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Statistics > Machine Learning

arXiv:2001.07819 (stat)
[Submitted on 22 Jan 2020 (v1), last revised 4 Apr 2022 (this version, v2)]

Title:Zeroth-Order Algorithms for Nonconvex Minimax Problems with Improved Complexities

Authors:Zhongruo Wang, Krishnakumar Balasubramanian, Shiqian Ma, Meisam Razaviyayn
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Abstract:In this paper, we study zeroth-order algorithms for minimax optimization problems that are nonconvex in one variable and strongly-concave in the other variable. Such minimax optimization problems have attracted significant attention lately due to their applications in modern machine learning tasks. We first consider a deterministic version of the problem. We design and analyze the Zeroth-Order Gradient Descent Ascent (\texttt{ZO-GDA}) algorithm, and provide improved results compared to existing works, in terms of oracle complexity. We also propose the Zeroth-Order Gradient Descent Multi-Step Ascent (\texttt{ZO-GDMSA}) algorithm that significantly improves the oracle complexity of \texttt{ZO-GDA}. We then consider stochastic versions of \texttt{ZO-GDA} and \texttt{ZO-GDMSA}, to handle stochastic nonconvex minimax problems. For this case, we provide oracle complexity results under two assumptions on the stochastic gradient: (i) the uniformly bounded variance assumption, which is common in traditional stochastic optimization, and (ii) the Strong Growth Condition (SGC), which has been known to be satisfied by modern over-parametrized machine learning models. We establish that under the SGC assumption, the complexities of the stochastic algorithms match that of deterministic algorithms. Numerical experiments are presented to support our theoretical results.
Comments: To appear in the Journal of Global Optimization
Subjects: Machine Learning (stat.ML); Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG); Optimization and Control (math.OC)
Cite as: arXiv:2001.07819 [stat.ML]
  (or arXiv:2001.07819v2 [stat.ML] for this version)
  https://doi.org/10.48550/arXiv.2001.07819

Submission history

From: Krishnakumar Balasubramanian [view email]
[v1] Wed, 22 Jan 2020 00:05:14 UTC (29 KB)
[v2] Mon, 4 Apr 2022 23:36:08 UTC (370 KB)
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