We propose and analyze exact and 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 algorithms generate iterates that remain within a bounded set and the averaged iterates converge to an $\epsilon$-saddle point within $O(\epsilon^{-2/3})$ iterations in terms of a restricted gap function. Our algorithms match the theoretically established lower bound in this context and our analysis provides a simple and intuitive convergence analysis for second-order methods without any boundedness requirements. Finally, we present a series of numerical experiments on synthetic and real data that demonstrate the efficiency of the proposed algorithms.

翻译：我们提出并分析了精确与非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，我们对二阶方法在极小极大优化中的理解相对有限，因为利用二阶信息获得全局收敛速率更为复杂。本文研究了如何在非精确性条件下利用二阶信息加速额外梯度方法。具体地，我们证明所提算法生成的迭代点始终保持在有界集内，且平均迭代点关于受限间隙函数在$O(\epsilon^{-2/3})$次迭代内收敛至$\epsilon$-鞍点。我们的算法匹配了该问题的理论下界，分析过程简洁直观，无需引入任何有界性假设。最后，我们在合成数据与真实数据上开展数值实验，验证了所提算法的有效性。