Abstract

Abstract Exponential inequalities are main tools in machine learning theory. To prove exponential inequalities for non i.i.d random variables allows to extend many learning techniques to these variables. Indeed, much work has been done both on inequalities and learning theory for time series, in the past 15 years. However, for the non independent case, almost all the results concern stationary time series. This excludes many important applications: for example any series with a periodic behaviour is nonstationary. In this paper, we extend the basic tools of [19] to nonstationary Markov chains. As an application, we provide a Bernsteintype inequality, and we deduce risk bounds for the prediction of periodic autoregressive processes with an unknown period.

Highlights

  • Exponential inequalities are corner stones of machine learning theory

  • We extend the basic tools of [19] to nonstationary Markov chains

  • Distribution free generalization bounds were proven by Vapnik and Cervonenkis based on Hoe ding’s inequality, see [48]

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Summary

Introduction

Exponential inequalities are corner stones of machine learning theory. For example, distribution free generalization bounds were proven by Vapnik and Cervonenkis based on Hoe ding’s inequality, see [48]. Model selection bounds in [31] rely on exponential moment inequalities To prove such inequalities in the non i.i.d setting is crucial to study the generalization ability of machine learning algorithms on time series. A Bernstein type inequality for α-mixing time series is proven in [36] This result is used by [45] to prove generalization bounds for general learning problems with α-mixing observations. We generalize the inequalities proven by [19] for time homogeneous Markov chains to nonhomogeneous chains This allows to study a large set of nonstationary processes.

We de ne
Application to periodic autoregressive models
Fix a maximal period max
It follows that
Similar derivations lead to
Remark that
The previous inequalities show that

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