Computational learning theory

<h2 id="overview">Overview</h2>
<p>Theoretical results in machine learning mainly deal with a type of inductive learning called <a href="/facts/Supervised_learning/JlNFNPFt">supervised learning</a>.  In supervised learning, an algorithm is given samples that are <a href="/facts/Labeled_data/o0doUSPA">labeled</a> 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.
</p><p>In addition to performance bounds, computational learning theory studies the time complexity and feasibility of learning. In
computational learning theory, a computation is considered feasible if it can be done in <a href="/facts/Polynomial_time/77T62gmf">polynomial time</a>. There are two kinds of time
complexity results:
</p>
<ul><li>Positive results – Showing that a certain class of functions is learnable in polynomial time.</li>
<li>Negative results – Showing that certain classes cannot be learned in polynomial time.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li></ul>
<p>Negative results often rely on commonly believed, but yet unproven assumptions, such as:
</p>
<ul><li>Computational complexity – <a href="/facts/P_versus_NP_problem/xoC7Ruda">P ≠ NP (the P versus NP problem)</a>;</li>
<li><a href="/facts/Cryptography/e94PEVau">Cryptographic</a> – <a href="/facts/One-way_function/rGtcumDW">One-way functions</a> exist.</li></ul>
<p>There are several different approaches to computational learning theory based on making different assumptions about the <a href="/facts/Inference/RBbe95bP">inference</a> principles used to generalise from limited data. This includes different definitions of <a href="/facts/Probability/fgYvMhwb">probability</a> (see <a href="/facts/Frequency_probability/1f1TjSey">frequency probability</a>, <a href="/facts/Bayesian_probability/Cvn7WWfG">Bayesian probability</a>) and different assumptions on the generation of samples. The different approaches include:
</p>
<ul><li>Exact learning, proposed by <a href="/facts/Dana_Angluin/9aEoCNKK">Dana Angluin</a>;</li>
<li><a href="/facts/Probably_approximately_correct_learning/vBQlvDV1">Probably approximately correct learning</a> (PAC learning), proposed by <a href="/facts/Leslie_Valiant/hA5okAxL">Leslie Valiant</a>;<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li>
<li><a href="/facts/VC_theory/jDST0Ulo">VC theory</a>, proposed by <a href="/facts/Vladimir_Vapnik/UQfpyUuN">Vladimir Vapnik</a> and <a href="/facts/Alexey_Chervonenkis/EAgIkPrg">Alexey Chervonenkis</a>;<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li>
<li><a href="/facts/Solomonoff%27s_theory_of_inductive_inference/4kecWjSB">Inductive inference</a> as developed by <a href="/facts/Ray_Solomonoff/2svRlto1">Ray Solomonoff</a>;<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li>
<li><a href="/facts/Algorithmic_learning_theory/1NMh5bn3">Algorithmic learning theory</a>, from the work of <a href="/facts/E._Mark_Gold/I1a7gfSq">E. Mark Gold</a>;<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li>
<li><a href="/facts/Online_machine_learning/4BoJlDfg">Online machine learning</a>, from the work of Nick Littlestone.</li></ul>
<p>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 <a href="/facts/Boosting_(meta-algorithm)/HgejTPPu">boosting</a>, VC theory led to <a href="/facts/Support_vector_machine/XobxpdBG">support vector machines</a>, and Bayesian inference led to <a href="/facts/Belief_networks/Ry31vq6T">belief networks</a>.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Error_tolerance_(PAC_learning)/R3cf9lfA">Error tolerance (PAC learning)</a></li>
<li><a href="/facts/Grammar_induction/Mx8ISTor">Grammar induction</a></li>
<li><a href="/facts/Information_theory/qNLH3U5P">Information theory</a></li>
<li><a href="/facts/Occam_learning/NzQpEbtK">Occam learning</a></li>
<li><a href="/facts/Stability_(learning_theory)/mreLQdwQ">Stability (learning theory)</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<p>A description of some of these publications is given at important publications in machine learning.
</p>
<h3>Surveys</h3>
<ul><li>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. <a href="http://portal.acm.org/citation.cfm?id=129712.129746">http://portal.acm.org/citation.cfm?id=129712.129746</a></li>
<li>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. <a href="http://citeseer.ist.psu.edu/haussler90probably.html">http://citeseer.ist.psu.edu/haussler90probably.html</a></li></ul>
<h3>Feature selection</h3>
