Academic Report of School of Mathematical Sciences [2026] No. 109

(Series Report for High-Level University Construction No. 1368)

Title：Stochastic Subgradient Descent at Sharply Repulsive Points: A Bounded Noise Counterexample and Gaussian Noise Avoidance

Speaker：Assistant Professor  (Tsinghua University)

Time：10:00-11:00, Nov. 23, 2026

Location：Room 601, Huiyan Building, Yuehai Campus

Abstract: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 converges with probability 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, whereas a global positive lower bound on the Fr\'echet 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.

Speaker Profile：Hu Jiang is an assistant professor at the Center for Mathematical Sciences at Tsinghua University. His primary research interests include mathematical optimization, machine learning, and AI4Math. He has published papers in journals and conferences such as the SIAM series, the IEEE series, Numerische Mathematik, MOR, JMLR, NeurIPS, and ICML, and has co-authored the textbooks Optimization: Modeling, Algorithms, and Theory and Computational Methods in Optimization. He received the sole Best Paper Award at ICASSP 2024, the flagship IEEE conference on signal processing.

Faculty and students are welcome to attend!

Invited by: Qu Zheng

School of Mathematical Sciences

November 3, 2026