Stochastic Belief Propagation: Low-Complexity Message-Passing with Guarantees

The sum-product or belief propagation (BP) algorithm is a widely-used message-passing algorithm for computing marginal distributions in graphical models with discrete variables. At the core of the BP updates, when applied to a graphical model with pairwise interactions, lies a matrix-vector product with complexity that is quadratic in the state dimension $d$, and requires transmission of a $(d-1)$-dimensional vector of real numbers (messages) to its neighbors. Since various applications involve very large state dimensions, such computation and communication complexities can be prohibitively complex. In this paper, we propose a low-complexity variant of belief propagation, referred to as stochastic belief propagation (SBP). As suggested by the name, it is an adaptively randomized version of the BP updates in which each node passes randomly chosen information to each of its neighbors. The SBP updates reduce the computational complexity (per iteration) from quadratic to linear in $d$, without assuming any particular structure of the potentials, and also reduce the communication complexity significantly, requiring only $\log d$ bits transmission per edge. Moreover, we establish a number of theoretical guarantees for the performance of SBP, showing that it converges almost surely to the BP fixed point for any tree-structured graph, and for graphical models with cycles satisfying a contractivity condition. In addition, we provide non-asymptotic upper bounds on the convergence rate, showing that it decays no slower than $O(1/\sqrt{t})$ with the number of iterations $t$ on trees and as $O(1/t)$ for general graphs.