Error Bounds of the Invariant Statistics in Machine Learning of Ergodic Itô Diffusions

Abstract

This paper studies the theoretical underpinnings of machine learning of ergodic Itô 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.

Stationary Density Estimation of Itô Diffusions Using Deep Learning

“…Theoretically, we deduced an error bound for the proposed approach for an SDE with global Lipschitz drift coefficients and constant diffusion matrix, accounting errors contributed by the discretization of the SDE in the training data, the regression of the drift terms using fully-connected ReLU networks with arbitrary width and layers, and the regression solution to the Fokker-Planck PDE using a fully-connected two-layer neural network with the ReLU 3 activation function. This error bound is deduced under various assumptions that underpin the perturbation theory result in [72], generalization errors in approximating Lipschitz continuous functions in [28] and in solving PDEs in [41].…”

“…The main goals of this theoretical study are to 1) understand under which mathematical assumptions can the density estimation problem be well-posed, 2) establish the convergence of the proposed scheme, and 3) identify the error in terms of training sample size, width/length of the neural-network models, discretization time step and noise amplitudes in the training data, and the dimension of the stochastic processes. In conjunction, we will also verify whether the perturbation theory [72] is valid. Particularly, we will check whether the stochastic process associated with the estimated drift and diffusion terms (obtained from deep learning regression in the first step) can indeed estimate the underlying invariant measure accurately.…”

“…Since the analysis depends on the results of the perturbation theory on the ergodic Itô diffusion in [72], we will briefly review the concepts of geometric ergodicity and other relevant results.…”

“…See e.g. Assumptions 2.2-2.3 in [72] for the detailed conditions to achieve the geometric ergodicity for the SDE driven by additive Brownian noises. One of the conditions that is important for our discussion is that there exists a Lyapunov function V :…”