Exact Recovery of Linear Dynamical Systems with Majority Corrupt Data

Core Concepts

The paper proposes two convex optimization-based estimators that can exactly recover the system dynamics of a linear time-invariant system even when more than half of the data is corrupted by adversarial attacks.

Abstract

The paper investigates the system identification problem for linear discrete-time systems under adversarial attacks. It analyzes two lasso-type estimators, one based on minimizing the sum of ℓ2 norms of the estimated disturbance vectors and the other based on minimizing the sum of ℓ1 norms. The key insights are: For autonomous systems with a deterministic periodic attack structure, the paper shows that both estimators can exactly recover the true system matrices when the system is stable and the number of samples is larger than the dimension of the states plus the attack period. For autonomous systems with a probabilistic attack structure where attacks occur at each time instance independently with probability p, the paper derives the required sample complexity for exact recovery with high probability. The sample complexity scales polynomially in the dimension of the states and the attack probability p. This result implies almost sure convergence to the true system dynamics under the asymptotic regime. The paper highlights that the attack vectors are allowed to be correlated with each other, and the estimators can still learn the system correctly even when more than half of the data is compromised. For non-autonomous systems with an input sequence, the paper extends the results and shows that the input sequence can be leveraged to accelerate the exact recovery. Overall, the 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.

Stats

The system dynamics is given by: xi+1 = Āxi + B̄ui + d̄i, i = 0,...,T-1, where Ā ∈ Rn×n and B̄ ∈ Rn×m are the unknown system matrices, and d̄i ∈ Rn are the unknown adversarial disturbance vectors.

Quotes

"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." "As a corollary to the previous result, we show that the estimators converge to true system matrices almost surely when the attack vectors are stealthy."

Key Insights Distilled From

Exact Recovery for System Identification with More Corrupt Data than Clean Data

by Baturalp Yal... at arxiv.org 04-26-2024

https://arxiv.org/pdf/2305.10506.pdf

Deeper Inquiries

How can the proposed estimators be extended to handle time-varying or nonlinear system dynamics

The proposed estimators can be extended to handle time-varying or nonlinear system dynamics by incorporating adaptive techniques and non-linear optimization algorithms. For time-varying systems, the system matrices can be updated iteratively based on the changing dynamics of the system. This can involve using recursive estimation methods such as Kalman filters or particle filters to adapt to the evolving system behavior. For nonlinear systems, the estimators can be modified to handle nonlinearity by employing techniques like kernel methods, neural networks, or Gaussian processes to model the nonlinear relationships between the inputs and outputs of the system. By incorporating these adaptive and non-linear modeling approaches, the estimators can effectively handle time-varying and nonlinear system dynamics.

What are the implications of the results on the design of secure and resilient control systems

The results of the study have significant implications for the design of secure and resilient control systems. By demonstrating the exact recovery of system dynamics even in the presence of adversarial attacks and a high level of corruption in the data, the research provides a foundation for developing robust control systems that can withstand malicious interference and data corruption. The ability to achieve exact recovery with a high probability, even when more than half of the data is compromised, highlights the potential for designing secure and resilient control systems that can operate effectively in challenging environments. These results can inform the development of defense mechanisms and anomaly detection strategies to enhance the security and resilience of control systems against adversarial attacks.

Can the techniques developed in this paper be applied to other robust online learning problems beyond system identification

The techniques developed in this paper for system identification under adversarial attacks and corrupt data can be applied to other robust online learning problems beyond system identification. The framework of using convex optimization and non-smooth estimators to achieve exact recovery in the presence of correlated data and adversarial attacks can be extended to various domains such as anomaly detection, fault diagnosis, and reinforcement learning. By adapting the proposed estimators and analysis techniques to different problem settings, it is possible to address a wide range of robust online learning challenges where data integrity and security are critical. The mathematical guarantees and sample complexity results obtained in this study can serve as a foundation for developing robust learning algorithms in diverse applications requiring resilience to adversarial interference.