A Poisson Corrector for Complexity Bounds of Moreau–Yosida Unadjusted Langevin Sampling

We study the classical Moreau–Yosida unadjusted Langevin algorithm (MYULA) of sampling from $\pi (\mathrm{d}x)\propto e^{-f(x)-g(x)}\mathrm{d}x$, where $f\in C^2({\mathbb{R}}^d)$ is $m$-strongly convex and has a $L_f$-Lipschitz gradient, and $g:{\mathbb{R}}^d\to \mathbb{R}$ is convex and globally $G$-Lipschitz. For the Moreau-smoothed target ${\pi }_{\lambda }$ and the MYULA invariant law ${\hat{\pi }}_{\lambda ,h}$, we prove $\sqrt{m}{W}_2({\pi }_{\lambda },{\hat{\pi }}_{\lambda ,h})=O(h)+\tilde{O}(h^{3/4})$ under $0<h(L_f+{\lambda }^{-1})\le c$, with only logarithmic dependence on ${\lambda }^{-1}$ in the error coefficients. Combining this estimate with the Moreau approximation bias yields $\tilde{O}({\epsilon }^{-4/3})$ iterations to achieve $\sqrt{m}{W}_2({\mu }_N,\pi )\le \epsilon$, 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.