Norm-Product Belief Propagation: Primal-Dual Message-Passing for Approximate Inference

Abstract

Inference problems in graphical models can be represented as a constrained optimization of a free-energy function. In this paper, we treat both forms of probabilistic inference, estimating marginal probabilities of the joint distribution and finding the most probable assignment, through a unified message-passing algorithm architecture. In particular we generalize the belief propagation (BP) algorithms of sum-product and max-product and tree-reweighted (TRW) sum and max product algorithms (TRBP) and introduce a new set of convergent algorithms based on “convex-free-energy” and linear-programming (LP) relaxation as a zero-temperature of a convex-free-energy. The main idea of this work arises from taking a general perspective on the existing BP and TRBP algorithms while observing that they all are reductions from the basic optimization formula of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> +Σ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ihi</i> where the function <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> is an extended-valued, strictly convex but nonsmooth and the functions <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hi</i> are extended-valued functions (not necessarily convex). We use tools from convex duality to present the “primal-dual ascent” algorithm which is an extension of the Bregman successive projection scheme and is designed to handle optimization of the general type <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> + Σ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ihi</i> . We then map the fractional-free-energy variational principle for approximate inference onto the optimization formula above and introduce the “norm-product” message-passing algorithm. Special cases of the norm-product include sum-product and max-product (BP algorithms), TRBP and NMPLP algorithms. When the fractional-free-energy is set to be convex (convex-free-energy) the norm-product is globally convergent for the estimation of marginal probabilities and for approximating the LP-relaxation. We also introduce another branch of the norm-product which arises as the “zero-temperature” of the convex-free-energy which we refer to as the “convex-max-product”. The convex-max-product is convergent (unlike max-product) and aims at solving the LP- relaxation.