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 πpropto e-f-g, where f∈ C2(ℝd) 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 Nth-iterate law μN satisfy √ m,W2(μN,π)≤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.