Adaptive Thermostats for Noisy Gradient Systems

TL;DR

This paper introduces an adaptive Langevin thermostat method that significantly improves numerical efficiency in high-dimensional stochastic systems, with proven superconvergence properties and applications to machine learning and molecular dynamics.

Contribution

A new adaptive Langevin thermostat method with enhanced efficiency and superconvergence properties for high-dimensional stochastic gradient systems.

Findings

Achieves fourth-order convergence to the invariant measure.

Demonstrates superior numerical efficiency over existing methods.

Validated through numerical experiments.

Abstract

We study numerical methods for sampling probability measures in high dimension where the underlying model is only approximately identified with a gradient system. Extended stochastic dynamical methods are discussed which have application to multiscale models, nonequilibrium molecular dynamics, and Bayesian sampling techniques arising in emerging machine learning applications. In addition to providing a more comprehensive discussion of the foundations of these methods, we propose a new numerical method for the adaptive Langevin/stochastic gradient Nos\'{e}--Hoover thermostat that achieves a dramatic improvement in numerical efficiency over the most popular stochastic gradient methods reported in the literature. We also demonstrate that the newly established method inherits a superconvergence property (fourth order convergence to the invariant measure for configurational quantities) recently demonstrated in the setting of Langevin dynamics. Our findings are verified by numerical experiments.