The Shape of Generalization through the Lens of Norm-based Capacity Control

Yichen Wang, Yudong Chen, Lorenzo Rosasco, Fanghui Liu

Keywords: generalization, norm-based capacity, deterministic equivalence

TL;DR: We provide a precise description on how the test risk scales with a suitable norm-based capacity when compared to a classical metric of model size.

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.

Student Paper: Yes

Submission Number: 7