Upper Bounds on the Generalization Error of Deep Learning Models via Local Robustness and Stability

Problem
The paper addresses the gap in existing generalization bounds for deep learning models, particularly in safety-critical applications, where robustness and generalization are paramount. Current robustness-based generalization bounds often yield vacuous estimates that do not reflect actual error rates, limiting their applicability in real-world scenarios. This issue is exacerbated by treating the robustness term as a global measure, neglecting its variability across different sub-regions of the input space. The authors propose a new approach to derive tighter upper bounds on generalization error, which is particularly relevant given the increasing deployment of deep learning models in critical domains. This work is a preprint and has not yet undergone peer review.

Method
The authors introduce a generalization bound that scales the robustness term based on the distribution of stable and unstable samples within input sub-regions. This approach allows for a more nuanced understanding of how local stability affects generalization performance. The proposed method incorporates both data-dependent and model-dependent factors, leading to tighter upper bounds on true error rates. The experiments utilize models trained on the ImageNet dataset, and the authors provide a detailed analysis of the robustness term’s contribution to the overall generalization bound. The training compute specifics are not disclosed, but the focus is on the theoretical framework and empirical validation of the bounds.

Results
The proposed bounds demonstrate significant improvements over existing methods, achieving the tightest estimates of generalization error. The authors report that their bounds remain consistently non-vacuous across various robust deep neural networks tested on the ImageNet dataset. While specific numerical results are not detailed in the abstract, the paper claims that the new bounds closely align with empirical performance, suggesting a substantial reduction in the gap between theoretical estimates and actual error rates compared to traditional approaches.

Limitations
The authors acknowledge that their approach, while improving upon existing bounds, may still be limited by the assumptions made regarding the distribution of stable and unstable samples. Additionally, the reliance on the 0-1 loss function may not generalize to all loss functions used in deep learning. The paper does not address potential computational overhead introduced by the proposed method, which could impact scalability in large-scale applications.

Why it matters
This work has significant implications for the development of more reliable deep learning models, particularly in safety-critical applications where understanding generalization and robustness is essential. By providing tighter bounds on generalization error, the proposed method enhances the interpretability and trustworthiness of deep learning systems. This advancement is crucial for future research in robust machine learning, as it lays the groundwork for more effective evaluation and deployment of models in real-world scenarios, as published in arXiv cs.AI.