The algorithm itself is a straightforward combination of Epoch-SGD for minimization and the well-known gradient-descent-ascent method for minimax problem. At the heart of the analysis is a new lemma that relates the duality gap to some distance measures between two points.

The main result in the paper extends a classical result of Hazan et al to a O(1/T) convergence bound for the duality gap of non-smooth strongly convex-strongly concave min-max problem (instead of the objective gap), proposing a min-max adaptation of the algorithm (Epoch-GDA). They also provide a related bound for finding approximate stationary points in weakly convex-strongly concave problems. Overall the reviewers found the contribution to be a significant and challenging extension over the existing result of Hazan et al. with specific challenges to be overcome in the duality gap version. The authors are strongly recommended to make the promised revisions in the rebutttal as they will embellish the paper.

Epoch-GD) proposed by (Hazan and Kale, 2011) was deemeda breakthrough for stochastic strongly convex minimization, which achieves theoptimal convergence rate of O(1/T) with T iterative updates for the objective gap. However, its extension to solving stochastic min-max problems with strong convexity and strong concavity still remains open, and it is still unclear whethera fast rate ofO(1/T)for theduality gapis achievable for stochastic min-max optimization under strong convexity and strong concavity. Although some re-cent studies have proposed stochastic algorithms with fast convergence rates formin-max problems, they require additional assumptions about the problem, e.g.,smoothness, bi-linear structure, etc. To the best of our knowledge, our result is the first one that shows Epoch-GDA can achieve the optimal rate ofO(1/T)for the duality gapof general SCSC min-max problems. We emphasize that such generalization of Epoch-GD for strongly convex minimization problems to Epoch-GDA for SCSC min-max problems is non-trivial and requires novel technical analysis. Moreover, we notice that the key lemma can also be used for proving the convergence of Epoch-GDA for weakly-convex strongly-concave min-max problems, leading to a nearly optimal complexity without resorting to smoothness or other structural conditions.

Minimax optimization problems have attracted significant attention in recent years due to their widespread application in numerous machine learning models. To solve the minimax optimization problem, a wide variety of stochastic optimization methods have been proposed. However, most of them ignore the distributed setting where the training data is distributed on multiple workers. In this paper, we developed a novel decentralized stochastic gradient descent ascent method for the finite-sum minimax optimization problem. In particular, by employing the variance-reduced gradient, our method can achieve $O(\frac{\sqrt{n}\kappa^3}{(1-\lambda)^2\epsilon^2})$ sample complexity and $O(\frac{\kappa^3}{(1-\lambda)^2\epsilon^2})$ communication complexity for the nonconvex-strongly-concave minimax optimization problem. As far as we know, our work is the first one to achieve such theoretical complexities for this kind of problem. At last, we apply our method to optimize the AUC maximization problem and the experimental results confirm the effectiveness of our method.

Epoch gradient descent method (a.k.a. Epoch-GD) proposed by (Hazan and Kale, 2011) was deemed a breakthrough for stochastic strongly convex minimization, which achieves the optimal convergence rate of O(1/T) with T iterative updates for the objective gap. However, its extension to solving stochastic min-max problems with strong convexity and strong concavity still remains open, and it is still unclear whether a fast rate of O(1/T) for the duality gap is achievable for stochastic min-max optimization under strong convexity and strong concavity. Although some recent studies have proposed stochastic algorithms with fast convergence rates for min-max problems, they require additional assumptions about the problem, e.g., smoothness, bi-linear structure, etc. In this paper, we bridge this gap by providing a sharp analysis of epoch-wise stochastic gradient descent ascent method (referred to as Epoch-GDA) for solving strongly convex strongly concave (SCSC) min-max problems, without imposing any additional assumptions about smoothness or its structure. To the best of our knowledge, our result is the first one that shows Epoch-GDA can achieve the fast rate of O(1/T) for the duality gap of general SCSC min-max problems. We emphasize that such generalization of Epoch-GD for strongly convex minimization problems to Epoch-GDA for SCSC min-max problems is non-trivial and requires novel technical analysis. Moreover, we notice that the key lemma can be also used for proving the convergence of Epoch-GDA for weakly-convex strongly-concave min-max problems, leading to the best complexity as well without smoothness or other structural conditions.