发表机构
MIRIAI; Innopolis University（MIRIAI; 因诺波利斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对具有双线性耦合的Hölder光滑凸-凹极小极大优化问题，提出基于单调变分不等式递归滑动方案的滑动方法，建立收敛保证并通过数值实验验证其有效性。
AI 中文摘要
我们研究具有双线性耦合的凸-凹极小极大优化问题，形式为$\u0026#10214;_{x\u2208 \u0026#12008;}\u0026#10215;_{y\u2208 \u0026#12009;} \u0026#160; f(x)+\u27e8 y,\u0026#11982;\u0026#11990;x\u0026#160;-g(y)$，其中函数$f$和$g$具有Hölder连续的（次）梯度。该设置涵盖了广泛的情形，从具有有界次梯度变化的非光滑问题到具有Lipschitz连续梯度的光滑问题；对于耦合诱导正则化器中使用的光滑分量，假设其Lipschitz梯度常数在环境空间中成立。我们提出一种滑动方法，该方法通过按由$f$、$g$和双线性耦合算子各自属性确定的规定频率查询相关神谕，利用问题的复合结构。该方法基于单调变分不等式的递归滑动方案。我们在Hölder连续性下建立了收敛保证，并表明所得的复杂度界如何显式依赖于Hölder指数、Hölder常数、强凸参数以及耦合矩阵的谱性质。我们的分析涵盖非强凸和部分强凸情形。对于随机问题，我们在退化情形下证明了统一的期望间隙界，并且在环境光滑性和正有效曲率下，证明了收敛到显式噪声 floor。数值实验重现了预测的Hölder指数，并确认每个函数所需的梯度评估数量按其自身的光滑度级别划分，而非取两者中的较差者。一个断层摄影基准表明，当梯度评估比额外的矩阵-向量乘积更昂贵时，运行时间会获得增益。
英文摘要
We study convex-concave minimax optimization problems with bilinear coupling of the form $\min_{x\in \mathcal X}\max_{y\in \mathcal Y} \; f(x)+\langle y,\mathbf{B}x\rangle-g(y),$ where the functions $f$ and $g$ have Hölder continuous (sub)gradients. This setting covers a broad range of regimes, from nonsmooth problems with bounded subgradient variation to smooth problems with Lipschitz continuous gradients; for a smooth component used in the coupling-induced regularizer, its Lipschitz-gradient constant is assumed to hold in the ambient space. We propose a sliding method that exploits the composite structure of the problem by querying the oracles associated with $f$, $g$, and the bilinear coupling operator at prescribed frequencies determined by their individual properties. The method is based on a recursive sliding scheme for monotone variational inequalities. We establish convergence guarantees under Hölder continuity and show how the resulting complexity bounds depend explicitly on the Hölder exponents, Hölder constants, strong convexity parameters, and spectral properties of the coupling matrix. Our analysis covers nonstrongly convex and partially strongly convex regimes. For stochastic function oracles, we prove componentwise Hölder guarantees with separate variance costs: a uniform expected-gap bound on bounded domains and, when both functions are strongly convex on their feasible sets, a restarted Lyapunov guarantee without an anchor noise floor. Numerical experiments reproduce the predicted Hölder exponents and confirm that the number of gradient evaluations required for each function separates according to its own smoothness level rather than the worse of the two. A tomographic benchmark shows runtime gains when gradient evaluations are more expensive than the additional matrix-vector products.
