nips nips2013 nips2013-191 knowledge-graph by maker-knowledge-mining

191 nips-2013-Minimax Optimal Algorithms for Unconstrained Linear Optimization

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Author: Brendan McMahan, Jacob Abernethy

Abstract: We design and analyze minimax-optimal algorithms for online linear optimization games where the player’s choice is unconstrained. The player strives to minimize regret, the difference between his loss and the loss of a post-hoc benchmark strategy. While the standard benchmark is the loss of the best strategy chosen from a bounded comparator set, we consider a very broad range of benchmark functions. The problem is cast as a sequential multi-stage zero-sum game, and we give a thorough analysis of the minimax behavior of the game, providing characterizations for the value of the game, as well as both the player’s and the adversary’s optimal strategy. We show how these objects can be computed efﬁciently under certain circumstances, and by selecting an appropriate benchmark, we construct a novel hedging strategy for an unconstrained betting game. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 The player strives to minimize regret, the difference between his loss and the loss of a post-hoc benchmark strategy. [sent-5, score-0.58]

2 While the standard benchmark is the loss of the best strategy chosen from a bounded comparator set, we consider a very broad range of benchmark functions. [sent-6, score-0.431]

3 The problem is cast as a sequential multi-stage zero-sum game, and we give a thorough analysis of the minimax behavior of the game, providing characterizations for the value of the game, as well as both the player’s and the adversary’s optimal strategy. [sent-7, score-0.323]

4 We show how these objects can be computed efﬁciently under certain circumstances, and by selecting an appropriate benchmark, we construct a novel hedging strategy for an unconstrained betting game. [sent-8, score-0.371]

5 Generally, these results use bounds on the value of the game (often based on the sequential Rademacher complexity) in order to construct efﬁcient algorithms. [sent-11, score-0.361]

6 In this work, we show that when the learner is unconstrained, it is often possible to efﬁciently compute an exact minimax strategy for both the player and nature. [sent-12, score-0.717]

7 , T , ﬁrst the learner selects xt 2 Rn , and then an adversary chooses gt 2 G ⇢ Rn , and the learner suffers loss gt · xt . [sent-17, score-1.801]

8 , gT ) (1) as the regret with respect to benchmark performance L (the L intended will be clear from context). [sent-22, score-0.255]

9 We view this interaction as a sequential zero-sum game played over T rounds, where the player strives to minimize Eq. [sent-33, score-0.773]

10 We provide a characterization of the value of the game Eq. [sent-44, score-0.38]

11 We provide a method for computing the player’s minimax optimal (deterministic) strategy in terms of a “discrete derivative. [sent-48, score-0.34]

12 For “coordinate-decomposable” games we give a natural and efﬁciently computable description of the value of the game and the player’s optimal strategy. [sent-51, score-0.488]

13 In particular, we show that constant-step-size gradient descent is minimax optimal for a quadratic , and an exponential L leads to a bounded-loss hedging algorithm that can still yield exponential reward on “easy” sequences. [sent-56, score-0.299]

14 , 2012] that can be described exactly in terms of a repeated game which ﬁts nicely into our framework. [sent-64, score-0.312]

15 The algorithm provides us with a tool for making potentially unconstrained bets/investments, but as we discuss it also leads to interesting regret bounds. [sent-68, score-0.271]

16 (3), though for different problems and not in the minimax setting. [sent-72, score-0.232]

17 [2012] consider the stronger notion of policy regret in the online experts and bandit settings, respectively. [sent-82, score-0.246]

18 [2006] 2 setting soft feasible set standard regret bounded-loss betting L(G) |G| minimax value 2 p exp(G/ T ) x 2 I(|x|  1) p p p T x log( T x) + T x update T 2 (x) G2 2 xt+1 = ! [sent-85, score-0.601]

19 Results are stated for the one-dimensional problem where gt 2 [ 1, 1]; Corollary 5 gives an extension to n dimensions. [sent-90, score-0.584]

20 2 applies to similar problems, but does not require a “no junk bonds” assumption, and is in fact minimax optimal for a natural benchmark. [sent-96, score-0.274]

21 Existing algorithms do offer bounds for unconstrained problems, generally of the form kx⇤ k/⌘ + P ⌘ t gt xt . [sent-97, score-0.831]

22 The ﬁeld has seen a number of minimax approaches to online learning. [sent-102, score-0.287]

23 3 studies the online linear game of Abernethy et al. [sent-106, score-0.395]

24 [2012] utilizes powerful tools for non-constructive analysis of online learning as a technique to design algorithms; our work differs in that we focus on cases where the exact minimax strategy can be computed. [sent-110, score-0.353]

25 Such a deﬁnition makes sense when the player by deﬁnition must select a strategy from some bounded set, for example a probability from the n-dimensional simplex, or a distribution on paths in a graph. [sent-112, score-0.459]

26 The player cannot do well in terms of the absolute loss t gt · xt for all sequences g1 , . [sent-119, score-1.176]

27 , gT ) is large and negative are those on which the player desires good performance, at the expense of allowing more loss (in absolute terms) on sequences where L(g1 , . [sent-126, score-0.437]

28 The value of the game V T tells us to what extent any online algorithm can hope to match the benchmark L. [sent-130, score-0.487]

29 We show that in certain cases the computation of the minimax value can be greatly simpliﬁed. [sent-133, score-0.255]

30 That is, the Regret when we ﬁx the plays on the ﬁrst t rounds, and then assume minimax optimal play for rounds t + 1 through T . [sent-150, score-0.394]

