nips nips2013 nips2013-248 knowledge-graph by maker-knowledge-mining

248 nips-2013-Point Based Value Iteration with Optimal Belief Compression for Dec-POMDPs

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Author: Liam C. MacDermed, Charles Isbell

Abstract: We present four major results towards solving decentralized partially observable Markov decision problems (DecPOMDPs) culminating in an algorithm that outperforms all existing algorithms on all but one standard inﬁnite-horizon benchmark problems. (1) We give an integer program that solves collaborative Bayesian games (CBGs). The program is notable because its linear relaxation is very often integral. (2) We show that a DecPOMDP with bounded belief can be converted to a POMDP (albeit with actions exponential in the number of beliefs). These actions correspond to strategies of a CBG. (3) We present a method to transform any DecPOMDP into a DecPOMDP with bounded beliefs (the number of beliefs is a free parameter) using optimal (not lossless) belief compression. (4) We show that the combination of these results opens the door for new classes of DecPOMDP algorithms based on previous POMDP algorithms. We choose one such algorithm, point-based valued iteration, and modify it to produce the ﬁrst tractable value iteration method for DecPOMDPs that outperforms existing algorithms. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract We present four major results towards solving decentralized partially observable Markov decision problems (DecPOMDPs) culminating in an algorithm that outperforms all existing algorithms on all but one standard inﬁnite-horizon benchmark problems. [sent-6, score-0.297]

2 (1) We give an integer program that solves collaborative Bayesian games (CBGs). [sent-7, score-0.228]

3 (2) We show that a DecPOMDP with bounded belief can be converted to a POMDP (albeit with actions exponential in the number of beliefs). [sent-9, score-0.493]

4 (3) We present a method to transform any DecPOMDP into a DecPOMDP with bounded beliefs (the number of beliefs is a free parameter) using optimal (not lossless) belief compression. [sent-11, score-1.088]

5 Unlike single agent POMDPs, DecPOMDPs suffer from a doubly-exponential curse of history [16]. [sent-15, score-0.296]

6 Not only do agents have to reason about the observations they see, but also about the possible observations of other agents. [sent-16, score-0.274]

7 This causes agents to view their world as non-Markovian because even if an agent returns to the same underlying state of the world, the dynamics of the world may appear to change due to other agent’s holding different beliefs and taking different actions. [sent-17, score-0.907]

8 Also, for POMDPs, a sufﬁcient belief space is the set of probability distributions over possible states. [sent-18, score-0.327]

9 In the case of DecPOMDPs an agent must reason about the beliefs of other agents (who are recursively reasoning about beliefs as well), leading to nested beliefs that can make it impossible to losslessly reduce an agent’s knowledge to less than its full observation history. [sent-19, score-1.657]

10 Even using policy methods, the curse of history is still a big problem, and current methods deal with it in a number of different ways. [sent-22, score-0.205]

11 Some use heuristics to prune (or never explore) particular belief regions [19, 17, 12, 11]. [sent-24, score-0.327]

12 There have also been approaches that attempt to operate directly on the inﬁnitely nested belief structure [4], but these are approximations 1 of unknown accuracy (if we stop at the nth nested belief the nth + 1 could dramatically change the outcome). [sent-29, score-0.654]

13 Our solution to the curse of history is simple: to assume that it doesn’t exist, or more precisely, that the number of possible beliefs at any point in time is bounded. [sent-31, score-0.42]

14 Our resulting algorithm is the ﬁrst true value-iteration algorithm for DecPOMDPs (where no policy information need be retained from iteration to iteration) and outperforms existing algorithms. [sent-37, score-0.193]

15 2 DecPOMDPs as a sequence of cooperative Bayesian Games Many current approaches for solving DecPOMDPs view the decision problem faced by agents as a sequence of CBGs [5]. [sent-38, score-0.368]

16 This view arises from ﬁrst noting that a complete policy must prescribe an action for every belief-state of an agent. [sent-39, score-0.243]

17 , through belief compression), then multiple histories can correspond to the same belief. [sent-53, score-0.393]

18 These beliefs are the types of the Bayesian game. [sent-55, score-0.369]

19 The rewards to the Bayesian game are ideally the immediate reward R = st θt R(st , πθt )P r[st , θt ] along with the utility of the best continuation policy. [sent-57, score-0.408]

