New Optimisation Methods for Machine Learning

Aaron Defazio

Published 2015-10-09Version 1

A thesis submitted for the degree of Doctor of Philosophy of The Australian National University. In this work we introduce several new optimisation methods for problems in machine learning. Our algorithms broadly fall into two categories: optimisation of finite sums and of graph structured objectives. The finite sum problem is simply the minimisation of objective functions that are naturally expressed as a summation over a large number of terms, where each term has a similar or identical weight. Such objectives most often appear in machine learning in the empirical risk minimisation framework in the non-online learning setting. The second category, that of graph structured objectives, consists of objectives that result from applying maximum likelihood to Markov random field models. Unlike the finite sum case, all the non-linearity is contained within a partition function term, which does not readily decompose into a summation. For the finite sum problem, we introduce the Finito and SAGA algorithms, as well as variants of each. For graph-structured problems, we take three complementary approaches. We look at learning the parameters for a fixed structure, learning the structure independently, and learning both simultaneously. Specifically, for the combined approach, we introduce a new method for encouraging graph structures with the "scale-free" property. For the structure learning problem, we establish SHORTCUT, a O(n^{2.5}) expected time approximate structure learning method for Gaussian graphical models. For problems where the structure is known but the parameters unknown, we introduce an approximate maximum likelihood learning algorithm that is capable of learning a useful subclass of Gaussian graphical models.