Efficient Higher-Order Optimization for Machine Learning

Abstract

In recent years, stochastic gradient descent (SGD) has taken center stage for training large-scale models in machine learning. Although some higher-order methods have improved iteration complexity in theory, the per-iteration costs render them unusable when faced with millions of parameters and training examples.

In this thesis, I will present several works which enable higher-order optimization to be as scalable as first-order methods. The first method is a stochastic second-order algorithm for convex optimization, called LiSSA, which uses Hessian information to construct an unbiased Newton step in time linear in the problem dimension. To bypass the typical efficiency barriers for second-order methods, we harness the ERM structure in standard machine learning tasks.

While convex problems allow for global convergence, recent state-of-the-art models, such as deep neural networks, highlight the importance of developing a better understanding of non-convex guarantees. In order to handle this challenging setting, I will present FastCubic, a Hessian-based method which provably converges to first-order critical points faster than gradient descent, while additionally converging to second-order critical points. Finally, I will establish how to leverage even higher-order derivative information by means of the FastQuartic algorithm, which achieves faster convergence for both smooth and non-smooth convex optimization problems by combining efficient tensor methods with near-optimal higher-order acceleration.