Hamiltonian Monte Carlo Methods in Machine Learning

222 Pages·2023·18.469 MB·English

by  TshilidziMarwala,Wilson TsakaneMongwe, Rendani Mbuvha

Hamiltonian Monte Carlo
Methods in Machine
Learning
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Hamiltonian Monte Carlo
Methods in Machine
Learning
Tshilidzi Marwala
SchoolofElectricalandElectronicEngineering
UniversityofJohannesburg
AucklandPark,SouthAfrica
Wilson Tsakane Mongwe
SchoolofElectricalandElectronicEngineering
UniversityofJohannesburg
AucklandPark,SouthAfrica
Rendani Mbuvha
SchoolofElectronicEngineeringandComputerScience
QueenMaryUniversityofLondon
London,UnitedKingdom
SchoolofStatisticsandActuarialScience
UniversityofWitwatersrand
Braamfontein,SouthAfrica
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Contents
List of figures xiii
List of tables xvii
Authors xxi
Foreword xxiii
Preface xxv
Nomenclature xxvii
List of symbols xxix
1. Introduction to Hamiltonian Monte Carlo 1
1.1. Introduction 1
1.2. BackgroundtoMarkovChainMonteCarlo 6
1.3. Metropolis-Hastingsalgorithm 8
1.4. MetropolisAdjustedLangevinalgorithm 10
1.5. HamiltonianMonteCarlo 11
1.6. MagneticHamiltonianMonteCarlo 14
1.7. Quantum-InspiredHamiltonianMonteCarlo 16
1.8. SeparableShadowHamiltonianHybridMonteCarlo 18
1.9. No-U-TurnSampleralgorithm 21
1.10.AntitheticHamiltonianMonteCarlo 24
1.11.Bookobjectives 26
1.12.Bookcontributions 27
1.13.Conclusion 29
2. Sampling benchmarks and performance metrics 31
2.1. Benchmarkproblemsanddatasets 31
vii
viii Contents
2.1.1. Bananashapeddistribution 31
2.1.2. MultivariateGaussiandistributions 31
2.1.3. Neal’sfunneldensity 32
2.1.4. Mertonjump-diffusionprocessmodel 32
2.1.5. Bayesianlogisticregression 33
2.1.6. Bayesianneuralnetworks 33
2.1.7. Benchmarkdatasets 33
2.1.8. Processingofthedatasets 35
2.2. Performancemetrics 35
2.2.1. Effectivesamplesize 36
2.2.2. Convergenceanalysis 37
2.2.3. Predictiveperformanceonunseendata 37
2.3. Algorithmparametertuning 38
2.4. Conclusion 39
3. Stochastic volatility Metropolis-Hastings 41
3.1. Proposedmethods 41
3.2. Experiments 43
3.3. Resultsanddiscussion 44
3.4. Conclusion 51
4. Quantum-inspired magnetic Hamiltonian Monte Carlo 53
4.1. Proposedalgorithm 53
4.2. Experimentdescription 56
4.2.1. Experimentsettings 56
4.2.2. Sensitivitytothevol-of-volparameter 57
4.3. Resultsanddiscussion 57
4.4. Conclusion 59
5. Generalised magnetic and shadow Hamiltonian Monte Carlo 63
5.1. Proposedpartialmomentumretentionalgorithms 63
Contents ix
5.2. Experimentdescription 65
5.2.1. Experimentsettings 65
5.2.2. Sensitivitytomomentumrefreshmentparameter 66
5.3. Resultsanddiscussion 67
5.4. Conclusion 71
6. Shadow Magnetic Hamiltonian Monte Carlo 73
6.1. Background 73
6.2. ShadowHamiltonianforMHMC 74
6.3. ProposedShadowMagneticalgorithm 75
6.4. Experimentdescription 77
6.4.1. Experimentsettings 77
6.4.2. Sensitivitytomomentumrefreshmentparameter 77
6.5. Resultsanddiscussion 78
6.6. Conclusion 81
7. Adaptive Shadow Hamiltonian Monte Carlo 83
7.1. Proposedadaptiveshadowalgorithm 83
7.2. Experimentdescription 86
7.3. Resultsanddiscussion 86
7.4. Conclusion 89
8. Adaptive noncanonical Hamiltonian Monte Carlo 91
8.1. Background 91
8.2. Proposedalgorithm 92
8.3. Experiments 95
8.4. Resultsanddiscussion 96
8.5. Conclusions 100
9. Antithetic Hamiltonian Monte Carlo techniques 101
9.1. Proposedantitheticsamplers 101