Applied Math Seminar: On Large Deviations for Large Neural Networks

Seminar | September 26 | 11 a.m.-12 p.m. | 891 Evans Hall

 Joan Bruna, Courant Institute

 Department of Mathematics

Virtually all modern deep learning systems are trained with some form of local descent algorithm over a high-dimensional parameter space. Despite its apparent simplicity, the mathematical picture of the resulting setup contains several mysteries that combine statistics, approximation theory and optimization, all intertwined in a curse of dimensionality.

In order to make progress, authors have focused in the so-called ‘overparametrised’ regime, which studies asymptotic properties of the algorithm as the number of neurons grows. In particular, neural networks with a large number of parameters admit a mean-field description, which has recently served as a theoretical explanation for its favorable training properties. In this regime, gradient descent obeys a deterministic partial differential equation (PDE) that converges to a globally optimal solution for networks with a single hidden layer under appropriate assumptions.

In this talk, we will review recent progress on this problem, and argue that such framework might provide crucial robustness against the curse of dimensionality. First, we will describe a non-local mass transport dynamics that leads to a modified PDE with the same minimizer, that can be implemented as a stochastic neuronal birth-death process, and such that it provably accelerates the rate of convergence in the mean-field limit. Next, such dynamics fit naturally within the framework of total-variation regularization, which following [Bach’17] have fundamental advantages in the high-dimensional regime. We will discuss a unified framework that controls both optimization, approximation and generalisation errors using large deviation principles, and discuss current open problems in this research direction.

Joint work with G. Rotskoff (NYU), and E. Vanden-Eijnden (NYU).