Non-Parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergence
Riccardo Bonalli, Alessandro Rudi
TL;DR
This work tackles non-parametric identification of drift $\,b\,$ and diffusion $\,a\,$ in multi-dimensional, nonlinear SDEs from discrete-time observations. It introduces a two-step RKHS-based framework: first estimate the law via a kernel-based density model $\hat p$, then fit finite-dimensional drift/diffusion models by enforcing the Fokker-Planck equation on $\hat p$. The authors establish non-asymptotic learning rates that tighten with higher regularity of $b$ and $a$, and show how offline kernel precomputation yields efficient computation while preserving accuracy. They derive a sequence of well-posed infinite- and finite-dimensional problems (LP and LP$_Q$) with explicit rates, and provide a scalable semidefinite-program formulation for the finite-dimensional step. The methodology offers rigorous guarantees for state-based observation and regulation of SDEs and opens avenues for controlled SDE identification with practical impact in engineering and economics.
Abstract
We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of multi-dimensional non-linear stochastic differential equations, which relies upon discrete-time observations of the state. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker-Planck equation to such observations, yielding theoretical estimates of non-asymptotic learning rates which, unlike previous works, become increasingly tighter when the regularity of the unknown drift and diffusion coefficients becomes higher. Our method being kernel-based, offline pre-processing may be profitably leveraged to enable efficient numerical implementation, offering excellent balance between precision and computational complexity.