Explicit Second-Order Min-Max Optimization Methods with Optimal Convergence Guarantee

• URL: http://arxiv.org/abs/2210.12860v4
• Date: Tue, 23 Apr 2024 15:40:29 GMT
• Title: Explicit Second-Order Min-Max Optimization Methods with Optimal Convergence Guarantee
• Authors: Tianyi Lin, Panayotis Mertikopoulos, Michael I. Jordan,
• Abstract summary: We propose and analyze inexact regularized Newton-type methods for finding a global saddle point of emphcon unconstrained min-max optimization problems. We show that the proposed methods generate iterates that remain within a bounded set and that the iterations converge to an $epsilon$-saddle point within $O(epsilon-2/3)$ in terms of a restricted function.
• Score: 86.05440220344755
• 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 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.

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