PAC-Bayesian Meta-Learning for Few-Shot Identification of Linear Dynamical Systems

Abstract:Identifying linear time-invariant (LTI) dynamical systems is challenging when trajectories are short, noisy, or high-dimensional. Traditional system identification typically treats each system independently and cannot exploit shared structure across related systems. We propose PBML-LTI, a PAC-Bayesian meta-learning framework for few-shot LTI system identification that learns a transferable prior over task-specific dynamics while preserving task heterogeneity. Each task corresponds to an unknown LTI system, and the meta-learner uses training trajectories to learn a data-dependent prior over transition matrices. For a new system with limited data, PBML-LTI performs Bayesian adaptation under this prior to obtain a task-specific posterior, providing accurate estimates and principled uncertainty quantification. A key challenge is temporal dependence, since LTI trajectories violate the i.i.d. assumptions underlying most PAC-Bayes meta-learning analyses. We address this with a martingale PAC-Bayes analysis for dependent trajectory losses and derive a support-query predictive-risk bound that motivates a fit-KL meta-training objective. The bound clarifies the roles of empirical fit, posterior complexity, and prior quality in few-shot adaptation under sequential dependence. We further derive corollaries for transition-matrix recovery and multi-step trajectory prediction, connecting uncertainty-aware meta-identification with finite-sample guarantees for dependent dynamical data.

Accepted at Transactions on Machine Learning Research (TMLR), 2026. J2C Certification

Machine Learning (cs.LG)

arXiv:2609.24117 [cs.LG]

(or arXiv:2609.24117v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.24117 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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