Concentration Inequalities

A Nonasymptotic Theory of Independence

Stéphane Boucheron, Gábor Lugosi, Pascal Massart (Autoren)

Buch | Softcover

496 Seiten

2016
Oxford University Press (Verlag)
9780198767657 (ISBN)

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An accessible account of the rich theory surrounding concentration inequalities in probability theory, with applications from machine learning and statistics to high-dimensional geometry. This book introduces key ideas and presents a detailed summary of the state-of-the-art in the area, making it ideal for independent learning and as a reference.

Concentration inequalities for functions of independent random variables is an area of probability theory that has witnessed a great revolution in the last few decades, and has applications in a wide variety of areas such as machine learning, statistics, discrete mathematics, and high-dimensional geometry. Roughly speaking, if a function of many independent random variables does not depend too much on any of the variables then it is concentrated in the sense that with high probability, it is close to its expected value. This book offers a host of inequalities to illustrate this rich theory in an accessible way by covering the key developments and applications in the field.
The authors describe the interplay between the probabilistic structure (independence) and a variety of tools ranging from functional inequalities to transportation arguments to information theory. Applications to the study of empirical processes, random projections, random matrix theory, and threshold phenomena are also presented.

A self-contained introduction to concentration inequalities, it includes a survey of concentration of sums of independent random variables, variance bounds, the entropy method, and the transportation method. Deep connections with isoperimetric problems are revealed whilst special attention is paid to applications to the supremum of empirical processes.

Written by leading experts in the field and containing extensive exercise sections this book will be an invaluable resource for researchers and graduate students in mathematics, theoretical computer science, and engineering.

Stéphane Boucheron is a Professor in the Applied Mathematics and Statistics Department at Université Paris-Diderot, France. ; Gábor Lugosi is ICREA Research Professor in the Department of Economics at the Pompeu Fabra University in Barcelona, Spain. ; Pascal Massart is a Professor in the Department of Mathematics at Université de Paris-Sud, France.

Michel Ledoux: Foreword
1: Introduction
2: Basic inequalities
3: Bounding the variance
4: Basic information inequalities
5: Logarithmic Sobolev inequalities
6: The entropy method
7: Concentration and isoperimetry
8: The transportation method
9: Influences and threshold phenomena
10: Isoperimetry on the hypercube and Gaussian spaces
11: The variance of suprema of empirical processes
12: Suprema of empirical processes: exponential inequalities
13: The expected value of suprema of empirical processes
14: *Q-entropies
15: Moment inequalities

Erscheinungsdatum	31.01.2016
Zusatzinfo	8 b/w line drawings
Verlagsort	Oxford
Sprache	englisch
Maße	155 x 233 mm
Gewicht	732 g
Themenwelt	Mathematik / Informatik ► Informatik ► Theorie / Studium
	Mathematik / Informatik ► Mathematik
	Technik
ISBN-13	9780198767657 / 9780198767657
Zustand	Neuware