The Wang-Landau Algorithm as Stochastic Optimization and Its Acceleration
This work provides incremental improvements to the Wang-Landau algorithm, a method used in statistical physics for sampling complex systems, by applying standard optimization techniques to enhance its performance.
The paper reformulates the Wang-Landau algorithm as a stochastic gradient descent optimization problem, enabling a new convergence rate analysis and accelerating it with momentum and adaptive learning rate methods, demonstrating improved efficiency on two-dimensional Ising and Potts models.
We show that the Wang-Landau algorithm can be formulated as a stochastic gradient descent algorithm minimizing a smooth and convex objective function, of which the gradient is estimated using Markov chain Monte Carlo iterations. The optimization formulation provides us a new way to establish the convergence rate of the Wang-Landau algorithm, by exploiting the fact that almost surely, the density estimates (on the logarithmic scale) remain in a compact set, upon which the objective function is strongly convex. The optimization viewpoint motivates us to improve the efficiency of the Wang-Landau algorithm using popular tools including the momentum method and the adaptive learning rate method. We demonstrate the accelerated Wang-Landau algorithm on a two-dimensional Ising model and a two-dimensional ten-state Potts model.