Outlier-Robust Linear System Identification Under Heavy-tailed Noise

TL;DR

This paper develops a robust method for linear system identification that works under heavy-tailed noise with only finite fourth moments, nearly matching results under light-tailed assumptions, and extends to adversarial data corruption scenarios.

Contribution

It introduces a novel robust identification algorithm and analysis that handle heavy-tailed noise with finite fourth moments, advancing control theory under realistic noise conditions.

Findings

Sample complexity bounds similar to sub-Gaussian noise cases.

The kurtosis of noise influences the number of trajectories needed.

Algorithm extends to adversarial data corruption scenarios.

Abstract

We consider the problem of estimating the state transition matrix of a linear time-invariant (LTI) system, given access to multiple independent trajectories sampled from the system. Several recent papers have conducted a non-asymptotic analysis of this problem, relying crucially on the assumption that the process noise is either Gaussian or sub-Gaussian, i.e., "light-tailed". In sharp contrast, we work under a significantly weaker noise model, assuming nothing more than the existence of the fourth moment of the noise distribution. For this setting, we provide the first set of results demonstrating that one can obtain sample-complexity bounds for linear system identification that are nearly of the same order as under sub-Gaussian noise. To achieve such results, we develop a novel robust system identification algorithm that relies on constructing multiple weakly-concentrated estimators, and then boosting their performance using suitable tools from high-dimensional robust statistics. Interestingly, our analysis reveals how the kurtosis of the noise distribution, a measure of heavy-tailedness, affects the number of trajectories needed to achieve desired estimation error bounds. Finally, we show that our algorithm and analysis technique can be easily extended to account for scenarios where an adversary can arbitrarily corrupt a small fraction of the collected trajectory data. Our work takes the first steps towards building a robust statistical learning theory for control under non-ideal assumptions on the data-generating process.