Explicit Second-Order Min-Max Optimization: Practical Algorithms and Complexity Analysis

• URL: http://arxiv.org/abs/2210.12860v5
• Date: Mon, 29 Sep 2025 03:16:15 GMT
• Title: Explicit Second-Order Min-Max Optimization: Practical Algorithms and Complexity Analysis
• Authors: Tianyi Lin, Panayotis Mertikopoulos, Michael I. Jordan,
• Abstract summary: We propose and analyze several inexact regularized Newton-type methods for finding a global saddle point of emphconcave unconstrained problems.<n>Our method improves the existing line-search-based min-max optimization by shaving off an $O(loglog(1/eps)$ factor in the required number of Schur decompositions.
• Score: 71.05708939639537
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 can be much more involved. In this paper, we examine how second-order information is used to speed up extra-gradient methods, even under inexactness. In particular, 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. 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 conduct experiments on synthetic and real data to demonstrate the efficiency of the proposed methods.

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