31 (5) xt+1 2R gt+1 2G xT 2R gT 2G s=t+1 Note the conditional value of the game before anything has been played, V0 (), is exactly V T . [sent-167, score-0.363]

32 The martingale characterization of the game The fundamental tool used in the rest of the paper is the following characterization of the conditional value of the game: Theorem 1. [sent-168, score-0.533]

33 , gt 2 G, we can write the conditional value of the game as Vt (g1 , . [sent-172, score-0.947]

34 , gt , G)], 0 G2 (G ),E[G]=0 where (G 0 ) is the set of random variables on G 0 . [sent-178, score-0.584]

35 Characterization of optimal strategies The result above gives a nice expression for the value of the game V T but unfortunately it does not lead directly to a strategy for the player. [sent-199, score-0.474]

36 We now dig a bit deeper and produce a characterization of the optimal player behavior. [sent-200, score-0.451]

37 The theorem makes a useful point about determining the player’s optimal strategy for games of this form. [sent-214, score-0.25]

38 If the player can determine a full-rank set of “best responses” {g 1 , . [sent-215, score-0.364]

39 We can express (5), the conditional value of the game Vt 1 , in recursive form as Vt 1 (g1 , . [sent-222, score-0.363]

40 , gt 1 , gt ), noting that the latter is convex in gt by Theorem 1, we see we have an immediate use of Theorem 3. [sent-232, score-1.787]

41 Consider the one-dimensional unconstrained game where the player selects xt 2 R and the adversary chooses gt 2 G ⇥= [ 1, 1], and L is concave in each of its arguments and bounded ⇤ on G T . [sent-236, score-1.739]

42 , gT ) where the expectation is over each gt chosen independently and uniformly from { 1, 1} (that is, the gt are Rademacher random variables). [sent-240, score-1.168]

43 Further, the conditional value of the game is ⇥ ⇤ Vt (g1 , . [sent-241, score-0.363]

44 1 Given Theorem 4, and the fact that the functions L of interest will generally depend only on g1:T , it will be useful to deﬁne BT to be the distribution of g1:T when each gt is drawn independently and uniformly from { 1, 1}. [sent-255, score-0.584]

45 Consider the game where the player chooses xt 2 Rn , the adversary chooses gt 2 PT Pn T [ 1, 1]n , and the payoff is t=1 gt · xt and i=1 L(g1:T,i ) for concave L. [sent-257, score-2.412]

46 , gt ) = n X i=1 E Gi ⇠BT t ⇥ ⇤ L(g1:t,i + Gi ) . [sent-261, score-0.584]

47 The proof follows by noting the constraints on both players’ strategies and the value of the game fully decompose on a per-coordinate basis. [sent-262, score-0.366]

48 A recipe for minimax optimal algorithms in one dimension Since Eq. [sent-263, score-0.274]

49 (5) gives the minimax value of the game if both players play optimally from round t + 1 forward, a minimax strategy for the learner on round t + 1 must be xt+1 = arg minx2R maxg2{ 1,1} g · x + Vt+1 (g1 , . [sent-264, score-1.059]

50 Thus, the player strategy is just the interpolation of the points ( 1, f ( 1)) and (1, f (1)), where we take f = Vt+1 , giving us 1 Vt+1 (g1 , . [sent-269, score-0.458]

51 , gt ), we will have an efﬁcient minimax-optimal algorithm. [sent-279, score-0.584]

52 5 the following sections we exactly compute the game values and unique minimax optimal strategies for a variety of interesting coordinate-decomposable games. [sent-282, score-0.617]

53 Even when such exact computations are not possible, any coordinate-decomposable game where L depends only on G = g1:T can be solved numerically in polynomial time. [sent-283, score-0.312]

54 We can also immediately provide a characterization of the potentially optimal player strategies in terms of the subgradients of L. [sent-290, score-0.482]

55 Then, on every round, the unique minimax optimal x⇤ satisﬁes x⇤ 2 L where L = [w2R @L(w). [sent-295, score-0.274]

56 Following Theorem 3, we know the minimax xt+1 interpolates (a, f (a)) and (b, f (b)), where we take f (g) = Vt+1 (g1 , . [sent-297, score-0.232]

57 Note that for standard regret, L(g) = inf x2X gx, we have @L(g) ✓ X , indicating that (in 1 dimension at least), the player never needs to play outside the comparator set X . [sent-310, score-0.554]

58 1 Constant step-size gradient descent can be minimax optimal Suppose we use a “soft” feasible set for the benchmark via a quadratic penalty, L(G) = min Gx + x 2 x2 = 1 2 G , 2 (11) for a constant > 0. [sent-313, score-0.371]

59 The value of this game is V T = EG⇠BT 21 G2 = 2 . [sent-317, score-0.335]

60 But this does not mean the minimax algorithm will be uninteresting. [sent-319, score-0.232]

61 To derive the minimax optimal algorithm, we compute conditional values (using similar techniques to Theorem 7), h 1 i 1 Vt (g1 , . [sent-320, score-0.302]

62 , gt ) = E (g1:t + G)2 = (g1:t )2 + (T t) , G⇠BT t 2 2 and so following Eq. [sent-323, score-0.584]

63 Note that for a ﬁxed , this is the optimal algorithm independent of T ; this is atypical, as usually the minimax optimal algorithm depends on the horizon (as we will see in the next two cases). [sent-325, score-0.316]

64 Note that the set L = R (from Theorem 6), and indeed the player could eventually play an arbitrary point in R (given large enough T ). [sent-326, score-0.406]

65 2 Non-stochastic betting with exponential upside and bounded worst-case loss A major advantage of the regret minimization framework is that the guarantees we can achieve are typically robust to arbitrary input sequences. [sent-328, score-0.412]

