Information-Theoretic Foundations for Machine Learning

The progress of machine learning over the past decade is undeniable. In retrospect, it is both remarkable and unsettling that this progress was achievable with little to no rigorous theory to guide experimentation. Despite this fact, practitioners have been able to guide their future experimentation via observations from previous large-scale empirical investigations. In this work, we propose a theoretical framework which attempts to provide rigor to existing practices in machine learning. To the theorist, we provide a framework which is mathematically rigorous and leaves open many interesting ideas for future exploration. To the practitioner, we provide a framework whose results are simple, and provide intuition to guide future investigations across a wide range of learning paradigms. Concretely, we provide a theoretical framework rooted in Bayesian statistics and Shannon's information theory which is general enough to unify the analysis of many phenomena in machine learning. Our framework characterizes the performance of an optimal Bayesian learner as it learns from a stream of experience. Unlike existing analyses that weaken with increasing data complexity, our theoretical tools provide accurate insights across diverse machine learning settings. Throughout this work, we derive theoretical results and demonstrate their generality by apply them to derive insights specific to settings. These settings range from learning from data which is independently and identically distributed under an unknown distribution, to data which is sequential, to data which exhibits hierarchical structure amenable to meta-learning, and finally to data which is not fully explainable under the learner's beliefs (misspecification). These results are particularly relevant as we strive to understand and overcome increasingly difficult machine learning challenges in this endlessly complex world.