Solving Convex-Concave Problems with $\tilde{\mathcal{O}}(\epsilon^{-4/7})$ Second-Order Oracle Complexity

TL;DR

This paper introduces a new second-order algorithm for convex-concave minimax problems that improves the oracle complexity from (^{-2/3}) to (^{-4/7}), advancing the efficiency of solving such problems.

Contribution

The paper presents an improved second-order method with (^{-4/7}) complexity for convex-concave minimax problems, extending optimal convex optimization techniques.

Findings

Achieved a new complexity bound of (^{-4/7}) for second-order oracle calls.

Demonstrated the method's applicability as a second-order (Catalyst) acceleration framework.

Extended the approach to lazy Hessian algorithms for minimax problems.

Abstract

Previous algorithms can solve convex-concave minimax problems $\min_{x \in \mathcal{X}} \max_{y \in \mathcal{Y}} f(x,y)$ with $\mathcal{O}(\epsilon^{-2/3})$ second-order oracle calls using Newton-type methods. This result has been speculated to be optimal because the upper bound is achieved by a natural generalization of the optimal first-order method. In this work, we show an improved upper bound of $\tilde{\mathcal{O}}(\epsilon^{-4/7})$ by generalizing the optimal second-order method for convex optimization to solve the convex-concave minimax problem. We further apply a similar technique to lazy Hessian algorithms and show that our proposed algorithm can also be seen as a second-order ``Catalyst'' framework (Lin et al., JMLR 2018) that could accelerate any globally convergent algorithms for solving minimax problems.