Fast Extra Gradient Methods for Smooth Structured Nonconvex-Nonconcave Minimax Problems

Abstract

Modern minimax problems, such as generative adversarial network and adversarial training, are often under a nonconvex-nonconcave setting, and developing an efficient method for such setting is of interest. Recently, two variants of the extragradient (EG) method are studied in that direction. First, a two-time-scale variant of the EG, named EG+, was proposed under a smooth structured nonconvexnonconcave setting, with a slow O(1/k) rate on the squared gradient norm, where k denotes the number of iterations. Second, another variant of EG with an anchoring technique, named extra anchored gradient (EAG), was studied under a smooth convex-concave setting, yielding a fast O(1/k 2 ) rate on the squared gradient norm. Built upon EG+ and EAG, this paper proposes a two-time-scale EG with anchoring, named fast extragradient (FEG), that has a fast O(1/k 2 ) rate on the squared gradient norm for smooth structured nonconvex-nonconcave problems. This paper further develops its backtracking line-search version, named FEG-A, for the case where the problem parameters are not available. The stochastic analysis of FEG is also provided.Preprint. Under review.1 Relations between the conditions on F considered in this paper is summarized in Figure 1.

On the Linear Convergence of Extra-Gradient Methods for Nonconvex-Nonconcave Minimax Problems

“…The following proposition establishes the equivalence of our assumptions and the negative comonotone setting recently considered by [14].…”

“…al [6] furnished the proof for the ergodic convergence of a special class of our damped EGM, that is, one with a damping parameter of λ = 1 2 , when applied to a nonconvex-nonconcave minimax problems. Lee and Kim [14] furnished the linear convergence of the so-called two-time-scale anchored extra-gradient method (FEG) to a stationary point (Definition 2.2) of an objective function with a negatively ρ-comonotone oracle. This is also known in literature as |ρ|-cohypomonotonicity (see [2], Remark 2.5 (ii)).…”

“…The first two conditions (Lipschitz continuity, smoothness, and weak convexity-concavity) are relatively standard. The third condition will be formalized in the following section (see Definition 2.1) and is equivalent to the settings considered for the proximal point method in [10] and the negative comonotone setting where accelerated, sublinear rates were recently derived by [14].…”

“…Recently, [14] presented algorithms with sublinear rate for nonconvex-nonconcave minimax optimization problems with ρ-comonotone gradient oracle for some negative ρ. A set-valued mapping T : R n+m ⇒ R n+m is said to be ρ-monotone on R n+m if for every z ∈ R n+m there exists a neighborhood V of z such that…”

Mirror frameworks for relatively Lipschitz and monotone-like variational inequalities

“…However, modern nonconvex-nonconcave saddle-point optimization problems such as those appeared in deep learning go beyond monotone VIs and hence the existing results fail to apply in non-monotone settings. This motivates a surge of interest to generalized VIs and its associated algorithms [12,13,16,23,25,26,45]. We restrict our attention to the line of research that relaxes the Lipschitz continuity and monotonicity assumptions.…”