66 On each round t, the world offers the player a betting opportunity on a coin toss, i. [sent-332, score-0.589]

67 The player may take either side of the bet, and selects a wager amount xt , where xt > 0 implies a bet on tails (gt = 1) and xt < 0 a bet on heads (gt = 1). [sent-335, score-1.072]

68 The world then announces whether the bet was won or lost, revealing gt . [sent-336, score-0.662]

69 The player’s wealth changes (additively) by gt xt (that is, the player strives to minimize loss gt xt ). [sent-337, score-1.981]

70 We assume that the player begins with some initial capital ↵ > 0, and at any time period the wager |xt | Pt 1 must not exceed ↵ s=1 gs xs , the initial capital plus the money earned thus far. [sent-338, score-0.58]

71 PT With the beneﬁt of hindsight, the gambler can see G = t=1 gt , the total number of heads minus the total number of heads. [sent-339, score-0.691]

72 A simple exercise shows that his wealth would become T Y T +G T G (1 + gt ) = (1 + ) 2 (1 ) 2 . [sent-343, score-0.641]

73 In other words, with knowledge of the ﬁnal G, a na¨ve betting strategy could have earned the gambler exponentially large winnings starting with ı constant capital. [sent-345, score-0.384]

74 We ask: does there exist an adaptive betting strategy that can compete with this hindsight benchmark, even if the gt are chosen fully adversarially? [sent-347, score-0.902]

75 We present a solution for the one-sided game, without the absolute value, so the player only aims for exponential wealth growth for large positive G. [sent-350, score-0.421]

76 Consider the game where G = [ 1, 1] with benchmark L(G) = exp(G/ T ). [sent-353, score-0.409]

77 That the value of the game here is of constant order is G critical, since it says that we can always achieve a payoff that is exponential in pT at a cost of no p more than e = O(1). [sent-357, score-0.379]

78 Notice we have said nothing thus far regarding the nature of our betting strategy; in particular we have not proved that the strategy satisﬁes the required condition that the gambler cannot bet more than ↵ plus the earnings thus far. [sent-358, score-0.4]

79 Consider a one dimensional game with G = [ 1, 1] with benchmark function L nonPt T positive on G T . [sent-360, score-0.409]

80 Then for the optimal betting strategy we have that |xt |  s=1 gs xs + V , and Pt T further V s=1 gs xs for any t and any sequence g1 , . [sent-361, score-0.508]

81 This implies that the starting capital ↵ required p “replicate” the payoff function is exactly the value2 of to the game V T . [sent-366, score-0.384]

82 Since this algorithm needs large a Reward when G is large and positive, we might expect that the minimax optimal algorithm only plays xt  0. [sent-375, score-0.468]

83 We can construct a minimax optimal algorithm for LC (G) by running two copies of the one-sided minimax algorithm simultaneously, switching the signs of the gradients and plays of the second copy. [sent-380, score-0.572]

84 In that work, the goal was to prove bounds on standard regret like Regret  p O(R T log ((1 + R)T )) simultaneously for any comparator x⇤ with |x⇤ | = R. [sent-383, score-0.263]

85 Stating their Theorem 1 in terms of losses, this traditional regret bound is achieved by any algorithm that guarantees ◆ ✓ T X |G| Loss = gt xt  exp p + O(1). [sent-384, score-0.933]

86 (13) T t=1 The symmetric algorithm (Appendix B) satisﬁes ✓ ◆ ✓ ◆ p G G Loss  exp p exp p +2 e T T exp ✓ |G| p T and so we also achieve a standard regret bound of the form given above. [sent-385, score-0.266]

87 3 ◆ p + 2 e, Optimal regret against hypercube adversaries Perhaps the simplest and best studied learning games are those that restrict both the player and adversary to a norm ball, and use the standard notion of regret. [sent-387, score-0.787]

88 We can derive results for the game where the adversary has an L1 constraint, the comparator set is also the L1 ball, and the player is unconstrained. [sent-388, score-0.902]

89 Consider the game between an adversary who chooses losses gt 2 [ 1, 1], and a player who chooses xt 2 R. [sent-391, score-1.602]

90 , xT , gT , the value to PT the adversary is t=1 gt xt |g1:T |. [sent-395, score-0.883]

91 Then, when T is even with T = 2M , the minimax value of this game is given by r 2M T ! [sent-396, score-0.567]

92 Then the minimax ⇡ optimal strategy for the player given the adversary has played Gt = g1:t is given by xt+1 = Pr(B < Gt ) Pr(B > Gt ) = 1 2 Pr(B > Gt ) 2 [ 1, 1]. [sent-404, score-0.851]

93 (14) p The fact that the limiting value of this game is 2T /⇡ was previously known, e. [sent-405, score-0.335]

94 [2009]; however, we believe this explicit form for the optimal player strategy is new. [sent-408, score-0.472]

95 It also follows from this expression that even though we allow the player to select xt+1 2 R, the minimax optimal algorithm always selects points from [ 1, 1], so our result applies to the case where the player is constrained to play from X . [sent-411, score-1.066]

96 [2008a] shows that for the linear game with n 3 where both the learner and p adversary select vectors from the unit sphere, the minimax value is exactly T . [sent-413, score-0.743]

97 Interestingly, in the p n = 1 case (where L2 and L1 coincide), the value of the game is lower, about 0. [sent-414, score-0.335]

