Almost sure convergence of the accelerated weight histogram algorithm

TL;DR

This paper proves the almost sure convergence of the accelerated weight histogram (AWH) algorithm, an iterative method used in statistical physics and biology for estimating free energy differences and expectations, by modeling it as a stochastic approximation.

Contribution

It provides the first rigorous proof of convergence for the AWH algorithm, linking it to stochastic approximation theory and analyzing its limit differential equation.

Findings

Proves almost sure convergence of AWH algorithm.

Establishes AWH as a stochastic approximation method.

Analyzes the associated limit ordinary differential equation.

Abstract

The accelerated weight histogram (AWH) algorithm is an iterative extended ensemble algorithm, developed for statistical physics and computational biology applications. It is used to estimate free energy differences and expectations with respect to Gibbs measures. The AWH algorithm is based on iterative updates of a design parameter, which is closely related to the free energy, obtained by matching a weight histogram with a specified target distribution. The weight histogram is constructed from samples of a Markov chain on the product of the state space and parameter space. In this paper almost sure convergence of the AWH algorithm is proved, for estimating free energy differences as well as estimating expectations with adaptive ergodic averages. The proof is based on identifying the AWH algorithm as a stochastic approximation and studying the properties of the associated limit ordinary differential equation.