A Short Information-Theoretic Analysis of Linear Auto-Regressive Learning

Learning the dynamics of a linear dynamical system is a classical problem in for instance signal processing, system identification and econometrics. It is also arguably one of the simplest examples of an auto-regressive learning problem, thereby rendering it an instance of self-supervised learning. Understanding the sample complexity--and which quantities are of relevance for it--of such learning problems is key in the current era of large language models. The traditional approach for analyzing sequential (self-) supervised learning problems operates via comparison of the empirical and population excess risk functionals. Recently, Jeon and Van Roy [2024] provided an information-theoretic proof approach for learning from dependent data eschewing any such direct comparison. However, their results only apply to the Bayesian setting.