98 We conjecture the minimax value for the L2 game with n = 2 lies somewhere in between. [sent-417, score-0.567]

99 Optimal strategies and minimax lower bounds for online convex games. [sent-424, score-0.353]

100 A stochastic view of optimal regret through minimax duality. [sent-431, score-0.432]

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Abstract: Learning the joint dependence of discrete variables is a fundamental problem in machine learning, with many applications including prediction, clustering and dimensionality reduction. More recently, the framework of copula modeling has gained popularity due to its modular parameterization of joint distributions. Among other properties, copulas provide a recipe for combining ﬂexible models for univariate marginal distributions with parametric families suitable for potentially high dimensional dependence structures. More radically, the extended rank likelihood approach of Hoff (2007) bypasses learning marginal models completely when such information is ancillary to the learning task at hand as in, e.g., standard dimensionality reduction problems or copula parameter estimation. The main idea is to represent data by their observable rank statistics, ignoring any other information from the marginals. 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Copulas have found numerous applications in statistics and machine learning since they provide a way of constructing ﬂexible multivariate distributions by mix-and-matching different copulas with different univariate marginals. For instance, one can combine ﬂexible univariate marginals Fi (·) with useful but more constrained high-dimensional copulas. We will not further motivate the use of copula models, which has been discussed at length in recent 1 machine learning publications and conference workshops, and for which comprehensive textbooks exist [e.g., 9]. For a recent discussion on the applications of copulas from a machine learning perspective, [4] provides an overview. [10] is an early reference in machine learning. The core idea dates back at least to the 1950s [16]. In the discrete case, copulas can be difﬁcult to apply: transforming a copula CDF into a probability mass function (PMF) is computationally intractable in general. For the continuous case, a common ˆ trick goes as follows: transform variables by deﬁning ai ≡ Fi (yi ) for an estimate of Fi (·) and then ﬁt a copula density c(·, . . . , ·) to the resulting ai [e.g. 9]. It is not hard to check this breaks down in the discrete case [7]. An alternative is to represent the CDF to PMF transformation for each data point by a continuous integral on a bounded space. Sampling methods can then be used. This trick has allowed many applications of the Gaussian copula to discrete domains. Readers familiar with probit models will recognize the similarities to models where an underlying latent Gaussian ﬁeld is discretized into observable integers as in Gaussian process classiﬁers and ordinal regression [18]. Such models can be indirectly interpreted as special cases of the Gaussian copula. In what follows, we describe in Section 2 the Gaussian copula and the general framework for constructing Bayesian estimators of Gaussian copulas by [7], the extended rank likelihood framework. This framework entails computational issues which are discussed. A recent general approach for MCMC in constrained Gaussian ﬁelds by [17] can in principle be directly applied to this problem as a blackbox, but at a cost that scales quadratically in sample size and as such it is not practical in general. Our key contribution is given in Section 4. An application experiment on the Bayesian Gaussian copula factor model is performed in Section 5. Conclusions are discussed in the ﬁnal section. 2 Gaussian copulas and the extended rank likelihood It is not hard to see that any multivariate Gaussian copula is fully deﬁned by a correlation matrix C, since marginal distributions have no free parameters. In practice, the following equivalent generative model is used to deﬁne a sample U according to a Gaussian copula GC(C): 1. Sample Z from a zero mean Gaussian with covariance matrix C 2. For each Zj , set Uj = Φ(zj ), where Φ(·) is the CDF of the standard Gaussian It is clear that each Uj follows a uniform distribution in [0, 1]. To obtain a model for variables {y1 , y2 , . . . , yp } with marginal distributions Fj (·) and copula GC(C), one can add the deterministic (n) (1) (1) (2) step yj = Fj−1 (uj ). Now, given n samples of observed data Y ≡ {y1 , . . . , yp , y1 , . . . , yp }, one is interested on inferring C via a Bayesian approach and the posterior distribution p(C, θF | Y) ∝ pGC (Y | C, θF )π(C, θF ) where π(·) is a prior distribution, θF are marginal parameters for each Fj (·), which in general might need to be marginalized since they will be unknown, and pGC (·) is the PMF of a (here discrete) distribution with a Gaussian copula and marginals given by θF . Let Z be the underlying latent Gaussians of the corresponding copula for dataset Y. Although Y is a deterministic function of Z, this mapping is not invertible due to the discreteness of the distribution: each marginal Fj (·) has jumps. Instead, the reverse mapping only enforces the constraints where (i ) (i ) (i ) (i ) yj 1 < yj 2 implies zj 1 < zj 2 . Based on this observation, [7] considers the event Z ∈ D(y), where D(y) is the set of values of Z in Rn×p obeying those constraints, that is (k) (k) D(y) ≡ Z ∈ Rn×p : max zj s.t. yj (i) < yj (i) (k) (i) (k) < zj < min zj s.t. yj < yj . Since {Y = y} ⇒ Z(y) ∈ D(y), we have pGC (Y | C, θF ) = pGC (Z ∈ D(y), Y | C, θF ) = pN (Z ∈ D(y) | C) × pGC (Y| Z ∈ D(y), C, θF ), (1) the ﬁrst factor of the last line being that of a zero-mean a Gaussian density function marginalized over D(y). 