  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.
