[2510.18120] Generalization Below the Edge of Stability: The Role of Data Geometry

Statistics > Machine Learning arXiv:2510.18120 (stat) [Submitted on 20 Oct 2025 (v1), last revised 5 Mar 2026 (this version, v2)] Title:Generalization Below the Edge of Stability: The Role of Data Geometry Authors:Tongtong Liang, Alexander Cloninger, Rahul Parhi, Yu-Xiang Wang View a PDF of the paper titled Generalization Below the Edge of Stability: The Role of Data Geometry, by Tongtong Liang and 3 other authors View PDF HTML (experimental) Abstract:Understanding generalization in overparameterized neural networks hinges on the interplay between the data geometry, neural architecture, and training dynamics. In this paper, we theoretically explore how data geometry controls this implicit bias. This paper presents theoretical results for overparametrized two-layer ReLU networks trained below the edge of stability. First, for data distributions supported on a mixture of low-dimensional balls, we derive generalization bounds that provably adapt to the intrinsic dimension. Second, for a family of isotropic distributions that vary in how strongly probability mass concentrates toward the unit sphere, we derive a spectrum of bounds showing that rates deteriorate as the mass concentrates toward the sphere. These results instantiate a unifying principle: When the data is harder to "shatter" with respect to the activation thresholds of the ReLU neurons, gradient descent tends to learn representations that capture shared patterns and thus finds solutions that generalize well. On the...