Learning Linear Dynamical Systems with Semi-Parametric Least Squares

TL;DR

This paper introduces a semi-parametric least squares method for estimating parameters of partially-observed linear dynamical systems, effectively handling long-term dependencies and biased noise, with proven consistency and improved rates.

Contribution

The paper presents the first algorithm with provable guarantees for estimating partially-observed linear systems that is robust to long-term dependencies and biased noise.

Findings

Mitigates variance from long-term dependencies

Achieves estimation rates independent of dependency decay

Proven consistent under marginal stability conditions

Abstract

We analyze a simple prefiltered variation of the least squares estimator for the problem of estimation with biased, semi-parametric noise, an error model studied more broadly in causal statistics and active learning. We prove an oracle inequality which demonstrates that this procedure provably mitigates the variance introduced by long-term dependencies. We then demonstrate that prefiltered least squares yields, to our knowledge, the first algorithm that provably estimates the parameters of partially-observed linear systems that attains rates which do not not incur a worst-case dependence on the rate at which these dependencies decay. The algorithm is provably consistent even for systems which satisfy the weaker marginal stability condition obeyed by many classical models based on Newtonian mechanics. In this context, our semi-parametric framework yields guarantees for both stochastic and worst-case noise.