Optimizing the diffusion coefficient of overdamped Langevin dynamics

TL;DR

This work introduces a principled framework to accelerate sampling from high-dimensional Gibbs measures by optimizing the diffusion matrix in overdamped Langevin dynamics. By formulating the problem as a convex optimization of the spectral gap with an $L^p_V$ normalization, the authors derive well-posedness, Euler–Lagrange conditions, and a smooth-min approximation to characterize optimal diffusions, including a sharp 1D homogenized limit where $D_{hom}(q)=e^{V(q)}$. They demonstrate that the optimal diffusion concentrates diffusion where the target is diffuse and slows diffusion in high-density regions, yielding faster convergence as reflected in larger spectral gaps and improved Monte Carlo performance. The paper also develops homogenization theory to connect periodic microstructures to an effective diffusion, proving commutation between optimization and homogenization and providing explicit 1D results; numerical experiments validate the theory and show practical gains for RWMH-based sampling. Overall, the work provides both theoretical foundations and practical algorithms for diffusion-based preconditioning to enhance sampling efficiency in Bayesian and statistical-physics contexts.

Abstract

Overdamped Langevin dynamics are reversible stochastic differential equations which are commonly used to sample probability measures in high-dimensional spaces, such as the ones appearing in computational statistical physics and Bayesian inference. By varying the diffusion coefficient, there are in fact infinitely many overdamped Langevin dynamics which are reversible with respect to 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 here, obtaining in particular necessary conditions on the optimal diffusion coefficient. We also derive an explicit expression of the optimal diffusion in some appropriate 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.

Figures (12)

• Figure 1: Results of the optimization procedure on the one-dimensional four-well potential $V(q) = \cos(8\pi q)$.
• Figure 2: Results of the optimization procedure on the one-dimensional one-well potential $V(q) = \cos(2\pi q)$.
• Figure 3: Results of the optimization procedure on the one-dimensional double-well potential $V(q) = \cos(4\pi q)$.
• Figure 4: Results of the optimization procedure on the one-dimensional double-well potential $V(q) = \sin(4\pi q) (2 + \sin(2\pi q))$.
• Figure 5: Results of the optimization procedure for various lower bounds $a$ and $V(q)=\sin(4\pi q)(2+\sin(2\pi q))$.

Theorems & Definitions (62)

• Remark 1: Degenerate diffusion matrices
• Remark 2: A numerical motivation for normalizing the diffusion matrix
• Remark 3: On the monotonicity assumption of $|\cdot|_{\mathrm{F}}$
• Remark 4: On the choice $p=+\infty$ in \ref{['eq:normLp']}
• Remark 5: On the choice of the weight in \ref{['eq:norm_L^p_mu_matrices']}
• Lemma 1
• proof
• Remark 6: On the importance of considering $a>0$ and/or $b>0$
• Proposition 2
• proof