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class="vector-toc-list"> <li id="toc-Surveys" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surveys"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Surveys</span> </div> </a> <ul id="toc-Surveys-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Feature_selection" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Feature_selection"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Feature selection</span> </div> </a> <ul id="toc-Feature_selection-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optimal_O_notation_learning" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optimal_O_notation_learning"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Optimal O notation learning</span> </div> </a> <ul id="toc-Optimal_O_notation_learning-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Negative_results" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Negative_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Negative results</span> </div> </a> <ul id="toc-Negative_results-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Error_tolerance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Error_tolerance"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Error tolerance</span> </div> </a> <ul id="toc-Error_tolerance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equivalence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalence"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Equivalence</span> </div> </a> <ul id="toc-Equivalence-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item 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.sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r886047488">.mw-parser-output .nobold{font-weight:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a><br />and <a href="/wiki/Data_mining" title="Data mining">data mining</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Paradigms</div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Supervised_learning" title="Supervised learning">Supervised learning</a></li> <li><a href="/wiki/Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a></li> <li><a href="/wiki/Semi-supervised_learning" class="mw-redirect" title="Semi-supervised learning">Semi-supervised learning</a></li> <li><a href="/wiki/Self-supervised_learning" title="Self-supervised learning">Self-supervised learning</a></li> <li><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></li> <li><a href="/wiki/Meta-learning_(computer_science)" title="Meta-learning (computer science)">Meta-learning</a></li> <li><a href="/wiki/Online_machine_learning" title="Online machine learning">Online learning</a></li> <li><a href="/wiki/Batch_learning" class="mw-redirect" title="Batch learning">Batch learning</a></li> <li><a href="/wiki/Curriculum_learning" title="Curriculum learning">Curriculum learning</a></li> <li><a href="/wiki/Rule-based_machine_learning" title="Rule-based machine learning">Rule-based learning</a></li> <li><a href="/wiki/Neuro-symbolic_AI" title="Neuro-symbolic AI">Neuro-symbolic AI</a></li> <li><a href="/wiki/Neuromorphic_engineering" class="mw-redirect" title="Neuromorphic engineering">Neuromorphic engineering</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Problems</div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Generative_model" title="Generative model">Generative modeling</a></li> <li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression</a></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Clustering</a></li> <li><a href="/wiki/Dimensionality_reduction" title="Dimensionality reduction">Dimensionality reduction</a></li> <li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li> <li><a href="/wiki/Anomaly_detection" title="Anomaly detection">Anomaly detection</a></li> <li><a href="/wiki/Data_cleaning" class="mw-redirect" title="Data cleaning">Data cleaning</a></li> <li><a href="/wiki/Automated_machine_learning" title="Automated machine learning">AutoML</a></li> <li><a href="/wiki/Association_rule_learning" title="Association rule learning">Association rules</a></li> <li><a href="/wiki/Semantic_analysis_(machine_learning)" title="Semantic analysis (machine learning)">Semantic analysis</a></li> <li><a href="/wiki/Structured_prediction" title="Structured prediction">Structured prediction</a></li> <li><a href="/wiki/Feature_engineering" title="Feature engineering">Feature engineering</a></li> <li><a href="/wiki/Feature_learning" title="Feature learning">Feature learning</a></li> <li><a href="/wiki/Learning_to_rank" title="Learning to rank">Learning to rank</a></li> <li><a href="/wiki/Grammar_induction" title="Grammar induction">Grammar induction</a></li> <li><a