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.

翻译：我们提出并分析了若干种不精确正则化牛顿型方法，用于求解无约束凸-凹极小极大优化问题的全局鞍点。与一阶方法相比，我们对二阶极小极大优化方法的理解相对有限，因为利用二阶信息获得全局收敛率要复杂得多。本文研究了如何在不精确条件下利用二阶信息加速外梯度方法。具体而言，我们证明所提方法生成的迭代点始终位于有界集内，且平均迭代点能在$O(\epsilon^{-2/3})$次迭代内收敛到$\epsilon$-鞍点（基于限制性间隙函数），这一结果与该问题的理论下界相匹配。我们还提供了求解每步子问题的简单方案：每次迭代仅需一次Schur分解，并在拟上三角系统中调用$O(\log\log(1/\epsilon))$次线性系统求解器。因此，我们的方法将现有基于线搜索的二阶极小极大优化方法所需的Schur分解次数降低了$O(\log\log(1/\epsilon))$因子。最后，我们在合成数据与真实数据上进行的数值实验验证了所提方法的有效性。