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Mathematics > Optimization and Control
Title: Explicit Second-Order Min-Max Optimization Methods with Optimal Convergence Guarantee
(Submitted on 23 Oct 2022 (v1), last revised 23 Apr 2024 (this version, v4))
Abstract: We propose and analyze several inexact regularized Newton-type methods for finding a global saddle point of \emph{convex-concave} unconstrained min-max optimization problems. Compared to first-order methods, our understanding of second-order methods for min-max optimization is relatively limited, as obtaining global rates of convergence with second-order information is much more involved. In this paper, we examine how second-order information can be used to speed up extra-gradient methods, even under inexactness. Specifically, we show that the proposed methods generate iterates that remain within a bounded set and that the averaged iterates converge to an $\epsilon$-saddle point within $O(\epsilon^{-2/3})$ iterations in terms of a restricted gap function. This matched the theoretically established lower bound in this context. We also provide a simple routine for solving the subproblem at each iteration, requiring a single Schur decomposition and $O(\log\log(1/\epsilon))$ calls to a linear system solver in a quasi-upper-triangular system. Thus, our method improves the existing line-search-based second-order min-max optimization methods by shaving off an $O(\log\log(1/\epsilon))$ factor in the required number of Schur decompositions. Finally, we present numerical experiments on synthetic and real data that demonstrate the efficiency of the proposed methods.
Submission history
From: Tianyi Lin [view email][v1] Sun, 23 Oct 2022 21:24:37 GMT (171kb)
[v2] Sun, 20 Nov 2022 02:29:15 GMT (172kb)
[v3] Wed, 6 Sep 2023 02:02:19 GMT (138kb)
[v4] Tue, 23 Apr 2024 15:40:29 GMT (139kb)
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