Faster Stochastic Algorithms for Minimax Optimization under Polyak-Lojasiewicz Conditions

This paper considers stochastic first-order algorithms for minimax optimization under Polyak-Lojasiewicz (PL) conditions. We propose SPIDER-GDA for solving the finite-sum problem of the form min(x) max(y) f(x, y) (sic) 1/n Sigma(n)(i=1) f(i)(x, y), where the objective function f(x, y) is mu(x)-PL in x and mu(y)-PL in y; and each f(i)(x, y) is L-smooth. We prove SPIDER-GDA could find an.-approximate solution within O((n + root n kappa(x)kappa(2)(y)) log(1/epsilon)) stochastic first-order oracle (SFO) complexity, which is better than the state-of-the-art method whose SFO upper bound is O((n + n(2/3)kappa(x)kappa(2)(y)) log(1/epsilon)), where kappa(x) (sic) L/mu(x) and kappa(y) (sic) L/mu(y). For the ill-conditioned case, we provide an accelerated algorithm to reduce the computational cost further. It achieves (O) over tilde (n + root n kappa(x)kappa(y)) log(2) (1/epsilon)) SFO upper bound when kappa(y) greater than or similar to root n. Our ideas also can be applied to the more general setting that the objective function only satisfies PL condition for one variable. Numerical experiments validate the superiority of proposed methods.