Machine learning approaches relying on such criteria as adversarial robustness or multi-agent settings have raised the need for solving game-theoretic equilibrium problems. Of particular relevance to these applications are methods targeting finite-sum structure, which generically arises in empirical variants of learning problems in these contexts. Further, methods with computable approximation errors are highly desirable, as they provide verifiable exit criteria. Motivated by these applications, we study finite-sum monotone inclusion problems, which model broad classes of equilibrium problems. Our main contributions are variants of the classical Halpern iteration that employ variance reduction to obtain improved complexity guarantees in which $n$ component operators in the finite sum are ``on average'' either cocoercive or Lipschitz continuous and monotone, with parameter $L$. The resulting oracle complexity of our methods, which provide guarantees for the last iterate and for a (computable) operator norm residual, is $\widetilde{\mathcal{O}}( n + \sqrt{n}L\varepsilon^{-1})$, which improves upon existing methods by a factor up to $\sqrt{n}$. This constitutes the first variance reduction-type result for general finite-sum monotone inclusions and for more specific problems such as convex-concave optimization when operator norm residual is the optimality measure. We further argue that, up to poly-logarithmic factors, this complexity is unimprovable in the monotone Lipschitz setting; i.e., the provided result is near-optimal.

翻译：依赖对抗鲁棒性或多智能体设置等准则的机器学习方法，激发了对求解博弈论均衡问题的需求。与这些应用尤为相关的是针对有限和结构的方法，这类结构普遍出现在这些情境下学习问题的经验变体中。此外，具有可计算近似误差的方法备受青睐，因为它们提供了可验证的退出准则。受这些应用驱动，我们研究有限和单调包含问题，这类问题可建模广泛类别的均衡问题。我们的主要贡献是经典Halpern迭代的变体，采用方差缩减以获得改进的复杂度保证，其中有限和中的$n$个分量算子“平均而言”要么是共余的，要么是Lipschitz连续且单调的，参数为$L$。所提方法为最后迭代点和（可计算的）算子范数残差提供保证，其预言复杂度为$\widetilde{\mathcal{O}}( n + \sqrt{n}L\varepsilon^{-1})$，相比现有方法最多可提升$\sqrt{n}$倍。这是首个针对一般有限和单调包含问题以及更具体问题（如以算子范数残差为最优性测度的凸-凹优化）的方差缩减类结果。我们进一步论证，除多对数因子外，该复杂度在单调Lipschitz设置下不可改进；即所得结果接近最优。