Title: Resource-Efficient Methods in Machine Learning
Description: In this thesis, we consider resource limitations on machine learning algorithms in a variety of settings. In the first two chapters, we study how to learn nonlinear model classes (monomials and neural nets) which are structured in various ways -- we consider sparse monomials and deep neural nets whose weight-matrices are low-rank respectively. These kinds of restrictions on the model class lead to gains in resource efficiency -- sparse and low-rank models are computationally easier to deploy and train. We prove that sparse nonlinear monomials are easier to learn (smaller sample complexity) while still remaining computationally efficient to both estimate and deploy, and we give both theoretical and empirical evidence for the benefit of novel nonlinear initialization schemes for low-rank deep networks. In both cases, we showcase a blessing of nonlinearity -- sparse monomials are in some sense easier to learn compared to a linear class, and the prior state-of-the-art linear low-rank initialization methods for deep networks are inferior to our proposed nonlinear method for initialization.

To achieve our theoretical results, we often make use of the theory of Hermite polynomials -- an orthogonal function basis over the Gaussian measure. In the last chapter, we consider resource limitations in an online streaming setting. In particular, we consider how many data points from an oblivious adversarial stream we must store from one pass over the stream to output an additive approximation to the Support Vector Machine (SVM) objective, and prove stronger lower bounds on the memory complexity.