As machine learning has developed, its methodologies have become increasingly mathematically sophisticated. For example, sampling and variational methods that were originally developed for application to mathematically diffcult problems in statistical mechanics are now commonplace in machine learning. Similarly, machine learning has co-opted many ideas from statistics, such as nonparametric Bayesian methods like Gaussian processes, Dirichlet processes, and completely random measures. In addition, graphical models and their associated inference techniques have emerged as a very important tool in a wide variety contexts. There are also interesting ideas that originated in machine learning rather than coming from other fields, ideas such as the kernelization of linear algorithms, and ideas in reinforcement and hierarchical reinforcement learning. This thesis reviews machine learning techniques of the types mentioned above that are of particular mathematical interest.
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