Stat 202C: Monte Carlo Methods for Optimization MW 2-3:15 pm, Spring 2017, Math Science 5128 [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 [Lect1.pdf] 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 [project_1.pdf] 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. Three interpretations of the SW method Chapter 7 Langevin Dynamics 1. Hamiltonian Monte Carlo 2. Langiven dynamics used in machine learning Gibbs Reaction and Diffusion equations, Alternative Back-propagation 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