Characterizing and improving generalized belief propagation algorithms on the 2D Edwards–Anderson model

We study the performance of different message passing algorithms in the two-dimensional Edwards–Anderson model. We show that the standard belief propagation (BP) algorithm converges only at high temperature to a paramagnetic solution. Then, we test a generalized belief propagation (GBP) algorithm, derived from a cluster variational method (CVM) at the plaquette level. We compare its performance with BP and with other algorithms derived under the same approximation: double loop (DL) and a two-way message passing algorithm (HAK). The plaquette-CVM approximation improves BP in at least three ways: the quality of the paramagnetic solution at high temperatures, a better estimate (lower) for the critical temperature, and the fact that the GBP message passing algorithm converges also to nonparamagnetic solutions. The lack of convergence of the standard GBP message passing algorithm at low temperatures seems to be related to the implementation details and not to the appearance of long range order. In fact, we prove that a gauge invariance of the constrained CVM free energy can be exploited to derive a new message passing algorithm which converges at even lower temperatures. In all its region of convergence this new algorithm is faster than HAK and DL by some orders of magnitude