Uniform convergence may be unable to explain generalization in deep learning (1902.04742v4)

Abstract: Aimed at explaining the surprisingly good generalization behavior of overparameterized deep networks, recent works have developed a variety of generalization bounds for deep learning, all based on the fundamental learning-theoretic technique of uniform convergence. While it is well-known that many of these existing bounds are numerically large, through numerous experiments, we bring to light a more concerning aspect of these bounds: in practice, these bounds can {\em increase} with the training dataset size. Guided by our observations, we then present examples of overparameterized linear classifiers and neural networks trained by gradient descent (GD) where uniform convergence provably cannot "explain generalization" -- even if we take into account the implicit bias of GD {\em to the fullest extent possible}. More precisely, even if we consider only the set of classifiers output by GD, which have test errors less than some small $\epsilon$ in our settings, we show that applying (two-sided) uniform convergence on this set of classifiers will yield only a vacuous generalization guarantee larger than $1-\epsilon$. Through these findings, we cast doubt on the power of uniform convergence-based generalization bounds to provide a complete picture of why overparameterized deep networks generalize well.

• The paper demonstrates that uniform convergence bounds can paradoxically increase with larger training sets, contradicting conventional learning theory.
• It uses theoretical and empirical analysis on overparameterized models, including linear classifiers and deep neural networks trained with gradient descent.
• The study implies that alternative generalization frameworks, such as algorithmic stability, may be necessary to properly understand deep learning performance.

Commentary on "Uniform convergence may be unable to explain generalization in deep learning"

The paper authored by Vaishnavh Nagarajan and J. Zico Kolter critically examines the effectiveness of uniform convergence-based techniques in elucidating the generalization capabilities of overparameterized deep neural networks. The paper is prompted by observing that despite substantial overparameterization, where deep networks can perfectly fit arbitrary labels, they still manage to generalize effectively on unseen real-world data. This phenomenon contradicts traditional learning theories which rely heavily on uniform convergence principles.

The authors initiate their discourse by empirically demonstrating a fundamental issue that challenges existing uniform convergence-based bounds. Specifically, it's observed that these bounds, counterintuitively, can increase with the size of the training dataset. This observation is significant as it challenges the typical expectation that generalization error bounds should improve, or at least remain stable, with additional training data. Through comprehensive experiments, they highlight this trend and question the applicability of such bounds in practical deep learning scenarios.

To concretively establish the limitations of uniform convergence, the authors present theoretical examples where generalization cannot be explained using this approach. They consider both overparameterized linear classifiers and deep neural networks trained with gradient descent. Through rigorous analysis, they demonstrate scenarios wherein the uniform convergence bounds fail to remain non-vacuous, irrespective of implicit bias considerations inherent in methods like gradient descent. This result is stark in that even amongst hypotheses with zero or minimal test error, the application of uniform convergence results in bounds that are effectively no better than random chance.

The implications of this research are profound for both theoretical and practical dimensions of machine learning. Theoretically, it challenges the efficacy of uniform convergence as a universal tool for generalization analysis in high-capacity models like deep neural networks. Practically, it suggests that reliance on uniform convergence to gauge model generalization may be misplaced for certain architectures or under specific training regimens. This has cascading effects on how future models should be assessed, especially considering the complexity and dimensionality intrinsic to deep learning tasks.

Looking forward, the findings of Nagarajan and Kolter open the door for alternative avenues to understand and guarantee generalization in deep learning. Recognition of the limitations of uniform convergence may advocate for a shift towards algorithmic stability or other theories potentially better suited to capture the nuances of overparameterized models in practical settings.

The paper is a reminder of the intricate relationship between model capacity, training data, and theoretical bounds. It underscores the need for flexible frameworks to adapt to the evolving landscape of deep learning, which continues to transcend traditional boundaries and assumptions of statistical learning theory. In this landscape, the foundational question remains: what new principles will emerge to accurately describe and predict the behavior of increasingly sophisticated models and datasets?

This paper provides a necessary critique on the existing methodologies while paving the way for innovative new strategies to approach this enduring challenge in machine learning.