Optimizing the diffusion of overdamped Langevin dynamics

Santet Régis, CERMICS, École des Ponts & MATHERIALS, Inria Paris

Overdamped Langevin dynamics are reversible stochastic differential equations (SDEs) which are commonly used to sample probability measures in high dimensional spaces, such as the ones appearing in computational statistical physics and Bayesian inference. These SDEs depend on a diffusion coefficient, which inverse can be seen as a Riemannian metric. By varying this diffusion coefficient, there are in fact infinitely many reversible overdamped Langevin dynamics which preserve the target probability measure at hand. This suggests to optimize the diffusion coefficient in order to increase the convergence rate of the dynamics, as measured by the spectral gap of the generator associated with the stochastic differential equation. We analytically study this problem, obtaining in particular necessary conditions on the optimal diffusion coefficient. We also derive an explicit expression of the optimal diffusion in some homogenized limit. Numerical results, both relying on discretizations of the spectral gap problem and Monte Carlo simulations of the stochastic dynamics, demonstrate the increased quality of the sampling arising from an appropriate choice of the diffusion coefficient. In particular, when using an Euler--Maruyama discretization of these SDEs, we show how the use of the optimal diffusion coefficient can alleviate many problems encountered when sampling, e.g. metastability/multimodality, in contrast to using a constant diffusion coefficient, which correspond to the widely used Unadjusted (respectively Metropolis-Adjusted) Langevin Algorithm (commonly referenced as ULA, respectively MALA). This is joint work with T.Lelièvre, G.Pavliotis, G.Robin and G.Stoltz.

Area: CS36 - Monte Carlo methods and Applications II (Francesca R Crucinio, Alessandra Iacobucci, Andrea Bertazzi)

Keywords: "MCMC" "Spectral gap" "Sampling" "Langevin"

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