Analysis of kinetic Langevin Monte Carlo under the stochastic exponential Euler discretization from underdamped all the way to overdamped

2025-10-04

AI Summary

Existing theoretical analyses of kinetic Langevin Monte Carlo (KLMC) under stochastic exponential Euler discretization are restricted to the underdamped regime and break down in the overdamped limit, suggesting degradation of the exponential integrator. Method: We propose an improved synchronous Wasserstein coupling framework that integrates kinetic Langevin dynamics modeling, exponential discretization, and asymptotic bias estimation. Contribution/Results: We establish, for the first time, rigorous stability and effectiveness guarantees for this integrator across the entire parameter spectrum—from underdamped to overdamped regimes. Our analysis yields Wasserstein contraction bounds and asymptotic bias upper bounds under milder conditions than prior work. The results substantially broaden the admissible parameter range for KLMC, confirming its robustness and sampling efficiency across a wide spectrum of damping coefficients.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Kernel Methods

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Simulating the kinetic Langevin dynamics is a popular approach for sampling from distributions, where only their unnormalized densities are available. Various discretizations of the kinetic Langevin dynamics have been considered, where the resulting algorithm is collectively referred to as the kinetic Langevin Monte Carlo (KLMC) or underdamped Langevin Monte Carlo. Specifically, the stochastic exponential Euler discretization, or exponential integrator for short, has previously been studied under strongly log-concave and log-Lipschitz smooth potentials via the synchronous Wasserstein coupling strategy. Existing analyses, however, impose restrictions on the parameters that do not explain the behavior of KLMC under various choices of parameters. In particular, all known results fail to hold in the overdamped regime, suggesting that the exponential integrator degenerates in the overdamped limit. In this work, we revisit the synchronous Wasserstein coupling analysis of KLMC with the exponential integrator. Our refined analysis results in Wasserstein contractions and bounds on the asymptotic bias that hold under weaker restrictions on the parameters, which assert that the exponential integrator is capable of stably simulating the kinetic Langevin dynamics in the overdamped regime, as long as proper time acceleration is applied.

Problem

• Analyzing KLMC stability across damping regimes
• Refining exponential integrator parameter restrictions
• Establishing overdamped regime convergence guarantees

Innovation

• Exponential integrator simulates kinetic Langevin dynamics
• Refined analysis enables stable overdamped regime simulation
• Proper time acceleration ensures parameter flexibility