Stochastic gradient descent and fast relaxation to thermodynamic equilibrium: a stochastic control approach

TL;DR

This paper investigates how controlled Langevin dynamics converge to equilibrium, revealing limits on accelerating convergence through temperature separation, with implications for stochastic optimization and molecular sampling.

Contribution

It introduces a stochastic control framework for Langevin equations, analyzing convergence behavior under different temperature regimes and limits.

Findings

Controlled dynamics converge to overdamped Langevin or gradient flow depending on temperature limits.

Ergodic limit and large temperature separation limit do not commute.

Increasing temperature separation does not speed up convergence to equilibrium.

Abstract

We study the convergence to equilibrium of an underdamped Langevin equation that is controlled by a linear feedback force. Specifically, we are interested in sampling the possibly multimodal invariant probability distribution of a Langevin system at small noise (or low temperature), for which the dynamics can easily get trapped inside metastable subsets of the phase space. We follow [Chen et al., J. Math. Phys. 56, 113302, 2015] and consider a Langevin equation that is simulated at a high temperature, with the control playing the role of a friction that balances the additional noise so as to restore the original invariant measure at a lower temperature. We discuss different limits as the temperature ratio goes to infinity and prove convergence to a limit dynamics. It turns out that, depending on whether the lower ("target") or the higher ("simulation") temperature is fixed, the controlled dynamics converges either to the overdamped Langevin equation or to a deterministic gradient flow. This implies that (a) the ergodic limit and the large temperature separation limit do not commute in general, and that (b) it is not possible to accelerate the speed of convergence to the ergodic limit by making the temperature separation larger and larger. We discuss the implications of these observation from the perspective of stochastic optimisation algorithms and enhanced sampling schemes in molecular dynamics.