href="/wiki/Ontology_learning" title="Ontology learning">Ontology learning</a></li> <li><a href="/wiki/Multimodal_learning" title="Multimodal learning">Multimodal learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Supervised_learning" title="Supervised learning">Supervised learning</a><br /><span class="nobold"><span style="font-size:85%;">(<b><a href="/wiki/Statistical_classification" title="Statistical classification">classification</a></b> • <b><a href="/wiki/Regression_analysis" title="Regression analysis">regression</a></b>)</span></span> </div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Apprenticeship_learning" title="Apprenticeship learning">Apprenticeship learning</a></li> <li><a href="/wiki/Decision_tree_learning" title="Decision tree learning">Decision trees</a></li> <li><a href="/wiki/Ensemble_learning" title="Ensemble learning">Ensembles</a> <ul><li><a href="/wiki/Bootstrap_aggregating" title="Bootstrap aggregating">Bagging</a></li> <li><a href="/wiki/Boosting_(machine_learning)" title="Boosting (machine learning)">Boosting</a></li> <li><a href="/wiki/Random_forest" title="Random forest">Random forest</a></li></ul></li> <li><a href="/wiki/K-nearest_neighbors_algorithm" title="K-nearest neighbors algorithm"><i>k</i>-NN</a></li> <li><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></li> <li><a href="/wiki/Naive_Bayes_classifier" title="Naive Bayes classifier">Naive Bayes</a></li> <li><a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural networks</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic regression</a></li> <li><a href="/wiki/Perceptron" title="Perceptron">Perceptron</a></li> <li><a href="/wiki/Relevance_vector_machine" title="Relevance vector machine">Relevance vector machine (RVM)</a></li> <li><a href="/wiki/Support_vector_machine" title="Support vector machine">Support vector machine (SVM)</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Cluster_analysis" title="Cluster analysis">Clustering</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/BIRCH" title="BIRCH">BIRCH</a></li> <li><a href="/wiki/CURE_algorithm" title="CURE algorithm">CURE</a></li> <li><a href="/wiki/Hierarchical_clustering" title="Hierarchical clustering">Hierarchical</a></li> <li><a href="/wiki/K-means_clustering" title="K-means clustering"><i>k</i>-means</a></li> <li><a href="/wiki/Fuzzy_clustering" title="Fuzzy clustering">Fuzzy</a></li> <li><a href="/wiki/Expectation%E2%80%93maximization_algorithm" title="Expectation–maximization algorithm">Expectation–maximization (EM)</a></li> <li><br /><a href="/wiki/DBSCAN" title="DBSCAN">DBSCAN</a></li> <li><a href="/wiki/OPTICS_algorithm" title="OPTICS algorithm">OPTICS</a></li> <li><a href="/wiki/Mean_shift" title="Mean shift">Mean shift</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Dimensionality_reduction" title="Dimensionality reduction">Dimensionality reduction</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">CCA</a></li> <li><a href="/wiki/Independent_component_analysis" title="Independent component analysis">ICA</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">LDA</a></li> <li><a href="/wiki/Non-negative_matrix_factorization" title="Non-negative matrix factorization">NMF</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">PCA</a></li> <li><a href="/wiki/Proper_generalized_decomposition" title="Proper generalized decomposition">PGD</a></li> <li><a href="/wiki/T-distributed_stochastic_neighbor_embedding" title="T-distributed stochastic neighbor embedding">t-SNE</a></li> <li><a href="/wiki/Sparse_dictionary_learning" title="Sparse dictionary learning">SDL</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Structured_prediction" title="Structured prediction">Structured prediction</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Graphical_model" title="Graphical model">Graphical models</a> <ul><li><a href="/wiki/Bayesian_network" title="Bayesian network">Bayes net</a></li> <li><a href="/wiki/Conditional_random_field" title="Conditional random field">Conditional random field</a></li> <li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Anomaly_detection" title="Anomaly detection">Anomaly detection</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Random_sample_consensus" title="Random sample consensus">RANSAC</a></li> <li><a href="/wiki/K-nearest_neighbors_algorithm" title="K-nearest neighbors algorithm"><i>k</i>-NN</a></li> <li><a href="/wiki/Local_outlier_factor" title="Local outlier factor">Local outlier factor</a></li> <li><a href="/wiki/Isolation_forest" title="Isolation forest">Isolation forest</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural network</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Autoencoder" title="Autoencoder">Autoencoder</a></li> <li><a