**Course description:**
This course will focus on theoretical aspects of machine learning. We
will examine questions such as: What kinds of guarantees can one prove
about learning algorithms? What are good algorithms for achieving
certain types of goals? Can we devise models that are both amenable to
mathematical analysis and make sense empirically? What can we say
about the inherent ease or difficulty of learning problems? Addressing
these questions will require pulling in notions and ideas from
statistics, complexity theory, information theory, cryptography, game
theory, and empirical machine learning research.
Grading will be based on
6 homework assignments, class
participation, a small class project, and a take-home final
(worth about 2 homeworks). Students from time
to time will also be asked to help with the grading of
assignments.
[2007 version]

**Text:**
*
An Introduction to Computational Learning Theory* by Michael Kearns
and Umesh Vazirani, plus papers and notes for topics not in the book.

- [Online learning survey] [Winnow paper] [Handout on tail inequalities] [SQ dimension paper]
- Homework 1 [ps, pdf]. Solutions.
- Homework 2 [ps, pdf] Solutions.
- Homework 3 [ps, pdf] Solutions.
- Homework 4 [ps, pdf] Solutions.
- Homework 4.5 [ps, pdf]
- Homework 5 [ps, pdf] Solutions.
- Homework 6 [ps, pdf] Solutions.
- Ideas for Projects.

- 01/14: Introduction. PAC model and Occam's razor.
- 01/16: The Mistake-Bound model. Combining expert advice. Connections to info theory and game theory.
- 01/23: The Winnow algorithm + applications.
- 01/28: The Perceptron Algorithm, Margins, and Kernel functions.
- 01/30: Uniform convergence, tail inequalities (Chernoff/Hoeffding), VC-dimension I. [more notes]
- 02/04: VC-dimension II.
- 02/06: Boosting I: weak vs strong learning, basic issues. [plus some slides]
- 02/11: Boosting II: Adaboost + connection to WM analysis + L_1 margin bounds.
- 02/13: Margins, random projection, kernels, and general similarity functions
- 02/18: Margin bounds, Support Vector Machines.
- 02/20: McDiarmid's inequality & Rademacher bounds.
- 02/25: Maxent and maximum-likelihood exponential models. Connection to winnow.
- 02/27: Cryptographic hardness results
- 03/03: Statistical Query model I
- 03/05: Statistical Query model II
- 03/17: Fourier-based algorithms
- 03/19: Membership Query algorithms
- 03/24: Membership Query algorithms II
- 03/26: Learning finite-state environments
- 03/31: MDPs and reinforcement learning I
- 04/02: MDPs and reinforcement learning II
- 04/07: Semi-supervised learning
- 04/09: Offline->online optimization
- 04/14: online learning and game theory I
- 04/16: online learning and game theory II
- 04/21: Active learning & course retrospective
- 04/23: [No Class Today]
- 04/28: Project presentations
- 04/30: Project presentations

**Books and tutorials:**

- O. Bousquet, S. Boucheron, and G. Lugosi, Introduction to Statistical Learning Theory.
- PASCAL video lectures.
- N. Cristianini and J. Shawe-Taylor, Kernel Methods for Pattern Analysis, 2004.
- N. Cristianini and J. Shawe-Taylor, An Introduction to Support Vector Machines (and other kernel-based learning methods), 2000.
- M. Anthony and P. Bartlett. Learning in Neural Networks : Theoretical Foundations. Cambridge University Press, 1999.
- V. Vapnik. Statistical Learning Theory. Wiley, 1998.
- L. Devroye, L. Györfi, G. Lugosi, A Probabilistic Theory of Pattern Recognition, Springer, New York, 1996.
- My FOCS'03 tutorial on Machine
Learning Theory

