Exact Recovery Guarantees for Parameterized Nonlinear System Identification Problem under Sparse Disturbances or Semi-Oblivious Attacks

2024-08-30

AI Summary

This paper addresses the finite-time exact identification of nonlinear dynamical systems under sparse or semi-blind adversarial perturbations. We consider a strong interference model where perturbations are temporally dependent, zero-mean, distribution-agnostic, and occur with arbitrary probability $ p $—including the extreme case $ p o 1 $. To tackle this, we propose a sparse optimization framework based on basis-function parameterization and a LASSO-type nonsmooth estimator. Under boundedness or Lipschitz conditions on the basis functions, we establish, for the first time, a finite-time exact recovery guarantee for nonlinear systems under such strong, structured interference—thereby relaxing the conventional i.i.d. assumption. Theoretically, our method achieves exact parameter recovery with high probability; the sample complexity depends explicitly on the sparsity level $ p $, and remains valid even as $ p o 1 $. This work provides a new paradigm for robust system identification in adversarial and highly corrupted regimes.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Search and Optimization: Adversarial Search
• Multiagent Systems: Adversarial Agents

Application Category

• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In this work, we study the problem of learning a nonlinear dynamical system by parameterizing its dynamics using basis functions. We assume that disturbances occur at each time step with an arbitrary probability $p$, which models the sparsity level of the disturbance vectors over time. These disturbances are drawn from an arbitrary, unknown probability distribution, which may depend on past disturbances, provided that it satisfies a zero-mean assumption. The primary objective of this paper is to learn the system's dynamics within a finite time and analyze the sample complexity as a function of $p$. To achieve this, we examine a LASSO-type non-smooth estimator, and establish necessary and sufficient conditions for its well-specifiedness and the uniqueness of the global solution to the underlying optimization problem. We then provide exact recovery guarantees for the estimator under two distinct conditions: boundedness and Lipschitz continuity of the basis functions. We show that finite-time exact recovery is achieved with high probability, even when $p$ approaches 1. Unlike prior works, which primarily focus on independent and identically distributed (i.i.d.) disturbances and provide only asymptotic guarantees for system learning, this study presents the first finite-time analysis of nonlinear dynamical systems under a highly general disturbance model. Our framework allows for possible temporal correlations in the disturbances and accommodates semi-oblivious adversarial attacks, significantly broadening the scope of existing theoretical results.

Problem

• Learning nonlinear dynamical systems with sparse disturbances
• Analyzing sample complexity for finite-time exact recovery
• Generalizing disturbance models beyond i.i.d. assumptions

Innovation

• Parameterizes dynamics using basis functions
• Uses LASSO-type non-smooth estimator
• Provides exact recovery guarantees