href="/wiki/Deep_learning" title="Deep learning">Deep learning</a></li> <li><a href="/wiki/Feedforward_neural_network" title="Feedforward neural network">Feedforward neural network</a></li> <li><a href="/wiki/Recurrent_neural_network" title="Recurrent neural network">Recurrent neural network</a> <ul><li><a href="/wiki/Long_short-term_memory" title="Long short-term memory">LSTM</a></li> <li><a href="/wiki/Gated_recurrent_unit" title="Gated recurrent unit">GRU</a></li> <li><a href="/wiki/Echo_state_network" title="Echo state network">ESN</a></li> <li><a href="/wiki/Reservoir_computing" title="Reservoir computing">reservoir computing</a></li></ul></li> <li><a href="/wiki/Boltzmann_machine" title="Boltzmann machine">Boltzmann machine</a> <ul><li><a href="/wiki/Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">Restricted</a></li></ul></li> <li><a href="/wiki/Generative_adversarial_network" title="Generative adversarial network">GAN</a></li> <li><a href="/wiki/Diffusion_model" title="Diffusion model">Diffusion model</a></li> <li><a href="/wiki/Self-organizing_map" title="Self-organizing map">SOM</a></li> <li><a href="/wiki/Convolutional_neural_network" title="Convolutional neural network">Convolutional neural network</a> <ul><li><a href="/wiki/U-Net" title="U-Net">U-Net</a></li> <li><a href="/wiki/LeNet" title="LeNet">LeNet</a></li> <li><a href="/wiki/AlexNet" title="AlexNet">AlexNet</a></li> <li><a href="/wiki/DeepDream" title="DeepDream">DeepDream</a></li></ul></li> <li><a href="/wiki/Neural_radiance_field" title="Neural radiance field">Neural radiance field</a></li> <li><a href="/wiki/Transformer_(machine_learning_model)" class="mw-redirect" title="Transformer (machine learning model)">Transformer</a> <ul><li><a href="/wiki/Vision_transformer" title="Vision transformer">Vision</a></li></ul></li> <li><a href="/wiki/Mamba_(deep_learning_architecture)" title="Mamba (deep learning architecture)">Mamba</a></li> <li><a href="/wiki/Spiking_neural_network" title="Spiking neural network">Spiking neural network</a></li> <li><a href="/wiki/Memtransistor" title="Memtransistor">Memtransistor</a></li> <li><a href="/wiki/Electrochemical_RAM" title="Electrochemical RAM">Electrochemical RAM</a> (ECRAM)</li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Q-learning" title="Q-learning">Q-learning</a></li> <li><a href="/wiki/State%E2%80%93action%E2%80%93reward%E2%80%93state%E2%80%93action" title="State–action–reward–state–action">SARSA</a></li> <li><a href="/wiki/Temporal_difference_learning" title="Temporal difference learning">Temporal difference (TD)</a></li> <li><a href="/wiki/Multi-agent_reinforcement_learning" title="Multi-agent reinforcement learning">Multi-agent</a> <ul><li><a href="/wiki/Self-play_(reinforcement_learning_technique)" class="mw-redirect" title="Self-play (reinforcement learning technique)">Self-play</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Learning with humans</div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Active_learning_(machine_learning)" title="Active learning (machine learning)">Active learning</a></li> <li><a href="/wiki/Crowdsourcing" title="Crowdsourcing">Crowdsourcing</a></li> <li><a href="/wiki/Human-in-the-loop" title="Human-in-the-loop">Human-in-the-loop</a></li> <li><a href="/wiki/Reinforcement_learning_from_human_feedback" title="Reinforcement learning from human feedback">RLHF</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Model diagnostics</div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li> <li><a href="/wiki/Confusion_matrix" title="Confusion matrix">Confusion matrix</a></li> <li><a href="/wiki/Learning_curve_(machine_learning)" title="Learning curve (machine learning)">Learning curve</a></li> <li><a href="/wiki/Receiver_operating_characteristic" title="Receiver operating characteristic">ROC curve</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Mathematical foundations</div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Kernel_machines" class="mw-redirect" title="Kernel machines">Kernel machines</a></li> <li><a href="/wiki/Bias%E2%80%93variance_tradeoff" title="Bias–variance tradeoff">Bias–variance tradeoff</a></li> <li><a class="mw-selflink selflink">Computational learning theory</a></li> <li><a href="/wiki/Empirical_risk_minimization" title="Empirical risk minimization">Empirical risk minimization</a></li> <li><a href="/wiki/Occam_learning" title="Occam learning">Occam learning</a></li> <li><a href="/wiki/Probably_approximately_correct_learning" title="Probably approximately correct learning">PAC learning</a></li> <li><a href="/wiki/Statistical_learning_theory" title="Statistical learning theory">Statistical learning</a></li> <li><a