均匀非退化条件下的一阶优化：几何、计算与信息
First-Order Optimization under Uniform Nondegeneracy: Geometry, Computation, and Information
- Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）
- School of Mathematical Sciences, Beijing International Center for Mathematical Research, Center for Quantitative Biology, Center for Machine Learning Research, Peking University（北京大学数学科学学院北京国际数学研究中心定量生物学中心机器学习研究中心）
- Academy for Multidisciplinary Studies, Capital Normal University（首都师范大学多学科研究院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出均匀非退化条件下的一阶优化理论，通过割线不等式和配对Moreau包络实现全局线性收敛，并引入认证可行性克服信息障碍。
AI中文摘要：
强凸最小化及其自然的非正定扩展到强凸-强凹极小极大问题，将曲率的定量控制与规定的曲率方向相结合。我们通过仅保留均匀非退化性来解开这两个作用：曲率保持与零均匀分离，但可能具有任一符号，没有规定的正负分裂。令人惊讶的是，熟悉理论的大部分内容仍然重新出现。我们首先通过一阶割线不等式推导出一个内在公式，使$\mathbb{R}^d$上的梯度成为全局双Lipschitz同胚，并产生唯一的驻点。然后，我们将带符号的Moreau包络配对，构建一个光滑的标量度量函数，在零阶和一阶上恢复缺失的下降几何，由此产生的配对近端下降方法实现了全局线性收敛，具有无维数的一阶预言机复杂度。同时，我们表明这种可处理性在受限域上可能会失效：仅假设域中存在驻点可能导致维数灾难，即使可以访问无限阶预言机。为了克服这一信息障碍，我们引入了认证可行性，这是一种可观测的局部化条件，能够实现可行的延拓。总之，这些结果建立了一个在均匀非退化条件下的一阶优化理论，涵盖了几何、计算和信息。
英文摘要：
Strongly convex minimization and its natural indefinite extension to strongly convex--strongly concave minimax problems combine quantitative control of curvature with a prescribed curvature orientation. We disentangle these two roles by retaining uniform nondegeneracy alone: curvature remains uniformly separated from zero but may have either sign, with no prescribed positive--negative splitting. Surprisingly, a large part of the familiar theory nevertheless re-emerges. We first derive an intrinsic formulation through first-order secant inequalities, making the gradient on $\mathbb{R}^d$ a global bi-Lipschitz homeomorphism and yielding a unique stationary point. We then pair signed Moreau envelopes to construct a smooth scalar merit that recovers the missing descent geometry at both zeroth and first order, and the resulting paired proximal descent method achieves global linear convergence with dimension-free first-order oracle complexity. Meanwhile, we show that this tractability can break down on restricted domains: merely assuming the existence of a stationary point in the domain may lead to the curse of dimensionality, even with access to an infinite-order oracle. To overcome this information barrier, we introduce certified feasibility, an observable localization condition that enables feasible continuation. Together, these results establish a first-order optimization theory under uniform nondegeneracy that spans geometry, computation, and information.
