Error Bounds of the Invariant Statistics in Machine Learning of Ergodic It\^o Diffusions

TL;DR

This paper provides theoretical error bounds for invariant statistics in machine learning of ergodic Itô diffusions, linking errors in estimating drift and diffusion coefficients to errors in invariant statistics, and discusses conditions on learning algorithms for linear error dependence.

Contribution

It introduces a theoretical framework connecting estimation errors of SDE parameters to invariant statistics, highlighting the importance of Lipschitz and consistency conditions in learning algorithms.

Findings

Error in invariant statistics depends linearly on drift and diffusion estimation errors.

L^2-norm alone is insufficient for guaranteeing linear error dependence.

Kernel spectral regression and shallow ReLU neural networks satisfy the necessary conditions.

Abstract

This paper studies the theoretical underpinnings of machine learning of ergodic It\^o diffusions. The objective is to understand the convergence properties of the invariant statistics when the underlying system of stochastic differential equations (SDEs) is empirically estimated with a supervised regression framework. Using the perturbation theory of ergodic Markov chains and the linear response theory, we deduce a linear dependence of the errors of one-point and two-point invariant statistics on the error in the learning of the drift and diffusion coefficients. More importantly, our study shows that the usual $L^2$-norm characterization of the learning generalization error is insufficient for achieving this linear dependence result. We find that sufficient conditions for such a linear dependence result are through learning algorithms that produce a uniformly Lipschitz and consistent estimator in the hypothesis space that retains certain characteristics of the drift coefficients, such as the usual linear growth condition that guarantees the existence of solutions of the underlying SDEs. We examine these conditions on two well-understood learning algorithms: the kernel-based spectral regression method and the shallow random neural networks with the ReLU activation function.