Scaling description of generalization with number of parameters in deep learning

We provide a description for the evolution of the generalization performance of fixed-depth fully-connected deep neural networks, as a function of their number of parameters $N$. In the setup where the number of data points is larger than the input dimension, as $N$ gets large, we observe that increasing $N$ at fixed depth reduces the fluctuations of the output function $f_N$ induced by initial conditions, with $|\!|f_N-{\bar f}_N|\!|\sim N^{-1/4}$ where ${\bar f}_N$ denotes an average over initial conditions. We explain this asymptotic behavior in terms of the fluctuations of the so-called Neural Tangent Kernel that controls the dynamics of the output function. For the task of classification, we predict these fluctuations to increase the true test error $\epsilon$ as $\epsilon_{N}-\epsilon_{\infty}\sim N^{-1/2} + \mathcal{O}( N^{-3/4})$. This prediction is consistent with our empirical results on the MNIST dataset and it explains in a concrete case the puzzling observation that the predictive power of deep networks improves as the number of fitting parameters grows. This asymptotic description breaks down at a so-called jamming transition which takes place at a critical $N=N^*$, below which the training error is non-zero. In the absence of regularization, we observe an apparent divergence $|\!|f_N|\!|\sim (N-N^*)^{-\alpha}$ and provide a simple argument suggesting $\alpha=1$, consistent with empirical observations. This result leads to a plausible explanation for the cusp in test error known to occur at $N^*$. Overall, our analysis suggests that once models are averaged, the optimal model complexity is reached just beyond the point where the data can be perfectly fitted, a result of practical importance that needs to be tested in a wide range of architectures and data set.