Performance Limitations of Flat Histogram Methods and Optimality of Wang-Landau Sampling

TL;DR

This paper analyzes the scaling behavior of flat-histogram methods, revealing that Wang-Landau sampling is optimal and that tunneling times scale differently across models, with some exhibiting exponential growth.

Contribution

It demonstrates the optimality of Wang-Landau sampling and characterizes the scaling of tunneling times in various Ising models using a perfect flat-histogram scheme.

Findings

Tunneling time scaling varies with model type.

Wang-Landau sampling matches the optimal scheme.

Exponential scaling observed in spin glass models.

Abstract

We determine the optimal scaling of local-update flat-histogram methods with system size by using a perfect flat-histogram scheme based on the exact density of states of 2D Ising models.The typical tunneling time needed to sample the entire bandwidth does not scale with the number of spins N as the minimal N^2 of an unbiased random walk in energy space. While the scaling is power law for the ferromagnetic and fully frustrated Ising model, for the +/- J nearest-neighbor spin glass the distribution of tunneling times is governed by a fat-tailed Frechet extremal value distribution that obeys exponential scaling. We find that the Wang-Landau algorithm shows the same scaling as the perfect scheme and is thus optimal.