Year: 2024
Type: preprint
Abstract: This paper addresses the problem of long-context linear system identification, where the state $x_t$ of a dynamical system at time $t$ depends linearly on previous states $x_s$ over a fixed context window of length $p$. We establish a sample complexity bound that matches the i.i.d. parametric rate up to logarithmic factors for a broad class of systems, extending previous works that considered only first-order dependencies. Our findings reveal a learning-without-mixing phenomenon, indicating that learning long-context linear autoregressive models is not hindered by slow mixing properties potentially associated with extended context windows. Additionally, we extend these results to (i) shared low-rank representations, where rank-regularized estimators improve the dependence of the rates on the dimensionality, and (ii) misspecified context lengths in strictly stable systems, where shorter contexts offer statistical advantages.
Source: arXiv (Cornell University)
Authors: Yüksel, Oğuz Kaan, Even, Mathieu, Flammarion, Nicolas
Language: English
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4403346652