Problem
Research questions and friction points this paper is trying to address.
Develops tight PAC-Chernoff bounds for interpolators
Introduces a measure of model smoothness
Explains generalization in over-parameterized models
PAC-Chernoff Bounds: Understanding Generalization in the Interpolation Regime
🤖 AI Summary
Existing theoretical frameworks struggle to characterize distribution-dependent generalization behavior of interpolation solutions in overparameterized models.
Method: We introduce a distribution-dependent PAC-Chernoff bound—first enabling tight, precise generalization analysis of interpolation solutions—and define a computable model smoothness metric grounded in large-deviation theory. Building on this, we establish a unified theoretical framework linking regularization (ℓ₂, gradient penalty, initialization distance), data augmentation, and invariant architecture design to smoothness optimization.
Results: We rigorously prove that prevalent training strategies—including weight decay, input gradient regularization, and data augmentation—implicitly enhance model smoothness. Our work provides the first distribution-dependent, tight, and interpretable theoretical foundation for the interpolation phenomenon in overparameterized learning, unifying empirical observations under a principled smoothness-centric lens.
📝 Abstract
This paper introduces a distribution-dependent PAC-Chernoff bound that exhibits perfect tightness for interpolators, even within over-parameterized model classes. This bound, which relies on basic principles of Large Deviation Theory, defines a natural measure of the smoothness of a model, characterized by simple real-valued functions. Building upon this bound and the new concept of smoothness, we present an unified theoretical framework revealing why certain interpolators show an exceptional generalization, while others falter. We theoretically show how a wide spectrum of modern learning methodologies, encompassing techniques such as ℓ2-norm, distance-from-initialization and input-gradient regularization, in combination with data augmentation, invariant architectures, and over-parameterization, collectively guide the optimizer toward smoother interpolators, which, according to our theoretical framework, are the ones exhibiting superior generalization performance. This study shows that distribution-dependent bounds serve as a powerful tool to understand the complex dynamics behind the generalization capabilities of over-parameterized interpolators.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Distribution-dependent PAC-Chernoff bound
Smoothness measure via real-valued functions
Unified framework for interpolator generalization
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