-276, score-0.035]

78 Here we also experimented with dropping the αij ≥ 0 constraints, which breaks the positive deﬁniteness of k, but might still give a reasonable similarity measure. [sent-277, score-0.066]

79 Finally, as a baseline, we trained an SVM over pairs of datapoints to predict yij , representing (xi , xj ) with a concatenated feature vector [xi , xj ] and using a Gaussian RBF between these concatenations. [sent-280, score-0.183]

80 We trained the system with m = 20 faces and considered all pairs of the training data-points (i. [sent-282, score-0.081]

81 400 constraints) to ﬁnd a kernel that predicted the labeling matrix. [sent-284, score-0.148]

82 When speed becomes an issue it often suﬃces to work with a subsample of the binary entries in the m × m label matrix and thus avoid having m2 constraints. [sent-285, score-0.017]

83 Using the learned kernel, we then test on 100 unseen faces and predict all their pairwise kernel evaluations, in other words, 104 predicted pair-wise labelings. [sent-287, score-0.201]

84 For both the baseline Gaussian RBF and the Gaussian hyperkernels we varied the σ parameter from 0. [sent-289, score-0.364]

85 For the Gaussian hyperkernel we also varied σh from 0 to 10σ. [sent-292, score-0.786]

86 Using a conic hyperkernel combination did best in labeling new faces. [sent-295, score-0.956]

87 75 while the Gaussian hyperkernel methods achieve an AUC of almost 0. [sent-298, score-0.786]

88 While the diﬀerence between the conic and linear hyperkernel methods is harder to see, across all settings of σ and σh , the conic combination outperformed the linear combination over 92% of the time. [sent-300, score-1.126]

89 The conic hyperkernel combination is also the only method of the three that guarantees a true Mercer kernel as an output which can then be converted into a valid metric. [sent-301, score-1.104]

90 6 Conclusions The main barrier to hyperkernels becoming more popular is their high computational demands (out of the box algorithms run in O(m6 ) time as opposed to O(m3 ) in regular learning). [sent-311, score-0.364]

91 In certain metric learning and on-line settings however this need not be forbidding, and is compensated for by the elegance and generality of the framework. [sent-312, score-0.073]

92 The Gaussian and Wishart hyperkernels presented in this paper are in a sense canonical, with intuitively appealing interpretations. [sent-313, score-0.384]

93 In the case of the Gaussian hyperkernel we even have a natural regularization scheme. [sent-314, score-0.812]

94 Preliminary experiments show that these new hyperkernels can capture the inherent structure of some input spaces. [sent-315, score-0.364]

