发表机构
Beijing International Center for Mathematical Research, Peking University; School of Mathematical Sciences, Beijing International Center for Mathematical Research, Center for Quantitative Biology, Center for Machine Learning Research, Institute for Artificial Intelligence, Peking University; Academy for Multidisciplinary Studies, Capital Normal University（北京大学北京国际数学研究中心; 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、人工智能研究院; 首都师范大学跨学科研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一致非退化性概念，通过下降-上升不等式刻画，并设计交替下降-上升算法，实现全局线性收敛，统一多种经典优化框架。
AI 中文摘要
经典模型，如光滑强凸最小化和强凸-强凹优化，结合了两种曲率信息：曲率与零保持均匀分离，且其正负方向是预先指定的。一致非退化性解耦了这些角色，保留与零的分离性，同时允许曲率方向在定义域内变化。我们识别出一族下降-上升不等式（DAI），该不等式刻画了一致非退化性，并自然引出了交替下降-上升（ADA）方法，这是一种简单的显式方法，交替进行全梯度下降和上升。ADA 从任意初始化出发全局且线性收敛到唯一驻点，具有无维数的一阶复杂度。该框架将光滑强凸情形作为极限恢复，并涵盖强凸-强凹和线性约束情形，突出一致非退化性作为各种经典机制中共同的一阶结构。
英文摘要
Classical models such as smooth strongly convex minimization and strongly convex--strongly concave optimization combine two types of curvature information: curvature remains uniformly separated from zero, and its positive and negative orientations are prescribed in advance. Uniform nondegeneracy disentangles these roles, retaining the separation from zero while allowing the curvature orientation to vary across the domain. We identify a family of descent--ascent inequalities (DAI) that characterizes uniform nondegeneracy and naturally gives rise to Alternating Descent Ascent (ADA), a simple explicit method that alternates full-gradient descent and ascent. ADA converges globally and linearly to the unique stationary point from arbitrary initialization, with dimension-free first-order complexity. The framework recovers the smooth strongly convex setting as a limit and encompasses the strongly convex--strongly concave and linearly constrained settings, highlighting uniform nondegeneracy as a common first-order structure across various classical regimes.
