发表机构
Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种热启动的两环随机镜像 Langevin 算法，通过外环递减步长、内环固定步长运行，实现约束采样，达到 O(T^{-1/2}) 的最优收敛速率。
AI 中文摘要
我们研究了从定义在凸集 $X\subseteq\mathbb R^d$ 上的目标分布 $\pi(x)\propto e^{-f(x)}$ 中进行采样的问题，其中势函数 $f$ 仅能通过随机一阶预言机访问。镜像 Langevin 算法通过将问题转换到无约束的对偶空间并对由此产生的镜像 Langevin 扩散进行离散化，为约束采样提供了一种自然的方法。然而，现有的实现通常使用固定的离散化步长，因此在任何固定步长下都会保留非消失的离散化偏差。此外，它们直接扩展到具有噪声梯度信息的设置时，需要同时控制离散化和随机预言机误差。我们研究了镜像 Langevin 算法（sFO-MLA）的随机一阶版本，并作为我们的主要贡献，开发了一种热启动的两环实现，其中外环逐步减小步长，而内环以固定步长运行 sFO-MLA 并采用适当选择的周期长度。该构造提供了一种原则性的调度，将步长和周期长度联系起来，使得连续的周期从越来越精确的分布热启动，而不是反复支付从冷启动混合的成本。我们为 sFO-MLA 建立了有限时间的 Wasserstein 保证，明确分离了混合、Euler-Maruyama 离散化和随机梯度误差。这些界产生了固定时间范围的 $\widetilde O(T^{-1/2})$ 速率，并表明两环方案消除了相关的对数惩罚，在几何步长调度和相应的周期长度下达到了规范的 $O(T^{-1/2})$ 速率。我们在两个统计上不同的设置中说明了该方法。
英文摘要
We study the problem of sampling from a target distribution $π(x)\propto e^{-f(x)}$ supported on a convex set $ X\subseteq\mathbb R^d$, when the potential $f$ is accessible only through a stochastic first-order oracle. Mirror Langevin algorithms provide a natural approach to constrained sampling by transporting the problem to an unconstrained dual space and discretizing the resulting Mirror Langevin diffusion. Existing implementations, however, typically use a fixed discretization step size and consequently retain a nonvanishing discretization bias at any fixed step size. Moreover, their direct extension to settings with noisy gradient information entails the challenge of controlling both discretization and stochastic-oracle error. We study a stochastic first-order version of the Mirror Langevin Algorithm (sFO-MLA) and, as our main contribution, develop a warm-started two-loop implementation in which an outer loop progressively decreases the step size while an inner loop runs sFO-MLA (with a fixed step size) for an appropriately chosen epoch length. The construction provides a principled schedule linking step sizes and epoch lengths, so that successive epochs warm-start from increasingly accurate distributions rather than repeatedly paying the cost of mixing from a cold start. We establish finite-time Wasserstein guarantees for sFO-MLA that explicitly separate mixing, Euler--Maruyama discretization, and stochastic-gradient errors. These bounds yield a fixed horizon rate of $\widetilde O(T^{-1/2})$ and show that the two-loop scheme removes the associated logarithmic penalty, attaining the canonical $O(T^{-1/2})$ rate under a geometric step-size schedule and corresponding epoch lengths. We illustrate the methodology in two statistically distinct settings.
