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. In computational learning theory, a computation is considered feasible if it can be done in polynomial time. There are two kinds of time complexity results:
Positive resultsShowing that a certain class of functions is learnable in polynomial time.
Negative resultsShowing that certain classes cannot be learned in polynomial time.
Negative results often rely on commonly believed, but yet unproven assumptions, such as:
There are several different approaches to computational learning theory based on making different assumptions about the inference principles used to generalize from limited data. This includes different definitions of probability and different assumptions on the generation of samples. The different approaches include:
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.
Surveys
Angluin, D. 1992. Computational learning theory: Survey and selected bibliography. In Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing, pages 351–369. http://portal.acm.org/citation.cfm?id=129712.129746
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
Inductive inference
Optimal O notation learning
Oded Goldreich, Dana Ron. . 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
Michael Kearns and Ming Li. Learning in the presence of malicious errors. SIAM Journal on Computing, 22:807–837, August 1993. http://citeseer.ist.psu.edu/kearns93learning.html
Kearns, M.. 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 M. Warmuth, Equivalence of models for polynomial learnability, Proc. 1st ACM Workshop on Computational Learning Theory, 42-55.
