T. Lin, P. Mertikopoulos, and M. I. Jordan. Working paper.
We propose and analyze exact and inexact regularized Newton-type methods for finding a global saddle point of a convex-concave unconstrained min-max optimization problem. Compared to their first-order counterparts, investigations of second-order methods for min-max optimization are relatively limited, as obtaining global rates of convergence with second-order information is much more involved. In this paper, we highlight how second-order information can be used to speed up the dynamics of dual extrapolation methods despite inexactness. Specifically, we show that the proposed algorithms generate iterates that remain within a bounded set and the averaged iterates converge to an $\varepsilon$-saddle point within $\mathcal{O}(\varepsilon^{-2/3})$ iterations in terms of a 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 requiring any compactness assumptions. Finally, we present a series of numerical experiments on synthetic and real data that demonstrate the efficiency of the proposed algorithms.
arXiv link: https://arxiv.org/abs/2210.12860