Hamiltonian Monte Carlo Methods in Machine Learning

Description

Hamiltonian Monte Carlo Methods in Machine Learning introduces methods for optimal tuning of HMC parameters, along with an introduction of Shadow and Non-canonical HMC methods with improvements and speedup. Lastly, the authors address the critical issues of variance reduction for parameter estimates of numerous HMC based samplers. The book offers a comprehensive introduction to Hamiltonian Monte Carlo methods and provides a cutting-edge exposition of the current pathologies of HMC-based methods in both tuning, scaling and sampling complex real-world posteriors. These are mainly in the scaling of inference (e.g., Deep Neural Networks), tuning of performance-sensitive sampling parameters and high sample autocorrelation. Other sections provide numerous solutions to potential pitfalls, presenting advanced HMC methods with applications in renewable energy, finance and image classification for biomedical applications. Readers will get acquainted with both HMC sampling theory and algorithm implementation. Provides in-depth analysis for conducting optimal tuning of Hamiltonian Monte Carlo (HMC) parameters Presents readers with an introduction and improvements on Shadow HMC methods as well as non-canonical HMC methods Demonstrates how to perform variance reduction for numerous HMC-based samplers Includes source code from applications and algorithms Front Cover Hamiltonian Monte Carlo Methods in Machine Learning Copyright Contents List of figures List of tables Authors Tshilidzi Marwala Wilson Tsakane Mongwe Rendani Mbuvha Foreword Preface Nomenclature List of symbols 1 Introduction to Hamiltonian Monte Carlo 1.1 Introduction 1.2 Background to Markov Chain Monte Carlo 1.3 Metropolis-Hastings algorithm 1.4 Metropolis Adjusted Langevin algorithm 1.5 Hamiltonian Monte Carlo 1.6 Magnetic Hamiltonian Monte Carlo 1.7 Quantum-Inspired Hamiltonian Monte Carlo 1.8 Separable Shadow Hamiltonian Hybrid Monte Carlo 1.9 No-U-Turn Sampler algorithm 1.10 Antithetic Hamiltonian Monte Carlo 1.11 Book objectives 1.12 Book contributions 1.13 Conclusion 2 Sampling benchmarks and performance metrics 2.1 Benchmark problems and datasets 2.1.1 Banana shaped distribution 2.1.2 Multivariate Gaussian distributions 2.1.3 Neal's funnel density 2.1.4 Merton jump-diffusion process model 2.1.5 Bayesian logistic regression 2.1.6 Bayesian neural networks 2.1.7 Benchmark datasets 2.1.8 Processing of the datasets 2.2 Performance metrics 2.2.1 Effective sample size 2.2.2 Convergence analysis 2.2.3 Predictive performance on unseen data 2.3 Algorithm parameter tuning 2.4 Conclusion 3 Stochastic volatility Metropolis-Hastings 3.1 Proposed methods 3.2 Experiments 3.3 Results and discussion 3.4 Conclusion 4 Quantum-inspired magnetic Hamiltonian Monte Carlo 4.1 Proposed algorithm 4.2 Experiment description 4.2.1 Experiment settings 4.2.2 Sensitivity to the vol-of-vol parameter 4.3 Results and discussion 4.4 Conclusion 5 Generalised magnetic and shadow Hamiltonian Monte Carlo 5.1 Proposed partial momentum retention algorithms 5.2 Experiment description 5.2.1 Experiment settings 5.2.2 Sensitivity to momentum refreshment parameter 5.3 Results and discussion 5.4 Conclusion 6 Shadow Magnetic Hamiltonian Monte Carlo 6.1 Background 6.2 Shadow Hamiltonian for MHMC 6.3 Proposed Shadow Magnetic algorithm 6.4 Experiment description 6.4.1 Experiment settings 6.4.2 Sensitivity to momentum refreshment parameter 6.5 Results and discussion 6.6 Conclusion 7 Adaptive Shadow Hamiltonian Monte Carlo 7.1 Proposed adaptive shadow algorithm 7.2 Experiment description 7.3 Results and discussion 7.4 Conclusion 8 Adaptive noncanonical Hamiltonian Monte Carlo 8.1 Background 8.2 Proposed algorithm 8.3 Experiments 8.4 Results and discussion 8.5 Conclusions 9 Antithetic Hamiltonian Monte Carlo techniques 9.1 Proposed antithetic samplers 9.2 Experiment description 9.3 Results and discussion 9.4 Conclusion 10 Bayesian neural network inference in wind speed nowcasting 10.1 Background 10.1.1 Automatic relevance determination 10.1.1.1 Inference of ARD hyperparameters 10.1.1.2 ARD committees 10.2 Experiment setup 10.2.1 WASA meteorological datasets 10.2.2 Relationship to wind power 10.2.3 Performance evaluation 10.2.4 Preliminary step size tuning runs 10.3 Results and discussion 10.3.1 Sampling performance 10.3.2 Predictive performance with ARD 10.3.3 ARD committees and feature importance 10.3.4 Re-training BNNs on relevant features 10.4 Conclusion 11 A Bayesian analysis of the efficacy of Covid-19 lockdown measures 11.1 Background 11.1.1 Review of compartment models for Covid-19 11.1.2 Lockdown alert levels 11.2 Methods 11.2.1 Infection data 11.2.2 The adjusted SIR model 11.2.3 Parameter inference using the No-U-Turn sampler 11.2.3.1 Prior distributions 11.3 Results and discussion 11.3.1 Spreading rate under various lockdown alert levels 11.3.1.1 No restrictions (alert level 0): 5 March 2020 – 18 March 2020 11.3.1.2 Initial restrictions (adjusted alert level 0): 18 March 2020 – 25 March 2020 11.3.1.3 Alert level 5: 26 March 2020 – 30 April 2020 11.3.1.4 Alert level 4: 1 May 2020 – 31 May 2020 11.3.1.5 Alert level 3: 1 June 2020 – 17 August 2020 11.3.1.6 Alert level 2: 18 August 2020 – 20 September 2020 11.3.1.7 Alert level 1: 21 September 2020 – 28 December 2020 11.3.1.8 Adjusted level 3: 29 December 2020 – 28 February 2021 11.3.1.9 Adjusted alert levels [1-4]: 1 March 2021 – 18 December 2021 11.3.1.10 Transitions between alert levels and efficacy of restrictions 11.3.1.11 Recovery rate 11.4 Conclusion 12 Probabilistic inference of equity option prices under jump-diffusion processes 12.1 Background 12.1.1 Merton jump diffusion option pricing model 12.2 Numerical experiments 12.2.1 Data description 12.2.2 Experiment description 12.3 Results and discussions 12.4 Conclusion 13 Bayesian inference of local government audit outcomes 13.1 Background 13.2 Experiment description 13.2.1 Data description 13.2.2 Financial ratio calculation 13.2.3 Bayesian logistic regression with ARD 13.3 Results and discussion 13.4 Conclusion 14 Conclusions 14.1 Summary of contributions 14.2 Ongoing and future work A Separable shadow Hamiltonian A.1 Derivation of separable shadow Hamiltonian A.2 S2HMC satisfies detailed balance A.3 Derivatives from non-canonical Poisson brackets B ARD posterior variances C ARD committee feature selection D Summary of audit outcome literature survey References Index Back Cover