Optimisation & Generalisation in Networks of Neurons
This work addresses foundational problems in machine learning theory for researchers, but it is incremental as it builds on existing optimization and generalization frameworks.
The thesis tackles the optimization and generalization foundations of neural networks by proposing a new theoretical framework for architecture-dependent optimization algorithms and establishing a correspondence between network ensembles and individual networks, leading to methods that transfer hyperparameters and PAC-Bayesian generalization theorems.
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.