Computational learning theory

{{see also|Statistical learning theory}} {{Short description|Theory of machine learning}}{{more citations needed|date=November 2018}} {{Machine learning|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.{{Cite web | url=http://www.learningtheory.org/ | title=ACL - Association for Computational Learning}}

==Overview== Theoretical results in machine learning often focus on a type of inductive learning known as [[supervised learning]]. In supervised learning, an algorithm is provided with [[Labeled data|labeled]] samples. For instance, the samples might be descriptions of mushrooms, with labels indicating whether they are edible or not. The algorithm uses these labeled samples to create a classifier. This classifier assigns labels to new samples, including those it has not previously encountered. The goal of the supervised learning algorithm is to optimize performance metrics, such as minimizing errors 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 computational learning theory, a computation is considered feasible if it can be done in [[polynomial time]].{{citation needed|date=October 2017}} There are two kinds of time complexity results:

• 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.{{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}}

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 ≠ 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:

• Exact learning, proposed by [[Dana Angluin]];{{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 }}{{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=337–350 | year=1978 }}
• [[Probably approximately correct learning]] (PAC learning), proposed by [[Leslie Valiant]];{{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 }}
• [[VC theory]], proposed by [[Vladimir Vapnik]] and [[Alexey Chervonenkis]];{{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}}
• [[Solomonoff's theory of inductive inference|Inductive inference]] as developed by [[Ray Solomonoff]];{{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 }}{{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}}
• [[Algorithmic learning theory]], from the work of [[E. Mark Gold]];{{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 }}
• [[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 tolerance (PAC learning)]]
• [[Grammar induction]]
• [[Information theory]]
• [[Occam learning]]
• [[Stability (learning theory)]]

==References== {{Reflist}}

==Further reading== A description of some of these publications is given at 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 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

===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

===Optimal O notation learning===

• [[Oded Goldreich]], [[Dana Ron]]. ''[http://www.wisdom.weizmann.ac.il/~oded/PS/ul.ps 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}}

===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 [[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 | doi = 10.1016/0022-0000(90)90028-J | doi-access = free }}

==External links==

• [http://research.microsoft.com/adapt/MSBNx/msbnx/Basics_of_Bayesian_Inference.htm Basics of Bayesian inference]

{{Differentiable computing}}

[[Category:Computational learning theory| ]] [[Category:Computational fields of study]]

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