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

• URL: http://arxiv.org/abs/2609.12594v1
• Date: Fri, 11 Sep 2026 08:47:33 GMT
• Title: Poisson-Corrector Complexity Bounds for Moreau--Yosida Unadjusted Langevin Sampling
• Abstract summary: We study the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA) for $(,mathrmd x)propto e-f(x)-g(x),mathrmd x$, where $fin C2(mathbbRd)$ is $m$-strongly convex with $L_f$-Lipschitz gradient and $g:mathbbRdtomathbbR$ is convex and globally $G$-Lipschitz
• Score: 9.454986540713655
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA) for $π(\,\mathrm{d} x)\propto e^{-f(x)-g(x)}\,\mathrm{d} x$, where $f\in C^2(\mathbb{R}^d)$ is $m$-strongly convex with $L_f$-Lipschitz gradient and $g:\mathbb{R}^d\to\mathbb{R}$ is convex and globally $G$-Lipschitz. For the Moreau-smoothed target $π_λ$ and the MYULA invariant law $\widehatπ_{λ,h}$, we prove \[ \sqrt m\,W_2(π_λ,\widehatπ_{λ,h}) =O(h)+\widetilde O(h^{3/4}) \] under $0<h(L_f+λ^{-1})\le c$, 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 $\sqrt m\,W_2(μ_N,π)\le\varepsilon$, 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.

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