Sharp dimensional analysis of midpoint methods for Langevin sampling

2026-10-05

AI Summary

This study addresses the efficiency bottleneck of Langevin sampling in high-dimensional spaces, where dimension complexity requires urgent optimization. To this end, it analyzes the performance of deterministic midpoint discretization within Langevin dynamics. By integrating Malliavin calculus, stationary cancellation, and smoothing techniques, the authors derive tight upper and lower error bounds. The primary contribution is the first theoretical proof that deterministic discretization can outperform its stochastic counterparts in high dimensions. Furthermore, this approach significantly reduces gradient query complexity; notably, Heun’s method achieves a complexity of O(κ^{4/3}d^{1/3}ε^{-2/3}), establishing a new benchmark for high-dimensional sampling.

Abstract

We study deterministic and randomized midpoint discretizations of Langevin dynamics for a target $\pi \propto e^{-V}$, where $0 \prec \alpha I\preceq\nabla^2V\preceq\beta I$ and $\kappa=\beta/\alpha$. To achieve $\sqrt\alpha\,W_2\leqslant\varepsilon$, we show that deterministic Heun uses at most $\widetilde O(\kappa^{4/3}d^{1/3}\varepsilon^{-2/3})$ gradient queries, and underdamped exponential midpoint uses $\widetilde O(\kappa^{5/4}d^{1/4}\varepsilon^{-1/2})$. The proofs exploit cancellation at stationarity and smoothing using techniques from Malliavin calculus, outperforming previous upper bounds based on standard couplings. At bounded condition number, a lower bound matches the $d$ and $\varepsilon$ powers of both deterministic methods. To contrast, for the randomized midpoint methods and Poisson midpoint with at least two grid points (both overdamped and underdamped variants), a simple Gaussian calculation yields a lower bound $d^{1/3}\varepsilon^{-1/3}$ to get an $\varepsilon$-close sample despite starting at a benign initialization. This shows surprisingly that in high dimensions, deterministic discretizations can outperform their random counterparts.

Problem

• Langevin sampling
• midpoint discretization
• dimensional analysis
• gradient complexity
• Wasserstein distance

Innovation

• Langevin sampling
• midpoint discretization
• Malliavin calculus
• dimensional analysis
• gradient complexity