Sufficient Conditions for Stability of Minimum-Norm Interpolating Deep ReLU Networks
arXiv:2602.13910v1 Announce Type: new Abstract: Algorithmic stability is a classical framework for analyzing the generalization error of learning algorithms. It predicts that an algorithm has small generalization error if it is insensitive to small perturbations in the training set such as the removal or replacement of a training point. While stability has been demonstrated for numerous well-known algorithms, this framework has had limited success in analyses of deep neural networks. In this paper we study the algorithmic stability of deep ReLU homogeneous neural networks that achieve zero training error using parameters with the smallest $L_2$ norm, also known as the minimum-norm interpolation, a phenomenon that can be observed in overparameterized models trained by gradient-based algorithms. We investigate sufficient conditions for such networks to be stable. We find that 1) such networks are stable when they contain a (possibly small) stable sub-network, followed by a layer with a
arXiv:2602.13910v1 Announce Type: new Abstract: Algorithmic stability is a classical framework for analyzing the generalization error of learning algorithms. It predicts that an algorithm has small generalization error if it is insensitive to small perturbations in the training set such as the removal or replacement of a training point. While stability has been demonstrated for numerous well-known algorithms, this framework has had limited success in analyses of deep neural networks. In this paper we study the algorithmic stability of deep ReLU homogeneous neural networks that achieve zero training error using parameters with the smallest $L_2$ norm, also known as the minimum-norm interpolation, a phenomenon that can be observed in overparameterized models trained by gradient-based algorithms. We investigate sufficient conditions for such networks to be stable. We find that 1) such networks are stable when they contain a (possibly small) stable sub-network, followed by a layer with a low-rank weight matrix, and 2) such networks are not guaranteed to be stable even when they contain a stable sub-network, if the following layer is not low-rank. The low-rank assumption is inspired by recent empirical and theoretical results which demonstrate that training deep neural networks is biased towards low-rank weight matrices, for minimum-norm interpolation and weight-decay regularization.
Executive Summary
The article 'Sufficient Conditions for Stability of Minimum-Norm Interpolating Deep ReLU Networks' explores the algorithmic stability of deep ReLU homogeneous neural networks, focusing on those that achieve zero training error with the smallest L2 norm parameters. The study identifies sufficient conditions for stability, notably the presence of a stable sub-network followed by a low-rank weight matrix layer. The research highlights that stability is not guaranteed if the subsequent layer is not low-rank, aligning with empirical and theoretical findings that deep neural networks tend towards low-rank weight matrices during training.
Key Points
- ▸ Algorithmic stability is crucial for analyzing generalization error in learning algorithms.
- ▸ Deep ReLU networks with minimum-norm interpolation can be stable under specific conditions.
- ▸ Stability requires a stable sub-network followed by a low-rank weight matrix layer.
- ▸ Low-rank assumptions are supported by recent empirical and theoretical results.
Merits
Theoretical Insight
The article provides a rigorous theoretical framework for understanding the stability of deep neural networks, which is a significant advancement in the field of machine learning.
Empirical Alignment
The findings align with recent empirical observations, enhancing the credibility and applicability of the theoretical results.
Demerits
Limited Scope
The study focuses on a specific type of neural network (ReLU homogeneous networks) and may not be generalizable to all types of deep learning models.
Complexity
The conditions for stability are complex and may be challenging to apply in practical scenarios, limiting their immediate utility.
Expert Commentary
The article makes a significant contribution to the understanding of algorithmic stability in deep neural networks. By identifying specific conditions under which stability can be achieved, the research provides a valuable theoretical foundation for future studies. The emphasis on low-rank weight matrices is particularly noteworthy, as it aligns with recent empirical findings and suggests a bias in the training process of deep neural networks. However, the practical application of these findings may be limited by the complexity of the conditions and the specificity of the network types considered. Future research should aim to extend these insights to a broader range of neural network architectures and training scenarios to enhance their practical utility.
Recommendations
- ✓ Further empirical studies should validate the theoretical findings in diverse real-world applications.
- ✓ Researchers should explore the generalizability of these conditions to other types of neural networks and learning algorithms.
