The complexity of nonconvex-strongly-concave minimax optimization

This paper studies the complexity for finding approximate stationary points of nonconvex-strongly-concave (NC-SC) smooth minimax problems, in both general and averaged smooth finite-sum settings. We establish nontrivial lower complexity bounds of $\Omega(\sqrt{\kappa}\Delta L\epsilon^{-2})$ and $\Omega(n+\sqrt{n\kappa}\Delta L\epsilon^{-2})$ for the two settings, respectively, where $\kappa$ is the condition number, $L$ is the smoothness constant, and $\Delta$ is the initial gap. Our result reveals substantial gaps between these limits and best-known upper bounds in the literature. To close these gaps, we introduce a generic acceleration scheme that deploys existing gradient-based methods to solve a sequence of crafted strongly-convex-strongly-concave subproblems. In the general setting, the complexity of our proposed algorithm nearly matches the lower bound; in particular, it removes an additional poly-logarithmic dependence on accuracy present in previous works. In the averaged smooth finite-sum setting, our proposed algorithm improves over previous algorithms by providing a nearly-tight dependence on the condition number

EXPERIMENTAL EVALUATION OF ITERATIVE METHODS FOR GAMES

Min-max optimization problems are a class of problems that are usually seen in game theory, machine learning, deep learning, and adversarial training. Deterministic gradient methods, such as gradient descent ascent (GDA), Extragradient (EG), and Hamiltonian Gradient Descent (HGD) are usually implemented to solve those problems. In large-scale setting, stochastic variants of those gradient methods are prefer because of their cheap per iteration cost. To further increase optimization efficiency, different improvements of deterministic and stochastic gradient methods are proposed, such as acceleration, variance reduction, and random reshuffling. In this work, we explore advanced iterative methods for solving min-max optimization problems, including deterministic gradient methods combined with accelerated methods and stochastic gradient methods combined with variance reduction and random reshuffling. We use experiments to evaluate the performance of the classical and advanced iterative methods on both bilinear and quadratic games. With an experimental approach, we show that the most advanced iterative methods in the deterministic and stochastic setting have improvements in iteration complexity