Metric Entropy Limits on Recurrent Neural Network Learning of Linear Dynamical Systems

TL;DR

This paper establishes theoretical limits on how well recurrent neural networks can learn linear dynamical systems, showing they can optimally identify stable LTI systems based on metric entropy analysis.

Contribution

It provides a quantitative framework for understanding RNN capabilities in learning linear systems, including a metric entropy-based optimality result for LTI system identification.

Findings

RNNs can learn stable LTI systems optimally.

The paper quantifies the complexity of LTI systems using metric entropy.

RNNs can identify difference equations from input-output data.

Abstract

One of the most influential results in neural network theory is the universal approximation theorem [1, 2, 3] which states that continuous functions can be approximated to within arbitrary accuracy by single-hidden-layer feedforward neural networks. The purpose of this paper is to establish a result in this spirit for the approximation of general discrete-time linear dynamical systems - including time-varying systems - by recurrent neural networks (RNNs). For the subclass of linear time-invariant (LTI) systems, we devise a quantitative version of this statement. Specifically, measuring the complexity of the considered class of LTI systems through metric entropy according to [4], we show that RNNs can optimally learn - or identify in system-theory parlance - stable LTI systems. For LTI systems whose input-output relation is characterized through a difference equation, this means that RNNs can learn the difference equation from input-output traces in a metric-entropy optimal manner.