Computational learning theory: Difference between revisions

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{{see also|Statistical learning theory}}

{{Short description|Theory of machine learning}}{{more citations needed|date=November 2018}}

{{Machine learning bar|Theory}}

In [[computer science]], '''computational learning theory''' (or just '''learning theory''') is a subfield of [[artificial intelligence]] devoted to studying the design and analysis of [[machine learning]] algorithms.<ref name="ACL">{{Cite web | url=http://www.learningtheory.org/ | title=ACL - Association for Computational Learning}}</ref>

==Overview==

Theoretical results in machine learning mainlyoften dealfocus withon a type of inductive learning calledknown as [[supervised learning]]. In supervised learning, an algorithm is givenprovided sampleswith that are[[Labeled data|labeled]] in some useful waysamples. For exampleinstance, the samples might be descriptions of mushrooms, and thewith labels could beindicating whether orthey notare theedible mushroomsor are ediblenot. The algorithm takesuses these previously labeled samples and uses them to inducecreate a classifier. This classifier is a function that assigns labels to new samples, including samplesthose thatit havehas not been seen previously by the algorithmencountered. The goal of the supervised learning algorithm is to optimize someperformance measures of performancemetrics, such as minimizing the number of mistakes madeerrors on new samples.

In addition to performance bounds, computational learning theory studies the time complexity and feasibility of learning.{{citation needed|date=October 2017}} In

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* Positive results{{spaced ndash}}Showing that a certain class of functions is learnable in polynomial time.

* Negative results{{spaced ndash}}Showing that certain classes cannot be learned in polynomial time.<ref>{{Cite book |last1=Kearns |first1=Michael |title=An Introduction to Computational Learning Theory |last2=Vazirani |first2=Umesh |date=August 15, 1994 |publisher=MIT Press |isbn=978-0262111935}}</ref>

Negative results often rely on commonly believed, but yet unproven assumptions,{{citation needed|date=October 2017}} such as:

* Computational complexity – [[P versus NP problem|P &ne;≠ NP (the P versus NP problem)]];

* [[cryptography|Cryptographic]] – [[One-way function]]s exist.

There are several different approaches to computational learning theory based on making different assumptions about the [[inference]] principles used to generalise from limited data. This includes different definitions of [[probability]] (see [[frequency probability]], [[Bayesian probability]]) and different assumptions on the generation of samples.{{citation needed|date=October 2017}} The different approaches include:

[[inference]] principles used to generalize from limited data. This includes different definitions of [[probability]] (see [[frequency probability]], [[Bayesian probability]]) and different assumptions on the generation of samples.{{citation needed|date=October 2017}} The different approaches include:{{citation needed|date=October 2017}}

* Exact learning, proposed by [[Dana Angluin]];<ref>{{cite thesis | type=Ph.D. thesis | author=Dana Angluin | title=An Application of the Theory of Computational Complexity to the Study of Inductive Inference | institution=University of California at Berkeley | year=1976 }}</ref><ref>{{cite journal | url=http://www.sciencedirect.com/science/article/pii/S0019995878906836 | author=D. Angluin | title=On the Complexity of Minimum Inference of Regular Sets | journal=Information and Control | volume=39 | number=3 | pages=337&ndash;350 | year=1978 }}</ref>

* Exact learning, proposed by [[Dana Angluin]];

* [[Probably approximately correct learning]] (PAC learning), proposed by [[Leslie Valiant]];<ref>{{cite journal |last1=Valiant |first1=Leslie |title=A Theory of the Learnable |journal=Communications of the ACM |date=1984 |volume=27 |issue=11 |pages=1134–1142 |doi=10.1145/1968.1972 |s2cid=12837541 |url=https://www.montefiore.ulg.ac.be/~geurts/Cours/AML/Readings/Valiant.pdf |ref=ValTotL |access-date=2022-11-24 |archive-date=2019-05-17 |archive-url=https://web.archive.org/web/20190517235548/http://www.montefiore.ulg.ac.be/~geurts/Cours/AML/Readings/Valiant.pdf |url-status=dead }}</ref>

* [[Probably approximately correct learning]] (PAC learning), proposed by [[Leslie Valiant]];

* [[VC theory]], proposed by [[Vladimir Vapnik]] and [[Alexey Chervonenkis]];<ref>{{cite journal |last1=Vapnik |first1=V. |last2=Chervonenkis |first2=A. |title=On the uniform convergence of relative frequencies of events to their probabilities |journal=Theory of Probability and Its Applications |date=1971 |volume=16 |issue=2 |pages=264–280 |doi=10.1137/1116025 |url=https://courses.engr.illinois.edu/ece544na/fa2014/vapnik71.pdf |ref=VCdim}}</ref>

* [[VC theory]], proposed by [[Vladimir Vapnik]] and [[Alexey Chervonenkis]];

