Machine learning is inevitably a technical subject that requires topics that are not always covered in the undergraduate CS curriculum.

The main goal of the proficiency exam at PiLab Sigma is to demonstrate that you have a sufficient understanding of the state of the art in Machine Learning and basic mathematical background, to pursue PhD work.

You will be assigned a jury of 4 faculty members.

The test has three parts

Written exam, Take home (about 5-7 days time) Example

Written exam, In class (about 3-4 hours)

Oral exam (1-2 hours)

Below is a list of topics that you must be familiar of, that is you should be able to explain what each term means and you should be able to have some experience with each one.

Foundations AL,BI

Probability distributions, Entropy, Expectation

Bayes Rule, conditional distributions

Bayesian model comparison

Statistics: Sampling, estimation, hypothesis testing

Models AL,BI

Mixture models / k-means

Factor Analysis / PCA

Matrix Factorisation models (ICA, NMF)

Hidden Markov models (HMMs)

State space models (SSMs)

Graphical models: directed, undirected, factor graphs

Algorithms AL,BI,R&N

Forward-backward

Kalman filtering, smoothing and extended Kalman filtering

Belief propagation

The EM Algorithm

Variational methods

Laplace approximation and the BIC

Monte Carlo, Rejection and Importance sampling

Markov chain Monte Carlo (MCMC) methods: Metropolis Hastings, Gibbs sampler

Sequential Monte Carlo, Particle filters

Supervised Learning: AL,BI

Linear regression

Logistic regression

Generalised Linear Models

Perceptrons

Neural networks (multi-layer perceptrons) and backpropagation

Gaussian processes

Support vector machines

Decision trees

Optimization: BI,B99

Linear Programming

Convex functions, Jensens inequality

Gradient Descent

Newtons method

Constrained optimization, Lagrange multipliers

Numerical Analysis and Linear Algebra : T&B,NR3

Interpolation

Fourier Transform

Numerical Integration, Gaussian quadrature

Matrix algebra and calculus

Least Squares

QR factorisation

Eigenvalues and Eigenvectors

Singular value decomposition

Numerical stability and floating point representation

Stochastic optimal control and Reinforcement Learning : (Specialisation) AL,R&N,B05

Value functions

Bellman's equation

Value iteration

Policy iteration

Q-Learning

TD(lambda)

(AL) Alpaydin, Ethem (2010). Introduction to Machine Learning (Second ed.)

(DB) David Barber, (2012). Bayesian Reasoning and Machine Learning, Cambridge University Press

(KM) Kevin Murphy(2012). Machine Learning, a Probabilistic Perspective. MIT Press

(BI) Bishop, Christopher (2006), Pattern recognition and Machine Learning

(R&N) Russell and Norvig (2001), Artificial Intelligence, a modern approach

(T&B) Trefethen and Bau (1996), Numerical Linear Algebra,

(NR3) Press, Teukolsky, Vetterling and Flannery (2007), Numerical Recipes 3rd Edition: The Art of Scientific Computing

(B99) Bertsekas, Dimitri P. (1999). Nonlinear Programming (Second ed.).

(B05) Bertsekas, Dimitri P. (2005).Dynamic Programming and Optimal Control, Vol 1

In addition to above references, if you want to improve yourself in the filed, below are some books and topics that are good for self study.

I personally think that everyone in machine learning should be (completely) familiar with essentially all of the material in the following intermediate-level statistics book:

1.) Casella, G. and Berger, R.L. (2001). “Statistical Inference” Duxbury Press.

For a slightly more advanced book that's quite clear on mathematical techniques, the following book is quite good:

2.) Ferguson, T. (1996). “A Course in Large Sample Theory” Chapman & Hall/CRC.

You'll need to learn something about asymptotics at some point, and a good starting place is:

3.) Lehmann, E. (2004). “Elements of Large-Sample Theory” Springer.

Those are all frequentist books. You should also read something Bayesian:

4.) Gelman, A. et al. (2003). “Bayesian Data Analysis” Chapman & Hall/CRC. and you should start to read about Bayesian computation:

5.) Robert, C. and Casella, G. (2005). “Monte Carlo Statistical Methods” Springer.

On the probability front, a good intermediate text is:

6.) Grimmett, G. and Stirzaker, D. (2001). “Probability and Random Processes” Oxford. At a more advanced level, a very good text is the following:

7.) Pollard, D. (2001). “A User's Guide to Measure Theoretic Probability” Cambridge.

The standard advanced textbook is Durrett, R. (2005). “Probability: Theory and Examples” Duxbury.

Machine learning research also reposes on optimization theory. A good starting book on linear optimization that will prepare you for convex optimization:

8.) Bertsimas, D. and Tsitsiklis, J. (1997). “Introduction to Linear Optimization” Athena. And then you can graduate to:

9.) Boyd, S. and Vandenberghe, L. (2004). “Convex Optimization” Cambridge.

Getting a full understanding of algorithmic linear algebra is also important. At some point you should feel familiar with most of the material in

10.) Golub, G., and Van Loan, C. (1996). “Matrix Computations” Johns Hopkins.

It's good to know some information theory. The classic is:

11.) Cover, T. and Thomas, J. “Elements of Information Theory” Wiley.

Finally, if you want to start to learn some more abstract math, you might want to start to learn some functional analysis (if you haven't already). Functional analysis is essentially linear algebra in infinite dimensions, and it's necessary for kernel methods, for nonparametric Bayesian methods, and for various other topics. Here's a book that I find very readable:

12.) Kreyszig, E. (1989). “Introductory Functional Analysis with Applications” Wiley.

I now tend to add some books that dig still further into foundational topics. In particular, I recommend

A. Tsybakov's book “Introduction to Nonparametric Estimation” a very readable source for the tools for obtaining lower bounds on estimators, and

Y. Nesterov's very readable “Introductory Lectures on Convex Optimization” as a way to start to understand lower bounds in optimization.

A. van der Vaart's “Asymptotic Statistics”, a book that we often teach from at Berkeley, as a book that shows how many ideas in inference (M estimation — which includes maximum likelihood and empirical risk minimization — the bootstrap, semiparametrics, etc) repose on top of empirical process theory.

I'd also include B. Efron's “Large-Scale Inference: Empirical Bayes Methods for Estimation, Testing, and Prediction”, as a thought-provoking book.