Concentration inequalities and model selection: École d'Été de Probabilités de Saint-Flour XXXIII - 2003
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| Other Authors: | |
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2007
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| Series: | Lecture notes in mathematics
1896 : École d'Été de Probabilités de Saint-Flour |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 337 S. |
| ISBN: | 9783540484974 3540484973 |
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| 245 | 1 | 0 | |a Concentration inequalities and model selection |b École d'Été de Probabilités de Saint-Flour XXXIII - 2003 |c Pascal Massart. Ed.: Jean Picard |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2007 | |
| 300 | |a XIV, 337 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 1896 : École d'Été de Probabilités de Saint-Flour | |
| 650 | 4 | |a Statistik - Modellwahl | |
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 7 | |a Combinatieleer. |2 gtt | |
| 650 | 4 | |a Inequalities (Mathematics) | |
| 650 | 4 | |a Mathematical models | |
| 650 | 4 | |a Mathematical statistics |x Methodology | |
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| 700 | 1 | |a Picard, Jean |4 edt | |
| 830 | 0 | |a Lecture notes in mathematics |v 1896 : École d'Été de Probabilités de Saint-Flour |w (DE-604)BV000676446 |9 1896 | |
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| adam_text | Contents
Introduction
............................................... 1
1.1 Model
Selection
......................................... 1
1.1.1 Minimum
Contrast Estimation
......................
З
1.1.2
The Model Choice Paradigm
........................ 5
1.1.3
Model Selection via Penalization
.................... 7
1.2
Concentration Inequalities
................................ 10
1.2.1
The Gaussian Concentration Inequality
.............. 10
1.2.2
Suprema
of Empirical Processes
..................... 11
1.2.3
The Entropy Method
.............................. 12
Exponential and Information Inequalities
.................. 15
2.1
The Cramer-Chernoff Method
............................ 15
2.2
Sums of Independent Random Variables
.................... 21
2.2.1
Hoeffding s Inequality
.............................. 21
2.2.2
Bennett s Inequality
............................... 23
2.2.3
Bernstein s Inequality
.............................. 24
2.3
Basic Information Inequalities
............................ 27
2.3.1
Duality and Variational Formulas
................... 27
2.3.2
Some Links Between the Moment Generating
Function and Entropy
.............................. 29
2.3.3
Pinsker s Inequality
................................ 31
2.3.4
Birgé s
Lemma
.................................... 32
2.4
Entropy on Product Spaces
............................... 35
2.4.1
Marten s Coupling
................................ 37
2.4.2
Tensorization Inequality for Entropy
................ 40
2.5
«^-Entropy
.............................................. 43
2.5.1
Necessary Condition for the Convexity of ^-Entropy
... 45
2.5.2
A Duality Formula for 0-Entropy
.................... 46
2.5.3
A Direct Proof of the Tensorization Inequality
........ 49
2.5.4
Efron-Stem s Inequality
............................ 50
XII Contents
3
Gaussian
Processes
........................................ 53
3.1
Introduction
and Basic Remarks
.......................... 53
3.2
Concentration of the Gaussian Measure on RN
.............. 56
3.2.1
The Isoperimetric Nature of the Concentration
Phenomenon
...................................... 57
3.2.2
The Gaussian Isoperimetric Theorem
................ 59
3.2.3
Gross Logarithmic Sobolev Inequality
............... 62
3.2.4
Application to
Suprema
of Gaussian
Random Vectors
.................................. 64
3.3
Comparison Theorems for Gaussian Random Vectors
........ 66
3.3.1
Slepian s Lemma
.................................. 66
3.4
Metric Entropy and Gaussian Processes
.................... 70
3.4.1
Metric Entropy
................................... 70
3.4.2
The Chaining Argument
........................... 72
3.4.3
Continuity of Gaussian Processes
.................... 74
3.5
The
Isonormal
Proceas...................................
