Zero-Temperature Limit of a Convergent Algorithm to Minimize the Bethe Free Energy

Abstract

After the discovery that xed points of loopy belief propagation coincide with stationary points of the Bethe free energy, several researchers proposed provably convergent algorithms to directly minimize the Bethe free energy. These algorithms were formulated only for non-zero temperature (thus nding xed points of the sum-product algorithm) and their possible extension to zero temperature is not obvious. We present the zero-temperature limit of the double-loop algorithm by Heskes, which converges a max-product xed point. The inner loop of this algorithm is max-sum di usion. Under certain conditions, the algorithm combines the complementary advantages of the max-product belief propagation and max-sum di usion (LP relaxation): it yields good approximation of both ground states and max-marginals.