改进随机单调变分不等式任意时间算法的最后迭代保证
Improving the Last-Iterate Guarantees of Anytime Algorithms for Stochastic Monotone Variational Inequalities
- University of British Columbia（不列颠哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究提出一种带Halpern锚定的单循环随机算法，用于约束凸-凹问题和单调变分不等式，实现任意时间最后迭代收敛速率$O(t^{-1/4})$，优于现有$O(t^{-1/5})$，并适用于无界可行集和无界方差场景。
AI中文摘要：
我们分析了一种带有Halpern锚定的随机算法，用于约束凸-凹问题和单调变分不等式。该算法是单循环且单调用的，因为它在每次迭代中使用梯度算子的一个无偏样本，适用于具有噪声反馈的单调博弈。以$t$表示迭代计数器，我们证明了梯度映射范数和受限间隙的任意时间最后迭代收敛速率为$O(t^{-1/4})$，改进了先前针对受限间隙函数获得的最佳已知速率$O(t^{-1/5})$。我们的速率涵盖了具有潜在无界可行集的约束问题以及一类无界方差的结构化随机预言机。
英文摘要：
We analyze a stochastic algorithm with Halpern-type anchoring for constrained convex-concave problems and monotone variational inequalities. This single-loop and single-call algorithm uses one unbiased sample of the gradient operator at every iteration, to be applicable to monotone games with noisy feedback. With $t$ denoting the iteration counter, we prove an anytime last-iterate convergence rate of $O(t^{-1/4})$ for both the gradient-mapping norm and restricted gap, bypassing the $O(t^{-1/5})$ constrained-anytime bottleneck in the literature. Specializing then to multi-point oracles, we use variance reduction to achieve the $O(t^{-1/2})$ rate with an anytime single-loop algorithm using $2$ samples per iteration. Our results allow constrained problems with a potentially unbounded feasible set; as well as a structured class of stochastic oracles whose variance need not be uniformly bounded.
