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This research establishes a theoretical bridge between over-parameterized deep neural networks and kernel methods, proving that gradient-based training can inherit optimal learning dynamics under sufficient width. By deriving the first minimax-optimal rates for excess population risk in regression tasks, the study resolves a key gap in understanding the generalization properties of deep architectures compared to shallow ones. For practitioners and theorists, this confirms that standard gradient descent and SGD on wide networks can achieve performance comparable to kernel methods without requiring specialized training regimes.
Generalization in Deep Neural Networks: Minimax Rates for Gradient Methods
arXiv:2606.06772v1 Announce Type: cross Abstract: Understanding the generalization performance of over-parameterized neural networks has become a central topic in deep learning theory. While recent advances, particularly works under the Neural Tangent Kernel (NTK) regime, have shed light on the behavior of shallow architectures, the statistical generalization properties of deep neural networks (DNNs), especially in regression tasks, remain far less understood. In this paper, we make significant progress toward closing this gap by providing a comprehensive generalization analysis of DNNs trained using gradient-based methods. First, we establish, for the first time, a crucial connection between the learning dynamics of a DNN with smooth activation functions trained via gradient-based methods and those of kernel methods, showing that gradient-based methods on over-parameterized DNNs can fully inherit the favorable learning dynamics of their kernel counterparts. Building on this connection and the well-established optimality of kernel methods, we derive the first known minimax-optimal rates for the excess population risk of both gradient descent (GD) and stochastic gradient descent (SGD), under the assumption that network width scales polynomially with the sample size. Our results demonstrate that, with sufficient width, DNNs trained by GD or SGD can achieve generalization performance comparable to kernel-based methods.
Abstract:Understanding the generalization performance of over-parameterized neural networks has become a central topic in deep learning theory. While recent advances, particularly works under the Neural Tangent Kernel (NTK) regime, have shed light on the behavior of shallow architectures, the statistical generalization properties of deep neural networks (DNNs), especially in regression tasks, remain far less understood. In this paper, we make significant progress toward closing this gap by providing a comprehensive generalization analysis of DNNs trained using gradient-based methods. First, we establish, for the first time, a crucial connection between the learning dynamics of a DNN with smooth activation functions trained via gradient-based methods and those of kernel methods, showing that gradient-based methods on over-parameterized DNNs can fully inherit the favorable learning dynamics of their kernel counterparts. Building on this connection and the well-established optimality of kernel methods, we derive the first known minimax-optimal rates for the excess population risk of both gradient descent (GD) and stochastic gradient descent (SGD), under the assumption that network width scales polynomially with the sample size. Our results demonstrate that, with sufficient width, DNNs trained by GD or SGD can achieve generalization performance comparable to kernel-based methods.
Machine Learning (stat.ML); Artificial Intelligence (cs.AI); Machine Learning (cs.LG)
arXiv:2606.06772 [stat.ML]
(or arXiv:2606.06772v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2606.06772 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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