Convergence analysis of Adaptive Biasing Potential methods for diffusion processes

TL;DR

This paper provides a rigorous mathematical analysis of Adaptive Biasing Potential methods for diffusion processes, demonstrating their consistency and efficiency in sampling invariant distributions through stochastic approximation techniques.

Contribution

It offers a detailed construction and proof of convergence for adaptive importance sampling algorithms applied to diffusions, extending stochastic approximation tools in this context.

Findings

Proves almost sure convergence of the sampling method.

Establishes the efficiency of the algorithms through qualitative and quantitative analysis.

Extends stochastic approximation techniques to self-interacting diffusions.

Abstract

This article is concerned with the mathematical analysis of a family of adaptive importance sampling algorithms applied to diffusion processes. These methods, referred to as Adaptive Biasing Potential methods, are designed to efficiently sample the invariant distribution of the diffusion process, thanks to the approximation of the associated free energy function (relative to a reaction coordinate). The bias which is introduced in the dynamics is computed adaptively; it depends on the past of the trajectory of the process through some time-averages. We give a detailed and general construction of such methods. We prove the consistency of the approach (almost sure convergence of well-chosen weighted empirical probability distribution). We justify the efficiency thanks to several qualitative and quantitative additional arguments. To prove these results , we revisit and extend tools from stochastic approximation applied to self-interacting diffusions, in an original context.