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Optimal Epoch Stochastic Gradient Descent Ascent Methods for Min-Max Optimization
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Optimal Epoch Stochastic Gradient Descent Ascent Methods for Min-Max Optimization

Yan Yan, Yi Xu, Qihang Lin, Wei Liu and Tianbao Yang
arXiv.org
Cornell University Library, arXiv.org
06/17/2020
DOI: 10.48550/arXiv.2002.05309
url
https://doi.org/10.48550/arXiv.2002.05309View
Preprint (Author's original)This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

Abstract

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.
 Algorithms Optimization Ascent Concavity Convergence Convexity Smoothness

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