刚看到一篇论文，把同样本和独立样本随机外梯度的收敛性区别讲透了，还有反例。
该论文研究随机外梯度（SEG）方法用于单调变分不等式。作者证明，同样本SEG（S-SEG）对逐样本Lipschitz参数敏感，仅靠均方Lipschitz和有界方差无法保证收敛。他们对可能无界的可行集建立了两种SEG变体的高概率受限间隙收敛，并指出这些结果无法再一般性改进。此外，论文构造了一个随机单调VIP，其中对I-SEG有效的非对称双步长会导致S-SEG几乎必然发散。
On Same-Sample and Independent-Sample Stochastic Extragradient for Monotone Variational Inequalities
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.