95 We hope that their introduction will give a boost to the whole hyperkernels ﬁeld. [sent-316, score-0.387]

96 Diﬀusion kernels on graphs and other discrete input spaces. [sent-349, score-0.033]

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Abstract: We describe an algorithmic framework for an abstract game which we term a convex repeated game. We show that various online learning and boosting algorithms can be all derived as special cases of our algorithmic framework. This uniﬁed view explains the properties of existing algorithms and also enables us to derive several new interesting algorithms. Our algorithmic framework stems from a connection that we build between the notions of regret in game theory and weak duality in convex optimization. 1 Introduction and Problem Setting Several problems arising in machine learning can be modeled as a convex repeated game. Convex repeated games are closely related to online convex programming (see [19, 9] and the discussion in the last section). A convex repeated game is a two players game that is performed in a sequence of consecutive rounds. On round t of the repeated game, the ﬁrst player chooses a vector wt from a convex set S. 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(2) implies that the average loss of the ﬁrst player forms a lower bound for a minimization problem. The notion of duality, commonly used in convex optimization theory, plays an important role in obtaining lower bounds for the minimal value of a minimization problem (see for example [14]). By generalizing the notion of Fenchel duality, we are able to derive a dual optimization problem, which can be optimized incrementally, as the game progresses. In order to derive explicit quantitative regret bounds we make an immediate use of the fact that dual objective lower bounds the primal objective. We therefore reduce the process of playing convex repeated games to the task of incrementally increasing the dual objective function. The amount by which the dual increases serves as a new and natural notion of progress. By doing so we are able to tie the primal objective value, the average loss of the ﬁrst player, and the increase in the dual. The rest of this paper is organized as follows. In Sec. 2 we establish our notation and point to a few mathematical tools that we use throughout the paper. Our main tool for deriving algorithms for playing convex repeated games is a generalization of Fenchel duality, described in Sec. 3. Our algorithmic framework is given in Sec. 4 and analyzed in Sec. 5. The generality of our framework allows us to utilize it in different problems arising in machine learning. Speciﬁcally, in Sec. 6 we underscore the applicability of our framework for online learning and in Sec. 7 we outline and analyze boosting algorithms based on our framework. We conclude with a discussion and point to related work in Sec. 8. Due to the lack of space, some of the details are omitted from the paper and can be found in [16]. 2 Mathematical Background We denote scalars with lower case letters (e.g. x and w), and vectors with bold face letters (e.g. x and w). The inner product between vectors x and w is denoted by x, w . Sets are designated by upper case letters (e.g. S). The set of non-negative real numbers is denoted by R+ . For any k ≥ 1, the set of integers {1, . . . , k} is denoted by [k]. A norm of a vector x is denoted by x . The dual norm is deﬁned as λ = sup{ x, λ : x ≤ 1}. For example, the Euclidean norm, x 2 = ( x, x )1/2 is dual to itself and the 1 norm, x 1 = i |xi |, is dual to the ∞ norm, x ∞ = maxi |xi |. We next recall a few deﬁnitions from convex analysis. The reader familiar with convex analysis may proceed to Lemma 1 while for a more thorough introduction see for example [1]. A set S is convex if for any two vectors w1 , w2 in S, all the line between w1 and w2 is also within S. That is, for any α ∈ [0, 1] we have that αw1 + (1 − α)w2 ∈ S. A set S is open if every point in S has a neighborhood lying in S. A set S is closed if its complement is an open set. A function f : S → R is closed and convex if for any scalar α ∈ R, the level set {w : f (w) ≤ α} is closed and convex. The Fenchel conjugate of a function f : S → R is deﬁned as f (θ) = supw∈S w, θ − f (w) . If f is closed and convex then the Fenchel