Stochastic Mirror Descent in Average Ensemble Models

TL;DR

This paper investigates the behavior of stochastic mirror descent in mean-field ensemble models, deriving a PDE for the parameter distribution evolution and analyzing the influence of mirror potentials on training dynamics and performance.

Contribution

It generalizes previous results for SGD to SMD, deriving a PDE for the continuous limit and analyzing the impact of mirror potentials on training dynamics.

Findings

Derived a nonlinear PDE for the distribution evolution in large networks

Showed the mirror potential influences training via a Riemannian metric

Numerical simulations illustrate the effect of mirror potentials on classification performance

Abstract

The stochastic mirror descent (SMD) algorithm is a general class of training algorithms, which includes the celebrated stochastic gradient descent (SGD), as a special case. It utilizes a mirror potential to influence the implicit bias of the training algorithm. In this paper we explore the performance of the SMD iterates on mean-field ensemble models. Our results generalize earlier ones obtained for SGD on such models. The evolution of the distribution of parameters is mapped to a continuous time process in the space of probability distributions. Our main result gives a nonlinear partial differential equation to which the continuous time process converges in the asymptotic regime of large networks. The impact of the mirror potential appears through a multiplicative term that is equal to the inverse of its Hessian and which can be interpreted as defining a gradient flow over an appropriately defined Riemannian manifold. We provide numerical simulations which allow us to study and characterize the effect of the mirror potential on the performance of networks trained with SMD for some binary classification problems.