Improved Guarantees for Langevin Monte Carlo with Average Smoothness

2026-05-29

AI Summary

This work addresses the looseness of existing non-asymptotic error bounds for Langevin Monte Carlo methods under strongly log-concave distributions, which overly rely on global smoothness constants and consequently deteriorate in high-dimensional or correlated-covariate settings. To overcome this limitation, the paper introduces a coordinate-wise averaged smoothness condition to characterize the potential function and combines synchronous coupling with Wasserstein distance analysis to derive substantially tighter bounds. The key innovation lies in replacing the global smoothness constant with an average coordinate-wise counterpart and employing a trace-type third-order smoothness quantity to weaken the Hessian-Lipschitz assumption. These improvements are extended to variable step sizes, Laplacian-smooth potentials, and finite-sum structures such as SGLD. Notably, the resulting bounds exhibit significantly improved dimension dependence in high-dimensional generalized linear models, especially under covariate correlation, offering broader theoretical applicability and outperforming current state-of-the-art results.

Abstract

We establish improved nonasymptotic bounds for Langevin Monte Carlo in the strongly log-concave setting, when the error is measured by the Wasserstein distance. The main result shows that the discretization error is governed by an average coordinate-wise smoothness constant, rather than by the usual global smoothness constant. The proof is short and probabilistic, and relies on a refined use of the synchronous coupling. We further show that the same ideas lead to improved bounds for variable step sizes, for potentials whose Laplacian is Lipschitz-continuous, and for finite-sum problems sampled by stochastic-gradient Langevin dynamics with fixed point control variates. In the Laplacian-smooth case, the usual Hessian-Lipschitz contribution is replaced by a weaker trace-type third-order smoothness quantity. In the finite-sum setting, the resulting SGLD bound improves the dependence on the root mean square smoothness of the component functions. Applications to generalized linear models with Gaussian design show that these refinements can yield substantial, dimension-dependent improvements over previously known bounds, especially for correlated covariates.

Problem

• Langevin Monte Carlo
• nonasymptotic bounds
• average smoothness
• Wasserstein distance
• strongly log-concave

Innovation

• Langevin Monte Carlo
• average smoothness
• Wasserstein distance
• synchronous coupling
• stochastic gradient Langevin dynamics