20 In each case, the solution to the Bayesian game is used as a heuristic to guide policy search. [sent-61, score-0.198]

21 Here we present a novel integer linear program that solves for an optimal pure strategy Bayes-Nash equilibrium (which always exists for games with common payoffs). [sent-64, score-0.299]

22 2 Our integer linear program for Bayesian game N, A, Θ, τ, R optimizes over Boolean variables xa,θ , one for each joint-action for each joint-type. [sent-67, score-0.186]

23 These represent the posterior probability that agent i, after becoming type θi , thinks other agents will take actions a−i when having types θ−i . [sent-72, score-0.552]

24 In order to ﬁnd the optimal solution for a game with distribution over types τ ∈ ∆(Θ) and rewards R : Θ × A → Rn we can solve the integer program: Maximize τθ R(θ, a)xa,θ over variables xa,θ ∈ {0, 1} subject to constraints (2). [sent-75, score-0.192]

25 We can view ANFCEs as having a mediator that each agent tells its type to and receives an action recommendation from. [sent-77, score-0.471]

26 An ANFCE is then a probability distribution across joint type/actions such that agents do not want to lie to the mediator nor deviate from the mediator’s recommendation. [sent-78, score-0.345]

27 4 Bounded Belief DecPOMDPs Here we show that we can convert a bounded belief DecPOMDP (BB-DecPOMDP) into an equivalent POMDP (that we call the belief-POMDP). [sent-83, score-0.381]

28 A BB-DecPOMDP is a DecPOMDP where each agent i has a ﬁxed upper bound |Θi | for the number of beliefs at each time-step. [sent-84, score-0.614]

29 The beliefPOMDP’s states are factored, containing each agent’s belief along with the DecPOMDP’s state. [sent-85, score-0.36]

30 [3] showed that a ﬁnite horizon DecPOMDP can be converted into a ﬁnite horizon POMDP where a probability distribution over histories is a sufﬁcient statistic that can be used as the POMDP’s state. [sent-88, score-0.218]

31 We extend this result to inﬁnite horizon problems when beliefs are bounded (note that a ﬁnite horizon problem always has bounded belief). [sent-89, score-0.501]

32 The main insight here is that we do not have to remember histories, only a distribution over belief-labels (without any a priori connection to the belief itself) as a sufﬁcient statistic. [sent-90, score-0.327]

33 In order to create the belief-POMDP we ﬁrst transform observations so that they correspond one-toone with beliefs for each agent. [sent-92, score-0.391]

34 This can be achieved naively by folding the previous belief into the new observation so that each agent receives a [previous-belief, observation] pair; however, because an agent has at most |Θi | beliefs we can partition these histories into at most |Θi | informationequivalent groups. [sent-93, score-1.316]

35 Each group corresponds to a distinct belief and instead of the [previous-belief, observation] pair we only need to provide the new belief’s label. [sent-94, score-0.327]

36 Third, recall that a belief is the sum of information that an agent uses to make decisions. [sent-97, score-0.572]

37 , by constructing a distributed policy together) then our modiﬁed state (which includes beliefs for each agent) fully determines the dynamics of the system. [sent-100, score-0.536]

38 Finally, its important to note that beliefs do not directly affect rewards or transitions. [sent-104, score-0.402]

39 We can therefore freely relabel and reorder beliefs without changing the decision problem. [sent-106, score-0.409]

40 This allows belief-observations in one timestep to use the same observation labels in the next time-step, even if the beliefs are different (in which case the distribution will be different). [sent-107, score-0.411]

41 We now formally deﬁne the belief-POMDP A , S , O , P , R , s (0) converted from BBDecPOMDP N, A, S, O, P, R, s(0) (with belief labels Θi for each agent). [sent-109, score-0.393]

42 The belief-POMDP has factored states ω, θ1 , · · · , θn ∈ S where ω ∈ S is the underlying state and θi ∈ Θi is n |Θi | agent i’s belief. [sent-110, score-0.354]

43 A = i=1 j=1 Ai is the set of actions (one action for each agent for each belief). [sent-112, score-0.433]

44 P (s |s, a), = [θ,o]=θ P (ω , o|ω, aθ1 , · · · , aθn ) (a sum over equivalent joint-beliefs) where aθi is the action agent i would take if holding belief θi . [sent-113, score-0.711]

