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

2026-09-11

AI Summary

本文研究了MYULA算法在特定条件下的复杂度边界，通过结合离散Poisson校正器等方法，证明了在给定条件下达到指定精度所需的迭代次数。

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.

Problem

• Poisson-Corrector
• Complexity Bounds
• Moreau--Yosida
• Unadjusted Langevin Sampling
• Wasserstein-2 Distance

Innovation

• Poisson Corrector
• Active-Trace Estimates
• Shared-Noise Bound