AI 中文总结
该研究针对有限最大值复合优化问题，开发确定性分支准则，证明可识别流形缺失的条件在过参数化鲁棒低秩恢复等场景中大概率成立，指出现代优化中可识别流形缺失或为常态。
AI 中文摘要
在非光滑优化中，可识别集描述了收敛到指定临界点的序列最终到达的局部区域。当该集合是目标函数限制为$C^2$函数的$C^2$流形时，它被称为可识别流形。其吸引力在于：许多一阶方法可在有限次迭代中识别这些流形，之后迭代进入问题实际光滑的区域，因此光滑优化的许多强大工具和保证可自然迁移到非光滑场景。由于这些特性，现有大量工作聚焦于刻画保证其存在的条件。本研究探讨互补问题：在何种条件下临界点不存在任何可识别流形？我们通过为一类广泛的有限最大值复合优化问题开发确定性分支准则来回答该问题，刻画临界点何时不存在可识别流形。该准则在某些问题类中出人意料地温和：对于过参数化的鲁棒低秩恢复问题，它以高概率成立；对于随机极小极大回归的常见插值器，它几乎必然成立，这表明在现代优化中，可识别流形的缺失可能是常态而非例外。
英文摘要
In nonsmooth optimization, identifiable sets describe the local region eventually reached by sequences converging to a prescribed critical point. When such a set is a $C^2$ manifold on which the objective restricts to a $C^2$ function, it is called an identifiable manifold. Their appeal lies in what they enable: many first-order methods identify these manifolds in finitely many iterations, after which the iterates enter a region in which the problem is effectively smooth. Consequently, many powerful tools and guarantees from smooth optimization transplant naturally to the nonsmooth setting. Owing to these properties, much existing work has focused on characterizing conditions that guarantee their existence. In this work, we study a complementary question: under what conditions is a critical point devoid of any identifiable manifold? We answer this by developing a deterministic branching criterion for a broad class of finite-max composite optimization problems, characterizing when a critical point admits no identifiable manifold. This criterion is surprisingly mild in certain classes of problems: it holds with high probability for overparameterized robust low-rank recovery and almost surely at common interpolators of random minimax regression, suggesting that the absence of identifiable manifolds may be the rule rather than the exception in modern optimization.
Comments57 pages, 6 figures
