Empirical risk minimization and complexity of dynamical models

TL;DR

This paper investigates how empirical risk minimization can be used to fit dynamical models to observed stochastic processes, establishing convergence results and conditions for effective signal-noise separation based on model complexity.

Contribution

It provides a general convergence theorem for minimum risk estimators in dynamical models and links model complexity to the ability to distinguish signal from noise.

Findings

Convergence of minimum risk estimators under ergodic observations.

Model complexity quantified through entropy affects noise separation.

Connections established between entropy and mean widths in stationary processes.

Abstract

A dynamical model consists of a continuous self-map $T: \mathcal{X} \to \mathcal{X}$ of a compact state space $\mathcal{X}$ and a continuous observation function $f: \mathcal{X} \to \mathbb{R}$. This paper considers the fitting of a parametrized family of dynamical models to an observed real-valued stochastic process using empirical risk minimization. The limiting behavior of the minimum risk parameters is studied in a general setting. We establish a general convergence theorem for minimum risk estimators and ergodic observations. We then study conditions under which empirical risk minimization can effectively separate the signal from the noise in an additive observational noise model. The key, necessary condition in the latter results is that the family of dynamical models has limited complexity, which is quantified through a notion of entropy for families of infinite sequences. Close connections between entropy and limiting average mean widths for stationary processes are established.