conjugate of f is f itself. The Fenchel-Young inequality states that for any w and θ we have that f (w) + f (θ) ≥ w, θ . A vector λ is a sub-gradient of a function f at w if for all w ∈ S we have that f (w ) − f (w) ≥ w − w, λ . The differential set of f at w, denoted ∂f (w), is the set of all sub-gradients of f at w. If f is differentiable at w then ∂f (w) consists of a single vector which amounts to the gradient of f at w and is denoted by f (w). Sub-gradients play an important role in the deﬁnition of Fenchel conjugate. In particular, the following lemma states that if λ ∈ ∂f (w) then Fenchel-Young inequality holds with equality. Lemma 1 Let f be a closed and convex function and let ∂f (w ) be its differential set at w . Then, for all λ ∈ ∂f (w ) we have, f (w ) + f (λ ) = λ , w . A continuous function f is σ-strongly convex over a convex set S with respect to a norm · if S is contained in the domain of f and for all v, u ∈ S and α ∈ [0, 1] we have 1 (3) f (α v + (1 − α) u) ≤ α f (v) + (1 − α) f (u) − σ α (1 − α) v − u 2 . 2 Strongly convex functions play an important role in our analysis primarily due to the following lemma. Lemma 2 Let · be a norm over Rn and let · be its dual norm. Let f be a σ-strongly convex function on S and let f be its Fenchel conjugate. Then, f is differentiable with f (θ) = arg maxx∈S θ, x − f (x). Furthermore, for any θ, λ ∈ Rn we have 1 f (θ + λ) − f (θ) ≤ f (θ), λ + λ 2 . 2σ Two notable examples of strongly convex functions which we use are as follows. 1 Example 1 The function f (w) = 2 w norm. Its conjugate function is f (θ) = 2 2 1 2 is 1-strongly convex over S = Rn with respect to the θ 2. 2 2 n 1 Example 2 The function f (w) = i=1 wi log(wi / n ) is 1-strongly convex over the probabilistic n simplex, S = {w ∈ R+ : w 1 = 1}, with respect to the 1 norm. Its conjugate function is n 1 f (θ) = log( n i=1 exp(θi )). 3 Generalized Fenchel Duality In this section we derive our main analysis tool. We start by considering the following optimization problem, T inf c f (w) + t=1 gt (w) , w∈S where c is a non-negative scalar. An equivalent problem is inf w0 ,w1 ,...,wT c f (w0 ) + T t=1 gt (wt ) s.t. w0 ∈ S and ∀t ∈ [T ], wt = w0 . Introducing T vectors λ1 , . . . , λT , each λt ∈ Rn is a vector of Lagrange multipliers for the equality constraint wt = w0 , we obtain the following Lagrangian T T L(w0 , w1 , . . . , wT , λ1 , . . . , λT ) = c f (w0 ) + t=1 gt (wt ) + t=1 λt , w0 − wt . The dual problem is the task of maximizing the following dual objective value, D(λ1 , . . . , λT ) = inf L(w0 , w1 , . . . , wT , λ1 , . . . , λT ) w0 ∈S,w1 ,...,wT = − c sup w0 ∈S = −c f −1 c w0 , − 1 c T t=1 T t=1 λt − λt − f (w0 ) − T t=1 gt (λt ) , T t=1 sup ( wt , λt − gt (wt )) wt where, following the exposition of Sec. 2, f , g1 , . . . , gT are the Fenchel conjugate functions of f, g1 , . . . , gT . Therefore, the generalized Fenchel dual problem is sup − cf λ1 ,...,λT −1 c T t=1 λt − T t=1 gt (λt ) . (4) Note that when T = 1 and c = 1, the above duality is the so called Fenchel duality. 4 A Template Learning Algorithm for Convex Repeated Games In this section we describe a template learning algorithm for playing convex repeated games. As mentioned before, we study convex repeated games from the viewpoint of the ﬁrst player which we shortly denote as P1. Recall that we would like our learning algorithm to achieve a regret bound of the form given in Eq. (2). We start by rewriting Eq. (2) as follows T m gt (wt ) − c L ≤ inf u∈S t=1 c f (u) + gt (u) , (5) t=1 √ where c = T . Thus, up to the sublinear term c L, the cumulative loss of P1 lower bounds the optimum of the minimization problem on the right-hand side of Eq. (5). In the previous section we derived the generalized Fenchel dual of the right-hand side of Eq. (5). Our construction is based on the weak duality theorem stating that any value of the dual problem is smaller than the optimum value of the primal problem. The algorithmic framework we propose is therefore derived by incrementally ascending the dual objective function. Intuitively, by ascending the dual objective we move closer to the optimal primal value and therefore our performance becomes similar to the performance of the best ﬁxed weight vector which minimizes the right-hand side of Eq. (5). Initially, we use