Concentration inequalities: a nonasymptotic theory of independence
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford
Oxford Univ. Press
2013
|
| Edition: | 1. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | X, 481 S. graph. Darst. |
| ISBN: | 9780199535255 9780198767657 |
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Record in the Search Index
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| adam_text | CONTENTS*
1
Introduction
........................................1
1.1
Sums of Independent Random Variables and the Martingale Method
2
1.2
The Concentratiorvof-Measure Phenomenon
4
1.3
The Entropy Method
9
1.4
The Transportation Method
12
1.5
Reading Guide
13
1.6
Acknowledgments
16
2
Basic Inequalities
....................................18
2.1
From Moments to Tails
19
2.2
The Cram
ér-Chemoff
Method
21
2.3
Sub-Gaussian Random Variables
24
2.4
Sub-Gamma Random Variables
27
2.5
AMaximallnequality
31
2.6
Hoeffding s Inequality
34
2.7
Bennett s Inequality
35
2.8
Bernstein s Inequality
36
2.9
Random Projections and the Johnson-Lindenstrauss Lemma
39
2.10
Association Inequalities
43
2.11
Minkowski
s
Inequality
44
2.12
Bibliographical Remarks
45
2.13
Exercises
46
3
Bounding the Variance
................................52
3.1
The Efron-Stein Inequality
53
3.2
Functions with Bounded Differences
56
3.3
Self-Bounding Functions
60
3.4
More Examples and Applications
63
3.5
A Convex
Poincaré
Inequality
66
3.6
Exponential Tail Bounds via the Efron-Stein Inequality
68
3.7
The Gaussian
Pomcaré
Inequality
72
3.8
A Proof of the
Eřron-Stein
Inequality Based on Duality
73
3.9
Bibliographical Remarks
76
3.10
Exercises
78
4
Basic Information Inequalities
...........................83
4.1
Shannon Entropy and Relative Entropy
84
4.2
Entropy on Product Spaces and the Chain Rule
85
4.3
Han s Inequality
86
Vili
I CONTENTS
4.4
Edge Isoperimetric Inequality on the Binary Hypercube
87
4.5
Combinatorial Entropies
89
4.6
Han s Inequality for Relative Entropies
91
4.7
Sub-Additivity of the Entropy
93
4.8
Entropy of Genera] Random Variables
96
4.9
Duality and Variational Formulas
97
4.10
A Transportation Lemma
101
4.11
Pinsker s Inequality
102
4.12
Birgé s
Inequality
103
4.13
Sub-Additivity of Entropy: The General Case
105
4.14
The Brunn-Minkowski Inequality
107
4.15
Bibliographical Remarks
110
4.16
Exercises
112
5
Logarithmic Sobolev Inequalities
........................117
5.1
Symmetric Bernoulli Distributions
118
5.2
Herbst s Argument: Concentration on the Hypercube
121
5.3
A Gaussian Logarithmic Sobolev Inequality
124
5.4
Gaussian Concentration: The Tsirelson-Ibragimov-Sudakov
Inequality
125
5.5
A Concentration Inequality for
Suprema
of Gaussian Processes
127
5.6
Gaussian Random Projections
128
5.7
A Performance Bound for the Lasso
132
5.8
Hypercontr activity: The
Bonami
-Beckner
Inequality
139
5.9
Gaussian Hyp ereontracti
vity
146
5.10
The Largest Eigenvalue of Random Matrices
147
5.11
Bibliographical Remarks
152
5.12
Exercises
154
6
The Entropy Method
................................168
6.1
The Bounded Differences Inequality
170
6.2
More on Bounded Differences
174
6.3
Modified Logarithmic Sobolev Inequalities
175
6.4
Beyond Bounded Differences
176
6.5
Inequalities for the Lower Tail
178
6.6
Concentration of Convex Lipschitz Functions
180
6.7
