Learning Linear Dynamical Systems via Spectral Filtering

TL;DR

This paper introduces a new spectral filtering algorithm for online prediction of symmetric linear dynamical systems, achieving near-optimal regret with an overparameterized convex approach.

Contribution

It proposes a novel spectral filtering method that overcomes non-convexity in LDS learning by overparameterization and eigenvector convolution, enabling efficient and provably optimal online learning.

Findings

Polynomial-time algorithm with near-optimal regret guarantees

Sample complexity bounds for agnostic learning

Effective spectral filtering technique based on Hankel matrix eigenvectors

Abstract

We present an efficient and practical algorithm for the online prediction of discrete-time linear dynamical systems with a symmetric transition matrix. We circumvent the non-convex optimization problem using improper learning: carefully overparameterize the class of LDSs by a polylogarithmic factor, in exchange for convexity of the loss functions. From this arises a polynomial-time algorithm with a near-optimal regret guarantee, with an analogous sample complexity bound for agnostic learning. Our algorithm is based on a novel filtering technique, which may be of independent interest: we convolve the time series with the eigenvectors of a certain Hankel matrix.