We are concerned with optimization in a broad sense through the lens of solving variational inequalities (VIs) -- a class of problems that are so general that they cover as particular cases minimization of functions, saddle-point (minimax) problems, Nash equilibrium problems, and many others. The key challenges in our problem formulation are the two-level hierarchical structure and finite-sum representation of the smooth operators in each level. For this setting, we are the first to prove convergence rates and complexity statements for variance-reduced stochastic algorithms approaching the solution of hierarchical VIs in Euclidean and Bregman setups.

翻译：我们通过求解变分不等式这一视角来关注广义优化问题——此类问题具有高度普适性，其特例涵盖函数最小化、鞍点（极小极大）问题、纳什均衡问题等诸多类型。本问题表述中的关键挑战在于双层分层结构以及每层光滑算子的有限和表示。针对此设定，我们首次证明了在欧几里得与布雷格曼框架下，逼近分层变分不等式解的随机方差缩减算法的收敛率与复杂度结论。