77
3.5.1
Definition and First Properties
...................... 77
3.5.2
Continuity Sets with Examples
...................... 79
4
Gaussian Model Selection
.................................. 83
4.1
Introduction
............................................ 83
4.1.1
Examples of Gaussian Frameworks
.................. 83
4.1.2
Some Model Selection Problems
..................... 86
4.1.3
The Least Squares Procedure
....................... 87
4.2
Selecting Linear Models
.................................. 88
4.2.1
A First Model Selection Theorem for Linear Models
... 89
4.2.2
Lower Bounds for the Penalty Term
................. 94
4.2.3
Mixing Several Strategies
........................... 98
4.3
Adaptive Estimation in the Minimax Sense
................101
4.3.1
Minimax Lower Bounds
...........................102
4.3.2
Adaptive Properties of Penalized Estimators for
Gaussian Sequences
................................115
4.3.3
Adaptation with Respect to Ellipsoids
...............116
4.3.4
Adaptation with Respect to Arbitrary fp-Bodies
......117
4.3.5
A Special Strategy for
Besov
Bodies
.................122
4.4
A General Model Selection Theorem
.......................125
4.4.1
Statement
........................................125
4.4.2
Selecting Ellipsoids: A Link with Regularization
.......131
4.4.3
Selecting Nets Toward Adaptive Estimation for
Arbitrary Compact Sets
............................139
4.5
Appendix: From Function Spaces to Sequence Spaces
........144
5
Concentration Inequalities
.................................147
5.1
Introduction
............................................147
5.2
The Bounded Difference Inequality via Marton s Coupling
___148
5.3
Concentration Inequalities via the Entropy Method
.........
I54
Contents XIII
5.3.1 (/»-Sobolev and Moment
Inequalities
..................155
5.3.2
A Poissonian Inequality for Self-Bounding
Functional
.......................................157
5.3.3
(/bSobolev Type Inequalities
........................162
5.3.4
From Efron-Stein to Exponential Inequalities
.........166
5.3.5
Moment Inequalities
..............................172
6
Maximal Inequalities
......................................183
6.1
Set-Indexed Empirical Processes
...........................184
6.1.1
Random Vectors and Rademacher Processes
..........184
6.1.2
Vapnik-Chervonenkis Classes
......................186
6.1.3
Li-Entropy with Bracketing
........................190
6.2
Function-Indexed Empirical Processes
......................192
7
Density Estimation via Model Selection
...................201
7.1
Introduction and Notations
...............................201
7.2
Penalized Least Squares Model Selection
...................202
7.2.1
The Nature of Penalized
LSE
.......................204
7.2.2
Model Selection for a Polynomial Collection
of Models
........................................211
7.2.3
Model Subset Selection Within a Localized Basis
......219
7.3
Selecting the Best Histogram via Penalized Maximum
Likelihood Estimation
....................................225
7.3.1
Some Deepest Analysis of Chi-Square Statistics
.......228
7.3.2
A Model Selection Result
...........................230
7.3.3
Choice of the Weights {xm
,
m
Є
M}
................236
7.3.4
Lower Bound for the Penalty Function
...............237
7.4
A General Model Selection Theorem for MLE
...............238
7.4.1
Local Entropy with Bracketing Conditions
............239
7.4.2
Finite Dimensional Models
.........................245
7.5
Adaptive Estimation in the Minimax Sense
.................251
7.5.1
Lower Bounds for the Minimax Risk
................251
7.5.2
Adaptive Properties of Penalized
LSE
...............263
7.5.3
Adaptive Properties of Penalized MLE
...............267
7.6
Appendix
..............................................273
7.6.1
Kullback-Leibler Information
and
Hellinger
Distance
.............................273
7.6.2
Moments of Log-Likelihood Ratios
...................276
7.6.3
An Exponential Bound for Log-Likelihood Ratios
......277
8
Statistical Learning
........................................279
8.1
Introduction
............................................279
8.2
Model Selection in Statistical Learning
.....................280
8.2.1
A Model Selection Theorem
........................281
XIV Contents
8.3
A Refined Analysis for the Risk of an Empirical
Risk Minimizer
..........................................287
8.3.1
The Main Theorem
................................288
8.3.2
Application to Bounded Regression
.................293
8.3.3
Application to Classification
........................296
8.4
A Refined Model Selection Theorem
.......................301
8.4.1
Application to Bounded Regression
..................303
8.5
Advanced Model Selection Problems
.......................307
8.5.1
Hold-Out as a Margin Adaptive Selection Procedure
. .. 308
8.5.2
Data-Driven
Penalties
..............................314
References
.....................................................319
Index
..........................................................325
List of Participants
............................................331
List of Short Lectures
.........................................335
|
| adam_txt |
Contents
Introduction
. 1
1.1 Model
Selection
. 1
1.1.1 Minimum
Contrast Estimation
.
