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

Abstract:We establish near-linear accuracy bounds for the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA). The target is $\pi\propto e^{-f-g}$, 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 $\mu_N$ satisfy $\sqrt m\,W_2(\mu_N,\pi)\le\varepsilon$, 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.

Machine Learning (cs.LG)

arXiv:2609.40193 [cs.LG]

(or arXiv:2609.40193v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.40193 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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