Outlier-Robust Linear System Identification Under Heavy-tailed Noise

2024-12-31

AI Summary

This paper addresses robust identification of the state transition matrix for linear time-invariant systems under heavy-tailed noise. Unlike conventional methods requiring Gaussian or sub-Gaussian noise assumptions, we only assume the existence of fourth-order moments—significantly relaxing distributional requirements. We propose a novel algorithm integrating high-dimensional robust statistics with weakly concentrated estimators, featuring multi-estimator aggregation and outlier suppression. Theoretically, our method achieves optimal-rate convergence under heavy-tailed noise and, for the first time, attains sample complexity nearly matching that under sub-Gaussian noise. We further quantify how noise kurtosis affects the required number of system trajectories. The approach is provably robust against adversarial data corruption. Empirical evaluations confirm strong robustness to both stochastic outliers and malicious contamination.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Adversarial Learning & Robustness
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• LTI System Identification
• Robust Estimation
• Noisy Environment

Innovation

• Robust System Identification
• Heavy-tailed Noise
• High-dimensional Robust Statistics