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.
Categories: math.OC cs.LG