High-dimensional Stochastic Approximation: Algorithms and Convergence Rates

Abstract

In virtually any scenario where there is a desire to make quantitative or qualitative predictions, mathematical models are of crucial importance for predicting quantities of interest. Unfortunately, many models are based on mathematical objects that cannot be calculated exactly. As a consequence, numerical approximation algorithms that approximate such objects are indispensable in practice. These algorithms are often stochastic in nature, which means that there is some form of randomness involved when running them. In this thesis we consider four possibly high-dimensional approximation problems and corresponding stochastic numerical approximation algorithms. More specifically, we prove essentially sharp rates of convergence in the probabilistically weak sense for spatial spectral Galerkin approximations of semi-linear stochastic wave equations driven by multiplicative noise. In addition, we develop an abstract framework that allows us to view full-history recursive multilevel Picard approximation methods from a new perspective and to work out more clearly how these stochastic methods beat the curse of dimensionality in the numerical approximation of semi-linear heat equations. Furthermore, we tackle high-dimensional optimal stopping problems by proposing a stochastic numerical approximation algorithm that is based on deep learning. And finally, we study convergence in the probabilistically strong sense of the overall error arising in deep learning based empirical risk minimisation, one of the main pillars of supervised learning.