45 (0) R (s, a) = R(ω, aθ1 , · · · , aθn ) and sω = s(0) is the initial state distribution with each agent having the same belief. [sent-114, score-0.28]

46 Actions in this belief-POMDP are pure strategies for each agent specifying what each agent should do for every belief they might have. [sent-115, score-0.849]

47 The action space thus has size i |Ai ||Θi | which is exponentially more actions than the number of joint-actions in the BB-DecPOMDP. [sent-117, score-0.188]

48 Both the transition and reward functions use the modiﬁed joint-action aθ1 , · · · , aθn which is the action that would be taken once agents see their beliefs and follow action a ∈ A. [sent-118, score-0.859]

49 This makes the single agent in the belief-POMDP act like a centralized mediator playing the sequence of Bayesian games induced by the BB-DecPOMDP. [sent-119, score-0.442]

50 At every time-step this centralized mediator must give a strategy to each agent (a solution to the current Bayesian game). [sent-120, score-0.421]

51 Given BB-DecPOMDP N, A, O, Θ, S, P, R, s(0) with policy π : ∆(S)×O → A and belief-POMDP A , S , O , P , R , s (0) as deﬁned above, with policy π : ∆(S ) → {O → A } then if π(s(t) )i,o = πi (s(t) , o), then Vπ (s(0) ) = Vπ (s (0) ) (the expected utility of the two policies are equal). [sent-126, score-0.401]

52 We can then use existing POMDP algorithms and replace action maximization with an integer linear program. [sent-133, score-0.209]

53 5 Optimal belief compression We present here a novel and optimal belief compression method that transforms any DecPOMDP into a BB-DecPOMDP. [sent-136, score-0.92]

54 The idea is to let agents themselves decide how they want to merge their beliefs and to add this decision directly into the problem’s structure. [sent-137, score-0.674]

55 This pushes the onus of belief compression onto the BB-DecPOMDP solver instead of an explicit approximation method. [sent-138, score-0.46]

56 We give agents the ability to optimally compress their own beliefs by interleaving each normal time-step 4 (where we fully expand each belief) with a compression time-step (where the agents must explicitly decide how to best merge beliefs). [sent-139, score-1.046]

57 We call these phases belief expansion and belief compression, respectively. [sent-140, score-0.654]

58 The ﬁrst phase acts like the original DecPOMDP without any belief compression: the observation given to each agent is its previous belief along with the DecPOMDP’s observation. [sent-141, score-0.985]

59 No information is lost during this phase; each observation for each agent-type (agents holding the same belief are the same type) results in a distinct belief. [sent-142, score-0.397]

60 This belief expansion occurs with the same transitions and rewards as the original DecPOMDP. [sent-143, score-0.36]

61 In this phase, each agent-type must choose its next belief but they only have a ﬁxed number of beliefs to choose from (the number of beliefs ti is a free parameter). [sent-146, score-1.089]

62 All agent-types that choose the same belief will be unable to distinguish themselves in the next time-step; the belief label in the next time-step will equal the action index they take in the belief compression phase. [sent-147, score-1.225]

63 This second phase can be seen as a purely mental phase and does not affect the environment beyond changes to beliefs although as a technical matter, we convert our discount factor to its square root to account for these new interleaved states. [sent-149, score-0.526]

64 The observation factor is ∅ at the expansion phase and the most recent observation, as given by the DecPOMDP, when starting the compression phase. [sent-154, score-0.219]

65 The probability of transitioning to a state where some agents have the empty set observation while others don’t is always zero. [sent-157, score-0.307]

66 The action set sizes may be different in the two phases, however we can easily get around this problem by mapping any action outside of the designated actions to an equivalent one inside the designated action. [sent-159, score-0.343]

67 Our value function representation is a standard convex and piecewise linear value-vector representation over the belief 5 simplex. [sent-166, score-0.35]

68 For a belief b ∈ R|S| its value as given by the value function Γ is VΓ (b) = maxα∈Γ α · b. [sent-171, score-0.373]

69 First, we sample common-knowledge beliefs and collect them into a belief set B. [sent-175, score-0.696]

70 This is done by taking random actions from a given starting belief and recording the resulting belief states. [sent-176, score-0.731]

