Moreau--Yosida 未调整 Langevin 采样的近线性精度界

中文摘要 AI 辅助

我们为经典的 Moreau--Yosida 未调整 Langevin 算法（MYULA）建立了近线性精度界。目标分布为 $\pi\propto e^{-f-g}$，其中 $f\in C^2(\mathbb{R}^d)$ 是 $m$-强凸且具有 Lipschitz 梯度的函数，$g$ 是凸且全局 Lipschitz 的函数。在显式的参数依赖步长条件下，我们将不变测度偏差相对于 Moreau 平滑目标的界限制为 $\widetilde O(h)$，且误差系数仅对逆平滑参数具有对数依赖。将该估计与 Moreau 近似偏差和 Wasserstein 收缩相结合，得到 $\widetilde O(\varepsilon^{-1})$ 次迭代即可使第 $N$ 次迭代的分布 $\mu_N$ 满足 $\sqrt m\\,W_2(\mu_N,\pi)\le\varepsilon$，其中模型参数和初始化固定。我们直接界定了平稳误差，无需假设三阶导数或 Lipschitz Hessian。每次迭代仅需一次梯度评估和一次精确近端评估。我们分析的关键思想是利用基于 Poisson 的估计将二阶平稳残差转化为 Wasserstein 界。

英文摘要

We establish near-linear accuracy bounds for the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA). The target is $π\propto e^{-f-g}$, where $f\in C^2(\mathbb{R}^d)$ is $m$-strongly convex with Lipschitz gradient and $g$ is convex and globally Lipschitz. Under an explicit parameter-dependent step-size condition, we bound the invariant-measure bias relative to the Moreau-smoothed target by $\widetilde O(h)$, with only logarithmic dependence on the inverse smoothing parameter in the error coefficient. Combining this estimate with the Moreau approximation bias and Wasserstein contraction gives $\widetilde O(\varepsilon^{-1})$ iterations to make the $N$th-iterate law $μ_N$ satisfy $\sqrt m\,W_2(μ_N,π)\le\varepsilon$, for fixed model parameters and initialization. We bound the stationary error directly, without assuming third derivatives or a Lipschitz Hessian. Each iteration uses one gradient evaluation and one exact proximal evaluation. The key idea in our analysis is to convert a second-order stationary residual into a Wasserstein bound using a Poisson-based estimate.

发表机构

• Peking University（北京大学）

机构由 AI 辅助整理，请以论文原文为准。