Computational Statistics (Stat G6104)

Spring 2015

This was a Ph.D.-level course in computational statistics. A link to this year's course is here.

Note: instructor permission is required to take this class for students outside of the Statistics Ph.D. program.

Professor: Liam Paninski; Office: Rm 1028, SSW (1255 Amsterdam Ave). Email: liam at stat dot columbia dot edu. Hours by appointment.
TA: Ben Reddy. Email: bmr2136 at columbia dot edu. Hours: W 3-5, 10th floor lounge area of SSW.
Time: Tu+Th, 10:10am-11:25am
Place: 903 SSW

Course goals: (partially adapted from the preface of Givens' and Hoeting's book): Computation plays a central role in modern statistics and machine learning. This course aims to cover topics needed to develop a broad working knowledge of modern computational statistics. We seek to develop a practical understanding of how and why existing methods work, enabling effective use of modern statistical methods. Achieving these goals requires familiarity with diverse topics in statistical computing, computational statistics, computer science, and numerical analysis. Our choice of topics reflects our view of what is central to this evolving field, and what will be interesting and useful. A key theme is scalability to problems of high dimensionality, which are of most interest to many recent applications.
Some important topics will be omitted because high-quality solutions are already available in most software. For example, the generation of pseudo-random numbers is a classic topic, but existing methods built in to standard software packages will suffice for our needs. On the other hand, we will spend a bit of time on some classical numerical linear algebra ideas, because choosing the right method for solving a linear equation (for example) can have a huge impact on the time it takes to solve a problem in practice, particularly if there is some special structure that we can exploit.

Audience: The course will be aimed at first- and second-year students in the Statistics Ph.D. program. Students from other departments or programs are welcome, space permitting; instructor permission required.

Background: The level of mathematics expected does not extend much beyond standard calculus and linear algebra. Breadth of mathematical training is more helpful than depth; we prefer to focus on the big picture of how algorithms work and to sweep under the rug some of the nitty-gritty numerical details. The expected level of statistics is equivalent to that obtained by a graduate student in his or her first year of study of the theory of statistics and probability. An understanding of maximum likelihood methods, Bayesian methods, elementary asymptotic theory, Markov chains, and linear models is most important.

Programming: With respect to computer programming, good students can learn as they go. We'll forgo much language-specific examples, algorithms, or coding; I won't be teaching much programming per se, but rather will focus on the overarching ideas and techniques. For the exercises and projects, I recommend you choose a high-level, interactive package that permits the flexible design of graphical displays and includes supporting statistics and probability functions, e.g., R or MATLAB.

Evaluation: Final grades will be based on class participation, a few short exercises, and a student project.

Deterministic optimization
- Newton-Raphson, conjugate gradients, preconditioning, quasi-Newton methods, Fisher scoring, EM and its various derivatives
- Numerical recipes for linear algebra: matrix inverse, LU, Cholesky decompositions, low-rank updates, SVD, banded matrices, Toeplitz matrices and the FFT, Kronecker products (separable matrices), sparse matrix solvers
- Convex analysis: convex functions, duality, KKT conditions, interior point methods, projected gradients, augmented Lagrangian methods, convex relaxations
- Applications: support vector machines, splines, kriging, isotonic regression, LASSO and LARS regression

Dynamic programming: hidden Markov models, forward-backward algorithm, Kalman filter, Markov random fields

Stochastic optimization: Robbins-Monro and Kiefer-Wolfowitz algorithms, simulated annealing, stochastic gradient methods

Deterministic integration: Gaussian quadrature, quasi-Monte Carlo. Application: expectation propagation

Monte Carlo methods
- Rejection sampling, importance sampling, variance reduction methods (Rao-Blackwellization, stratified sampling)
- MCMC methods: Gibbs sampling, Metropolis-Hastings, Langevin methods, Hamiltonian Monte Carlo, slice sampling. Implementation issues: burnin, monitoring convergence
- Sequential Monte Carlo (particle filtering)

