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Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems

École d’Été de Probabilités de Saint-Flour XXXVIII-2008

  • Book
  • © 2011

Overview

Authors:
  1. Vladimir Koltchinskii

    1. School of Mathematics, Georgia Institute of Technology, Atlanta, USA

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  • Provides a unified framework for machine learning problems (such as large margin
  • classification), sparse recovery and low rank matrix problems
  • Develops a variety of probabilistic inequalities for empirical processes needed to obtain error bounds
  • in machine learning and sparse recovery
  • Develops a comprehensive theory of excess risk bounds and oracle inequalities for penalized empirical
  • risk minimization
  • Includes supplementary material: sn.pub/extras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2033)

Part of the book sub series: École d'Été de Probabilités de Saint-Flour (LNMECOLE)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-ix
    Download chapter PDF

  2. Introduction

    • Vladimir Koltchinskii
    Pages 1-16

  3. Empirical and Rademacher Processes

    • Vladimir Koltchinskii
    Pages 17-32

  4. Bounding Expected Sup-Norms of Empirical and Rademacher Processes

    • Vladimir Koltchinskii
    Pages 33-57

  5. Excess Risk Bounds

    • Vladimir Koltchinskii
    Pages 59-79

  6. Examples of Excess Risk Bounds in Prediction Problems

    • Vladimir Koltchinskii
    Pages 81-97

  7. Penalized Empirical Risk Minimization and Model Selection Problems

    • Vladimir Koltchinskii
    Pages 99-119

  8. Linear Programming in Sparse Recovery

    • Vladimir Koltchinskii
    Pages 121-149

  9. Convex Penalization in Sparse Recovery

    • Vladimir Koltchinskii
    Pages 151-189

  10. Low Rank Matrix Recovery: Nuclear Norm Penalization

    • Vladimir Koltchinskii
    Pages 191-234

  11. Back Matter

    Pages 235-254
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Keywords

About this book

The purpose of these lecture notes is to provide an introduction to the general theory of empirical risk minimization with an emphasis on excess risk bounds and oracle inequalities in penalized problems. In recent years, there have been new developments in this area motivated by the study of new classes of methods in machine learning such as large margin classification methods (boosting, kernel machines). The main probabilistic tools involved in the analysis of these problems are concentration and deviation inequalities by Talagrand along with other methods of empirical processes theory (symmetrization inequalities, contraction inequality for Rademacher sums, entropy and generic chaining bounds). Sparse recovery based on l_1-type penalization and low rank matrix recovery based on the nuclear norm penalization are other active areas of research, where the main problems can be stated in the framework of penalized empirical risk minimization, and concentration inequalities and empirical processes tools have proved to be very useful.

Reviews

From the reviews:

“The book is an introduction to the general theory of empirical risk minimization with an emphasis on excess risk bounds and oracle inequalities in penalized problems. … The book is interesting and useful for students as well as for professionals in the field of probability theory, statistics, and their applications.” (Pavel Stoynov, Zentralblatt MATH, Vol. 1223, 2011)

Authors and Affiliations

  • School of Mathematics, Georgia Institute of Technology, Atlanta, USA

    Vladimir Koltchinskii

Bibliographic Information

  • Book Title: Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems

  • Book Subtitle: École d’Été de Probabilités de Saint-Flour XXXVIII-2008

  • Authors: Vladimir Koltchinskii

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-642-22147-7

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2011

  • Softcover ISBN: 978-3-642-22146-0Published: 29 July 2011

  • eBook ISBN: 978-3-642-22147-7Published: 29 July 2011

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: IX, 254

  • Topics: Probability Theory and Stochastic Processes

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