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We propose and analyze exact and inexact regularized Newton-type methods forfinding a global saddle point of \emph{convex-concave} unconstrained min-maxoptimization problems. Compared to first-order methods, our understanding ofsecond-order methods for min-max optimization is relatively limited, asobtaining global rates of convergence with second-order information is muchmore involved. In this paper, we examine how second-order information can beused to speed up extra-gradient methods, even under inexactness. Specifically,we show that the proposed algorithms generate iterates that remain within abounded set and the averaged iterates converge to an $\epsilon$-saddle pointwithin $O(\epsilon^{-2/3})$ iterations in terms of a restricted gap function.Our algorithms match the theoretically established lower bound in this contextand our analysis provides a simple and intuitive convergence analysis forsecond-order methods without any boundedness requirements. Finally, we presenta series of numerical experiments on synthetic and real data that demonstratethe efficiency of the proposed algorithms.

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