# The Wang-Landau Algorithm as Stochastic Optimization and Its Acceleration

## TL;DR

This paper reformulates the Wang-Landau algorithm as a stochastic gradient descent method, providing new insights into its convergence and enabling acceleration techniques like momentum and adaptive learning rates, demonstrated on Ising and Potts models.

## Contribution

It introduces an optimization perspective to the Wang-Landau algorithm, establishing convergence rates and proposing accelerated variants using modern optimization tools.

## Key findings

- The algorithm can be viewed as a stochastic gradient descent minimizing a convex function.
- Acceleration techniques improve convergence speed in practical models.
- The accelerated algorithm performs well on Ising and Potts models.

## Abstract

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.