Online Stochastic Gradient Descent Learns Linear Dynamical Systems from A Single Trajectory

TL;DR

This paper demonstrates that online stochastic gradient descent can efficiently and linearly converge to the true weight matrices of a stable linear dynamical system from a single noisy trajectory, under certain conditions.

Contribution

It introduces the first linear convergence analysis of online and offline SGD methods for estimating linear dynamical system weights from a single trajectory.

Findings

SGD converges linearly in expectation to the true weights.

The methods outperform existing state-of-the-art techniques.

Numerical experiments confirm theoretical results.

Abstract

This work investigates the problem of estimating the weight matrices of a stable time-invariant linear dynamical system from a single sequence of noisy measurements. We show that if the unknown weight matrices describing the system are in Brunovsky canonical form, we can efficiently estimate the ground truth unknown matrices of the system from a linear system of equations formulated based on the transfer function of the system, using both online and offline stochastic gradient descent (SGD) methods. Specifically, by deriving concrete complexity bounds, we show that SGD converges linearly in expectation to any arbitrary small Frobenius norm distance from the ground truth weights. To the best of our knowledge, ours is the first work to establish linear convergence characteristics for online and offline gradient-based iterative methods for weight matrix estimation in linear dynamical systems from a single trajectory. Extensive numerical tests verify that the performance of the proposed methods is consistent with our theory, and show their superior performance relative to existing state of the art methods.