the elementary dual solution λ1 = 0 for all t. We assume that inf w f (w) = 0 and t for all t inf w gt (w) = 0 which imply that D(λ1 , . . . , λ1 ) = 0. We assume in addition that f is 1 T σ-strongly convex. Therefore, based on Lemma 2, the function f is differentiable. At trial t, P1 uses for prediction the vector wt = f −1 c T i=1 λt i . (6) After predicting wt , P1 receives the function gt and suffers the loss gt (wt ). Then, P1 updates the dual variables as follows. Denote by ∂t the differential set of gt at wt , that is, ∂t = {λ : ∀w ∈ S, gt (w) − gt (wt ) ≥ λ, w − wt } . (7) The new dual variables (λt+1 , . . . , λt+1 ) are set to be any set of vectors which satisfy the following 1 T two conditions: (i). ∃λ ∈ ∂t s.t. D(λt+1 , . . . , λt+1 ) ≥ D(λt , . . . , λt , λ , λt , . . . , λt ) 1 1 t−1 t+1 T T (ii). ∀i > t, λt+1 = 0 i . (8) In the next section we show that condition (i) ensures that the increase of the dual at trial t is proportional to the loss gt (wt ). The second condition ensures that we can actually calculate the dual at trial t without any knowledge on the yet to be seen loss functions gt+1 , . . . , gT . We conclude this section with two update rules that trivially satisfy the above two conditions. The ﬁrst update scheme simply ﬁnds λ ∈ ∂t and set λt+1 = i λ λt i if i = t if i = t . (9) The second update deﬁnes (λt+1 , . . . , λt+1 ) = argmax D(λ1 , . . . , λT ) 1 T λ1 ,...,λT s.t. ∀i = t, λi = λt . i (10) 5 Analysis In this section we analyze the performance of the template algorithm given in the previous section. Our proof technique is based on monitoring the value of the dual objective function. The main result is the following lemma which gives upper and lower bounds for the ﬁnal value of the dual objective function. Lemma 3 Let f be a σ-strongly convex function with respect to a norm · over a set S and assume that minw∈S f (w) = 0. Let g1 , . . . , gT be a sequence of convex and closed functions such that inf w gt (w) = 0 for all t ∈ [T ]. Suppose that a dual-incrementing algorithm which satisﬁes the conditions of Eq. (8) is run with f as a complexity function on the sequence g1 , . . . , gT . Let w1 , . . . , wT be the sequence of primal vectors that the algorithm generates and λT +1 , . . . , λT +1 1 T be its ﬁnal sequence of dual variables. Then, there exists a sequence of sub-gradients λ1 , . . . , λT , where λt ∈ ∂t for all t, such that T 1 gt (wt ) − 2σc t=1 T T λt 2 ≤ D(λT +1 , . . . , λT +1 ) 1 T t=1 ≤ inf c f (w) + w∈S gt (w) . t=1 Proof The second inequality follows directly from the weak duality theorem. Turning to the left most inequality, denote ∆t = D(λt+1 , . . . , λt+1 ) − D(λt , . . . , λt ) and note that 1 1 T T T D(λ1 +1 , . . . , λT +1 ) can be rewritten as T T t=1 D(λT +1 , . . . , λT +1 ) = 1 T T t=1 ∆t − D(λ1 , . . . , λ1 ) = 1 T ∆t , (11) where the last equality follows from the fact that f (0) = g1 (0) = . . . = gT (0) = 0. The deﬁnition of the update implies that ∆t ≥ D(λt , . . . , λt , λt , 0, . . . , 0) − D(λt , . . . , λt , 0, 0, . . . , 0) for 1 t−1 1 t−1 t−1 some subgradient λt ∈ ∂t . Denoting θ t = − 1 j=1 λj , we now rewrite the lower bound on ∆t as, c ∆t ≥ −c (f (θ t − λt /c) − f (θ t )) − gt (λt ) . Using Lemma 2 and the deﬁnition of wt we get that 1 (12) ∆t ≥ wt , λt − gt (λt ) − 2 σ c λt 2 . Since λt ∈ ∂t and since we assume that gt is closed and convex, we can apply Lemma 1 to get that wt , λt − gt (λt ) = gt (wt ). Plugging this equality into Eq. (12) and summing over t we obtain that T T T 1 2 . t=1 ∆t ≥ t=1 gt (wt ) − 2 σ c t=1 λt Combining the above inequality with Eq. (11) concludes our proof. The following regret bound follows as a direct corollary of Lemma 3. T 1 Theorem 1 Under the same conditions of Lemma 3. Denote L = T t=1 λt w ∈ S we have, T T c f (w) 1 1 + 2L c . t=1 gt (wt ) − T t=1 gt (w) ≤ T T σ √ In particular, if c = T , we obtain the bound, 1 T 6 T t=1 gt (wt ) − 1 T T t=1 gt (w) ≤ f (w)+L/(2 σ) √ T 2 . Then, for all . Application to Online learning In Sec. 1 we cast the task of online learning as a convex repeated game. We now demonstrate the applicability of our algorithmic framework for the problem of instance ranking. We analyze this setting since several prediction problems, including binary classiﬁcation, multiclass prediction, multilabel prediction, and