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Concentration of Measure for the Analysis of Randomised Algorithms

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Concentration of Measure for the Analysis of Randomised Algorithms Concentration of Measure for the Analysis of Randomised Algorithms Devdatt P. Dubhashi Alessandro Panconesi October 21, 2005 DRAFT 2 DRAFT Contents 1 Chernoff–Hoeffding Bounds 17 1.1 Whatis“ConcentrationofMeasure”?. 17 1.2 TheBinomialDistribution . 19 1.3 TheChernoffBound ......................... 19 1.4 HeterogeneousVariables . 21 1.5 TheHoeffdingExtension . 22 1.6 UsefulFormsoftheBound. 22 1.7 AVarianceBound .......................... 24 1.8 BibliographicNotes. .. .. 26 1.9 Problems................................ 27 2 Interlude: Probabilistic Recurrences 33 3 Applying the CH-bounds 39 3.1 Probabilistic Amplification . 39 3.2 LoadBalancing ............................ 40 3.3 DataStructures:SkipLists . 41 3.3.1 SkipLists: TheDataStructure . 41 3.3.2 Skip Lists: Randomization makes it easy . 42 3 DRAFT 4 CONTENTS 3.3.3 Quicksort ........................... 45 3.4 PacketRouting ............................ 47 3.5 RandomizedRounding . 49 3.6 BibliographicNotes. .. .. 49 3.7 Problems................................ 50 4 Chernoff-Hoeffding Bounds in Dependent Settings 55 4.1 NegativeDependence. 55 4.2 LocalDependence........................... 59 4.3 Janson’sInequality . .. .. 61 4.4 LimitedIndependence . 64 4.5 MarkovDependence ......................... 67 4.5.1 Definitions........................... 67 4.5.2 StatementoftheBound . 68 4.5.3 Application: Probability Amplification . 68 4.6 BibliographicNotes. .. .. 70 4.7 Problems................................ 71 5 Martingales and Azuma’s Inequality 73 5.1 Review of Conditional Probabilities and Expectations . ...... 74 5.2 Martingales and Azuma’s Inequality. 76 5.3 Generalizing Martingales and Azuma’s Inequality. ..... 81 5.4 The Method of Bounded Differences . 83 5.5 BibliographicNotes. .. .. 87 5.6 Problems................................ 88 DRAFT CONTENTS 5 6 The Method of Bounded Differences 91 6.1 Chernoff–HoeffdingRevisited . 91 6.2 Stochastic Optimization: Bin Packing . 92 6.3 Game Theory and Blackwell’s Approachability Theorem . ... 92 6.4 BallsandBins............................. 93 6.5 DistributedEdgeColouring . 94 6.5.1 Topvertices .......................... 96 6.5.2 The Difficulty with the Other Vertices . 97 6.6 Problems................................ 98 7 Averaged Bounded Differences 101 7.1 HypergeometricDistribution . 102 7.2 OccupancyinBallsandBins. 103 7.3 StochasticOptimization: TSP . 104 7.4 Coupling................................ 105 7.5 DistributedEdgeColouring . 108 7.5.1 PreliminaryAnalysis . 108 7.5.2 GeneralVertexinalgorithmI . 109 7.5.3 BottomVertexinAlgorithmP . 110 7.6 HandlingRareBadEvents . 112 7.7 Quicksort ............................... 114 7.8 Problems................................ 116 8 The Method of Bounded Variances 119 8.1 A Variance Bound for Martingale Sequences . 120 DRAFT 6 CONTENTS 8.2 Applications.............................. 123 8.2.1 BoundingtheVariance . 125 8.2.2 Dealing with Unlikely Circumstances . 127 8.3 BibliographicNotes. 130 8.4 Problems................................ 130 9 The Infamous Upper Tail 135 9.1 Motivation: Non-Lipschitz Functions . 135 9.2 Concentration of Multivariate Polynomials . 136 9.3 TheDeletionMethod. 137 9.4 Problems................................ 139 10 Isoperimetric Inequalities and Concentration 141 10.1 Isoperimetric inequalities . 141 10.2 IsoperimetryandConcentration . 142 10.2.1 Concentration of Lipschitz functions . 142 10.3 Examples: Classical and Discrete . 144 10.3.1 Euclidean Space with Lebesgue Measure . 144 10.3.2 TheEuclideanSphere . 144 10.3.3 Euclidean Space with Gaussian Measure . 145 10.3.4 TheHammingCube . 145 10.4 Martingales and Isoperimetric inequalities . ..... 146 10.5 BibliographicNotes. 147 10.6Problems................................ 148 11 Talagrand’s Isoperimetric Inequality 151 DRAFT CONTENTS 7 11.1 Statementoftheinequality . 151 11.2 Hereditary Functions of of Index Sets . 153 11.2.1 AGeneralFramework . 153 11.2.2 Increasing subsequences . 154 11.2.3 BallsandBins. 155 11.2.4 Discrete Isoperimetric Inequalities . 155 11.3 CertifiableFunctions . 155 11.3.1 EdgeColouring . 158 11.4 The Method of Non–uniformly Bounded Differences . 159 11.4.1 Chernoff–HoeffdingBounds . 160 11.4.2 BallsandBins. 161 11.4.3 StochasticTSP . 161 11.4.4 First Birth Times in Branching Processes . 162 11.5 BibliographicNotes. 163 11.6Problems................................ 163 12 Isoperimetric Inequalities and Concentration via Transportation Cost Inequalities 167 12.1 Distance Between Probability Distributions . ..... 167 12.1.1 Distance in Product Spaces with Hamming Metric . 168 12.2 TC Inequalities Imply Isoperimetric Inequalities and Concentration 169 12.3 TC Inequality in Product Spaces with Hamming Distance . 