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

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 $\tilde{O}(h)$, 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 $\tilde{O}({\epsilon }^{-1})$ iterations to make the $N$th-iterate law ${\mu }_N$ satisfy $\sqrt{m}{W}_2({\mu }_N,\pi )\le \epsilon$, 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.