Non-asymptotic mixing of the MALA algorithm

TL;DR

This paper analyzes the convergence properties of the MALA algorithm for SDEs with nonglobally Lipschitz drifts, showing it converges exponentially fast up to exponentially small errors in finite time.

Contribution

It provides a non-asymptotic analysis of MALA's mixing behavior, reconciling its ergodicity with the lack of a spectral gap in certain cases.

Findings

MALA converges exponentially fast to the invariant measure up to small errors.

The convergence rate is exponential up to terms exponentially small in stepsize.

MALA accurately approximates the SDE's transition probabilities on finite intervals.

Abstract

The Metropolis-Adjusted Langevin Algorithm (MALA), originally introduced to sample exactly the invariant measure of certain stochastic differential equations (SDE) on infinitely long time intervals, can also be used to approximate pathwise the solution of these SDEs on finite time intervals. However, when applied to an SDE with a nonglobally Lipschitz drift coefficient, the algorithm may not have a spectral gap even when the SDE does. This paper reconciles MALA's lack of a spectral gap with its ergodicity to the invariant measure of the SDE and finite time accuracy. In particular, the paper shows that its convergence to equilibrium happens at exponential rate up to terms exponentially small in time-stepsize. This quantification relies on MALA's ability to exactly preserve the SDE's invariant measure and accurately represent the SDE's transition probability on finite time intervals.