Near-Linear Accuracy Bounds for Moreau--Yosida Unadjusted Langevin Sampling

2026-09-30

AI Summary

This study addresses the unclear accuracy bounds of the Moreau-Yosida Unadjusted Langevin Algorithm (MYULA) for non-smooth target functions by proposing a convergence analysis framework that eliminates third-order derivative assumptions. By integrating Moreau-Yosida regularization with unadjusted Langevin sampling, this work employs Poisson estimation to transform second-order residuals into Wasserstein contraction bounds, thereby achieving logarithmic dependence on the smoothing parameter and directly bounding the stationary error. Theoretically, it is proven that attaining ε-accuracy requires only O(1/ε) iterations, establishing a near-linear complexity bound. This result significantly improves the computational efficiency of non-smooth sampling.

Abstract

We establish near-linear accuracy bounds for the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA). The target is $π\propto e^{-f-g}$, where $f\in C^2(\mathbb{R}^d)$ is $m$-strongly convex with Lipschitz gradient and $g$ is convex and globally Lipschitz. Under an explicit parameter-dependent step-size condition, we bound the invariant-measure bias relative to the Moreau-smoothed target by $\widetilde O(h)$, with only logarithmic dependence on the inverse smoothing parameter in the error coefficient. Combining this estimate with the Moreau approximation bias and Wasserstein contraction gives $\widetilde O(\varepsilon^{-1})$ iterations to make the $N$th-iterate law $μ_N$ satisfy $\sqrt m\,W_2(μ_N,π)\le\varepsilon$, for fixed model parameters and initialization. We bound the stationary error directly, without assuming third derivatives or a Lipschitz Hessian. Each iteration uses one gradient evaluation and one exact proximal evaluation. The key idea in our analysis is to convert a second-order stationary residual into a Wasserstein bound using a Poisson-based estimate.

Problem

• Moreau-Yosida unadjusted Langevin algorithm
• accuracy bounds
• non-smooth sampling
• Wasserstein distance
• proximal evaluation

Innovation

• Moreau-Yosida unadjusted Langevin algorithm
• near-linear accuracy bounds
• Wasserstein distance
• Poisson equation
• proximal operator