Stat 202C: Monte Carlo Methods for Optimization MW 2-3:15 pm, Spring 2018, Math Science 5128 [syllabus.pdf]
This graduate level course introduces Monte Carlo methods for simulation, optimization, estimation, learning and complex landscape visualization, 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; Hamiltonian and Langevin Monte Carlo; Equi-energy and multi-domain sampler; and Techniques for mapping complex 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% Final exam 35%
Tentative List of Topics
Chapter 1, Introduction to Monte Carlo Methods [Lect1.pdf] PDF files will be distriibuted through CCLE 1, Monte Carlo methods in science and enginnering -- Simulation, estimation, sampling, optimization, learning, and visualization. 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: particle filtering in object tracking, Monte Carlo Tree Search 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. Attraction-Diffusion Algorithm 4. Mapping the energy landscape and case studies 5. Visualization of object recognition and the image universe 6. Landscapes for curriculum learning