Huber-based Robust System Identification with Near-Optimal Guarantees Across Independent and Adversarial Regimes

Dynamical systems can confront one of two extreme types of disturbances: persistent zero-mean independent noise, and sparse nonzero-mean adversarial attacks, depending on the specific scenario being modeled. While mean-based estimators like least-squares are well-suited for the former, a median-based approach such as the $\ell_1$-norm estimator is required for the latter. In this paper, we propose a Huber-based estimator, characterized by a threshold constant $\mu$, to identify the governing matrix of a linearly parameterized nonlinear system from a single trajectory of length $T$. This formulation bridges the gap between mean- and median-based estimation, achieving provably robust error in both extreme disturbance scenarios under mild assumptions. In particular, for persistent zero-mean noise with a positive probability density around zero, the proposed estimator achieves an $\mathcal{O}(1/\sqrt{T})$ error rate if the disturbance is symmetric or the basis functions are linear. For arbitrary nonzero-mean attacks that occur at each time with probability smaller than 0.5, the error is bounded by $\mathcal{O}(\mu)$. We validate our theoretical results with experiments illustrating that integrating our approach into frameworks like SINDy yields robust identification of discrete-time systems.