Givens and Hoeting (2005) Computational statistics
Robert and Casella (2004) Monte Carlo Statistical Methods
Boyd and Vandenberghe (2004), Convex Optimization.
Press et al, Numerical Recipes
Sun and Yuan (2006), Optimization theory and methods
Fletcher (2000) Practical methods of optimization
Searle (2006) Matrix Algebra Useful for Statistics
Spall (2003), Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control
Shewchuk (1994), An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
Boyd et al (2011), Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers


Date Topic Reading Notes
Jan 17 Introduction; linesearch Ch. 1-2 of Givens+Hoeting See Sun and Yuan (2006) for further details on convergence analysis. HW: derive the convergence rate of the secant method.
Jan 22 Choosing search directions: Newton, generalized linear models, inexact Newton, quasi-Newton, Fisher scoring, BFGS See Vandenberghe's notes for some further background
Jan 27 Snow day
Jan 29 Exploiting special structure to solve Newton linear equations more efficiently: banded, sparse, low-rank, block-structured (etc.) matrices HW: density estimation via convex optimization and banded Hessians.
Feb 3 Conjugate gradients. Preconditioning. Toeplitz and circulant matrices Shewchuk (1994); see Chan and Ng (1996) on PCG for Toeplitz systems.
Feb 5 Gaussian process regression Notes by John Cunningham. See Rasmussen and Williams (2006) for more background on GP regression. HW: code up a GP regression.
Feb 10 Expectation maximization See Dempster et al (1977) and Neal and Hinton (1999) for further reading.
Feb 12, 17 Constrained and non-smooth optimization: convex functions; interior point methods Boyd and Vandenberghe, ch. 3-4
Feb 19, 24 No class HW: Read and do some problems from Boyd and Vandenberghe
Feb 26, Mar 3 Linear, quadratic, and semidefinite programs. LASSO methods Efron et al (2004), Zou et al (2007), Friedman et al (2010), Bradley et al (2011), Tibshirani et al (2012)
Mar 5 Convex duality, KKT conditions. Some advanced topics: proximal methods, dual decomposition, convex relaxation, and branch-and-bound Boyd and Vandenberghe, ch. 5 Background: Bach et al (2011), Boyd et al (2011), Luo et al (2010); branch-and-bound notes by Rahul Mazumder
Mar 10 Graphical models; dynamic programming; message passing; LP relaxations Rabiner tutorial, Jordan (2004) Background: Wainwright and Jordan (2008), MP and AMP notes by A. Maleki, LP relaxation notes by Y.-W. Teh
Mar 12 Short informal project presentations
Mar 17, 19 No class: spring break
Mar 24 Monte Carlo basics. Rejection sampling; Metropolis-Hastings; Gibbs sampling Ch. 1-7 of Robert and Casella Background: Devroye (1986).
Mar 26 Hamiltonian Monte Carlo Neal (2010)
Guest lecture by Ari Pakman. Further background: Hoffman and Gelman (2012), Pakman and Paninski (2013). Also, see this nice video.
Mar 31 Guest lecture by Arian Maleki on message passing and approximate message passing notes
Apr 2, 7 More on Monte Carlo. Adaptive rejection sampling; importance sampling; slice sampling. MCMC diagnostics. Rao-Blackwellization. Control variates. Adaptive simulated tempering. Background: Park and Casella (2008), Neal (2003), Doucet (2010), Dellaportas and Kontoyiannis (2012), Ranganath et al (2014), Salakhutdinov (2010)
Apr 7, 9 Sequential Monte Carlo Doucet and Johansen (2011), Pitt and Shephard (1999) Further reading collected by A. Doucet; Kantas et al (2014)
Apr 14 Stochastic gradient methods; Bayesian optimization Book chapter by Lauren Hannah; Duchi et al (2011), Lacoste-Julien et al (2012), Snoek et al (2012)
Apr 16, 23 Variational inference Ormerod and Wand (2010) Additional reading: Hoffman et al (2013), Emtiyaz Khan et al (2013), Ranganath et al (2014)
Apr 21 No class
Apr 23 More deterministic approximations for posteriors: expectation propagation, Gaussian quadrature Sudderth (2002)
Apr 28 Introduction to Dirichlet process methods Neal (2000), Teh (2010) See also the lecture notes on Peter Orbanz's page.
Apr 30 No class
May 5 Project presentations Send me your report as a .pdf by May 10.