Breaking the Complexity Barrier in Compositional Minimax Optimization

Compositional minimax optimization is a pivotal yet under-explored challenge across machine learning, including distributionally robust training and policy evaluation for reinforcement learning. Current techniques exhibit suboptimal complexity or rely heavily on large batch sizes. This paper proposes Nested STOchastic Recursive Momentum (NSTORM), attaining the optimal sample complexity of $O(\kappa^3/\epsilon^3)$ for finding an $\epsilon$-accurate solution. However, NSTORM requires low learning rates, potentially limiting applicability. Thus we introduce ADAptive NSTORM (ADA-NSTORM) with adaptive learning rates, proving it achieves the same sample complexity while experiments demonstrate greater effectiveness. Our methods match lower bounds for minimax optimization without large batch requirements, validated through extensive experiments. This work significantly advances compositional minimax optimization, a crucial capability for distributional robustness and policy evaluation