Computational learning theory

See also: Statistical learning theory

Short description: Theory of machine learning

Machine learning and
data mining

Problems

* Classification
* Clustering
* Regression
* Anomaly detection
* AutoML
* Association rules
* Reinforcement learning
* Structured prediction
* Feature engineering
* Feature learning
* Online learning
* Semi-supervised learning
* Unsupervised learning
* Learning to rank
* Grammar induction

Supervised learning
(classification • regression)

* Decision trees
* Ensembles
* Bagging
* Boosting
* Random forest
* k-NN
* Linear regression
* Naive Bayes
* Artificial neural networks
* Logistic regression
* Perceptron
* Relevance vector machine (RVM)
* Support vector machine (SVM)

Clustering

* BIRCH
* CURE
* Hierarchical
* k-means
* Expectation–maximization (EM)

DBSCAN
* OPTICS
* Mean-shift

Dimensionality reduction

* Factor analysis
* CCA
* ICA
* LDA
* NMF
* PCA
* t-SNE

Structured prediction

* Graphical models
* Bayes net
* Conditional random field
* Hidden Markov

Anomaly detection

* k-NN
* Local outlier factor

Artificial neural networks

* Autoencoder
* Deep learning
* DeepDream
* Multilayer perceptron
* RNN
* LSTM
* GRU
* Restricted Boltzmann machine
* GAN
* SOM
* Convolutional neural network
* U-Net

Reinforcement learning

* Q-learning
* SARSA
* Temporal difference (TD)

Theory

* Bias–variance dilemma
* Computational learning theory
* Empirical risk minimization
* Occam learning
* PAC learning
* Statistical learning
* VC theory

Machine-learning venues

* NeurIPS
* ICML
* ML
* JMLR
* ArXiv:cs.LG

Glossary of artificial intelligence

* Glossary of artificial intelligence

Related articles

* List of datasets for machine-learning research
* Outline of machine learning

* v
* t
* e

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.[1]

## Contents

* 1 Overview
* 2 See also
* 3 References
* 4 Further reading
* 4.1 Surveys
* 4.2 Feature selection
* 4.3 Optimal O notation learning
* 4.4 Negative results
* 4.5 Error tolerance
* 4.6 Equivalence
* 5 External links

## 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 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. 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 results - Showing that a certain class of functions is learnable in polynomial time.
* Negative results - Showing that certain classes cannot be learned in polynomial time.[2]

Negative results often rely on commonly believed, but yet unproven assumptions, 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. The different approaches include:

* Exact learning, proposed by Dana Angluin;[3][4]
* Probably approximately correct learning (PAC learning), proposed by Leslie Valiant;[5]
* VC theory, proposed by Vladimir Vapnik and Alexey Chervonenkis;[6]
* Inductive inference as developed by Ray Solomonoff;[7][8]
* Algorithmic learning theory, from the work of E. Mark Gold;[9]
* Online machine learning, from the work of Nick Littlestone .

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.

## See also

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

## References

1. ↑ "ACL - Association for Computational Learning". http://www.learningtheory.org/.
2. ↑ Kearns, Michael; Vazirani, Umesh (August 15, 1994). An Introduction to Computational Learning Theory. MIT Press. ISBN 978-0262111935.
3. ↑ Dana Angluin (1976). An Application of the Theory of Computational Complexity to the Study of Inductive Inference (Ph.D. thesis). University of California at Berkeley.
4. ↑ D. Angluin (1978). "On the Complexity of Minimum Inference of Regular Sets". Information and Control 39 (3): 337-350. http://www.sciencedirect.com/science/article/pii/S0019995878906836.
5. ↑ Valiant, Leslie (1984). "A Theory of the Learnable". Communications of the ACM 27 (11): 1134–1142. doi:10.1145/1968.1972. https://www.montefiore.ulg.ac.be/~geurts/Cours/AML/Readings/Valiant.pdf. Retrieved 2022-11-24.
6. ↑ Vapnik, V.; Chervonenkis, A. (1971). "On the uniform convergence of relative frequencies of events to their probabilities". Theory of Probability and Its Applications 16 (2): 264–280. doi:10.1137/1116025. https://courses.engr.illinois.edu/ece544na/fa2014/vapnik71.pdf.
7. ↑ 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.
8. ↑ 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.
9. ↑ Gold, E. Mark (1967). "Language identification in the limit". Information and Control 10 (5): 447–474. doi:10.1016/S0019-9958(67)91165-5. http://web.mit.edu/~6.863/www/spring2009/readings/gold67limit.pdf.

## 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. 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[|permanent dead link|dead link}}]

### 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 M. Warmuth, Equivalence of models for polynomial learnability, Proc. 1st ACM Workshop on Computational Learning Theory, (1988) 42-55.
* Pitt, L.; Warmuth, M. K. (1990). "Prediction-Preserving Reducibility". Journal of Computer and System Sciences 41 (3): 430–467. doi:10.1016/0022-0000(90)90028-J.

## External links

* Basics of Bayesian inference

* v
* t
* e

Differentiable computing

General|

* Differentiable programming
* Neural Turing machine
* Differentiable neural computer
* Automatic differentiation
* Neuromorphic engineering

Concepts|

* Gradient descent
* Cable theory
* Cluster analysis
* Regression analysis
* Pattern recognition
* Adversarial machine learning
* Computational learning theory

Programming languages|

* Python
* Julia

Application|

* Machine learning
* Artificial neural network
* Scientific computing
* Artificial Intelligence

Hardware|

* TPU
* VPU
* Memristor
* SpiNNaker

Software library|

* TensorFlow
* PyTorch

Implementation|

| Audio-visual|

* AlexNet
* WaveNet
* Human image synthesis
* HWR
* OCR
* Speech synthesis
* Speech recognition
* Facial recognition system

|

Verbal|

* Word2vec
* Transformer
* BERT
* NMT
* Project Debater
* Watson
* GPT-3

Decisional|

* AlphaGo
* Q-learning
* SARSA
* OpenAI Five

People|

* Alex Graves
* Ian Goodfellow
* Yoshua Bengio
* Geoffrey Hinton
* Yann LeCun
* Andrew Ng
* Demis Hassabis

* Portals
* Computer programming
* Technology
* Category
* Artificial neural networks
* Machine learning

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