Variational inference via decomposable transports : algorithms for Bayesian filtering and smoothing

We describe a variational inference method that approximates an intractable target measure as the pushforward of a tractable distribution (e.g., a Gaussian) through a transport map. We then show how such transport maps can be decomposed—i.e., factorized into the composition of finitely many low-dimensional maps. We use the notion of decomposable transports to derive new deterministic online algorithms for Bayesian filtering and smoothing in nonlinear/non-Gaussian state-space models with static parameters, and illustrate the theory on a stochastic volatility model. 1 Measure transport and variational inference Let Z be a random variable on R endowed with an intractable continuous density π that we wish to simulate. We assume that π is available only up to a normalizing constant. For instance, π may represent the posterior density of a Bayesian inference problem, where the goal is to approximate integrals of the form ∫ g(z) · π(z) dz for some measurable g : R → R. One possible approach to the problem of sampling is to seek a deterministic (transport) map T : R → R that couples a tractable reference random variable X of density η (e.g., a standard normal) with Z [15]. The coupling ensures that T (X) = Z in distribution [30], or, equivalently, that T pushes forward η to π, i.e., T]η = π, where T]η denotes the pushforward density of η by T . (For any invertible map T , we have T]η(z) = η(T−1(z)) · |det∇T−1(z)|, where ∇T (z)−1 ∈ Rn×n denotes the gradient of the inverse map at z. ) Thus, if X1, . ,Xn is an independent and identically distributed (iid) sample from η, then T (X1), . , T (Xn) is an iid sample from π. In other words, T enables the generation of cheap, independent, and unweighted samples from π by pushing forward a collection of reference samples through the map. Clearly, a transport map between a tractable density η and the target π turns π into a tractable distribution and solves, at least formally, the problem of sampling. A transport map between random variables on R exists under very weak conditions. For instance, in the example above it suffices that the law of X vanishes on subsets of (Hausdorff) dimension n − 1 [16]. As shown in [18], the transport map can be computed via deterministic optimization by minimizing the Kullback–Leibler (KL) divergence DKL(T]η ||π ) over a suitable function space for the map, i.e., for T ∈ T . At optimality, we have T]η = π. In practice, we need to represent the transport. The approach adopted in [18, 20] seeks a parametric transport map within a finite dimensional approximation space, Th ⊂ T . The resulting variational problem reads as: min T∈Th DKL(T]η ||π ). (1.1) We can interpret (1.1) as seeking a density q that minimizes DKL( q ||π ) over a finite dimensional class of tractable distributions, Q, which consists of distributions q = T]η that can be written as the pushforward of η by a map in Th. Thus, (1.1) defines a particular variational inference method [6, 31, 2]: one that uses measure transport to characterize the class of approximating distributions Q (see [29, 22] for related methods). The richer the function space for the map, the richer Q. In 30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain. particular, a general parameterization of the map can capture arbitrary probabilistic interactions [18], well beyond the usual mean-field approximation [19, 21, 27]. A key feature of this approach is that it produces a transport map T and not just an approximation, T]η, to the target density. This idea becomes very useful when T is only approximate. In this case, if the bias of approximating π with T]η is unacceptable, one can simply evaluate (possibly up to a normalizing constant) the pullback density T π, defined as T π(x) = π(T (x)) · | det∇T (x)|, and rewrite the integral ∫ g(z) · π(z) dz as ∫ g(T (x)) · T π(x) dx. One possibility is then to use a stochastic sampling technique, like MCMC [23], to probe T π, which, by virtue of (1.1), will be closer (in KL divergence) to the reference density η. In particular, if η is Gaussian, then we could interpret pulling back π by T as a “Gaussianization” of the target [13], which can remove the correlations that make sampling a challenging task. Thus, we can regard an approximate T as a preconditioner for existing sampling schemes [20, 17, 32]. There are infinitely many transports that push forward one density to another [30]. An important transport for our analysis is the Knothe-Rosenblatt (KR) rearrangement in R [24, 10]. For a pair of continuous densities, η and π, the KR rearrangement is the unique monotone increasing (lower) triangular transport that pushes forward η to π [3]. A lower triangular transport T : R → R is a multivariate map whose kth component depends only on the first k input variables, i.e., T (x) =  T (x1) T (x1, x2) . T(x1, x2, . . . xn)  ∀x = (x1, . . . , xn) ∈ R, (1.2) where T k denotes the kth output of the map. The KR rearrangement is general—it can couple arbitrary continuous densities—and enjoys many attractive computational features. As shown in [18, 15], it can be characterized as the unique minimizer of DKL(T]η ||π ) over the cone T4 of triangular maps that are monotone increasing with respect to the lexicographic order on R. In this case, (1.1) is equivalent to: min −Eη[log π̄(T (X)) + ∑ k log ∂xkT (X)− log η(X)] (1.3) s.t. T ∈ T4,h ⊂ T4; dim(T4,h) <∞ where π̄ denotes the unnormalized target density. In particular, by using monotone parameterizations for T , we can regard (1.3) as an unconstrained stochastic program [15, 9, 12, 28]. 2 Decomposable transports The key observation of this work is that a transport map is not just any multivariate function on R. There exist transports which inherit low-dimensional parameterizations from the Markov structure [14, 11] of the underlying target density. By considering recursive graph decompositions of a (noncomplete) Markov network for π, it is possible to prove the existence of transports T that factorize exactly as the composition of k low-dimensional maps, T = T1 ◦ · · · ◦ Tk, for some finite k, where each map Tj differs from the identity function only along few components and is triangular up to a permutation of the input and output space. We call such transports decomposable. Clearly, a decomposable transport is easier to parameterize than a regular one. Moreover, the decomposition T = T1 ◦ · · · ◦Tk suggests that the computation of T may be broken into multiple simpler steps, each associated with the computation of a low-dimensional map Tj that accounts only for local features of π. Instead of detailing the general theory of decomposable transports, due to the length constraints of this manuscript we will explore this theory in the context of sequential Bayesian inference for state-space models (see Section 3). Our analysis, in this setting, will suggest new and powerful variational algorithms for the Bayesian filtering and smoothing problems. 3 Online algorithms for Bayesian filtering and smoothing We consider the problem of sequential Bayesian inference for a discrete time, continuous, nonlinear, non-Gaussian state-space model [4] in a very general formulation that includes hyperparameters (i.e.,