label ranking, can be cast as special cases of the instance ranking problem. Recall that on each online round, the learner receives a question-answer pair. In instance ranking, the question is encoded by a matrix Xt of dimension kt × n and the answer is a vector yt ∈ Rkt . The semantic of yt is as follows. For any pair (i, j), if yt,i > yt,j then we say that yt ranks the i’th row of Xt ahead of the j’th row of Xt . We also interpret yt,i − yt,j as the conﬁdence in which the i’th row should be ranked ahead of the j’th row. For example, each row of Xt encompasses a representation of a movie while yt,i is the movie’s rating, expressed as the number of stars this movie has received by a movie reviewer. The predictions of the learner are determined ˆ based on a weight vector wt ∈ Rn and are deﬁned to be yt = Xt wt . Finally, let us deﬁne two loss functions for ranking, both generalize the hinge-loss used in binary classiﬁcation problems. Denote by Et the set {(i, j) : yt,i > yt,j }. 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The following theorem provides us with a sufﬁcient condition under which the regret bound from Thm. 1 holds for ranking as well. Theorem 2 Let f be a σ-strongly convex function over S with respect to a norm · . Denote by Lt the maximum over (i, j) ∈ Et of xt,i − xt,j 2 . Then, for both gt (w) = avg (w; (Xt , yt )) and ∗ gt (w) = max (w; (Xt , yt )), the following regret bound holds ∀u ∈ S, 7 1 T T t=1 gt (wt ) − 1 T T t=1 gt (u) ≤ 1 f (u)+ T PT t=1 Lt /(2 σ) √ T . The Boosting Game In this section we describe the applicability of our algorithmic framework to the analysis of boosting algorithms. A boosting algorithm uses a weak learning algorithm that generates weak-hypotheses whose performances are just slightly better than random guessing to build a strong-hypothesis which can attain an arbitrarily low error. 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Therefore, the exp-loss function, deﬁned as exp(−y hf (x)), is a smooth upper bound of the zero-one error, which equals to 1 if y hf (x) ≤ 0 and to 0 otherwise. Thus, we can restate the goal of boosting as minimizing the average exp-loss of hf over the training set with respect to the variables α1 , . . . , αT . To simplify our derivation in the sequel, we prefer to say that boosting maximizes the negation of the loss, that is, T m 1 (14) max − m i=1 exp −yi t=1 αt ht (xi ) . α1 ,...,αT In this view, boosting is an optimization procedure which iteratively maximizes Eq. (14) with respect to the variables α1 , . . . , αT . This view of boosting, enables the hypotheses returned by the weak learner to be general functions into the reals, ht : X → R (see for instance [15]). In this paper we view boosting as a convex repeated game between a booster and a weak learner. To motivate our construction, we would like to note that boosting algorithms deﬁne weights in two different domains: the vectors wt ∈ Rm which assign weights to examples and the weights {αt : t ∈ [T ]} over weak-hypotheses. In the terminology used throughout this paper, the weights wt ∈ Rm are primal vectors while (as we show in the sequel) each weight αt of the hypothesis ht is related to a dual vector λt . In particular, we show that Eq. (14) is exactly the Fenchel dual of a primal problem for a convex repeated game, thus the algorithmic framework described thus far for playing games naturally ﬁts the problem of iteratively solving Eq. (14). To derive the primal problem whose Fenchel dual is the problem given in Eq. (14) let us ﬁrst denote by vt the vector in Rm whose ith element is vt,i = yi ht (xi ). For all t, we set gt to be the function gt (w) = [ w, vt ]+ . Intuitively, gt penalizes vectors w which assign large weights to examples which are predicted accurately, that is yi ht (xi ) > 0. In particular, if ht (xi ) ∈ {+1, −1} and wt is a distribution over the m examples (as is the case in AdaBoost), gt (wt ) reduces to 1 − 2 t (see Eq. (13)). In this case, minimizing gt is equivalent to maximizing the error of the individual T hypothesis ht over the examples. Consider the problem of minimizing c f (w) + t=1 gt (w) where f (w) is the relative entropy given in Example 2 and c = 1/(2 γ) (see Eq. (13)). To derive its Fenchel dual, we note that gt (λt ) = 0 if there exists βt ∈ [0, 1] such that λt = βt vt and otherwise gt (λt ) = ∞ (see [16]). In addition, let us deﬁne αt = 2 γ βt . Since our goal is to maximize the αt dual, we can restrict λt to take the form λt = βt vt = 2 γ vt , and get that D(λ1 , . . . , λT ) = −c f − 1 c T βt vt t=1 =− 1 log 2γ 1 m m e− PT t=1 αt yi ht (xi ) . (15) i=1 Minimizing the exp-loss of the strong hypothesis is therefore the dual problem of the following primal minimization problem: ﬁnd a distribution over the examples, whose relative entropy to the uniform distribution is as small as possible while the correlation of the distribution with each vt is as small as possible. Since the correlation of w with vt is inversely proportional to the error of ht with respect to w, we obtain that in the primal problem we are trying to maximize the error of each individual hypothesis, while in the dual problem we minimize the global error of the strong hypothesis. The intuition of ﬁnding distributions which in retrospect result in large error rates of individual hypotheses was also alluded in [15, 8]. We can now apply our algorithmic framework from Sec. 4 to boosting. We describe the game αt with the parameters αt , where αt ∈ [0, 2 γ], and underscore that in our case, λt = 2 γ vt . At the beginning of the game the booster sets all dual variables to be zero, ∀t αt = 0. At trial t of the boosting game, the booster ﬁrst constructs a primal weight vector wt ∈ Rm , which assigns importance weights to the examples in the training set. The primal vector wt is constructed as in Eq. (6), that is, wt = f (θ t ), where θ t = − i αi vi . Then, the weak learner responds by presenting the loss function gt (w) = [ w, vt ]+ . Finally, the booster updates the dual variables so as to increase the dual objective function. It is possible to show that if the range of ht is {+1, −1} 1 then the update given in Eq. (10) is equivalent to the update αt = min{2 γ, 2 log((1 − t )/ t )}. We have thus obtained a variant of AdaBoost in which the weights αt are capped above by 2 γ. A disadvantage of this variant is that we need to know the parameter γ. We would like to note in passing that this limitation can be lifted by a different deﬁnition of the functions gt . We omit the details due to the lack of space. To analyze our game of boosting, we note that the conditions given in Lemma 3 holds T and therefore the left-hand side inequality given in Lemma 3 tells us that t=1 gt (wt ) − T T +1 T +1 1 2 , . . . , λT ) . The deﬁnition of gt and the weak learnability ast=1 λt ∞ ≤ D(λ1 2c sumption given in Eq. (13) imply that wt , vt ≥ 2 γ for all t. Thus, gt (wt ) = wt , vt ≥ 2 γ which also implies that λt = vt . Recall that vt,i = yi ht (xi ). Assuming that the range of ht is [+1, −1] we get that λt ∞ ≤ 1. Combining all the above with the left-hand side inequality T given in Lemma 3 we get that 2 T γ − 2 c ≤ D(λT +1 , . . . , λT +1 ). Using the deﬁnition of D (see 1 T Eq. (15)), the value c = 1/(2 γ), and rearranging terms we recover the original bound for AdaBoost PT 2 m 1 −yi t=1 αt ht (xi ) ≤ e−2 γ T . i=1 e m 8 Related Work and Discussion We presented a new framework for designing and analyzing algorithms for playing convex repeated games. Our framework was used for the analysis of known algorithms for both online learning and boosting settings. The framework also paves the way to new algorithms. In a previous paper [17], we suggested the use of duality for the design of online algorithms in the context of mistake bound analysis. The contribution of this paper over [17] is three fold as we now brieﬂy discuss. First, we generalize the applicability of the framework beyond the speciﬁc setting of online learning with the hinge-loss to the general setting of convex repeated games. The setting of convex repeated games was formally termed “online convex programming” by Zinkevich [19] and was ﬁrst presented by Gordon in [9]. There is voluminous amount of work on unifying approaches for deriving online learning algorithms. We refer the reader to [11, 12, 3] for work closely related to the content of this paper. By generalizing our previously studied algorithmic framework [17] beyond online learning, we can automatically utilize well known online learning algorithms, such as the EG and p-norm algorithms [12, 11], to the setting of online convex programming. We