# Computational learning theory

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In theoretical computer science, **computational learning theory** is a mathematical field related to the analysis of machine learning algorithms.

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 that a certain class of functions 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 that are common in negative results are:

- Computational complexity - P≠NP
- 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:

- 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).

*See also:*

## 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.ist.psu.edu/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. http://citeseer.ist.psu.edu/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.ist.psu.edu/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.ist.psu.edu/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.ist.psu.edu/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.ist.psu.edu/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.

## 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

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