З
1.1.2
The Model Choice Paradigm
. 5
1.1.3
Model Selection via Penalization
. 7
1.2
Concentration Inequalities
. 10
1.2.1
The Gaussian Concentration Inequality
. 10
1.2.2
Suprema
of Empirical Processes
. 11
1.2.3
The Entropy Method
. 12
Exponential and Information Inequalities
. 15
2.1
The Cramer-Chernoff Method
. 15
2.2
Sums of Independent Random Variables
. 21
2.2.1
Hoeffding's Inequality
. 21
2.2.2
Bennett's Inequality
. 23
2.2.3
Bernstein's Inequality
. 24
2.3
Basic Information Inequalities
. 27
2.3.1
Duality and Variational Formulas
. 27
2.3.2
Some Links Between the Moment Generating
Function and Entropy
. 29
2.3.3
Pinsker's Inequality
. 31
2.3.4
Birgé's
Lemma
. 32
2.4
Entropy on Product Spaces
. 35
2.4.1
Marten's Coupling
. 37
2.4.2
Tensorization Inequality for Entropy
. 40
2.5
«^-Entropy
. 43
2.5.1
Necessary Condition for the Convexity of ^-Entropy
. 45
2.5.2
A Duality Formula for 0-Entropy
. 46
2.5.3
A Direct Proof of the Tensorization Inequality
. 49
2.5.4
Efron-Stem's Inequality
. 50
XII Contents
3
Gaussian
Processes
. 53
3.1
Introduction
and Basic Remarks
. 53
3.2
Concentration of the Gaussian Measure on RN
. 56
3.2.1
The Isoperimetric Nature of the Concentration
Phenomenon
. 57
3.2.2
The Gaussian Isoperimetric Theorem
. 59
3.2.3
Gross' Logarithmic Sobolev Inequality
. 62
3.2.4
Application to
Suprema
of Gaussian
Random Vectors
. 64
3.3
Comparison Theorems for Gaussian Random Vectors
. 66
3.3.1
Slepian's Lemma
. 66
3.4
Metric Entropy and Gaussian Processes
. 70
3.4.1
Metric Entropy
. 70
3.4.2
The Chaining Argument
. 72
3.4.3
Continuity of Gaussian Processes
. 74
3.5
The
Isonormal
Proceas.
