This paper introduces a family of stochastic extragradient-type algorithms for a class of nonconvex-nonconcave problems characterized by the weak Minty variational inequality (MVI). Unlike existing results on extragradient methods in the monotone setting, employing diminishing stepsizes is no longer possible in the weak MVI setting. This has led to approaches such as increasing batch sizes per iteration which can however be prohibitively expensive. In contrast, our proposed methods involves two stepsizes and only requires one additional oracle evaluation per iteration. We show that it is possible to keep one fixed stepsize while it is only the second stepsize that is taken to be diminishing, making it interesting even in the monotone setting. Almost sure convergence is established and we provide a unified analysis for this family of schemes which contains a nonlinear generalization of the celebrated primal dual hybrid gradient algorithm.

翻译：本文针对一类由弱Minty变分不等式刻画的非凸-非凹问题，提出了一类随机外梯度型算法。与单调设定下外梯度方法的现有结果不同，在弱Minty变分不等式设定中，递减步长策略不再可行。现有方法通过逐次增加迭代批次大小来解决该问题，但这可能导致计算成本过高。相比之下，本文所提方法仅需采用两步长策略，且每次迭代仅需额外一次oracle评估。我们证明，在仅令第二步长递减而保持第一步长固定的条件下，算法仍能有效运行——这一特性即使在单调设定下也具有重要价值。本文建立了几乎必然收敛性，并为该算法族提供了统一分析框架，该族算法包含著名原始-对偶混合梯度算法的非线性推广形式。