In statistics, computational learning theory is a mathematical field related to the analysis of machine learning algorithms. An example of statistics used in educational assessment. ... ...

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:

1. Positive results --- Showing that a certain class of functions is learnable in polynomial time.
2. Negative results - Showing that certain classes cannot be learned in polynomial time.

Negative results are proven only by assumption. The assumptions that are common in negative results are:

• Computational complexity - PNP
• Cryptographic - One-way functions exist.

There are several different approaches to 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. The incompatibility also arises from differing definitions of probability (see frequency probability, Bayesian probability). The different approaches include: Diagram of complexity classes provided that P ≠ NP. If P = NP, then all three classes are equal. ... Diagram of complexity classes provided that P ≠ NP. If P = NP, then all three classes are equal. ... Cryptography has had a long and colourful history. ... Unsolved problems in computer science: Do one-way functions exist? A one-way function is a function that is easy to calculate but hard to invert — it is difficult to calculate the input to the function given its output. ... ... The word probability derives from the Latin probare (to prove, or to test). ... Statistical regularity has motivated the development of the relative frequency concept of probability. ... Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of a statement. ...

• Probably approximately correct learning (PAC learning), proposed by Leslie Valiant;
• VC 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, VC theory led to support vector machines, and Bayesian inference led to belief networks (by Judea Pearl). Probably approximately correct learning (PAC learning) is a framework of learning that was proposed by Leslie Valiant in his paper A theory of the learnable. ... Leslie Valiant was educated at Kings College, Cambridge, Imperial College, London; and at Warwick University where he received his Ph. ... Vapnik Chervonenkis theory (also known as VC theory) was developed during 1960-1990 by Vladimir Vapnik and Alexey Chervonenkis. ... Vladimir Naumovich Vapnik is one of the main developers of Vapnik Chervonenkis theory. ... Bayesian inference is a statistical inference in which probabilities are interpreted not as frequencies or proportions or the like, but rather as degrees of belief. ... Thomas Bayes Reverend Thomas Bayes (c. ... Algorithmic learning theory (or inductive inference) is a framework for machine learning. ... Boosting is a machine learning meta-algorithm for performing supervised learning. ... Support vector machines (SVMs) are a set of related supervised learning methods used for classification and regression. ... A Bayesian network or Bayesian belief network is a directed acyclic graph of nodes representing variables and arcs representing dependence relations among the variables. ... Judea Pearl is a computer science professor at UCLA. He was one of the pioneers of Bayesian networks and the probabilistic approach to artificial intelligence. ...

See also:

• information theory

Information theory is the mathematical theory of data communication and storage, generally considered to have been founded in 1948 by Claude E. Shannon. ...

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.

The VC dimension (for Vapnik Chervonenkis dimension) is a measure of the capacity of a classification algorithm. ...

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

Boosting is a machine learning meta-algorithm for performing supervised learning. ...

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.

Occams Razor (also spelled Ockhams Razor), is a principle attributed to the 14th-century English logician and Franciscan friar, William of Ockham. ...

Probably approximately correct learning

• L. Valiant. A Theory of the Learnable. Communications of the ACM, 27(11):1134--1142, 1984.

Probably approximately correct learning (PAC learning) is a framework of learning that was proposed by Leslie Valiant in his paper A theory of the learnable. ...

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 publications is given at important publications in machine learning. This is a list of important publications in computer science, organized by field. ...

External links

• Computational learning theory web site
• On-line book: Information Theory, Inference, and Learning Algorithms, by David MacKay, gives a detailed account of the Bayesian approach to machine learning.
• Review of An Introduction to Computational Learning Theory
• Review of The Nature of Statistical Learning Theory
• Basics of Bayesian inference