Geometrical view on mean-field approximation for solving optimization problems

When one wishes to solve optimization problems by simulated annealing, the naive mean-field approximation provides a practical way of doing it. Extensions of the naive approximation by including higher-order terms have been proposed in the prospect of improving accuracy of the approximation. It has been reported, however, that higher-order approximations do not work well, especially in low temperature regions. We present an analytical argument and a geometrical view on this contradictory observation based on information-geometry and give an intuitive explanation as to why the naive approximation does work well when it is applied to solving optimization problems.