would like to note that the algorithms presented in [19] can be derived as special cases of our algorithmic framework 1 by setting f (w) = 2 w 2 . Parallel and independently to this work, Gordon [10] described another algorithmic framework for online convex programming that is closely related to the potential based algorithms described by Cesa-Bianchi and Lugosi [3]. Gordon also considered the problem of deﬁning appropriate potential functions. Our work generalizes some of the theorems in [10] while providing a somewhat simpler analysis. Second, the usage of generalized Fenchel duality rather than the Lagrange duality given in [17] enables us to analyze boosting algorithms based on the framework. Many authors derived unifying frameworks for boosting algorithms [13, 8, 4]. Nonetheless, our general framework and the connection between game playing and Fenchel duality underscores an interesting perspective of both online learning and boosting. We believe that this viewpoint has the potential of yielding new algorithms in both domains. Last, despite the generality of the framework introduced in this paper, the resulting analysis is more distilled than the earlier analysis given in [17] for two reasons. (i) The usage of Lagrange duality in [17] is somehow restricted while the notion of generalized Fenchel duality is more appropriate to the general and broader problems we consider in this paper. (ii) The strongly convex property we employ both simpliﬁes the analysis and enables more intuitive conditions in our theorems. There are various possible extensions of the work that we did not pursue here due to the lack of space. For instanc, our framework can naturally be used for the analysis of other settings such as repeated games (see [7, 19]). The applicability of our framework to online learning can also be extended to other prediction problems such as regression and sequence prediction. Last, we conjecture that our primal-dual view of boosting will lead to new methods for regularizing boosting algorithms, thus improving their generalization capabilities. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] J. Borwein and A. Lewis. Convex Analysis and Nonlinear Optimization. Springer, 2006. S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004. N. Cesa-Bianchi and G. Lugosi. Prediction, learning, and games. Cambridge University Press, 2006. M. Collins, R.E. Schapire, and Y. Singer. Logistic regression, AdaBoost and Bregman distances. Machine Learning, 2002. K. Crammer, O. Dekel, J. Keshet, S. Shalev-Shwartz, and Y. Singer. Online passive aggressive algorithms. JMLR, 7, Mar 2006. Y. Freund and R.E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. In EuroCOLT, 1995. Y. Freund and R.E. Schapire. Game theory, on-line prediction and boosting. In COLT, 1996. J. Friedman, T. Hastie, and R. Tibshirani. Additive logistic regression: a statistical view of boosting. Annals of Statistics, 28(2), 2000. G. Gordon. Regret bounds for prediction problems. In COLT, 1999. G. Gordon. No-regret algorithms for online convex programs. In NIPS, 2006. A. J. Grove, N. Littlestone, and D. Schuurmans. General convergence results for linear discriminant updates. Machine Learning, 43(3), 2001. J. Kivinen and M. Warmuth. Relative loss bounds for multidimensional regression problems. Journal of Machine Learning, 45(3),2001. L. Mason, J. Baxter, P. Bartlett, and M. Frean. Functional gradient techniques for combining hypotheses. In Advances in Large Margin Classiﬁers. MIT Press, 1999. Y. Nesterov. Primal-dual subgradient methods for convex problems. Technical report, Center for Operations Research and Econometrics (CORE), Catholic University of Louvain (UCL), 2005. R. E. Schapire and Y. Singer. Improved boosting algorithms using conﬁdence-rated predictions. Machine Learning, 37(3):1–40, 1999. S. Shalev-Shwartz and Y. Singer. Convex repeated games and fenchel duality. Technical report, The Hebrew University, 2006. S. Shalev-Shwartz and Y. Singer. Online learning meets optimization in the dual. In COLT, 2006. J. Weston and C. Watkins. Support vector machines for multi-class pattern recognition. In ESANN, April 1999. M. Zinkevich. Online convex programming and generalized inﬁnitesimal gradient ascent. In ICML, 2003.

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