Statistical Error of Numerical Integrators for Underdamped Langevin Dynamics with Deterministic And Stochastic Gradients
Xuda Ye, Zhennan Zhou
TL;DR
This work develops a discrete Poisson equation framework to quantify the statistical error of numerical integrators for underdamped Langevin dynamics under both deterministic and stochastic gradients. By constructing a discrete Poisson solution $\phi_h$ from the Kolmogorov equation, the authors derive explicit mean-square error rates that depend on the integrator's strong order $p$ and the convexity properties of the potential $U(x)$, showing $\mathbb{E}[e^2(N,h)] = O\big(h^{2p-1}+\frac{1}{Nh}\big)$ when $U$ is strongly convex outside a ball and $O\big(h^{2p}+\frac{1}{Nh}\big)$ when $U$ is globally strongly convex. In the stochastic-gradient setting, with $U(x)$ globally convex, the error scales as $O\big(h^{\min\{2p,2\}}+\frac{1}{Nh}\big)$, and specifically $O(h^2+1/(Nh))$ for common schemes such as SG-EM and SG-UBU. The approach relies only on geometric ergodicity of the continuous process and yields relaxed time-step constraints, with numerical experiments validating the predicted rates. This provides a practical, broadly applicable toolkit for assessing and tuning integrators in Langevin-based sampling and optimization contexts.
Abstract
We propose a novel discrete Poisson equation approach to estimate the statistical error of a broad class of numerical integrators for the underdamped Langevin dynamics. The statistical error refers to the mean square error of the estimator to the exact ensemble average with a finite number of iterations. With the proposed error analysis framework, we show that when the potential function $U(x)$ is strongly convex in $\mathbb R^d$ and the numerical integrator has strong order $p$, the statistical error is $O(h^{2p}+\frac1{Nh})$, where $h$ is the time step and $N$ is the number of iterations. Besides, this approach can be adopted to analyze integrators with stochastic gradients, and quantitative estimates can be derived as well. Our approach only requires the geometric ergodicity of the continuous-time underdamped Langevin dynamics, and relaxes the constraint on the time step.