Lower-bound analyses for nonconvex strongly-concave minimax optimization
problems have shown that stochastic first-order algorithms require at least
$\mathcal{O}(\varepsilon^{-4})$ oracle complexity to find an
$\varepsilon$-stationary point. Some works indicate that this complexity can be
improved to $\mathcal{O}(\varepsilon^{-3})$ when the loss gradient is Lipschitz
continuous. The question of achieving enhanced convergence rates under distinct
conditions, remains unresolved. In this work, we address this question for
optimization problems that are nonconvex in the minimization variable and
strongly concave or Polyak-Lojasiewicz (PL) in the maximization variable. We
introduce novel bias-corrected momentum algorithms utilizing efficient
Hessian-vector products. We establish convergence conditions and demonstrate a
lower iteration complexity of $\mathcal{O}(\varepsilon^{-3})$ for the proposed
algorithms. The effectiveness of the method is validated through applications
to robust logistic regression using real-world datasets.

针对非凸优化中最小最大优化问题，本研究提出了利用高效的 Hessian - 向量乘积的新型修正动量算法，建立了收敛条件并证明了所提算法的迭代复杂度为 O (ε^{-3})。通过在实际数据集上进行鲁棒的逻辑回归的应用验证了该方法的有效性。