**Online Learning:**

- Nick Littlestone, Learning Quickly
when Irrelevant Attributes Abound: A New Linear-threshold Algorithm.
*Machine Learning*2:285--318, 1987. (The version pointed to here is the tech report UCSC-CRL-87-28.) This is the paper that first defined the Mistake-bound model, and also introduced the Winnow algorithm. A great paper. - Littlestone and Warmuth,
The Weighted Majority Algorithm.
*Information and Computation*108(2):212-261, 1994. Introduces the weighted majority algorithm, along with a number of variants. Also a great paper. - Nicolò Cesa-Bianchi, Yoav Freund, David Haussler, David
Helmbold, Robert Schapire, and Manfred Warmuth, How
to use expert advice, Journal of the ACM, 44(3):427-485, May 1997.
Yoav Freund and Robert Schapire, Adaptive game playing using
multiplicative weights, Games and Economic Behavior, 29:79-103,
1999.
Continuing on with line of research in the [LW] paper, these give
tighter analyses of multiplicative-weighting expert algorithms and
give a game-theoretic perspective, as well as address a number of other issues.
- Adam Kalai and Santosh Vempala, Efficient algorithms for the online decision problem, COLT '03. Martin Zinkevich,
Online convex
programming and generalized infinitesimal gradient ascent, ICML '03.
These papers give efficient algorithms for a broad class of settings that one
can view as having exponentially many "experts", but which are represented
in an implicit compact way.
- Peter Auer, Nicolò Cesa-Bianchi, Yoav Freund, Robert Schapire: The Nonstochastic Multiarmed Bandit Problem, SIAM J. Comput. 32(1): 48-77 (2002). Brendan McMahan and Avrim Blum: Online Geometric Optimization
in the Bandit Setting Against an Adaptive Adversary, COLT '04.
Abie Flaxman, Adam Tauman Kalai, and Brendan McMahan: Online Convex Optimization in the Bandit Setting: Gradient Descent Without a Gradient, SODA '2005. These papers extend above results to the bandit setting, in which only the loss or gain of the action actually played can be observed at each time step.
- Survey articles:
Avrim Blum, On-Line
Algorithms in Machine Learning. From "Online Algorithms: the
state of the art", Fiat and Woeginger eds., LNCS #1442, 1998.
Avrim Blum and Yishay Mansour, Learning, Regret Minimization, and Equilibria, Chapter 4 in "Algorithmic Game Theory," Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay Vazirani, eds. (2007).

- David Haussler Chapter on PAC learning model, and decision-theoretic generalizations, with applications to neural nets. From Mathematical
Perspectives on Neural Networks, Lawrence Erlbaum Associates, 1995, containing reprinted material from "Decision Theoretic
Generalizations of the PAC Model for Neural Net and Other Learning Applications", Information and Computation, Vol. 100,
September, 1992, pp. 78-150. This is a really nice survey of the PAC
model and various sample-complexity results.
- David Williamson, John Shawe-Taylor, Bernhard Schölkopf, Alex Smola Sample Based Generalization Bounds. Gives tighter generalization bounds where instead of using "the maximum number of ways of labeling a set of 2m points" you can use "the number of ways of labeling your actual sample".

- The original paper: Robert E. Schapire, The strength of weak learnability. Machine Learning, 5(2):197-227, 1990.
- The Adaboost paper: Yoav Freund and Rob Schapire, A decision-theoretic generalization of on-line learning and an application to boosting, Journal of Computer and System Sciences, 55(1):119-139, 1997.
- An overview by Rob Schapire: The boosting approach to machine learning: An overview. In MSRI Workshop on Nonlinear Estimation and Classification, 2002.

**More on Kernels:**

- Nati Srebro and Shai Ben-David, Learning Bounds for Support Vector Machines with Learned Kernels, COLT 2006.

**Fourier analysis, weak learning, SQ learning:**

- Avrim Blum, Merrick Furst, Jeffrey Jackson, Michael Kearns, Yishay Mansour, and Steven Rudich, Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis, STOC '94 pp. 253--262.
- Y. Mansour. Learning
Boolean Functions via the Fourier Transform. Survey article in
``Theoretical Advances in Neural Computation and Learning", 391--424
(1994).
- A. Blum, C. Burch, and J. Langford, On
Learning Monotone Boolean Functions. Proceedings of the
39th Annual Symposium on Foundations of Computer Science (FOCS '98).
- V. Feldman, P. Gopalan, S. Khot, A. Ponnuswami. New Results for Learning Noisy Parities and Halfspaces, FOCS 2006.

**Computational hardness results:**

- Feldman, Optimal Hardness Results for Maximizing Agreements with Monomials, CCC 2006.
- Kharitonov, Cryptographic hardness of distribution-specific learning, STOC 93.