The $\varphi$ Curve: The Shape of Generalization through the Lens of Norm-based Capacity Control

TL;DR

This paper investigates how norm-based capacity measures explain the shape of learning curves in over-parameterized models, revealing a phase transition without double descent, contrasting classical theories.

Contribution

It provides a precise analysis of norm concentration in random features estimators and demonstrates the phase transition in test error without double descent behavior.

Findings

Learning curves exhibit a phase transition from under- to over-parameterization.

Classical U-shaped test error behavior is recovered with norm-based capacity measures.

No double descent behavior is observed in the studied models.

Abstract

Understanding how the test risk scales with model complexity is a central question in machine learning. Classical theory is challenged by the learning curves observed for large over-parametrized deep networks. Capacity measures based on parameter count typically fail to account for these empirical observations. To tackle this challenge, we consider norm-based capacity measures and develop our study for random features based estimators, widely used as simplified theoretical models for more complex networks. In this context, we provide a precise characterization of how the estimator's norm concentrates and how it governs the associated test error. Our results show that the predicted learning curve admits a phase transition from under- to over-parameterization, but no double descent behavior. This confirms that more classical U-shaped behavior is recovered considering appropriate capacity measures based on models norms rather than size. From a technical point of view, we leverage deterministic equivalence as the key tool and further develop new deterministic quantities which are of independent interest.