Uncertainty Quantification From Scaling Laws in Deep Neural Networks

Quantifying the uncertainty from machine learning analyses is critical to their use in the physical sciences. In this work we focus on uncertainty inherited from the initialization distribution of neural networks. We compute the mean μL and variance σL2 of the test loss L for an ensemble of multi-layer perceptrons (MLPs) with neural tangent kernel (NTK) initialization in the infinite-width limit, and compare empirically to the results from finite-width networks for three example tasks: MNIST classification, CIFAR classification and calorimeter energy regression. We observe scaling laws as a function of training set size ND for both μL and σL, but find that the coefficient of variation εL equiv σL/μL becomes independent of ND at both infinite and finite width for sufficiently large ND. This implies that the coefficient of variation of a finite-width network may be approximated by its infinite-width value, and may in principle be calculable using finite-width perturbation theory.