Poisson-Corrector Complexity Bounds for Moreau--Yosida Unadjusted Langevin Sampling (arxiv.org)
arXiv:2609.12594v1 Announce Type: new
Abstract: We study the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA) for $\pi(\,\mathrm{d} x)\propto e^{-f(x)-g(x)}\,\mathrm{d} x$, where $f\in C^2(\mathbb{R}^d)$ is $m$-strongly convex with $L_f$-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 $\widehat\pi_{\lambda,h}$, we prove \[
\sqrt m\,W_2(\pi_\lambda,\widehat\pi_{\lambda,h})
=O(h)+\widetilde O(h^{3/4}) \] 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 $\widetilde O(\varepsilon^{-4/3})$ iterations to achieve $\sqrt m\,W_2(\mu_N,\pi)\le\varepsilon$, 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.
Abstract: We study the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA) for $\pi(\,\mathrm{d} x)\propto e^{-f(x)-g(x)}\,\mathrm{d} x$, where $f\in C^2(\mathbb{R}^d)$ is $m$-strongly convex with $L_f$-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 $\widehat\pi_{\lambda,h}$, we prove \[
\sqrt m\,W_2(\pi_\lambda,\widehat\pi_{\lambda,h})
=O(h)+\widetilde O(h^{3/4}) \] 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 $\widetilde O(\varepsilon^{-4/3})$ iterations to achieve $\sqrt m\,W_2(\mu_N,\pi)\le\varepsilon$, 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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