2 The extended rank likelihood is deﬁned by the ﬁrst factor of (1). With this likelihood, inference for C is given simply by marginalizing p(C, Z | Y) ∝ I(Z ∈ D(y)) pN (Z| C) π(C), (2) the ﬁrst factor of the right-hand side being the usual binary indicator function. Strictly speaking, this is not a fully Bayesian method since partial information on the marginals is ignored. Nevertheless, it is possible to show that under some mild conditions there is information in the extended rank likelihood to consistently estimate C [13]. It has two important properties: ﬁrst, in many applications where marginal distributions are nuisance parameters, this sidesteps any major assumptions about the shape of {Fi (·)} – applications include learning the degree of dependence among variables (e.g., to understand relationships between social indicators as in [7] and [13]) and copula-based dimensionality reduction (a generalization of correlation-based principal component analysis, e.g., [5]); second, MCMC inference in the extended rank likelihood is conceptually simpler than with the joint likelihood, since dropping marginal models will remove complicated entanglements between C and θF . Therefore, even if θF is necessary (when, for instance, predicting missing values of Y), an estimate of C can be computed separately and will not depend on the choice of estimator for {Fi (·)}. The standard model with a full correlation matrix C can be further reﬁned to take into account structure implied by sparse inverse correlation matrices [2] or low rank decompositions via higher-order latent variable models [13], among others. We explore the latter case in section 5. An off-the-shelf algorithm for sampling from (2) is full Gibbs sampling: ﬁrst, given Z, the (full or structured) correlation matrix C can be sampled by standard methods. More to the point, sampling (i) Z is straightforward if for each variable j and data point i we sample Zj conditioned on all other variables. The corresponding distribution is an univariate truncated Gaussian. This is the approach used originally by Hoff. However, mixing can be severely compromised by the sampling of Z, and that is where novel sampling methods can facilitate inference. 3 Exact HMC for truncated Gaussian distributions (i) Hoff’s algorithm modiﬁes the positions of all Zj associated with a particular discrete value of Yj , conditioned on the remaining points. As the number of data points increases, the spread of the hard (i) boundaries on Zj , given by data points of Zj associated with other levels of Yj , increases. This (i) reduces the space in which variables Zj can move at a time. To improve the mixing, we aim to sample from the joint Gaussian distribution of all latent variables (i) Zj , i = 1 . . . n , conditioned on other columns of the data, such that the constraints between them are satisﬁed and thus the ordering in the observation level is conserved. Standard Gibbs approaches for sampling from truncated Gaussians reduce the problem to sampling from univariate truncated Gaussians. Even though each step is computationally simple, mixing can be slow when strong correlations are induced by very tight truncation bounds. In the following, we brieﬂy describe the methodology recently introduced by [17] that deals with the problem of sampling from log p(x) ∝ − 1 x Mx + r x , where x, r ∈ Rn and M is positive 2 deﬁnite, with linear constraints of the form fj x ≤ gj , where fj ∈ Rn , j = 1 . . . m, is the normal vector to some linear boundary in the sample space. Later in this section we shall describe how this framework can be applied to the Gaussian copula extended rank likelihood model. More importantly, the observed rank statistics impose only linear constraints of the form xi1 ≤ xi2 . We shall describe how this special structure can be exploited to reduce the runtime complexity of the constrained sampler from O(n2 ) (in the number of observations) to O(n) in practice. 3.1 Hamiltonian Monte Carlo for the Gaussian distribution Hamiltonian Monte Carlo (HMC) [15] is a MCMC method that extends the sampling space with auxiliary variables so that (ideally) deterministic moves in the joint space brings the sampler to 3 potentially far places in the original variable space. Deterministic moves cannot in general be done, but this is possible in the Gaussian case. The form of the Hamiltonian for the general d-dimensional Gaussian case with mean µ and precision matrix M is: 1 1 H = x Mx − r x + s M−1 s , (3) 2 2 where M is also known in the present context as the mass matrix, r = Mµ and s is the velocity. Both x and s are Gaussian distributed so this Hamiltonian can be seen (up to a constant) as the negative log of the product of two independent Gaussian random variables. The physical interpretation is that of a sum of potential and kinetic energy terms, where the total energy of the system is conserved. In a system where this Hamiltonian function is constant, we can exactly compute its evolution through the pair of differential equations: ˙ x= sH = M−1 s , ˙ s=− xH = −Mx + r . (4) These are solved exactly by x(t) = µ + a sin(t) + b cos(t) , where a and b can be identiﬁed at initial conditions (t = 0) : ˙ a = x(0) = M−1 s , b = x(0) − µ . (5) Therefore, the exact HMC algorithm can be summarised as follows: • Initialise the allowed travel time T and some initial position x0 . • Repeat for HMC samples k = 1 . . . N 1. Sample sk ∼ N (0, M) 2. Use sk and xk to update a and b and store the new position at the end of the trajectory xk+1 = x(T ) as an HMC sample. It can be easily shown that the Markov chain of sampled positions has the desired equilibrium distribution N µ, M−1 [17]. 3.2 Sampling with linear constraints Sampling from multivariate Gaussians does not require any method as sophisticated as HMC, but the plot thickens when the target distribution is truncated by linear constraints of the form Fx ≤ g . Here, F ∈ Rm×n is a constraint matrix whose every row is the normal vector to a linear boundary in the sample space. This is equivalent to sampling from a Gaussian that is conﬁned in the (not necessarily bounded) convex polyhedron {x : Fx ≤ g}. In general, to remain within the boundaries of each wall, once a new velocity has been sampled one must compute all possible collision times with the walls. The smallest of all collision times signiﬁes the wall that the particle should bounce from at that collision time. Figure 1 illustrates the concept with two simple examples on 2 and 3 dimensions. The collision times can be computed analytically and their equations can be found in the supplementary material. We also point the reader to [17] for a more detailed discussion of this implementation. Once the wall to be hit has been found, then position and velocity at impact time are computed and the velocity is reﬂected about the boundary normal1 . The constrained HMC sampler is summarized follows: • Initialise the allowed travel time T and some initial position x0 . • Repeat for HMC samples k = 1 . . . N 1. Sample sk ∼ N (0, M) 2. Use sk and xk to update a and b . 