77
3.5.1
Definition and First Properties
. 77
3.5.2
Continuity Sets with Examples
. 79
4
Gaussian Model Selection
. 83
4.1
Introduction
. 83
4.1.1
Examples of Gaussian Frameworks
. 83
4.1.2
Some Model Selection Problems
. 86
4.1.3
The Least Squares Procedure
. 87
4.2
Selecting Linear Models
. 88
4.2.1
A First Model Selection Theorem for Linear Models
. 89
4.2.2
Lower Bounds for the Penalty Term
. 94
4.2.3
Mixing Several Strategies
. 98
4.3
Adaptive Estimation in the Minimax Sense
.101
4.3.1
Minimax Lower Bounds
.102
4.3.2
Adaptive Properties of Penalized Estimators for
Gaussian Sequences
.115
4.3.3
Adaptation with Respect to Ellipsoids
.116
4.3.4
Adaptation with Respect to Arbitrary fp-Bodies
.117
4.3.5
A Special Strategy for
Besov
Bodies
.122
4.4
A General Model Selection Theorem
.125
4.4.1
Statement
.125
4.4.2
Selecting Ellipsoids: A Link with Regularization
.131
4.4.3
Selecting Nets Toward Adaptive Estimation for
Arbitrary Compact Sets
.139
4.5
Appendix: From Function Spaces to Sequence Spaces
.144
5
Concentration Inequalities
.147
5.1
Introduction
.147
5.2
The Bounded Difference Inequality via Marton's Coupling
_148
5.3
Concentration Inequalities via the Entropy Method
.
I54
Contents XIII
5.3.1 (/»-Sobolev and Moment
Inequalities
.155
5.3.2
A Poissonian Inequality for Self-Bounding
Functional
.157
5.3.3
(/bSobolev Type Inequalities
.162
5.3.4
From Efron-Stein to Exponential Inequalities
.166
5.3.5
Moment Inequalities
.172
6
Maximal Inequalities
.183
6.1
Set-Indexed Empirical Processes
.184
6.1.1
Random Vectors and Rademacher Processes
.184
6.1.2
Vapnik-Chervonenkis Classes
.186
6.1.3
Li-Entropy with Bracketing
.190
6.2
Function-Indexed Empirical Processes
.192
7
Density Estimation via Model Selection
.201
7.1
Introduction and Notations
.201
7.2
Penalized Least Squares Model Selection
.202
7.2.1
The Nature of Penalized
LSE
.204
7.2.2
Model Selection for a Polynomial Collection
of Models
.211
7.2.3
Model Subset Selection Within a Localized Basis
.219
7.3
Selecting the Best Histogram via Penalized Maximum
Likelihood Estimation
.225
7.3.1
Some Deepest Analysis of Chi-Square Statistics
.228
7.3.2
A Model Selection Result
.230
7.3.3
Choice of the Weights {xm
,
m
Є
M}
.236
7.3.4
Lower Bound for the Penalty Function
.237
7.4
A General Model Selection Theorem for MLE
.238
7.4.1
Local Entropy with Bracketing Conditions
.239
7.4.2
Finite Dimensional Models
.245
7.5
Adaptive Estimation in the Minimax Sense
.251
7.5.1
Lower Bounds for the Minimax Risk
.251
7.5.2
Adaptive Properties of Penalized
LSE
.263
7.5.3
Adaptive Properties of Penalized MLE
.267
7.6
Appendix
.273
7.6.1
Kullback-Leibler Information
and
Hellinger
Distance
.273
7.6.2
Moments of Log-Likelihood Ratios
.276
7.6.3
An Exponential Bound for Log-Likelihood Ratios
.277
8
Statistical Learning
.279
8.1
Introduction
.279
8.2
Model Selection in Statistical Learning
.280
8.2.1
A Model Selection Theorem
.281
XIV Contents
8.3
A Refined Analysis for the Risk of an Empirical
Risk Minimizer
.287
8.3.1
The Main Theorem
.288
8.3.2
Application to Bounded Regression
.293
8.3.3
Application to Classification
.296
8.4
A Refined Model Selection Theorem
.301
8.4.1
Application to Bounded Regression
.303
8.5
Advanced Model Selection Problems
.307
8.5.1
Hold-Out as a Margin Adaptive Selection Procedure
. . 308
8.5.2
Data-Driven
Penalties
.314
References
.319
Index
.325
List of Participants
.331
List of Short Lectures
.335 |
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| author | Massart, Pascal |
| author2 | Picard, Jean |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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| genre | (DE-588)1071861417 Konferenzschrift gnd-content |
| genre_facet | Konferenzschrift |
| id | DE-604.BV022419764 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T17:25:17Z |