Exponential inequalities for Self-Bounding Functions
181
6.8
Symmetrized Modified Logarithmic Sobolev Inequalities
184
6.9
Exponential
E
fron—
Stein Inequalities
185
6.10
A
Modilied
Logarithmic Sobolev Inequality for the
Poisson
Distribution
188
6.11
Weakly Self-Bounding Functions
189
6.12
Proof
o f
Lemma
6.22 196
6.13
Some Variations
199
6.14
Jansons
Inequality
204
6.15
Bibliographical Remarks
207
6.16
Exercises
209
CONTENTS
I
ІХ
7
Concentration and Isoperimetry
.........................215
7.1
Levy s Inequalities
215
7.2
The Classical Isoperimetric Theorem
218
7.3
Vertex Isoperimetric Inequality in the Hypercube
222
7.4
Convex Distance Inequality
224
7.5
Convex
Lipschití
Functions Revisited
229
7.6
Bin Packing
230
7.7
Bibliographical Remarks
232
7.8
Exercises
233
8
The Transportation Method
...........................237
8.1
The Bounded Differences Inequality Revisited
239
8.2
Bounded Differences in Quadratic Mean
241
8.3
Applications of Marten s Conditional Transportation Inequality
247
8.4
The Convex Distance Inequality Revisited
249
8.5
Talagrands
Gaussian Transportation Inequality
251
8.6
Appendix: A General Induction Lemma
256
8.7 Bibliograph
i ca]
Remarks
259
8.8
Exercises
260
9
Influences and Threshold Phenomena
.....................262
9.1
Influences
263
9.2
Some Fundamental Inequalities for Influences
264
9.3
Local Concentration
271
9.4
Discrete Fourier Analysis and a Variance Inequality
273
9.5
Monotone Set?
277
9.6
Threshold Phenomena
279
9.7
Bibliographical Remarks
286
9.8
Exercises
287
10
Isoperimetry on the Hypercube and Gaussian Spaces
...........290
10.1
Bobkov s Inequality for Functions on the Hypercube
291
10.2
An Isoperimetric Inequality on the Binary Hypercube
297
10.3
Asymmetric Bernoulli Distributions and Threshold Phenomena
298
10.4
The Gaussian Isoperimetric Theorem
303
10.5
Lipschitz Functions of Gaussian Random Variables
307
10.6
Bibliographical Remarks
308
10.7
Exercises
309
11
The Variance of Sup
rema
of Empirical Processes
..............312
11.1
General Upper Bounds tor the Variance
315
11.2
Nemirovskľs
Inequality
317
11.3
The Symmetrixation and Contraction Principles
322
11.4
Weak and Wimpy Variances
327
11.5
Unbounded
Summ
ands
330
11.6
Bibliographical Remarks
335
11.7
Exercises
336
Χ Ι
CONTENTS
12
Suprema
of Empirical Processes: Exponential Inequalities
........341
12.1
An Extension of Hoeffding s Inequality
342
12.2
A
Bernstein
-Туре
Inequality for Bounded Processes
342
12.3
A Symmetrization Argument
344
12.4
Bousquet s Inequality for
Suprema
of Empirical Processes
347
12.5
Non-Identically Distributed Summands and Left-Tail Inequalities
351
12.6
Chi-Square Statistics and Quadratic Forms
353
12.7
Bibliographical Remarks
354
12.8
Exercises
355
13
The Expected Value of
Suprema
of Empirical Processes
..........362
13.1
Classical Chaining
363
13.2
Lower Bounds for Gaussian Processes
366
13-3
Chaining and VC-Classes
371
13.4
Gaussian and
Rademach er
Averages of Symmetric Matrices
374
13.5
Variations of Nemirovski s Inequality
377
13.6
Random Projections of Sparse and Large Sets
379
13.7
Normalized Processes: Slicing and Reweighting
387
13.8
Relative Deviations tor L2 Distances
391
13.9
Risk Bounds in Classification
392
13.10
Bibliographical Remarks
395
13.11
Exercises
397
14
Ф
-Entropies
......................................
