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 $ε$-saddle point within $O(ε^{-2/3})$ 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 $O(\log\log(1/ε))$ 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 $O(\log\log(1/ε))$ 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.

翻译：针对凸凹无约束极小极大优化问题的全局鞍点求解，本文提出并分析了几种非精确正则化牛顿型方法。与一阶方法相比，我们对极小极大优化二阶方法的理解仍相对有限，因为利用二阶信息获得全局收敛速率往往更为复杂。本文研究了在非精确条件下，如何利用二阶信息加速外梯度方法。特别地，我们证明了所提方法生成的迭代点始终位于有界集合内，且平均迭代点通过限制间隙函数在$O(ε^{-2/3})$次迭代内收敛至$ε$-鞍点。同时，我们为每次迭代的子问题求解提供了简洁的计算流程，仅需单次Schur分解与$O(\log\log(1/ε))$次拟上三角线性方程组求解。因此，本方法通过消除现有基于线搜索的二阶极小极大优化方法中所需的$O(\log\log(1/ε))$次Schur分解因子，实现了计算效率提升。最后，我们在合成数据与真实数据上进行了实验，验证了所提方法的有效性。