The Price of Robustness: Stable Classifiers Need Overparameterization

TL;DR

This paper explores the trade-off between overparameterization and stability in classifiers, showing that high stability requires substantial overparameterization and providing theoretical bounds supported by experiments.

Contribution

The paper establishes a generalization bound linking class stability to overparameterization and extends robustness results to discontinuous classifiers.

Findings

Stability increases with model size and correlates with test performance.

Overparameterization is necessary for high stability in classifiers.

Traditional norm-based measures are less informative than margin-based stability.

Abstract

The relationship between overparameterization, stability, and generalization remains incompletely understood in the setting of discontinuous classifiers. We address this gap by establishing a generalization bound for finite function classes that improves inversely with class stability, defined as the expected distance to the decision boundary in the input domain (margin). Interpreting class stability as a quantifiable notion of robustness, we derive as a corollary a law of robustness for classification that extends the results of Bubeck and Sellke beyond smoothness assumptions to discontinuous functions. In particular, any interpolating model with $p \approx n$ parameters on $n$ data points must be unstable, implying that substantial overparameterization is necessary to achieve high stability. We obtain analogous results for parameterized infinite function classes by analyzing a stronger robustness measure derived from the margin in the codomain, which we refer to as the normalized co-stability. Experiments support our theory: stability increases with model size and correlates with test performance, while traditional norm-based measures remain largely uninformative.