Poisson-Corrector Complexity Bounds for Moreau--Yosida Unadjusted Langevin Sampling

We study the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA) for π(,d x)propto e-f(x)-g(x),d x, where f∈ C2(ℝd) is m-strongly convex with Lf-Lipschitz gradient and g:ℝd→ℝ is convex and globally G-Lipschitz. For the Moreau-smoothed target πλ and the MYULA invariant law widehatπλ,h, we prove [ √ m,W2(πλ,widehatπλ,h) =O(h)+widetilde O(h3/4) ] under 0<h(Lf+λ-1)≤ c, with only logarithmic dependence on λ-1 in the error coefficients. Combining this estimate with the Moreau approximation bias yields widetilde O(varepsilon-4/3) iterations to achieve √ m,W2(μN,π)≤varepsilon, for fixed model parameters and initialization. The proof combines a discrete Poisson corrector with active-trace estimates and a shared-noise bound for the exact--Euler two-point curvature.