Computational learning theory: Difference between revisions

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In [[statistics]], '''computational learning theory''' is a mathematical field related to the analysis of [[machine learning]] algorithms.
#REDIRECT [[learning theory (statistics)]]
 
Machine learning algorithms take a training set, form hypotheses or models, and make predictions about the future. Because the training set is finite and the future is uncertain, learning theory usually does not yield absolute guarantees of performance of the algorithms. Instead, probabilistic bounds on the performance of machine learning algorithms are quite common.
 
In addition to performance bounds, computational learning theorists study the time complexity and feasibility of learning. In computational learning theory, a computation is considered feasible if it can be done in polynomial time. There are two kinds of time complexity results:
#Positive results --- Showing the a certain class of function is learnable in polynomial time.
#Negative results - Showing that certain classes cannot be learned in polynomial time.
Negative results are proven only by assumption. The assumptions the are common in negative results are:
* Computational complexity - [[Complexity classes P and NP|P]]<math>\neq</math>[[Complexity classes P and NP|NP]]
* [[cryptography|Cryptographic]] - [[One-way function]]s exist.
 
There are several difference branches of computational learning theory, which are often mathematically incompatible. This incompatibility arises from using different [[inference]] principles: principles which tell you how to generalize from limited data.
 
Examples of different branches of computational learning theory include:
* [[Probably approximately correct learning]] (PAC learning), proposed by [[Leslie Valiant]];
* [[Statistical learning theory]], proposed by [[Vladimir Vapnik]];
* [[Bayesian inference]], arising from work first done by [[Thomas Bayes]].
* [[Algorithmic learning theory]], from the work of E. M. Gold.
 
Computational learning theory has led to practical algorithms. For example, PAC theory inspired [[boosting]], statistical learning theory led to [[support vector machine]]s, and Bayesian inference led to [[belief networks]] (by [[Judea Pearl]]).
 
''See also:''
* [[information theory]]
 
== References ==
===Surveys===
* Angluin, D. 1992. Computational learning theory: Survey and selected bibliography. In Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing (May 1992), pp. 351--369.
* D. Haussler. Probably approximately correct learning. In AAAI-90 Proceedings of the Eight National Conference on Artificial Intelligence, Boston, MA, pages 1101--1108. American Association for Artificial Intelligence, 1990. http://citeseer.nj.nec.com/haussler90probably.html
===[[VC dimension]]===
* V. Vapnik and A. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16(2):264--280, 1971.
===Feature selection===
* A. Dhagat and L. Hellerstein. PAC learning with irrelevant attributes. In Proceedings of the IEEE Symp. on Foundation of Computer Science, 1994. To appear. http://citeseer.nj.nec.com/dhagat94pac.html
===Inductive inference===
* E. M. Gold. Language identification in the limit. Information and Control, 10:447--474, 1967.
===Optimal O notation learning===
* O. Goldreich, D. Ron. On universal learning algorithms. http://citeseer.nj.nec.com/69804.html
===Negative results===
* M. Kearns and L. G. Valiant. 1989. Cryptographic limitations on learning boolean formulae and finite automata. In Proceedings of the 21st Annual ACM Symposium on Theory of Computing, pages 433--444, New York. ACM. http://citeseer.ist.psu.edu/kearns89cryptographic.html
===[[Boosting]]===
* Robert E. Schapire. The strength of weak learnability. Machine Learning, 5(2):197--227, 1990 http://citeseer.nj.nec.com/schapire90strength.html
===[[Occam's Razor]]===
* Blumer, A.; Ehrenfeucht, A.; Haussler, D.; Warmuth, M. K. "Occam's razor" Inf.Proc.Lett. 24, 377-380, 1987.
* A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth. Learnability and the Vapnik-Chervonenkis dimension. Journal of the ACM, 36(4):929--865, 1989.
===[[Probably approximately correct learning]]===
* L. Valiant. A Theory of the Learnable. Communications of the ACM, 27(11):1134--1142, 1984.
===Error tolerance===
* Michael Kearns and Ming Li. Learning in the presence of malicious errors. SIAM Journal on Computing, 22(4):807--837, August 1993. http://citeseer.nj.nec.com/kearns93learning.html
* Kearns, M. (1993). Efficient noise-tolerant learning from statistical queries. In Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, pages 392--401. http://citeseer.nj.nec.com/kearns93efficient.html
===Equivalence===
* D.Haussler, M.Kearns, N.Littlestone and M.Warmuth, Equivalence of models for polynomial learnability, Proc. 1st ACM Workshop on Computational Learning Theory, (1988) 42-55.
* L. Pitt and M. K. Warmuth: Prediction preserving reduction, Journal of Computer System and Science 41, 430--467, 1990.
 
A description of some of these publictions is given at [[list of important publications in computer science#Machine learning|important publications in machine learning]].
 
== External links ==
* [http://www.learningtheory.org Computational learning theory web site]
* [http://www.inference.phy.cam.ac.uk/mackay/itila/ On-line book: Information Theory, Inference, and Learning Algorithms], by [[David MacKay]], gives a detailed account of the Bayesian approach to machine learning.
* [http://www.santafe.edu/~shalizi/reviews/kearns-vazirani/ Review of ''An Introduction to Computational Learning Theory'']
* [http://www.santafe.edu/~shalizi/reviews/vapnik-nature/ Review of ''The Nature of Statistical Learning Theory'']
* [http://research.microsoft.com/adapt/MSBNx/msbnx/Basics_of_Bayesian_Inference.htm Basics of Bayesian inference]
 
[[Category:Machine learning]]