Optimisation & Generalisation in Networks of Neurons

TL;DR

This thesis develops new theoretical foundations for optimization and generalization in neural networks, proposing architecture-dependent algorithms and a Bayesian perspective on network ensembles to improve understanding and transferability.

Contribution

It introduces a novel framework for architecture-dependent optimization algorithms and establishes a new link between network ensembles and individual networks for generalization analysis.

Findings

Optimization methods transfer hyperparameters across problems.

Large networks and margins relate to Bayesian 'Bayes point' models.

Provides a new perspective on regularization in overparameterized networks.

Abstract

The goal of this thesis is to develop the optimisation and generalisation theoretic foundations of learning in artificial neural networks. On optimisation, a new theoretical framework is proposed for deriving architecture-dependent first-order optimisation algorithms. The approach works by combining a "functional majorisation" of the loss function with "architectural perturbation bounds" that encode an explicit dependence on neural architecture. The framework yields optimisation methods that transfer hyperparameters across learning problems. On generalisation, a new correspondence is proposed between ensembles of networks and individual networks. It is argued that, as network width and normalised margin are taken large, the space of networks that interpolate a particular training set concentrates on an aggregated Bayesian method known as a "Bayes point machine". This correspondence provides a route for transferring PAC-Bayesian generalisation theorems over to individual networks. More broadly, the correspondence presents a fresh perspective on the role of regularisation in networks with vastly more parameters than data.