In this course we will study several techniques developed in the last 30 years to sample from sophisticated probability distributions of exponential size. Approximately half of the course will focus on techniques based on Markov Chain Monte Carlo techniques. In the rest of the course we will see more modern results based on Correlation decay and geometry of Polynomials.
Administrative Information
- Instructor: Shayan Oveis Gharan
- Office Hours: By appointment, email me at shayan at cs dot washington dot edu.
- Lectures: Wednesday - Friday 3:00 - 4:20 at Low 116
Assignments
- Problem Set 1, due October 27th in canvas. solution
- Problem Set 2, due December 9th in canvas.
List of Reading
- Dyer: Approximate counting by dynamic programming
- Peters, Regts On a conjecture of Sokal concerning roots of the independence polynomial
- Barvinok: Concentration of the Mixed Discriminant of well-conditioned Matrcies
- Efthymiou, Hayes, Stefankovic, Vigoda, Yin: Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model
- Barvinok, Luria, Samorodnitsky, Yong: An approximation algorithm for counting contingency tables
- Cryan, Dyer, Goldberg, Jerrum, Martin: RAPIDLY MIXING MARKOV CHAINS FOR SAMPLING CONTINGENCY TABLES WITH A CONSTANT NUMBER OF ROWS
- Bezakova, Galanis, Goldberg, Guo, Stefankovic: Approximation via Correlation Decay when Strong Spatial Mixing Fails
- Hayes, Vigoda: A Non-Markovian Coupling for Randomly Sampling Colorings
- Moitra: Approximate Counting, the Lovasz Local Lemma and Inference in Graphical Models
- Jerrum, Sinclair, Vigoda: An FPRAS for Permanent of Nonnegative Matrices
Related Courses
Related Books/Monographs
- Persi Diaconis's Monograph on the mathematics of mixing things up
- Persi's Monograph on The Markov Chain Monte Carlo Revolution
- Markov Chains and Mixing Times by David Levin, Yuval Peres, and Elizabeth Wilmer. Also, take a look at the 2nd edition
- Combinatorics and Complexity of Partition Functions by Sasha Barvinok
- Algebraic Combinatorics by Chris Godsil
| Lecture | Topic | Notes | Bonus Materials | |
|---|---|---|---|---|
| Lecture 1 (09/27/17) | Intro, Equivalence of Counting and Sampling | |||
| Lecture 2 (09/29/17) | DNF Counting and Karger's Algorithm for Network Reliability | Faster Algorithm | ||
| Lecture 3 (10/04/17) | Markov chains Intro, Metropolis Rule, Glauber Dynamics, Fundamental Theorem of Markov Chains | Bayer-Diaconis | ||
| Lecture 4 (10/06/17) | Introduction to Coupling | |||
| Lecture 5 (10/11/17) | Strong Stationary Time | Diaconis-Fill | ||
| Lecture 6 (10/13/17) | Path Coupling: Graph Coloring | Vigoda's bound Coupling for Partial Order | ||
| Lecture 7 (10/18/17) | Advanced Coupling and Mixing Time via Eigenvalues | Hayes-Vigoda | ||
| Lecture 8 (10/20/17) | Poincare Constant, bounding Mixing time by Flows | Mixing Time by Conductance curve | ||
| Lecture 9 (10/25/17) | Counting Matchings | Heilman-Lieb: Theory of Monomer-Dimer | ||
| Lecture 10 (10/27/17) | Counting Matchings/Conductance | Jerrum-Sinclair-Vigoda on Permanent | ||
| Lecture 11 (11/01/17) | Lower Bounds on Mixing | Mixing of Ising above critical temperature | ||
| Lecture 12 (11/03/17) | Correlation Decay for counting Matchings | Correlation Decay for Expanders | ||
| Lecture 13 (11/08/17) | Weitz Algorithm for Counting Independent Sets | Inapprox above t_c Indepdence Polyn in Shearer Region | ||
| (11/10/17) Veternas Day | ||||
| Lecture 14 (11/15/17) | Barvinok Polynomial Method, Approximating Permanent | Approximating Permanent of Real Matrices | ||
| Lecture 15 (11/17/17) | Barvinok Strip Technique, Patel-Regts Result | Patel-Regts | ||
| Lecture 16 (11/22/17) | More on Matching Polynomial, Roots, Moments of the Roots | Chapter 6 of Algebraic Combinatorics | ||
| (11/24/17) Thanksgiving day | ||||
| Lecture 17 (11/29/17) | D-stable Polynomials and Approximating Partition Functions of Ising Model | FPTAS for Ising on Bounded Degree Graphs | ||
| Lecture 18 (12/01/17) | Permanent and Gurvits' technique | 2^n Approx to Permanent | ||
| Lecture 19 (12/06/17) | Log Concavity in Counting and Optimization | |||
| Lecture 20 (12/08/17) | Stability Perservers, Matching Polynomial and Generalization of Gurvits's Technique | Generalization of Gurvits' Technique H-stability and Nash-Welfare Problem | ||
Several of the notes on analysis of Markov chains are heavily borrowed from Sinclair's Courese Notes.