* [[Solomonoff's theory of inductive inference|Inductive inference]] as developed by [[Ray Solomonoff]];<ref>{{cite journal |last1=Solomonoff |first1=Ray |title=A Formal Theory of Inductive Inference Part 1 |journal=Information and Control |date=March 1964 |volume=7 |issue=1 |pages=1–22 |doi=10.1016/S0019-9958(64)90223-2|doi-access=free }}</ref><ref>{{cite journal |last1=Solomonoff |first1=Ray |title=A Formal Theory of Inductive Inference Part 2 |journal=Information and Control |date=1964 |volume=7 |issue=2 |pages=224–254 |doi=10.1016/S0019-9958(64)90131-7}}</ref>

* [[Bayesian inference]];

* [[Algorithmic learning theory]], from the work of [[E. Mark Gold]];<ref>{{Cite journal | last1 = Gold | first1 = E. Mark | year = 1967 | title = Language identification in the limit | journal = Information and Control | volume = 10 | issue = 5 | pages = 447–474 | jstor = | doi = 10.1016/S0019-9958(67)91165-5 | url=http://web.mit.edu/~6.863/www/spring2009/readings/gold67limit.pdf | formatdoi-access = | accessdate =free }}</ref>

* [[Algorithmic learning theory]], from the work of [[E. Mark Gold]];

* [[Online machine learning]], from the work of Nick Littlestone{{citation needed|date=October 2017}}.

While its primary goal is to understand learning abstractly, computational learning theory has led to the development of practical algorithms. For example, PAC theory inspired [[Boosting (meta-algorithm)|boosting]], VC theory led to [[support vector machine]]s, and Bayesian inference led to [[belief networks]].

==See also==

* [[Error Tolerancetolerance (PAC learning)]]

* [[Grammar induction]]

* [[Information theory]]

* [[Category:MachineOccam learning]]

* [[Stability (learning theory)]]

* [[Error Tolerance (PAC learning)]]

==References==

{{Reflist}}

==Further reading==

A description of some of these publications is given at [[list of important publications in computer science#Machine learning|important publications in machine learning]].

===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), pages&nbsp;351–369. http://portal.acm.org/citation.cfm?id=129712.129746

* 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. [https://courses.engr.illinois.edu/ece544na/fa2014/vapnik71.pdf 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===

* {{Cite journal | last1 = Gold | first1 = E. Mark | year = 1967 | title = Language identification in the limit | journal = Information and Control | volume = 10 | issue = 5 | pages = 447–474 | jstor = | doi = 10.1016/S0019-9958(67)91165-5 | url=http://web.mit.edu/~6.863/www/spring2009/readings/gold67limit.pdf | format = | accessdate = }}

===Optimal O notation learning===

* [[Oded Goldreich]], [[Dana Ron]]. ''[http://www.engwisdom.tauweizmann.ac.il/~danaroded/Public-pdfPS/ul.pdfps On universal learning algorithms]''. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.2224

===Negative results===

* M. Kearns and [[Leslie 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{{dl|date=August 2024}}

===[[Boosting (machine learning)]]===

* Robert E. Schapire. The strength of weak learnability. Machine Learning, 5(2):197–227, 1990 http://citeseer.ist.psu.edu/schapire90strength.html

===[[Occam Learning]]===

* Blumer, A.; Ehrenfeucht, A.; Haussler, D.; Warmuth, M. K. [http://www.cse.buffalo.edu/~hungngo/classes/2008/694/papers/occam.pdf Occam's razor] Inf.Proc.Lett. 24, 377–380, 1987.

* Blumer, A.; Ehrenfeucht, A.; Haussler, D.; Warmuth, M. K. [http://www.trhvidsten.com/docs/classics/Blumer-1989.pdf Learnability and the Vapnik-Chervonenkis dimension]. Journal of the ACM, 36(4):929–865, 1989.

===[[Probably approximately correct learning]]===

* L. Valiant. [http://www.montefiore.ulg.ac.be/~geurts/Cours/AML/Readings/Valiant.pdf A Theory of the Learnable]. Communications of the ACM, 27(11):1134–1142, 1984.

===Error tolerance===

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===Equivalence===

* D.Haussler, M.Kearns, N.Littlestone and [[Manfred K. Warmuth|M. Warmuth]], Equivalence of models for polynomial learnability, Proc. 1st ACM Workshop on Computational Learning Theory, (1988) 42-55.

* {{Cite journal | last1 = Pitt | first1 = L. | last2 = Warmuth | first2 = M. K. | year = 1990 | title = Prediction-Preserving Reducibility | journal = Journal of Computer and System Sciences | volume = 41 | issue = 3| pages = 430–467 | jstor = | doi = 10.1016/0022-0000(90)90028-J | doi-access = free }}

A description of some of these publications is given at [[list of important publications in computer science#Machine learning|important publications in machine learning]].

===[[Distribution learning theory|Distribution Learning Theory]]===

==External links==

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[[Category:Computational learning theory| ]]

[[Category:Theoretical computer science]]

[[Category:Machine learning]]

[[Category:Computational fields of study]]