发表机构
Johns Hopkins University（约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对单调变分不等式问题，分析同一样本与独立样本随机超梯度方法的收敛特性，指出同一样本方法的敏感性及步长选择的失效情况，完善了相关理论空白。
AI 中文摘要
我们研究用于求解可行集上单调变分不等式问题（VIP）的随机超梯度（SEG）方法。尽管超梯度是求解VIP的基础算法，其确定性收敛理论已发展完善，但随机超梯度的对应理论仍不够清晰。现有大多数分析聚焦于独立样本随机超梯度（I-SEG），且假设要么定义域是紧集，要么随机算子的方差一致有界。同一样本随机超梯度（S-SEG）作为具有显著不同性质的自然变体，其行为受到的关注少得多。本研究旨在弥补文献中的这些空白：首先，我们证明S-SEG对逐样本Lipschitz参数敏感，即使在紧集上，仅均值Lipschitz性和有界方差也无法保证收敛；其次，针对可能无界的定义域，我们在一组放宽的假设下，为每种SEG变体建立了高概率受限间隙收敛性，并证明对这些结果的某些根本性改进在一般情况下是不可能的；最后，我们证明一种已知的非对称双步长选择（该选择可保证I-SEG的几乎必然最后迭代收敛）对S-SEG可能失效，存在某一随机单调VIP，即使采用修改后的步长，S-SEG仍几乎必然发散。
英文摘要
We study stochastic extragradient (SEG) methods for solving monotone variational inequality problems (VIPs) over a feasible set. Although extragradient is a foundational algorithm for VIPs and its deterministic convergence theory is well developed, its stochastic counterpart remains less understood. Most existing analyses focus on independent-sample SEG (I-SEG) and assume either that the domain is compact or that the variance of the stochastic operator is uniformly bounded. The behavior of same-sample SEG (S-SEG), a natural variant with materially different properties, has received far less attention. In this work, we address these gaps in the literature. We first show that S-SEG is sensitive to samplewise Lipschitz parameters: mean Lipschitzness and bounded variance alone do not ensure convergence, even on a compact set. Then, for possibly unbounded domains, we establish a high-probability restricted-gap convergence for each SEG variant under a relaxed set of assumptions, and show that certain fundamental improvements to these results are impossible in general. Finally, we show that a known asymmetric double step-size selection that guarantees almost sure last-iterate convergence for I-SEG can fail for S-SEG: there exists a stochastic monotone VIP for which S-SEG diverges almost surely even under the modified step-sizes.
