Global Minima of Overparameterized Neural Networks

Abstract

We explore some mathematical features of the loss landscape of overparameterized neural networks. A priori, one might imagine that the loss function looks like a typical function from ℝd to ℝ, in particular, that it has discrete global minima. In this paper, we prove that in at least one important way, the loss function of an overparameterized neural network does not look like a typical function. If a neural net has d parameters and is trained on n data points (xi, yi) ε ℝs × ℝr, with d > rn, we show that the locus M of global minima of L is usually not discrete but rather an (d-rn)-dimensional submanifold of ℝd. In practice, neural nets commonly have orders of magnitude more parameters than data points, so this observation implies that M is typically a very-high-dimensional submanifold of ℝd