  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 {\it 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
{\it 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 assumption about
smoothness or the function's structure. To the best of our knowledge, our
result is the first one that shows Epoch-GDA can achieve the optimal 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 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.
