Theoretical properties of the global optimizer of two-layer Neural Network

Digvijay Boob, Guanghui Lan

Abstract: In this paper, we study the problem of optimizing a two-layer artificial neural network that best fits a training dataset. We look at this problem in the setting where the number of parameters is greater than the number of sampled points. We show that for a wide class of differentiable activation functions (this class involves most nonlinear functions and excludes piecewise linear functions), we have that arbitrary first-order optimal solutions satisfy global optimality provided the hidden layer is non-singular. We essentially show that these non-singular hidden layer matrix satisfy a ``"good" property for these big class of activation functions. Techniques involved in proving this result inspire us to look at a new algorithmic, where in between two gradient step of hidden layer, we add a stochastic gradient descent (SGD) step of the output layer. In this new algorithmic framework, we extend our earlier result and show that for all finite iterations the hidden layer satisfies the``good" property mentioned earlier therefore partially explaining success of noisy gradient methods and addressing the issue of data independency of our earlier result. Both of these results are easily extended to hidden layers given by a flat matrix from that of a square matrix. Results are applicable even if network has more than one hidden layer provided all inner hidden layers are arbitrary, satisfy non-singularity, all activations are from the given class of differentiable functions and optimization is only with respect to the outermost hidden layer. Separately, we also study the smoothness properties of the objective function and show that it is actually Lipschitz smooth, i.e., its gradients do not change sharply. We use smoothness properties to guarantee asymptotic convergence of $O(1/\text{number of iterations})$ to a first-order optimal solution.

TL;DR: This paper talks about theoretical properties of first-order optimal point of two layer neural network in over-parametrized case

Keywords: Non-convex optimization, Two-layer Neural Network, global optimality, first-order optimality