Information Theoretically Optimal Sample Complexity of Learning Dynamical Directed Acyclic Graphs

Mishfad Shaikh Veedu, Deepjyoti Deka, Murti V. Salapaka

Research output: Contribution to journalConference articlepeer-review


In this article, the optimal sample complexity of learning the underlying interactions or dependencies of a Linear Dynamical System (LDS) over a Directed Acyclic Graph (DAG) is studied. We call such a DAG underlying an LDS as dynamical DAG (DDAG). In particular, we consider a DDAG where the nodal dynamics are driven by unobserved exogenous noise sources that are wide-sense stationary (WSS) in time but are mutually uncorrelated, and have the same power spectral density (PSD). Inspired by the static DAG setting, a metric and an algorithm based on the PSD matrix of the observed time series are proposed to reconstruct the DDAG. It is shown that the optimal sample complexity (or length of state trajectory) needed to learn the DDAG is n = Θ(qlog(p/q)), where p is the number of nodes and q is the maximum number of parents per node. To prove the sample complexity upper bound, a concentration bound for the PSD estimation is derived, under two different sampling strategies. A matching min-max lower bound using generalized Fano’s inequality also is provided, thus showing the order optimality of the proposed algorithm. The codes used in the paper are available at

Original languageEnglish (US)
Pages (from-to)4636-4644
Number of pages9
JournalProceedings of Machine Learning Research
StatePublished - 2024
Event27th International Conference on Artificial Intelligence and Statistics, AISTATS 2024 - Valencia, Spain
Duration: May 2 2024May 4 2024

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