15-859(B) MACHINE LEARNING THEORY
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.
- Instructor: Avrim Blum
- Time: MW 1:30-2:50
- Place: Wean 5409
An Introduction to Computational Learning Theory by Michael Kearns
and Umesh Vazirani, plus papers and notes for topics not in the book.
Office hours: Just stop by my office or make an appointment.
The final exam is Tuesday 5/16 1:00-4:00pm in Wean
6423 (not 4623)
- General course information
- Handout on tail inequalities
- Homework 1 [ps,pdf].
- Homework 2 [ps,pdf].
- Homework 3 [ps,pdf].
- Homework 4 [ps,pdf].
- Homework 4.5 [ps,pdf].
- Homework 5 [ps,pdf].
- Homework 6 [ps,pdf].
Lecture Notes & tentative plan
- 01/18: Introduction, basic definitions, the consistency model.
- 01/23: The Mistake-bound model.
- 01/25: The Winnow algorithm + applications.
- 01/30: The Perceptron Algorithm, Margins,
and Kernel functions.
- 02/01: Combining expert advice, Weighted-Majority, Regret-bounds
and connections to game theory.
- 02/06: Kalai-Vempala algorithm.
- 02/08: PAC model I: basic results and Occam's razor.
- 02/13: PAC model II: Chernoff/Hoeffding
bounds, MB->PAC [pdf].
Handout on tail inequalities
- 02/15: VC-dimension I: Proof of Sauer's lemma. slides and notes.
- 02/20: [no class today]
- 02/22: VC-dimension II.
- 02/27: Boosting I: weak vs strong learning, basic issues.
- 03/01: Boosting II: Adaboost.
- 03/06: Cryptographic hardness results.
- 03/08: Maxent and maximum-likelihood exponential models. Connection to winnow.
- 03/20: Learning from noisy data. The Statistical Query model.
- 03/22: Characterizing SQ learnability with Fourier analysis.
- 03/27: Finish SQ hardness. Using Fourier for learning.
- 03/29: Membership queries I: basic
algorithms, KM algorithm.
- 04/03: Membership queries II: Fourier spectrum of DTs & DNFs, Bshouty's alg.
- 04/05: Membership queries III: Angluin's algorithm for learning DFAs. Additional notes.
- 04/10: Learning finite-state environments without a reset.
- 04/12: MDPs and reinforcement learning.
- 04/17: MDPs and reinforcement learning II.
- 04/19: Semi-supervised learning, active learning.
- 04/24: Boosting and margins
- 04/26: Kernels and similarity functions
- 05/01: Project presentations
- 05/03: Project presentations
Additional Readings & More Information
Very new results:
Presentations on the Web:
- N. Cristianini and J. Shawe-Taylor,
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.
- 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. (The version pointed to here is the tech report UCSC-CRL-91-28.)
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,
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,
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.
- Avrim Blum, On-Line
Algorithms in Machine Learning (a survey). From "Online
Algorithms: the state of the art", Fiat
and Woeginger eds., LNCS #1442, 1998.
- 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
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".
- My FOCS'03 tutorial on Machine
Fourier analysis, weak learning, SQ learning:
Web sites on maxent:
- 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
- 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).