The noise level in linear regression with dependent data

Ingvar Ziemann, Stephen Tu, George J. Pappas, Nikolai Matni

Keywords: Learning Theory, Learning with dependent data, Time-Series

TL;DR: We provide the first non-asymptotic instance-optimal analysis of misspecified random design linear regression with dependent data---our leading order term is not deflated by slow mixing

Abstract: We derive upper bounds for random design linear regression with dependent ($\beta$-mixing) data absent any realizability assumptions. In contrast to the strictly realizable martingale noise regime, no sharp \emph{instance-optimal} non-asymptotics are available in the literature. Up to constant factors, our analysis correctly recovers the variance term predicted by the Central Limit Theorem---the noise level of the problem---and thus exhibits graceful degradation as we introduce misspecification. Past a burn-in, our result is sharp in the moderate deviations regime, and in particular does not inflate the leading order term by mixing time factors.

Submission Number: 14664