A Generalized Wang–Landau Algorithm for Monte Carlo Computation

Inference for a complex system with a rough energy landscape is a central topic in Monte Carlo computation. Motivated by the successes of the Wang–Landau algorithm in discrete systems, we generalize the algorithm to continuous systems. The generalized algorithm has some features that conventional Monte Carlo algorithms do not have. First, it provides a new method for Monte Carlo integration based on stochastic approximation; second, it is an excellent tool for Monte Carlo optimization. In an appropriate setting, the algorithm can lead to a random walk in the energy space, and thus it can sample relevant parts of the sample space, even in the presence of many local energy minima. The generalized algorithm can be conveniently used in many problems of Monte Carlo integration and optimization, for example, normalizing constant estimation, model selection, highest posterior density interval construction, and function optimization. Our numerical results show that the algorithm outperforms simulated annealing and parallel tempering in optimization for the system with a rough energy landscape. Some theoretical results on the convergence of the algorithm are provided.