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

TMLR Paper7277 Authors

Abstract:

Identifying linear time-invariant (LTI) dynamical systems from data is especially challenging when trajectories are short, noisy, or high-dimensional. Traditional system identification methods typically treat each system in isolation and therefore fail to exploit shared structure across related systems. We propose a PAC-Bayesian meta-learning framework for few-shot LTI system identification (PBML-LTI), which learns a transferable prior over task-specific dynamics while preserving task-level heterogeneity. Each task corresponds to an unknown LTI system, and a meta-learner uses a collection of training trajectories to learn a data-dependent prior over transition matrices. Given a new system with limited trajectory data, PBML-LTI performs Bayesian adaptation under the learned prior to produce a task-specific posterior, yielding both accurate point estimates and principled uncertainty quantification in the few-shot regime.

A key technical challenge is temporal dependence: trajectories generated by LTI systems violate the i.i.d. assumptions underlying most existing PAC-Bayes analyses for meta-learning. To address this, we develop a martingale PAC-Bayes analysis for dependent trajectory losses and use it to motivate a fit-KL surrogate objective for meta-training. The resulting support--query predictive-risk bound clarifies how empirical fit, posterior complexity, and prior quality interact in few-shot adaptation under sequential dependence. We further show how this predictive bound induces problem-specific corollaries for transition-matrix recovery and multi-step trajectory prediction. Together, these results connect uncertainty-aware meta-identification with finite-sample analysis for dependent dynamical data.

Submission Type: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Fredrik_Daniel_Johansson1

Submission Number: 7277