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We study the performance of different message passing algorithms in the twodimensional Edwards Anderson model. We show that the standard BeliefPropagation (BP) algorithm converges only at high temperature to a paramagneticsolution. Then, we test a Generalized Belief Propagation (GBP) algorithm,derived from a Cluster Variational Method (CVM) at the plaquette level. Wecompare its performance with BP and with other algorithms derived under thesame approximation: Double Loop (DL) and a two-ways message passing algorithm(HAK). The plaquette-CVM approximation improves BP in at least three ways: thequality of the paramagnetic solution at high temperatures, a better estimate(lower) for the critical temperature, and the fact that the GBP message passingalgorithm converges also to non paramagnetic solutions. The lack of convergenceof the standard GBP message passing algorithm at low temperatures seems to berelated to the implementation details and not to the appearance of long rangeorder. In fact, we prove that a gauge invariance of the constrained CVM freeenergy can be exploited to derive a new message passing algorithm whichconverges at even lower temperatures. In all its region of convergence this newalgorithm is faster than HAK and DL by some orders of magnitude.

Characterizing and Improving Generalized Belief Propagation Algorithms on the 2D Edwards-Anderson Model