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.

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 21 | Introduction; linesearch | Ch. 1-2 of Givens+Hoeting | See Sun and Yuan (2006) for further details on convergence analysis |

Jan 23 | Choosing search directions: Newton, generalized linear models, inexact Newton, quasi-Newton, Fisher scoring, BFGS | See Vandenberghe's notes for some further background | |

Jan 28 | Exploiting special structure to solve Newton linear equations more efficiently: banded, sparse, low-rank (etc.) matrices | ||

Jan 30 | Conjugate gradients | Shewchuk (1994) | |

Feb 4 | Preconditioning. Toeplitz and circulant matrices | See Chan and Ng (1996) on PCG for Toeplitz systems. | |

Feb 6 | Expectation maximization | See Dempster et al (1977) and Neal and Hinton (1999) for further reading. | |

Feb 11 | Gaussian process regression | Guest lecture by John Cunningham. See notes here, and Rasmussen and Williams (2006) for more background on GP regression | |

Feb 13 | Snow day - class cancelled | ||

Feb 18 | Constrained and non-smooth optimization: convex functions; interior point methods | Boyd and Vandenberghe, ch. 3-4 | |

Feb 20 | Linear, quadratic, and semidefinite programs | ||

Feb 25 | LASSO methods | Efron et al (2004), Zou et al (2007), Friedman et al (2010), Bradley et al (2011) | |

Feb 27 | Convex duality, KKT conditions. Support vector methods | Boyd and Vandenberghe, ch. 5 | |

Mar 4 | A brief tour of some advanced topics: proximal methods, dual decomposition, and convex relaxation | Background: Bach et al (2011), Boyd et al (2011), Luo et al (2010) | |

Mar 6, 11 | Graphical models; dynamic programming; message passing; LP relaxations | Rabiner tutorial, Jordan (2004) | Background: Wainwright and Jordan (2008), Smith et al (2012), MP and AMP notes by A. Maleki, LP relaxation notes by Y.-W. Teh |

Mar 13 | Monte Carlo basics. Rejection and importance sampling; adaptive rejection sampling | Ch. 1-7 of Robert and Casella | Background: Devroye (1986), Doucet (2010). |

Mar 18-20 | No class | Spring break | |

Mar 25 | Short project presentations | ||

Mar 27 | Guest lecture by Rahul Mazumder on integer programming via branch-and-bound | notes | |

Apr 1-3, 10 | Metropolis-Hastings. Gibbs sampling: slice sampling, Bayesian lasso, spike-and-slab, hit and run. MCMC diagnostics. Rao-Blackwellization. Adaptive simulated tempering. | Background: Park and Casella (2008), Neal (2003), Doucet (2010), Mohamed et al. (2011), Salakhutdinov (2010) | |

Apr 8 | Hamiltonian Monte Carlo | Neal (2010) |
Guest lecture by Ari Pakman. Further background: Hoffman and Gelman (2012), Pakman and Paninski (2013) |

Apr 15 | Sequential Monte Carlo | Doucet and Johansen (2011), Pitt and Shephard (1999) | Further reading collected by A. Doucet |

Apr 17 | Nonparametric Bayes methods | Guest lecture by Peter Orbanz | |

Apr 22, 24 | Deterministic approximations for posteriors: variational Bayes, expectation propagation, Gaussian quadrature | Ormerod and Wand (2010), Sudderth (2002) | Additional reading: Hoffman et al (2013) |

Apr 29 | Stochastic approximation methods | Ch. 4-5 in Spall (2003) | Guest lecture by Lauren Hannah; notes |

May 1 | Project presentations | Send me your report as a .pdf by May 9. |