On the Sharp Input-Output Analysis of Nonlinear Systems under Adversarial Attacks

TL;DR

This paper develops a sharp input-output analysis framework for nonlinear dynamical systems under adversarial disturbances, extending beyond traditional Gaussian noise assumptions to include correlated, nonzero-mean attacks.

Contribution

It introduces a novel reformulation as a basis function linear combination and proves the effectiveness of an $ ext{l}_2$-norm estimator under adversarial attack probability constraints.

Findings

Estimator's error bound decays with input memory length

Bound is proven optimal through constructed worst-case instances

Framework applies to general nonlinear, partially observed systems

Abstract

This paper is concerned with learning the input-output mapping of general nonlinear dynamical systems. While the existing literature focuses on Gaussian inputs and benign disturbances, we significantly broaden the scope of admissible control inputs and allow correlated, nonzero-mean, adversarial disturbances. With our reformulation as a linear combination of basis functions, we prove that the $\ell_2$-norm estimator overcomes the challenges as long as the probability that the system is under adversarial attack at a given time is smaller than a certain threshold. We provide an estimation error bound that decays with the input memory length and prove its optimality by constructing a problem instance that suffers from the same bound under adversarial attacks. Our work provides a sharp input-output analysis for a generic nonlinear and partially observed system under significantly generalized assumptions compared to existing works.