Abstract:Recent advances have significantly improved our understanding of the generalization performance of gradient descent (GD) methods in deep neural networks. A natural and fundamental question is whether GD can achieve generalization rates comparable to the minimax optimal rates established in the kernel setting. Existing results either yield suboptimal rates of $O(1/\sqrt{n})$, or focus on networks with smooth activation functions, incurring exponential dependence on network depth $L$. In this work, we establish optimal generalization rates for GD with deep ReLU networks by carefully trading off optimization and generalization errors, achieving only polynomial dependence on depth. Specifically, under the assumption that the data are NTK separable from the margin $\gamma$, we prove an excess risk rate of $\widetilde{O}(L^6 / (n \gamma^2))$, which aligns with the optimal SVM-type rate $\widetilde{O}(1 / (n \gamma^2))$ up to depth-dependent factors. A key technical contribution is our novel control of activation patterns near a reference model, enabling a sharper Rademacher complexity bound for deep ReLU networks trained with gradient descent.
From: Yuanfan Li [view email]
[v1]
Fri, 3 Oct 2025 07:22:36 UTC (39 KB)
[v2]
Mon, 24 Nov 2025 13:14:48 UTC (38 KB)
[v3]
Sat, 11 Apr 2026 02:29:48 UTC (159 KB)
[v4]
Tue, 2 Jun 2026 12:35:02 UTC (159 KB)
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