412
14.1
Ф
-Entropy
and its
Sub-Additivi
ty
412
14.2
From
Ф
-Entropíes
to
Φ
-Sobolev Inequalities
419
14.3
Ф
-Sobolev
Inequalities for Bernoulli Random Variables
423
14.4
Bibliographical Remarks
427
14.5
Exercises
428
15
Moment Inequalities
................................430
15.1
Generalized Efron-Stein Inequalities
431
15.2
Moments of Functions of Independent Random Variables
432
15.3
Some Variants and Corollaries
436
15.4
Sums of Random Variables
440
15.5
Suprema
of Empirical Processes
443
15.6
Conditional Rademacher Averages
446
15.7
Bibliographical Remarks
447
15.8
Exercises
449
References
451
Author Index
473
Subject Index
477
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 or 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
and Business at Pompeu Fabra University, Spain
Pascal
Massart
is a Professor in the Department of Mathematics at
Université
de Paris-Sud,
France
OXFORD
UNIVERSITY PRESS
www.crap.com
Cover image
©
David Torrents www.torrents.info
ISBN
978-0-19-953525-5
9 780
99 535255
|
| any_adam_object | 1 |
| author | Boucheron, Stéphane Lugosi, Gábor 1964- Massart, Pascal |
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| author_facet | Boucheron, Stéphane Lugosi, Gábor 1964- Massart, Pascal |
| author_role | aut aut aut |
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| building | Verbundindex |
| bvnumber | BV040904487 |
| classification_rvk | SK 800 |
| classification_tum | MAT 600f |
| ctrlnum | (OCoLC)840601285 (DE-599)HEB318097176 |
| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. ed. |
| format | Book |
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| id | DE-604.BV040904487 |
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| language | English |
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| spelling | Boucheron, Stéphane Verfasser (DE-588)1036520757 aut Concentration inequalities a nonasymptotic theory of independence Stéphane Boucheron, Gábor Lugosi, Pascal Massart 1. ed. Oxford Oxford Univ. Press 2013 X, 481 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Stochastische Ungleichung (DE-588)4700404-6 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Stochastische Ungleichung (DE-588)4700404-6 s Wahrscheinlichkeitstheorie (DE-588)4079013-7 s DE-604 Lugosi, Gábor 1964- Verfasser (DE-588)17173677X aut Massart, Pascal Verfasser (DE-588)103652082X aut Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025883949&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025883949&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Boucheron, Stéphane Lugosi, Gábor 1964- Massart, Pascal Concentration inequalities a nonasymptotic theory of independence Stochastische Ungleichung (DE-588)4700404-6 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4700404-6 (DE-588)4079013-7 |
| title | Concentration inequalities a nonasymptotic theory of independence |
| title_auth | Concentration inequalities a nonasymptotic theory of independence |
| title_exact_search | Concentration inequalities a nonasymptotic theory of independence |
| title_full | Concentration inequalities a nonasymptotic theory of independence Stéphane Boucheron, Gábor Lugosi, Pascal Massart |
| title_fullStr | Concentration inequalities a nonasymptotic theory of independence Stéphane Boucheron, Gábor Lugosi, Pascal Massart |
| title_full_unstemmed | Concentration inequalities a nonasymptotic theory of independence Stéphane Boucheron, Gábor Lugosi, Pascal Massart |
| title_short | Concentration inequalities |
| title_sort | concentration inequalities a nonasymptotic theory of independence |
| title_sub | a nonasymptotic theory of independence |
| topic | Stochastische Ungleichung (DE-588)4700404-6 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Stochastische Ungleichung Wahrscheinlichkeitstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025883949&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025883949&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT boucheronstephane concentrationinequalitiesanonasymptotictheoryofindependence AT lugosigabor concentrationinequalitiesanonasymptotictheoryofindependence AT massartpascal concentrationinequalitiesanonasymptotictheoryofindependence |