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

TL;DR

This paper introduces practical second-order algorithms for convex-concave min-max problems, achieving faster convergence rates and reduced computational complexity compared to existing methods, with demonstrated empirical efficiency.

Contribution

It develops inexact regularized Newton-type methods that improve convergence and computational efficiency for min-max optimization, with novel analysis and routines.

Findings

Iterates remain bounded during optimization.

Convergence to an $ ext{epsilon}$-saddle point within $O( ext{epsilon}^{-2/3})$ iterations.

Reduced number of Schur decompositions needed, improving existing methods.

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.