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

Jihun Kim, Javad Lavaei

Abstract

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 $μ$, 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 $\mathcal{O}(μ)$. We validate our theoretical results with experiments illustrating that integrating our approach into frameworks like SINDy yields robust identification of discrete-time systems.

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

Abstract

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 -norm estimator is required for the latter. In this paper, we propose a Huber-based estimator, characterized by a threshold constant , to identify the governing matrix of a linearly parameterized nonlinear system from a single trajectory of length . 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 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 . We validate our theoretical results with experiments illustrating that integrating our approach into frameworks like SINDy yields robust identification of discrete-time systems.

Paper Structure

This paper contains 8 sections, 8 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Core Assumptions
  3. Huber-based System Identification
  4. Main Results: Near the Best of Both Worlds
  5. Numerical Experiments
  6. Proof of Theorem \ref{['noisescenario']}
  7. Proof of Theorem \ref{['attackscenario']}
  8. Conclusion

Key Result

Proposition 1

The problem hubermin is equivalent to the problem where $A$ is a matrix whose rows are $a_i^T$ for $i\in\{1,\dots,n\}$.

Figures (3)

  • Figure 2: Estimation error over Trajectory Length
  • Figure 3: Stability of trajectories reconstructed based on different estimators: Persistent zero-mean independent noise
  • Figure 4: Stability of trajectories reconstructed based on different estimators: Sparse nonzero-mean adversarial attack

Theorems & Definitions (18)

  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Theorem 2
  • Remark 5
  • Lemma 1: juditsky2008large
  • ...and 8 more