arXiv：cs.AI· Guy Smorodinsky, Sveta Gimpleson, Itay Safran·· 13 小时前
非线性神经网络中 GD 收敛速率的对抗鲁棒性视角研究
On the Rate of Convergence of GD in Non-linear Neural Networks: An Adversarial Robustness Perspective
AI 导读
研究证明，即便在仅含两个神经元 ReLU 网络和两个训练样本的最简二分类设定下，梯度下降（GD）虽能收敛到最优鲁棒性间隔，但收敛速率极慢，严格为 Θ(1/ln(t))。这是首个针对非线性模型鲁棒性间隔收敛速率的显式下界，仿真显示该失效模式在多种自然网络初始化下普遍存在且速率一致。
正文
Abstract:We study the convergence dynamics of Gradient Descent (GD) in a minimal binary classification setting, consisting of a two-neuron ReLU network and two training instances. We prove that even under these strong simplifying assumptions, while GD successfully converges to an optimal robustness margin, effectively maximizing the distance between the decision boundary and the training points, this convergence occurs at a prohibitively slow rate, scaling strictly as $\Theta(1/\ln(t))$. To the best of our knowledge, this establishes the first explicit lower bound on the convergence rate of the robustness margin in a non-linear model. Through empirical simulations, we further demonstrate that this inherent failure mode is pervasive, exhibiting the exact same tight convergence rate across multiple natural network initializations. Our theoretical guarantees are derived via a rigorous analysis of the GD trajectories across the distinct activation patterns of the model. Specifically, we develop tight control over the system's dynamics to bound the trajectory of the decision boundary, overcoming the primary technical challenge introduced by the non-linear nature of the architecture.
| Subjects: | Machine Learning (cs.LG); Artificial Intelligence (cs.AI) |
| Cite as: | arXiv:2603.02095 [cs.LG] |
| (or arXiv:2603.02095v2 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2603.02095 arXiv-issued DOI via DataCite |
Submission history
From: Guy Smorodinsky [view email]
[v1]
Mon, 2 Mar 2026 17:13:33 UTC (38 KB)
[v2]
Thu, 8 Oct 2026 18:50:56 UTC (38 KB)
来源：arXiv：cs.AI · arxiv.org