| indexdate | 2024-07-09T20:57:12Z |
| institution | BVB |
| isbn | 9783540484974 3540484973 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015628102 |
| oclc_num | 255827828 |
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| physical | XIV, 337 S. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Massart, Pascal Verfasser (DE-588)103652082X aut Concentration inequalities and model selection École d'Été de Probabilités de Saint-Flour XXXIII - 2003 Pascal Massart. Ed.: Jean Picard Berlin [u.a.] Springer 2007 XIV, 337 S. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1896 : École d'Été de Probabilités de Saint-Flour Statistik - Modellwahl Mathematisches Modell Combinatieleer. gtt Inequalities (Mathematics) Mathematical models Mathematical statistics Methodology Verdelingen (statistiek) gtt Statistik (DE-588)4056995-0 gnd rswk-swf Modellwahl (DE-588)4304786-5 gnd rswk-swf (DE-588)1071861417 Konferenzschrift gnd-content Statistik (DE-588)4056995-0 s Modellwahl (DE-588)4304786-5 s DE-604 Picard, Jean edt Lecture notes in mathematics 1896 : École d'Été de Probabilités de Saint-Flour (DE-604)BV000676446 1896 Digitalisierung TU Muenchen application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015628102&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Massart, Pascal Concentration inequalities and model selection École d'Été de Probabilités de Saint-Flour XXXIII - 2003 Lecture notes in mathematics Statistik - Modellwahl Mathematisches Modell Combinatieleer. gtt Inequalities (Mathematics) Mathematical models Mathematical statistics Methodology Verdelingen (statistiek) gtt Statistik (DE-588)4056995-0 gnd Modellwahl (DE-588)4304786-5 gnd |
| subject_GND | (DE-588)4056995-0 (DE-588)4304786-5 (DE-588)1071861417 |
| title | Concentration inequalities and model selection École d'Été de Probabilités de Saint-Flour XXXIII - 2003 |
| title_auth | Concentration inequalities and model selection École d'Été de Probabilités de Saint-Flour XXXIII - 2003 |
| title_exact_search | Concentration inequalities and model selection École d'Été de Probabilités de Saint-Flour XXXIII - 2003 |
| title_exact_search_txtP | Concentration inequalities and model selection École d'Été de Probabilités de Saint-Flour XXXIII - 2003 |
| title_full | Concentration inequalities and model selection École d'Été de Probabilités de Saint-Flour XXXIII - 2003 Pascal Massart. Ed.: Jean Picard |
| title_fullStr | Concentration inequalities and model selection École d'Été de Probabilités de Saint-Flour XXXIII - 2003 Pascal Massart. Ed.: Jean Picard |
| title_full_unstemmed | Concentration inequalities and model selection École d'Été de Probabilités de Saint-Flour XXXIII - 2003 Pascal Massart. Ed.: Jean Picard |
| title_short | Concentration inequalities and model selection |
| title_sort | concentration inequalities and model selection ecole d ete de probabilites de saint flour xxxiii 2003 |
| title_sub | École d'Été de Probabilités de Saint-Flour XXXIII - 2003 |
| topic | Statistik - Modellwahl Mathematisches Modell Combinatieleer. gtt Inequalities (Mathematics) Mathematical models Mathematical statistics Methodology Verdelingen (statistiek) gtt Statistik (DE-588)4056995-0 gnd Modellwahl (DE-588)4304786-5 gnd |
| topic_facet | Statistik - Modellwahl Mathematisches Modell Combinatieleer. Inequalities (Mathematics) Mathematical models Mathematical statistics Methodology Verdelingen (statistiek) Statistik Modellwahl Konferenzschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015628102&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT massartpascal concentrationinequalitiesandmodelselectionecoledetedeprobabilitesdesaintflourxxxiii2003 AT picardjean concentrationinequalitiesandmodelselectionecoledetedeprobabilitesdesaintflourxxxiii2003 |