Long-Context Linear System Identification

Oğuz Kaan Yüksel, Mathieu Even, Nicolas Flammarion

Keywords: autoregressive, linear, statistics, low rank, mispecification

TL;DR: This paper provides sample complexity for long-context linear dynamical systems.

Abstract: This paper addresses the problem of long-context linear system identification, where the state $x_t$ of the 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 work 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 feature representations, where rank-regularized estimators improve rates with respect to dimensionality, and _(ii)_ misspecified context lengths in strictly stable systems, where shorter contexts offer statistical advantages.

Primary Area: learning on time series and dynamical systems

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Submission Number: 1861