Infinite-dimensional optimization and Bayesian nonparametric learning of stochastic differential equations

TL;DR

This paper develops a theoretical framework for infinite-dimensional optimization in Hilbert spaces and applies it to Bayesian nonparametric learning of SDE drift functions, enabling efficient sparse learning and uncertainty quantification.

Contribution

It introduces a generalization of the representer theorem for infinite-dimensional optimization and integrates it into a Bayesian approach for learning SDE drifts with sparse priors.

Findings

The approach accurately learns SDE drift functions in examples.

Incorporates shrinkage priors for sparse Bayesian learning.

Provides a systematic method with uncertainty quantification.

Abstract

The paper has two major themes. The first part of the paper establishes certain general results for infinite-dimensional optimization problems on Hilbert spaces. These results cover the classical representer theorem and many of its variants as special cases and offer a wider scope of applications. The second part of the paper then develops a systematic approach for learning the drift function of a stochastic differential equation by integrating the results of the first part with Bayesian hierarchical framework. Importantly, our Baysian approach incorporates low-cost sparse learning through proper use of shrinkage priors while allowing proper quantification of uncertainty through posterior distributions. Several examples at the end illustrate the accuracy of our learning scheme.