171 12.3.1 Onedimension . 171 12.3.2 Higherdimensions . 172 12.4 An Extension to Non-Product Measures . 174 DRAFT 8 CONTENTS 12.5Problems................................ 176 12.6 BibliographicNotes. 178 13 Quadratic Transportation Cost and Talagrand’s Inequality 179 13.1 Introduction.............................. 179 13.2 ReviewandRoadmap . 180 13.3 A L2 (Pseudo)MetriconDistributions . 182 13.3.1 OneDimension . 182 13.3.2 Tensorization to Higher Dimensions . 183 13.4 QuadraticTransportationCost . 183 13.4.1 Basecase: OneDimension . 184 13.4.2 Tensorization to Higher Dimensions . 185 13.5 Talagrand’s Inequality via Quadratic Transportation Cost . 186 13.6 Method of Bounded Differences Revisited . 187 13.7 Extension to Dependent Processes . 189 13.8 BibliographicNotes. 190 13.9Problems................................ 190 14 Log-Sobolev Inequalities and Concentration 191 14.1 Introduction.............................. 191 14.2 A Discrete Log-Sobolev Inequality on the Hamming Cube . 192 14.3 Concentration: The Herbst Argument . 192 14.4 Tensorization ............................. 194 14.5 Modified Log-Sobolev Inequalities in Product Spaces . ..... 195 14.6 The Method of Bounded Differences Revisited . 197 DRAFT CONTENTS 9 14.7 Talagrand’s Inequality Revisited . 198 14.8Problems................................ 200 14.9 BibliographicNotes. 201 DRAFT 10 CONTENTS DRAFT Preface The aim of this monograph is to make a body of tools for establishing concen- tration of measure accessible to researchers working in the design and analysis of randomized algorithms. Concentration of measure refers to the phenomenon that a function of a large number of random variables tends to concentrate its values in a relatively narrow range (under certain conditions of smoothness of the function and under certain conditions on the dependence amongst the set of random variables). Such a result is of obvious importance to the analysis of randomized algorithms: for instance, the running time of such an algorithm can then be guaranteed to be concentrated around a pre-computed value. More generally, various other parameters measur- ing the performance of randomized algorithms can be provided tight guarantees via such an analysis. In a sense, the subject of concentration of measure lies at the core of modern probability theory as embodied in the laws of large numbers,denH00the Central Limit Theorem and, in particular, the theory of Large Deviations [15]. However, these results are asymptotic – they refer to the limit as the number of variables n, goes to infinity, for example. In the analysis of algorithms, we typically require quan- titative estimates that are valid for finite (though large) values of n. The earliest such results can be traced back to the work of Azuma, Chernoff and Hoeffding in the 1950s. Subsequently there have been steady advances, particularly in the classical setting of martingales. In the last couple of decades, these methods have taken on renewed interest driven by applications in algorithms and optimization. Also several new techniques have been developed. Unfortunately, much of this material is scattered in the literature, and also rather forbidding for someone entering the field from a Computer Science/Algorithms background. Often this is because the methods are couched in the technical language of analysis and/or measure theory. While this may be strictly necessary to develop results in their full generality, it is not needed when the method is used in computer science applications (where the probability spaces are often 11 DRAFT 12 CONTENTS finite and discrete), and indeed may only serve as a distraction or barrier. Our main goal here is to give an exposition of the basic methods for measure con- centration in a manner which makes it accessible to the researcher in randomized algorithms and enables him/her to quickly start putting them to work in his/her application. Our approach is as follows: 1. Motivate the need for a concentration tool by picking an application in the form of the analysis of a randomized algorithm or probabilistic combina- torics. 2. Give only the main outline of the application, suppressing details and iso- lating the abstraction that relates to the concentration of measure analysis. The reader can go back to the details of the specific application following the references or the complementary book used (see below for suggestions). 3. State and prove the results not necessarily in the most general or sharpest form possible, but rather in the form that is clearest to understand and convenient as well as sufficient to use for the application at hand. 4. Return to the same example or abstraction several times using different tools to illustrate their relative strengths and weaknesses and ease of use – a particular tool works better than another in some situations, worse in others. Other significant
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