Editing Computational learning theory (section)

==Overview==
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 data|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.<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 ≠ 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]]{{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=1–22 |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>
* [[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}}.

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]].