# Adaptive and optimal online linear regression on $\ell^1$-balls

2 CLASSIC - Computational Learning, Aggregation, Supervised Statistical, Inference, and Classification
DMA - Département de Mathématiques et Applications - ENS Paris, ENS Paris - École normale supérieure - Paris, Inria Paris-Rocquencourt
Abstract : We consider the problem of online linear regression on individual sequences. The goal in this paper is for the forecaster to output sequential predictions which are, after $T$ time rounds, almost as good as the ones output by the best linear predictor in a given $\ell^1$-ball in $\\R^d$. We consider both the cases where the dimension~$d$ is small and large relative to the time horizon $T$. We first present regret bounds with optimal dependencies on $d$, $T$, and on the sizes $U$, $X$ and $Y$ of the $\ell^1$-ball, the input data and the observations. The minimax regret is shown to exhibit a regime transition around the point $d = \sqrt{T} U X / (2 Y)$. Furthermore, we present efficient algorithms that are adaptive, \ie, that do not require the knowledge of $U$, $X$, $Y$, and $T$, but still achieve nearly optimal regret bounds.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-00594399
Contributor : Sébastien Gerchinovitz <>
Submitted on : Monday, January 14, 2019 - 11:23:17 PM
Last modification on : Friday, April 12, 2019 - 10:18:10 AM

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GY14-OnlineRegressionL1balls.p...
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### Identifiers

• HAL Id : hal-00594399, version 4
• ARXIV : 1105.4042

### Citation

Sébastien Gerchinovitz, Jia Yuan Yu. Adaptive and optimal online linear regression on $\ell^1$-balls. Theoretical Computer Science, Elsevier, 2014, 519, pp.4-28. ⟨hal-00594399v4⟩

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