强凸-强凹极小极大优化的近最优纯单循环外梯度方法
Near-Optimal Single-Loop Predictor--Corrector Extragradient Method for Strongly Convex--Strongly Concave Minimax Optimization
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中文总结 AI 辅助
针对强凸-强凹极小极大优化，提出一种固定参数的纯单循环阻尼外梯度方法，实现最后迭代线性收敛，达到最优条件数阶复杂度，数值实验验证其有效性。
中文摘要 AI 辅助
我们研究确定性无约束设置下具有一般非线性耦合的光滑强凸-强凹极小极大优化问题。我们提出了一种纯单循环阻尼外梯度方法，该方法使用固定参数，并在一次初始化查询后每次迭代进行两次新的全梯度评估。该方法采用辅助反馈递归，不需要内层求解、精度调度或分阶段重启。我们建立了最后迭代线性收敛性，并表明将到鞍点的平方欧氏距离减小到其初始值的$\varepsilon$比例需要$O(\sqrt{\kappa_x\kappa_y}\log(2\kappa_x\kappa_y/\varepsilon))$次全梯度查询，其中$\kappa_x=L/\mu_x$和$\kappa_y=L/\mu_y$。该界限通过固定的显式更新达到了最优条件数阶（至多相差对数因子）。数值实验证明了该方法的有效性。
英文摘要
We study smooth strongly convex--strongly concave minimax optimization in the deterministic unconstrained setting, without assuming a bilinear or separable structure. Although existing multi-loop methods attain near-optimal condition-number dependence, standard single-loop methods generally exhibit a substantial complexity gap. To close this gap, we propose the Single-Loop Predictor--Corrector Extragradient Method with Damped Momentum (PCE-DM), which combines an extragradient prediction--correction scheme with a novel auxiliary feedback recursion for the weaker-curvature variable. PCE-DM uses fixed parameters and two new full-gradient evaluations per iteration after one initialization query, while requiring no inner solves, accuracy schedules, or staged restarts. We develop a Lyapunov analysis that controls the predictor--corrector mismatch through corrected-gradient increments and establish last-iterate linear convergence. Specifically, PCE-DM computes an $\varepsilon$-accurate relative solution, measured by the squared Euclidean distance to the saddle point, within $\mathcal{O}\!\left(\sqrt{κ_xκ_y} \log(2κ_xκ_y/\varepsilon)\right)$ full-gradient queries. This result closes the condition-number complexity gap between standard single-loop methods and near-optimal multi-loop methods, matching the known lower-bound order up to logarithmic factors while retaining fixed, explicit single-loop updates. Numerical experiments on regularized linear regression and AUC maximization demonstrate the computational efficiency of PCE-DM.
