Scaling description of generalization with number of parameters in deep learning

Supervised deep learning involves the training of neural networks with a large number N of parameters. For large enough N, in the so-called over-parametrized regime, one can essentially fit the training data points. Sparsity-based arguments would suggest that the generalization error increases as N grows past a certain threshold N ^∗. Instead, empirical studies have shown that in the over-parametrized regime, generalization error keeps decreasing with N. We resolve this paradox through a new framework. We rely on the so-called Neural Tangent Kernel, which connects large neural nets to kernel methods, to show that the initialization causes finite-size random fluctuations f_N-(fN) ∼ N^-1/4 of the neural net output function f_N around its expectation. These affect the generalization error for classification: Under natural assumptions, it decays to a plateau value in a power-law fashion ∼N ^-1/2. This description breaks down at a so-called jamming transition N = N ^∗. At this threshold, we argue that diverges. This result leads to a plausible explanation for the cusp in test error known to occur at N ^∗. Our results are confirmed by extensive empirical observations on the MNIST and CIFAR image datasets. Our analysis finally suggests that, given a computational envelope, the smallest generalization error is obtained using several networks of intermediate sizes, just beyond N ^∗, and averaging their outputs.