Computational learning theory

Theoretical results in machine learning mainly deal with a type of inductive learning called supervised learning. In supervised learning, an algorithm is given samples that are labeled in some useful way. For example, the samples might be descriptions of mushrooms, and the labels could be whether or not the mushrooms are edible. The algorithm takes these previously labeled samples and uses them to induce a classifier. This classifier is a function that assigns labels to samples, including samples that have not been seen previously by the algorithm. The goal of the supervised learning algorithm is to optimize some measure of performance such as minimizing the number of mistakes made 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 ≠ NP (the P versus NP problem);
• Cryptographic – One-way functions 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{{citation needed|date=October 2017}};
• 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}}
• 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, VC theory led to support vector machines, and Bayesian inference led to belief networks.

• Error tolerance (PAC learning)
• Grammar induction
• Information theory
• Occam learning
• Stability (learning theory)

{{Reflist}}

A description of some of these publications is given at important publications in machine learning.

• 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

• 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

• 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

• 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}}

• 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

• 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.
• {{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 }}

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

{{Differentiable computing}}

Category:Computational fields of study

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