Fast rates in learning with dependent observations

TL;DR

This paper demonstrates that for time series with phi-mixing properties, the learning rate for prediction errors can be improved from the standard 1/√n to the optimal 1/n, matching iid scenarios.

Contribution

It establishes the first conditions under which fast 1/n learning rates are achievable for dependent time series data, specifically with the least squares loss and phi-mixing assumptions.

Findings

Achieves 1/n learning rate for phi-mixing time series.

Shows optimality of the method for sparse linear predictor aggregation.

Extends iid learning theory results to dependent data under certain conditions.

Abstract

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.