Abstract 

The performance of neural networks on high-dimensional data distributions suggests that it may be possible to parameterize a representation of a given highdimensional function with controllably small errors, potentially outperforming standard interpolation methods. We demonstrate, both theoretically and numerically, that this is indeed the case. We map the parameters of a neural network to a system of particles relaxing with an interaction potential determined by the loss function. We show that in the limit that the number of parameters n is large, the landscape of the mean-squared error becomes convex and the representation error in the function scales as In this limit, we prove a dynamical variant of the universal approximation theorem showing that the optimal representation can be attained by stochastic gradient descent, the algorithm ubiquitously used for parameter optimization in machine learning. In the asymptotic regime, we study the fluctuations around the optimal representation and show that they arise at a scale These fluctuations in the landscape identify the natural scale for the noise in stochastic gradient descent. Our results apply to both single and multi-layer neural networks, as well as standard kernel methods like radial basis functions.