Concentration Inequalities A Nonasymptotic Theory of Independence

ISBN-10: 0199535256

ISBN-13: 9780199535255

Edition: 2013

List price: $77.00
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Book details

List price: $77.00
Copyright year: 2013
Publisher: Oxford University Press
Publication date: 2/7/2013
Binding: Hardcover
Pages: 480
Size: 6.75" wide x 9.50" long x 1.00" tall
Weight: 1.936
Language: English

Prof. Massart has received the bronze medal of the CNRS (in mathematics and theoretical physics) in 1988 and the COPPS Presidents' award in 1998.

Introduction
Sums of Independent Random Variables and the Martingale Method
The Concentration-of-Measure Phenomenon
The Entropy Method
The Transportation Method
Reading Guide
Acknowledgments
Basic Inequalities
From Moments to Tails
The Cram�r-Chernoff Method
Sub-Gaussian Random Variables
Sub-Gamma Random Variables
A Maximal Inequality
Hoeffding's Inequality
Bennett's Inequality
Bernstein's Inequality
Random Projections and the Johnson-Lindenstrauss Lemma
Association Inequalities
Minkowski's Inequality
Bibliographical Remarks
Exercises
Bounding the Variance
The Efron-Stein Inequality
Functions with Bounded Differences
Self-Bounding Functions
More Examples and Applications
A Convex Poincar� Inequality
Exponential Tail Bounds via the Efron-Stein Inequality
The Gaussian Poincar� Inequality
A Proof of the Efron-Stein Inequality Based on Duality
Bibliographical Remarks
Exercises
Basic Information Inequalities
Shannon Entropy and Relative Entropy
Entropy on Product Spaces and the Chain Rule
Han's Inequality
Edge Isoperirnetric Inequality on the Binary Hypercube
Combinatorial Entropies
Han's Inequality for Relative Entropies
Sub-Additivity of the Entropy
Entropy of General Random Variables
Duality and Variational Formulas
A Transportation Lemma
Pinsker's Inequality
Birg�'s Inequality
Sub-Additivity of Entropy: The General Case
The Brunn-Minkowski Inequality
Bibliographical Remarks
Exercises
Logarithmic Sobolev Inequalities
Symmetric Bernoulli Distributions
Herbst's Argument: Concentration on the Hypercube
A Gaussian Logarithmic Sobolev Inequality
Gaussian Concentration: The Tsirelson-Ibragimov-Sudakov Inequality
A Concentration Inequality for Suprema of Gaussian Processes
Gaussian Random Projections
A Performance Bound for the Lasso
Hypercontractivity: The Bonami-Beckner Inequality
Gaussian Hypercontractivity
The Largest Eigenvalue of Random Matrices
Bibliographical Remarks
Exercises
The Entropy Method
The Bounded Differences Inequality
More on Bounded Differences
Modified Logarithmic Sobolev Inequalities
Beyond Bounded Differences
Inequalities for the Lower Tail
Concentration of Convex Lipschitz Functions
Exponential Inequalities for Self-Bounding Functions
Symmetrized Modified Logarithmic Sobolev Inequalities
Exponential Efron-Stein Inequalities
A Modified Logarithmic Sobolev Inequality for the Poisson Distribution
Weakly Self-Bounding Functions
Proof of Lemma 6.22
Some Variations
Janson's Inequality
Bibliographical Remarks
Exercises
Concentration and Isoperimetry
L�vy's Inequalities
The Classical Isoperirnetric Theorem
Vertex Isoperirnetric Inequality in the Hypercube
Convex Distance Inequality
Convex Lipschitz Functions Revisited
Bin Packing
Bibliographical Remarks
Exercises
The Transportation Method
The Bounded Differences Inequality Revisited
Bounded Differences in Quadratic Mean
Applications of Marton's Conditional Transportation Inequality
The Convex Distance Inequality Revisited
Talagrand's Gaussian Transportation Inequality
Appendix: A General Induction Lemma
Bibliographical Remarks
Exercises
Influences and Threshold Phenomena
Influences
Some Fundamental Inequalities for Influences
Local Concentration
Discrete Fourier Analysis and a Variance Inequality
Monotone Sets
Threshold Phenomena
Bibliographical Remarks
Exercises
Isoperimetry on the Hypercube and Gaussian Spaces
Bobkov's Inequality for Functions on the Hypercube
An Isoperimetric Inequality on the Binary Hypercube
Asymmetric Bernoulli Distributions and Threshold Phenomena
The Gaussian Isoperimetric Theorem
Lipschitz Functions of Gaussian Random Variables
Bibliographical Remarks
Exercises
The Variance of Suprema of Empirical Processes
General Upper Bounds for the Variance
Nemirovski's Inequality
The Symmetrization and Contraction Principles
Weak and Wimpy Variances
Unbounded Summands
Bibliographical Remarks
Exercises
Suprema of Empirical Processes: Exponential Inequalities
An Extension of Hoeffding's Inequality
A Bernstein-Type Inequality for Bounded Processes
A Symmetrization Argument
Bousquet's Inequality for Suprema of Empirical Processes
Non-Identically Distributed Summands and Left-Tail Inequalities
Chi-Square Statistics and Quadratic Forms
Bibliographical Remarks
Exercises
The Expected Value of Suprema of Empirical Processes
Classical Chaining
Lower Bounds for Gaussian Processes
Chaining and VC-Classes
Gaussian and Rademacher Averages of Symmetric Matrices
Variations of Nemirovski's Inequality
Random Projections of Sparse and Large Sets
Normalized Processes: Slicing and Reweighting
Relative Deviations for L<sub>2</sub> Distances
Risk Bounds in Classification
Bibliographical Remarks
Exercises
�-Entropies
�-Entropy and its Sub-Additivity
From �-Entropies to �-Sobolev Inequalities
�-Sobolev Inequalities for Bernoulli Random Variables
Bibliographical Remarks
Exercises
Moment Inequalities
Generalized Efron-Stein Inequalities
Moments of Functions of Independent Random Variables
Some Variants and Corollaries
Sums of Random Variables
Suprema of Empirical Processes
Conditional Rademacher Averages
Bibliographical Remarks
Exercises
References
Author Index
Subject Index
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