Gradient Descent Learns Linear Dynamical Systems

TL;DR

This paper proves that stochastic gradient descent can efficiently learn unknown linear dynamical systems from noisy data, providing the first polynomial guarantees for this classical problem.

Contribution

It establishes polynomial convergence and sample complexity bounds for gradient descent in linear system identification, a longstanding open problem.

Findings

SGD converges to the global optimum efficiently

Provides polynomial bounds on running time and samples

First such guarantees for this classical problem

Abstract

We prove that stochastic gradient descent efficiently converges to the global optimizer of the maximum likelihood objective of an unknown linear time-invariant dynamical system from a sequence of noisy observations generated by the system. Even though the objective function is non-convex, we provide polynomial running time and sample complexity bounds under strong but natural assumptions. Linear systems identification has been studied for many decades, yet, to the best of our knowledge, these are the first polynomial guarantees for the problem we consider.