<ul><li>A. Dhagat and L. Hellerstein, "PAC learning with irrelevant attributes", in 'Proceedings of the IEEE Symp. on Foundation of Computer Science', 1994. <a href="http://citeseer.ist.psu.edu/dhagat94pac.html">http://citeseer.ist.psu.edu/dhagat94pac.html</a></li></ul>
<h3>Optimal O notation learning</h3>
<ul><li><a href="/facts/Oded_Goldreich/Ufn1bqW7">Oded Goldreich</a>, <a href="/facts/Dana_Ron/Wmo3cEdS">Dana Ron</a>. <i><a href="http://www.wisdom.weizmann.ac.il/~oded/PS/ul.ps">On universal learning algorithms</a></i>. <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.2224">http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.2224</a></li></ul>
<h3>Negative results</h3>
<ul><li>M. Kearns and <a href="/facts/Leslie_Valiant/hA5okAxL">Leslie Valiant</a>. 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. <a href="http://citeseer.ist.psu.edu/kearns89cryptographic.html">http://citeseer.ist.psu.edu/kearns89cryptographic.html</a>[<i>dead link</i>‍]</li></ul>
<h3>Error tolerance</h3>
<ul><li>Michael Kearns and Ming Li. Learning in the presence of malicious errors. SIAM Journal on Computing, 22(4):807–837, August 1993. <a href="http://citeseer.ist.psu.edu/kearns93learning.html">http://citeseer.ist.psu.edu/kearns93learning.html</a></li>
<li>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. <a href="http://citeseer.ist.psu.edu/kearns93efficient.html">http://citeseer.ist.psu.edu/kearns93efficient.html</a></li></ul>
<h3>Equivalence</h3>
<ul><li>D.Haussler, M.Kearns, N.Littlestone and <a href="/facts/Manfred_K._Warmuth/G1bBxcVX">M. Warmuth</a>, Equivalence of models for polynomial learnability, Proc. 1st ACM Workshop on Computational Learning Theory, (1988) 42-55.</li>
<li>Pitt, L.; Warmuth, M. K. (1990). <a href="https://doi.org/10.1016%2F0022-0000%2890%2990028-J">"Prediction-Preserving Reducibility"</a>. <i>Journal of Computer and System Sciences</i>. 41 (3): 430–467. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0022-0000%2890%2990028-J">10.1016/0022-0000(90)90028-J</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://research.microsoft.com/adapt/MSBNx/msbnx/Basics_of_Bayesian_Inference.htm">Basics of Bayesian inference</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>"ACL - Association for Computational Learning". <a href="http://www.learningtheory.org/" target="_blank">http://www.learningtheory.org/</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Kearns, Michael; Vazirani, Umesh (August 15, 1994). An Introduction to Computational Learning Theory. MIT Press. ISBN 978-0262111935. <a href="978-0262111935" target="_blank">978-0262111935</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Valiant, Leslie (1984). "A Theory of the Learnable" (PDF). Communications of the ACM. 27 (11): 1134–1142. doi:10.1145/1968.1972. S2CID 12837541. Archived from the original (PDF) on 2019-05-17. Retrieved 2022-11-24. <a href="https://web.archive.org/web/20190517235548/http://www.montefiore.ulg.ac.be/~geurts/Cours/AML/Readings/Valiant.pdf" target="_blank">https://web.archive.org/web/20190517235548/http://www.montefiore.ulg.ac.be/~geurts/Cours/AML/Readings/Valiant.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Vapnik, V.; Chervonenkis, A. (1971). "On the uniform convergence of relative frequencies of events to their probabilities" (PDF). Theory of Probability and Its Applications. 16 (2): 264–280. doi:10.1137/1116025. <a href="https://courses.engr.illinois.edu/ece544na/fa2014/vapnik71.pdf" target="_blank">https://courses.engr.illinois.edu/ece544na/fa2014/vapnik71.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Solomonoff, Ray (March 1964). "A Formal Theory of Inductive Inference Part 1". Information and Control. 7 (1): 1–22. doi:10.1016/S0019-9958(64)90223-2. <a href="https://doi.org/10.1016%2FS0019-9958%2864%2990223-2" target="_blank">https://doi.org/10.1016%2FS0019-9958%2864%2990223-2</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Solomonoff, Ray (1964). "A Formal Theory of Inductive Inference Part 2". Information and Control. 7 (2): 224–254. doi:10.1016/S0019-9958(64)90131-7. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Gold, E. Mark (1967). "Language identification in the limit" (PDF). Information and Control. 10 (5): 447–474. doi:10.1016/S0019-9958(67)91165-5. <a href="http://web.mit.edu/~6.863/www/spring2009/readings/gold67limit.pdf" target="_blank">http://web.mit.edu/~6.863/www/spring2009/readings/gold67limit.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
</ol>

Computational learning theory open-in-new

Computational learning theory