The Wang-Landau algorithm reaches the flat histogram criterion in finite time

TL;DR

This paper proves that certain variations of the Wang-Landau algorithm can reach the flat histogram criterion in finite time, while others may never do so, clarifying its convergence properties in specific settings.

Contribution

It establishes finite-time convergence results for some versions of the Wang-Landau algorithm in simple, bounded contexts, advancing understanding of its theoretical behavior.

Findings

Some variations reach the flat histogram criterion in finite time.

Other variations may never reach the flat histogram criterion.

Results are demonstrated in simple, bounded space settings.

Abstract

The Wang-Landau algorithm aims at sampling from a probability distribution, while penalizing some regions of the state space and favoring others. It is widely used, but its convergence properties are still unknown. We show that for some variations of the algorithm, the Wang-Landau algorithm reaches the so-called flat histogram criterion in finite time, and that this criterion can be never reached for other variations. The arguments are shown in a simple context - compact spaces, density functions bounded from both sides - for the sake of clarity, and could be extended to more general contexts.