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Fast rates in learning with dependent observations

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Date
2012
Publisher city
Paris
Publisher
Université Paris-Dauphine
Link to item file
http://hal.archives-ouvertes.fr/hal-00671979
Dewey
Probabilités et mathématiques appliquées
Sujet
PAC-Bayesian bounds; Mixing processes; Fast rates Sparsity; Oracle inequalities; Time series prediction; Statistical learning theory
URI
https://basepub.dauphine.fr/handle/123456789/8309
Collections
  • CEREMADE : Publications
Metadata
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Author
Wintenberger, Olivier
Alquier, Pierre
Type
Document de travail / Working paper
Item number of pages
15
Abstract (EN)
In this paper we tackle the problem of fast rates in time series forecasting from a statistical learning perspective. In a serie of papers (e.g. Meir 2000, Modha and Masry 1998, Alquier and Wintenberger 2012) it is shown that the main tools used in learning theory with iid observations can be extended to the prediction of time series. The main message of these papers is that, given a family of predictors, we are able to build a new predictor that predicts the series as well as the best predictor in the family, up to a remainder of order $1/\sqrt{n}$. It is known that this rate cannot be improved in general. In this paper, we show that in the particular case of the least square loss, and under a strong assumption on the time series (phi-mixing) the remainder is actually of order $1/n$. Thus, the optimal rate for iid variables, see e.g. Tsybakov 2003, and individual sequences, see \cite{lugosi} is, for the first time, achieved for uniformly mixing processes. We also show that our method is optimal for aggregating sparse linear combinations of predictors.

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