1 Also equivalent to transforming the velocity with a Householder reﬂection matrix about the bounding hyperplane. 4 1 2 3 4 1 2 3 4 Figure 1: Left: Trajectories of the ﬁrst 40 iterations of the exact HMC sampler on a 2D truncated Gaussian. A reﬂection of the velocity can clearly be seen when the particle meets wall #2 . Here, the constraint matrix F is a 4 × 2 matrix. Center: The same example after 40000 samples. The coloring of each sample indicates its density value. Right: The anatomy of a 3D Gaussian. The walls are now planes and in this case F is a 2 × 3 matrix. Figure best seen in color. 3. Reset remaining travel time Tleft ← T . Until no travel time is left or no walls can be reached (no solutions exist), do: (a) Compute impact times with all walls and pick the smallest one, th (if a solution exists). (b) Compute v(th ) and reﬂect it about the hyperplane fh . This is the updated velocity after impact. The updated position is x(th ) . (c) Tleft ← Tleft − th 4. Store the new position at the end of the trajectory xk+1 as an HMC sample. In general, all walls are candidates for impact, so the runtime of the sampler is linear in m , the number of constraints. This means that the computational load is concentrated in step 3(a). Another consideration is that of the allocated travel time T . Depending on the shape of the bounding polyhedron and the number of walls, a very large travel time can induce many more bounces thus requiring more computations per sample. On the other hand, a very small travel time explores the distribution more locally so the mixing of the chain can suffer. What constitutes a given travel time “large” or “small” is relative to the dimensionality, the number of constraints and the structure of the constraints. Due to the nature of our problem, the number of constraints, when explicitly expressed as linear functions, is O(n2 ) . Clearly, this restricts any direct application of the HMC framework for Gaussian copula estimation to small-sample (n) datasets. More importantly, we show how to exploit the structure of the constraints to reduce the number of candidate walls (prior to each bounce) to O(n) . 4 HMC for the Gaussian Copula extended rank likelihood model Given some discrete data Y ∈ Rn×p , the task is to infer the correlation matrix of the underlying Gaussian copula. Hoff’s sampling algorithm proceeds by alternating between sampling the continu(i) (i) ous latent representation Zj of each Yj , for i = 1 . . . n, j = 1 . . . p , and sampling a covariance matrix from an inverse-Wishart distribution conditioned on the sampled matrix Z ∈ Rn×p , which is then renormalized as a correlation matrix. From here on, we use matrix notation for the samples, as opposed to the random variables – with (i) Zi,j replacing Zj , Z:,j being a column of Z, and Z:,\j being the submatrix of Z without the j-th column. In a similar vein to Hoff’s sampling algorithm, we replace the successive sampling of each Zi,j conditioned on Zi,\j (a conditional univariate truncated Gaussian) with the simultaneous sampling of Z:,j conditioned on Z:,\j . This is done through an HMC step from a conditional multivariate truncated Gaussian. The added beneﬁt of this HMC step over the standard Gibbs approach, is that of a handle for regulating the trade-off between exploration and runtime via the allocated travel time T . Larger travel times potentially allow for larger moves in the sample space, but it comes at a cost as explained in the sequel. 5 4.1 The Hough envelope algorithm The special structure of constraints. Recall that the number of constraints is quadratic in the dimension of the distribution. This is because every Z sample must satisfy the conditions of the event Z ∈ D(y) of the extended rank likelihood (see Section 2). In other words, for any column Z:,j , all entries are organised into a partition L(j) of |L(j) | levels, the number of unique values observed for the discrete or ordinal variable Y (j) . Thereby, for any two adjacent levels lk , lk+1 ∈ L(j) and any pair i1 ∈ lk , i2 ∈ lk+1 , it must be true that Zli ,j < Zli+1 ,j . Equivalently, a constraint f exists where fi1 = 1, fi2 = −1 and g = 0 . It is easy to see that O(n2 ) of such constraints are induced by the order statistics of the j-th variable. To deal with this boundary explosion, we developed the Hough Envelope algorithm to search efﬁciently, within all pairs in {Z:,j }, in practically linear time. Recall in HMC (section 3.2) that the trajectory of the particle, x(t), is decomposed as xi (t) = ai sin(t) + bi cos(t) + µi , (6) and there are n such functions, grouped into a partition of levels as described above. The Hough envelope2 is found for every pair of adjacent levels. We illustrate this with an example of 10 dimensions and two levels in Figure 2, without loss of generalization to any number of levels or dimensions. Assume we represent trajectories for points in level lk with blue curves, and points in lk+1 with red curves. Assuming we start with a valid state, at time t = 0 all red curves are above all blue curves. The goal is to ﬁnd the smallest t where a blue curve meets a red curve. This will be our collision time where a bounce will be necessary. 5 3 1 2 Figure 2: The trajectories xj (t) of each component are sinusoid functions. The right-most green dot signiﬁes the wall and the time th of the earliest bounce, where the ﬁrst inter-level pair (that is, any two components respectively from the blue and red level) becomes equal, in this case the constraint activated being xblue2 = xred2 . 