Re-examining Double Descent and Scaling Laws under Norm-based Capacity via Deterministic Equivalence

We investigate double descent and scaling laws in terms of weights rather than the number of parameters. Specifically, we analyze linear and random features models using the deterministic equivalence approach from random matrix theory. We precisely characterize how the weights norm concentrate around deterministic quantities and elucidate the relationship between the expected test error and the norm-based capacity (complexity). Our results rigorously answer whether double descent exists under norm-based capacity and reshape the corresponding scaling laws. Moreover, they prompt a rethinking of the data-parameter paradigm - from under-parameterized to over-parameterized regimes - by shifting the focus to norms (weights) rather than parameter count.