Exact Recovery for System Identification With More Corrupt Data Than Clean Data

This paper investigates the system identification problem for linear discrete-time systems under adversaries and analyzes two lasso-type estimators. We examine non-asymptotic properties of these estimators in two separate scenarios, corresponding to deterministic and stochastic models for the attack times. We prove that when the system is stable and attacks are injected periodically, the sample complexity for exact recovery of the system dynamics is linear in terms of the dimension of the states. When adversarial attacks occur at each time instance with probability $p$, the required sample complexity for exact recovery scales polynomially in the dimension of the states and the probability $p$. This result implies almost sure convergence to the true system dynamics under the asymptotic regime. As a by-product, our estimators still learn the system correctly even when more than half of the data is compromised. We emphasize that the attack vectors are allowed to be correlated with each other in this work. This paper provides the first mathematical guarantee in the literature on learning from correlated data for dynamical systems in the case when there is less clean data than corrupt data.