发表机构
Qiuzhen College Yau Mathematical Sciences Center Tsinghua University; Tsinghua University（清华大学丘成桐数学科学中心; 清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对非弱凸可定义优化，本文证明有界噪声不足以保证规避尖锐排斥点，并构造反例；同时证明高斯噪声可确保几乎必然规避任意有限尖锐排斥点集。
AI 中文摘要
受Bianchi、Hachem和Schechtman提出的开放问题（Bianchi等人，2024年，备注4）的启发，我们证明在非弱凸的可定义设定中，具有条件四阶矩控制的全向噪声并不足以保证对尖锐排斥临界点的普遍几乎必然规避。我们通过构造一个全局Lipschitz、强制、可定义但非弱凸的目标函数来展示这一失败。对于该目标函数及其足够小的线性扰动，独立噪声均匀分布在$\mathbb{R}^2$中一个球上的随机次梯度下降，从指定球内的任意初始点出发，以概率1收敛到一个非最小的尖锐排斥临界点。一个二循环论证建立了重新缩放迭代的一致界，从而得出收敛到该尖锐排斥临界点的结论。相反，对于局部Lipschitz可定义目标函数，高斯噪声确保同时几乎必然规避任何有限集合的尖锐排斥点。通过构造差商函数，我们证明以正概率收敛到任何此类点将产生一个具有期望下降0的平稳分布，而极限函数的Fréchet次梯度范数的全局正下界在该分布下强制期望下降严格为正。这一矛盾证明了规避性，其中处处为正的高斯密度发挥了关键作用。
英文摘要
Motivated by the open question of Bianchi, Hachem, and Schechtman (Bianchi et al., 2024, Remark 4), we show that in the non-weakly convex definable setting, omnidirectional noise with conditional fourth-moment control does not suffice for universal almost-sure avoidance of sharply repulsive critical points. We demonstrate this failure by constructing a globally Lipschitz, coercive, definable objective that is not weakly convex. For this objective and its sufficiently small linear perturbations, stochastic subgradient descent with independent noise uniformly distributed on a ball in $\mathbb{R}^2$ converges with probability $1$ to a nonminimal sharply repulsive critical point from every initial point in a specified ball. A two-cycle argument establishes uniform bounds on the rescaled iterates, yielding convergence to the sharply repulsive critical point. In contrast, for locally Lipschitz definable objectives, Gaussian noise ensures simultaneous almost-sure avoidance of any finite collection of sharply repulsive points. Through a construction of difference quotient functions, we show that convergence to any such point with positive probability would produce a stationary distribution with expected descent $0$, whereas a global positive lower bound on the Fréchet subgradient norms of the limiting functions forces strictly positive expected descent under the same distribution. This contradiction proves avoidance, with the everywhere positive Gaussian density playing a key role.
