Lower Bounds for Nonparametric Estimation of Ordinary Differential Equations

TL;DR

This paper establishes minimax lower bounds for nonparametric estimation of ODE vector fields from noisy observations, revealing the optimal convergence rates depending on smoothness, dimension, and observation regimes.

Contribution

It introduces a master theorem for lower bounds in nonparametric regression, and derives optimal error rates for estimating ODE vector fields under different observation models.

Findings

Lower bounds depend on smoothness, dimension, and observation regime.

Optimal convergence rate characterized by exponent -2β/(2(β+1)+d).

Master theorem simplifies derivation of lower bounds in nonparametric regression.

Abstract

We noisily observe solutions of an ordinary differential equation $\dot u = f(u)$ at given times, where $u$ lives in a $d$-dimensional state space. The model function $f$ is unknown and belongs to a H\"older-type smoothness class with parameter $\beta$. For the nonparametric problem of estimating $f$, we provide lower bounds on the error in two complementary model specifications: the snake model with few, long observed solutions and the stubble model with many short ones. The lower bounds are minimax optimal in some settings. They depend on various parameters, which in the optimal asymptotic regime leads to the same rate for the squared error in both models: it is characterized by the exponent $-2\beta/(2(\beta+1)+d)$ for the total number of observations $n$. To derive these results, we establish a master theorem for lower bounds in general nonparametric regression problems, which makes the proofs more comparable and seems to be a useful tool for future use.