线性动力系统少样本识别的PAC-Bayesian元学习
PAC-Bayesian Meta-Learning for Few-Shot Identification of Linear Dynamical Systems
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中文总结 AI 辅助
针对短轨迹、噪声大的LTI系统识别，提出PBML-LTI元学习框架，学习可迁移先验并进行贝叶斯自适应，通过鞅PAC-Bayes分析提供有限样本保证。
中文摘要 AI 辅助
当轨迹短、噪声大或维度高时，识别线性时不变（LTI）动力系统具有挑战性。传统系统识别通常独立处理每个系统，无法利用相关系统间的共享结构。我们提出PBML-LTI，一种用于少样本LTI系统识别的PAC-Bayesian元学习框架，它在保留任务异质性的同时，学习任务特定动力学上的可迁移先验。每个任务对应一个未知的LTI系统，元学习器利用训练轨迹学习转移矩阵上的数据相关先验。对于数据有限的新系统，PBML-LTI在该先验下进行贝叶斯自适应以获得任务特定的后验，提供准确的估计和原理性的不确定性量化。一个关键挑战是时间依赖性，因为LTI轨迹违反了大多数PAC-Bayes元学习分析所基于的独立同分布假设。我们通过针对依赖轨迹损失的鞅PAC-Bayes分析解决这一问题，并推导出支持-查询预测风险界，该界激发了fit-KL元训练目标。该界阐明了在序列依赖下少样本自适应中经验拟合、后验复杂性和先验质量的作用。我们进一步推导了转移矩阵恢复和多步轨迹预测的推论，将不确定性感知的元识别与依赖动态数据的有限样本保证联系起来。
英文摘要
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.
发表机构
- University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
机构由 AI 辅助整理，请以论文原文为准。
