Computational learning theory: Difference between revisions

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==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 measure 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:

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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:

* 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]]{{citation needed|date=October 2017}};

* [[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>

* [[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>

* [[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=1-221–22 |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=224-254224–254 |doi=10.1016/S0019-9958(64)90131-7}}</ref>

* [[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 | doi = 10.1016/S0019-9958(67)91165-5 | url=http://web.mit.edu/~6.863/www/spring2009/readings/gold67limit.pdf | doi-access = free }}</ref>

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

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==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

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===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}}

===Error tolerance===