4 4 5 1 2 3 0.2 0.4 0.6 t 0.8 1 1.2 1.4 1. First we ﬁnd the largest component bluemax of the blue level at t = 0. This takes O(n) time. Clearly, this will be the largest component until its sinusoid intersects that of any other component. 2. To ﬁnd the next largest component, compute the roots of xbluemax (t) − xi (t) = 0 for all components and pick the smallest (earliest) one (represented by a green dot). This also takes O(n) time. 3. Repeat this procedure until a red sinusoid crosses the highest running blue sinusoid. When this happens, the time of earliest bounce and its constraint are found. In the worst-case scenario, n such repetitions have to be made, but in practice we can safely assume an ﬁxed upper bound h on the number of blue crossings before a inter-level crossing occurs. In experiments, we found h << n, no more than 10 in simulations with hundreds of thousands of curves. Thus, this search strategy takes O(n) time in practice to complete, mirroring the analysis of other output-sensitive algorithms such as the gift wrapping algorithm for computing convex hulls [8]. Our HMC sampling approach is summarized in Algorithm 1. 2 The name is inspired from the fact that each xi (t) is the sinusoid representation, in angle-distance space, of all lines that pass from the (ai , bi ) point in a − b space. A representation known in image processing as the Hough transform [3]. 6 Algorithm 1 HMC for GCERL # Notation: T MN (µ, C, F) is a truncated multivariate normal with location vector µ, scale matrix C and constraints encoded by F and g = 0 . # IW(df, V0 ) is an inverse-Wishart prior with degrees of freedom df and scale matrix V0 . Input: Y ∈ Rn×p , allocated travel time T , a starting Z and variable covariance V ∈ Rp×p , df = p + 2, V0 = df Ip and chain size N . Generate constraints F(j) from Y:,j , for j = 1 . . . p . for samples k = 1 . . . N do # Resample Z as follows: for variables j = 1 . . . p do −1 −1 2 Compute parameters: σj = Vjj − Vj,\j V\j,\j V\j,j , µj = Z:,\j V\j,\j V\j,j . 2 Get one sample Z:,j ∼ T MN µj , σj I, F(j) efﬁciently by using the Hough Envelope algorithm, see section 4.1. end for Resample V ∼ IW(df + n, V0 + Z Z) . Compute correlation matrix C, s.t. Ci,j = Vi,j / Vi,i Vj,j and store sample, C(k) ← C . end for 5 An application on the Bayesian Gausian copula factor model In this section we describe an experiment that highlights the beneﬁts of our HMC treatment, compared to a state-of-the-art parameter expansion (PX) sampling scheme. During this experiment we ask the important question: “How do the two schemes compare when we exploit the full-advantage of the HMC machinery to jointly sample parameters and the augmented data Z, in a model of latent variables and structured correlations?” We argue that under such circumstances the superior convergence speed and mixing of HMC undeniably compensate for its computational overhead. Experimental setup In this section we provide results from an application on the Gaussian copula latent factor model of [13] (Hoff’s model [7] for low-rank structured correlation matrices). We modify the parameter expansion (PX) algorithm used by [13] by replacing two of its Gibbs steps with a single HMC step. We show a much faster convergence to the true mode with considerable support on its vicinity. We show that unlike the HMC, the PX algorithm falls short of properly exploring the posterior in any reasonable ﬁnite amount of time, even for small models, even for small samples. Worse, PX fails in ways one cannot easily detect. Namely, we sample each row of the factor loadings matrix Λ jointly with the corresponding column of the augmented data matrix Z, conditioning on the higher-order latent factors. This step is analogous to Pakman and Paninski’s [17, sec.3.1] use of HMC in the context of a binary probit model (the extension to many levels in the discrete marginal is straightforward with direct application of the constraint matrix F and the Hough envelope algorithm). The sampling of the higher level latent factors remains identical to [13]. Our scheme involves no parameter expansion. We do however interweave the Gibbs step for the Z matrix similarly to Hoff. This has the added beneﬁt of exploring the Z sample space within their current boundaries, complementing the joint (λ, z) sampling which moves the boundaries jointly. The value of such ”interweaving” schemes has been addressed in [19]. Results We perform simulations of 10000 iterations, n = 1000 observations (rows of Y), travel time π/2 for HMC with the setups listed in the following table, along with the elapsed times of each sampling scheme. These experiments were run on Intel COREi7 desktops with 4 cores and 8GB of RAM. Both methods were parallelized across the observed variables (p). Figure p (vars) k (latent factors) M (ordinal levels) elapsed (mins): HMC PX 3(a) : 20 5 2 115 8 3(b) : 10 3 2 80 6 10 3 5 203 16 3(c) : Many functionals of the loadings matrix Λ can be assessed. We focus on reconstructing the true (low-rank) correlation matrix of the Gaussian copula. In particular, we summarize the algorithm’s 7 outcome with the root mean squared error (RMSE) of the differences between entries of the ground-truth correlation matrix and the implied correlation matrix at each iteration of a MCMC scheme (so the following plots looks like a time-series of 10000 timepoints), see Figures 3(a), 3(b) and 3(c) . (a) (b) (c) Figure 3: Reconstruction (RMSE per iteration) of the low-rank structured correlation matrix of the Gaussian copula and its histogram (along the left side). (a) Simulation setup: 20 variables, 5 factors, 5 levels. HMC (blue) reaches a better mode faster (in iterations/CPU-time) than PX (red). Even more importantly the RMSE posterior samples of PX are concentrated in a much smaller region compared to HMC, even after 10000 iterations. This illustrates that PX poorly explores the true distribution. (b) Simulation setup: 10 vars, 3 factors, 2 levels. We observe behaviors similar to Figure 3(a). Note that the histogram counts RMSEs after the burn-in period of PX (iteration #500). (c) Simulation setup: 