71 We then start with a poor approximation of the value function and improve it over successive iterations by performing a one-step backup for each belief b ∈ B. [sent-177, score-0.407]

72 Each backup produces a policy which yields a valuevector to improve the value function. [sent-178, score-0.212]

73 PERSEUS improves standard PBVI during each iteration by skipping beliefs already improved by another backup. [sent-179, score-0.402]

74 In order to backup a particular belief point we must maximize the utility of a strategy x. [sent-182, score-0.519]

75 The utility is computed using the immediate reward combined with our value-function’s current estimate of a chosen continuation policy that has value vector α. [sent-183, score-0.43]

76 Thus, a resulting belief b will achieve estimated value s b (s)α(s). [sent-184, score-0.35]

77 The resulting belief b after taking action a from belief b is b (s ) = s b(s)P (s |s, a). [sent-185, score-0.765]

78 Every iteration then attempts to improve the value function at each belief in our belief set B. [sent-193, score-0.71]

79 A random common-knowledge belief b ∈ B is selected and we compute an improved policy for that belief by performing a one step backup. [sent-194, score-0.786]

80 This backup involves ﬁnding the best immediate strategy-proﬁle (an action for each observation of each agent) at belief b along with the best continuation policy from Γt . [sent-195, score-0.788]

81 We then compute the value of the resulting strategy + continuation policy (which is itself a policy) and insert this new α-vector into Γt+1 . [sent-196, score-0.289]

82 Any belief that is improved by α (including b) is removed from B. [sent-197, score-0.327]

83 We then select a new common-knowledge belief and iterate until every belief in B has been improved. [sent-198, score-0.654]

84 Therefore, at every iteration at least one of the beliefs must improve by at least Γ . [sent-202, score-0.402]

85 We ran the algorithm on all six benchmark problems using the dynamic belief compression approximation scheme to convert each of the DecPOMDP problems into BB-DecPOMDPs. [sent-214, score-0.519]

86 For each problem we converted them into a BB-DecPOMDP with one, two, three, four, and ﬁve dynamic beliefs (the value of ti ). [sent-215, score-0.482]

87 We sampled 3,000 belief points to a maximum depth of 36. [sent-218, score-0.327]

88 9 for our dynamic approximation (recall that an agent visits two states for every one of the original problem). [sent-221, score-0.278]

89 To compensate for this low discount factor in this domain, we sampled 30,000 beliefs to a depth of 360. [sent-224, score-0.407]

90 The ﬁnal value reported is the value of the computed decentralized policy on the original DecPOMDP run to a horizon which pushed the utility error below the reported precision. [sent-228, score-0.454]

91 This lack of improvement with extra beliefs is strong evidence that our BB-DecPOMDP approximation is quite powerful and that the policies found are near optimal. [sent-268, score-0.41]

92 It also suggests that these problems do not have terribly complicated optimal policies and new benchmark problems should be proposed that require a richer belief set. [sent-269, score-0.396]

93 A large state-space also requires a greater number of belief samples to adequately cover and represent the value-function; with more states it becomes increasingly likely that a random walk will fail to traverse a desirable region of the state-space. [sent-274, score-0.36]

94 This problem is not nearly as bad as it would be for a normal POMDP because much of the belief-space is unreachable and a belief-POMDP’s value function has a great deal of symmetry due to the label invariance of beliefs (a relabeling of beliefs will still have the same utility). [sent-275, score-0.761]

95 Second, we showed how a DecPOMDP with bounded belief can be converted into a POMDP. [sent-279, score-0.416]

96 Almost all methods bound the beliefs in some way (through belief compression or ﬁnite horizons), and viewing these problems as POMDPs with large action spaces could precipitate new approaches. [sent-280, score-0.94]

97 Third, we showed how to achieve optimal belief compression by allowing agents themselves to decide how best to merge beliefs. [sent-281, score-0.725]

98 In Proceedings of the 6th international joint conference on Autonomous agents and multiagent systems, page 220. [sent-372, score-0.292]

99 Periodic ﬁnite state controllers for efﬁcient pomdp and dec-pomdp planning. [sent-385, score-0.263]

100 In Proceedings of the fourth international joint conference on Autonomous agents and multiagent systems, pages 786–793. [sent-406, score-0.292]

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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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