Stochastic belief propagation: Low-complexity message-passing with guarantees

The sum-product or belief propagation (BP) algorithm is widely used to compute exact or approximate marginals in graphical models. However, for graphical models with continuous or high-dimensional discrete states and/or high degree factors, it can be computationally expensive to update messages. We propose the stochastic belief propagation algorithm (SBP) as a low-complexity alternative. It is a randomized variant of BP that passes only stochastically chosen information at each round, thereby reducing the complexity per iteration by an order of magnitude. We prove that it enjoys a number of rigorous convergence guarantees: for any tree-structured graph, the SBP updates converge almost surely to the BP fixed point, and we provide non-asymptotic bounds on the mean absolute error. For general graphs that satisfy a standard contraction condition, we establish almost sure convergence to the unique BP fixed point, as well as non-asymptotic guarantees on the mean squared error, showing that it decays as 1/t with the number of iterations t. We also provide high probability bounds on the actual error.