Online Min-max Optimization: Nonconvexity, Nonstationarity, and Dynamic Regret

Online min-max optimization has recently gained considerable interest due to its rich applications to game theory, multi-agent reinforcement learning, online robust learning, etc. Theoretical understanding in this field has been mainly focused on convex-concave settings. Online min-max optimization with nonconvex geometries, which captures various online deep learning problems, has yet been studied so far. In this paper, we make the first effort and investigate online nonconvex-strongly-concave min-max optimization in the nonstationary environment. We first introduce a natural notion of dynamic Nash equilibrium (NE) regret, and then propose a novel algorithm coined SODA to achieve the optimal regret. We further generalize our study to the setting with stochastic first-order feedback, and show that a variation of SODA can also achieve the same optimal regret in expectation. Our theoretical results and the superior performance of the proposed method are further validated by empirical experiments. To our best knowledge, this is the first exploration of efficient online nonconvex min-max optimization.