Stat 202C: Monte Carlo Methods for Optimization MW 4-5:15 pm, Spring 2016, Boelter Hall 5273 [syllabus.pdf]
This graduate level course introduces Monte Carlo methods for optimization, estimation and learning, including: Importance sampling; Sequential importance sampling; Markov chain Monte Carlo (MCMC) sampling techniques including Gibbs samplers, Metropolis/Hastings and various improvements; Simulated annealing; Exact sampling techniques; Convergence analysis; Data augmentation; Cluster sampling, such as Swendsen-Wang and SW-cuts; Equi-energy and multi-domain sampler; and Mapping the energy landscapes.
Textbooks The lectures will be based on the following book draft.
Grading Plan: 4 units, letter grades
The grade will be based on four parts 2 homework 20% 3 small projects 45% Project 1: Importance sampling for counting the number of SAWs in a lattice (15%) Project 2: Exact sampling of Potts model with Gibbs sampler (15%) Project 3: Cluster sampling for Potts model using Swerndsen-Wang method (15%) Final exam 35%
Tentative List of Topics
Chapter 1, Introduction to Monte Carlo Methods 1, Monte Carlo methods in science and enginnering -- Simulation, estimation, sampling, optimization and learning. 2, Topics and issues in Monte Carlo methods Chapter 2, Sequential Monte Carlo 1. Importance sampling and weighted samples 2. Advanced importance sampling techniques 3. Framework for sequential Monte Carlo (selection, pruning, resampling, ...) 4, Application on learning log-linear/Gibbs models 5. Application: particle filtering in object tracking Chapter 3, Backgrounds on Markov Chains 1. The transition matrix 2. Topology of transition matrix: communication and period 3. Positive recurrence and invariant measures 4. Ergodicity theorem Chapter 4, Metropolis methods and its variants 1. Metropolis algorithm and the Hastings's generalization 2. Special case: Metropolized independence sampler 3. Reversible jumps and trans-dimensional MCMC Chapter 5 Gibbs sampler and its variants 1. Gibbs sampler 2. generalizations: Hit-and-run, Multi-grid, generalized Gibbs, Metropolized Gibbs, 3. Data association and data augmentation 4. Slice sampling Chapter 6 Clustering sampling 1. Ising/Potts models 2. Swendsen-Wang and clustering sampling 3, Original papers by Swendson-Wang and Edwards-Sokal in Physics Review Letters. 4, Three interpretations of the SW method Chapter 7 Monte Carlo for Bayesian statistics 1. Bayesian hierarchical modeling 2. Missing data imputation Chapter 8 Convergence analysis 1. Monitoring and diagnosing convergence 2*. Contraction coefficient 3. Puskin's order 4*. Eigen-structures of the transition matrix (Perron-Frobenius theorem, spectral theorem) 5. Geometric bounds 6*. Exact analysis on independence Metropolised Sampler (IMS) 7*. First hitting time analysis and bounds for IMS (paper) 8. Path coupling techniques. Bounds for Gibbs sampler and Swendson-Wang algorithm (paper). * discussed in previous Chapters. Chapter 9 Exact sampling 1. Coupling from the past CFTP 2. Bounding chains Chapter 10 Advanced topics 1. Equi-energy and mult-domain sampler 2. Wang-Landau algorithm 3. Stochastic gradient 4. Mapping the energy landscape and case studies 5. Comparing the clustering algorithms 6. Landscapes for curriculum learning