10 vars, 3 factors, 5 levels. We observe behaviors similar to Figures 3(a) and 3(b) but with a thinner tail for HMC. Note that the histogram counts RMSEs after the burn-in period of PX (iteration #2000). Main message HMC reaches a better mode faster (iterations/CPUtime). Even more importantly the RMSE posterior samples of PX are concentrated in a much smaller region compared to HMC, even after 10000 iterations. This illustrates that PX poorly explores the true distribution. As an analogous situation we refer to the top and bottom panels of Figure 14 of Radford Neal’s slice sampler paper [14]. If there was no comparison against HMC, there would be no evidence from the PX plot alone that the algorithm is performing poorly. This mirrors Radford Neal’s statement opening Section 8 of his paper: “a wrong answer is obtained without any obvious indication that something is amiss”. The concentration on the posterior mode of PX in these simulations is misleading of the truth. PX might seen a bit simpler to implement, but it seems one cannot avoid using complex algorithms for complex models. We urge practitioners to revisit their past work with this model to ﬁnd out by how much credible intervals of functionals of interest have been overconﬁdent. Whether trivially or severely, our algorithm offers the ﬁrst principled approach for checking this out. 6 Conclusion Sampling large random vectors simultaneously in order to improve mixing is in general a very hard problem, and this is why clever methods such as HMC or elliptical slice sampling [12] are necessary. We expect that the method here developed is useful not only for those with data analysis problems within the large family of Gaussian copula extended rank likelihood models, but the method itself and its behaviour might provide some new insights on MCMC sampling in constrained spaces in general. Another direction of future work consists of exploring methods for elliptical copulas, and related possible extensions of general HMC for non-Gaussian copula models. Acknowledgements The quality of this work has beneﬁted largely from comments by our anonymous reviewers and useful discussions with Simon Byrne and Vassilios Stathopoulos. Research was supported by EPSRC grant EP/J013293/1. 8 References [1] Y. Bishop, S. Fienberg, and P. Holland. Discrete Multivariate Analysis: Theory and Practice. MIT Press, 1975. [2] A. Dobra and A. Lenkoski. Copula Gaussian graphical models and their application to modeling functional disability data. Annals of Applied Statistics, 5:969–993, 2011. [3] R. O. Duda and P. E. Hart. Use of the Hough transformation to detect lines and curves in pictures. Communications of the ACM, 15(1):11–15, 1972. [4] G. Elidan. Copulas and machine learning. Proceedings of the Copulae in Mathematical and Quantitative Finance workshop, to appear, 2013. [5] F. Han and H. Liu. Semiparametric principal component analysis. Advances in Neural Information Processing Systems, 25:171–179, 2012. [6] G. Hinton and R. Salakhutdinov. Reducing the dimensionality of data with neural networks. Science, 313(5786):504–507, 2006. [7] P. Hoff. Extending the rank likelihood for semiparametric copula estimation. Annals of Applied Statistics, 1:265–283, 2007. [8] R. Jarvis. On the identiﬁcation of the convex hull of a ﬁnite set of points in the plane. Information Processing Letters, 2(1):18–21, 1973. [9] H. Joe. Multivariate Models and Dependence Concepts. Chapman-Hall, 1997. [10] S. Kirshner. Learning with tree-averaged densities and distributions. Neural Information Processing Systems, 2007. [11] S. Lauritzen. Graphical Models. Oxford University Press, 1996. [12] I. Murray, R. Adams, and D. MacKay. Elliptical slice sampling. JMLR Workshop and Conference Proceedings: AISTATS 2010, 9:541–548, 2010. [13] J. Murray, D. Dunson, L. Carin, and J. Lucas. Bayesian Gaussian copula factor models for mixed data. Journal of the American Statistical Association, to appear, 2013. [14] R. Neal. Slice sampling. The Annals of Statistics, 31:705–767, 2003. [15] R. Neal. MCMC using Hamiltonian dynamics. Handbook of Markov Chain Monte Carlo, pages 113–162, 2010. [16] R. Nelsen. An Introduction to Copulas. Springer-Verlag, 2007. [17] A. Pakman and L. Paninski. Exact Hamiltonian Monte Carlo for truncated multivariate Gaussians. arXiv:1208.4118, 2012. [18] C. Rasmussen and C. Williams. Gaussian Processes for Machine Learning. MIT Press, 2006. [19] Y. Yu and X. L. Meng. To center or not to center: That is not the question — An ancillaritysufﬁciency interweaving strategy (ASIS) for boosting MCMC efﬁciency. Journal of Computational and Graphical Statistics, 20(3):531–570, 2011. 9

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Abstract: In this paper, we theoretically study the problem of binary classiﬁcation in the presence of random classiﬁcation noise — the learner, instead of seeing the true labels, sees labels that have independently been ﬂipped with some small probability. Moreover, random label noise is class-conditional — the ﬂip probability depends on the class. We provide two approaches to suitably modify any given surrogate loss function. First, we provide a simple unbiased estimator of any loss, and obtain performance bounds for empirical risk minimization in the presence of iid data with noisy labels. If the loss function satisﬁes a simple symmetry condition, we show that the method leads to an efﬁcient algorithm for empirical minimization. Second, by leveraging a reduction of risk minimization under noisy labels to classiﬁcation with weighted 0-1 loss, we suggest the use of a simple weighted surrogate loss, for which we are able to obtain strong empirical risk bounds. This approach has a very remarkable consequence — methods used in practice such as biased SVM and weighted logistic regression are provably noise-tolerant. On a synthetic non-separable dataset, our methods achieve over 88% accuracy even when 40% of the labels are corrupted, and are competitive with respect to recently proposed methods for dealing with label noise in several benchmark datasets.

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