href="/wiki/Vapnik%E2%80%93Chervonenkis_theory" title="Vapnik–Chervonenkis theory">VC theory</a></li> <li><a href="/wiki/Topological_deep_learning" title="Topological deep learning">Topological deep learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Journals and conferences</div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/ECML_PKDD" title="ECML PKDD">ECML PKDD</a></li> <li><a href="/wiki/Conference_on_Neural_Information_Processing_Systems" title="Conference on Neural Information Processing Systems">NeurIPS</a></li> <li><a href="/wiki/International_Conference_on_Machine_Learning" title="International Conference on Machine Learning">ICML</a></li> <li><a href="/wiki/International_Conference_on_Learning_Representations" title="International Conference on Learning Representations">ICLR</a></li> <li><a href="/wiki/International_Joint_Conference_on_Artificial_Intelligence" title="International Joint Conference on Artificial Intelligence">IJCAI</a></li> <li><a href="/wiki/Machine_Learning_(journal)" title="Machine Learning (journal)">ML</a></li> <li><a href="/wiki/Journal_of_Machine_Learning_Research" title="Journal of Machine Learning Research">JMLR</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Related articles</div><div class="sidebar-list-content mw-collapsible-content hlist" style="background-color: #FFFFFF;"> <ul><li><a href="/wiki/Glossary_of_artificial_intelligence" title="Glossary of artificial intelligence">Glossary of artificial intelligence</a></li> <li><a href="/wiki/List_of_datasets_for_machine-learning_research" title="List of datasets for machine-learning research">List of datasets for machine-learning research</a> <ul><li><a href="/wiki/List_of_datasets_in_computer_vision_and_image_processing" title="List of datasets in computer vision and image processing">List of datasets in computer vision and image processing</a></li></ul></li> <li><a href="/wiki/Outline_of_machine_learning" title="Outline of machine learning">Outline of machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Machine_learning" title="Template:Machine learning"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Machine_learning" title="Template talk:Machine learning"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Machine_learning" title="Special:EditPage/Template:Machine learning"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Computer_science" title="Computer science">computer science</a>, <b>computational learning theory</b> (or just <b>learning theory</b>) is a subfield of <a href="/wiki/Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a> devoted to studying the design and analysis of <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a> algorithms.<sup id="cite_ref-ACL_1-0" class="reference"><a href="#cite_note-ACL-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_learning_theory&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Theoretical results in machine learning mainly deal with a type of inductive learning called <a href="/wiki/Supervised_learning" title="Supervised learning">supervised learning</a>. In supervised learning, an algorithm is given samples that are <a href="/wiki/Labeled_data" title="Labeled data">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.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2017)">citation needed</span></a></i>]</sup> In computational learning theory, a computation is considered feasible if it can be done in <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2017)">citation needed</span></a></i>]</sup> 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.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li></ul> <p>Negative results often rely on commonly believed, but yet unproven assumptions,<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2017)">citation needed</span></a></i>]</sup> such as: </p> <ul><li>Computational complexity – <a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P ≠ NP (the P versus NP problem)</a>;</li> <li><a href="/wiki/Cryptography" title="Cryptography">Cryptographic</a> – <a href="/wiki/One-way_function" title="One-way function">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="/wiki/Inference" title="Inference">inference</a> principles used to generalise from limited data. This includes different definitions of <a href="/wiki/Probability" title="Probability">probability</a> (see <a href="/wiki/Frequency_probability" class="mw-redirect" title="Frequency probability">frequency probability</a>, <a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a>) and different assumptions on the generation of samples.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2017)">citation needed</span></a></i>]</sup> The different approaches include: </p> <ul><li>Exact learning, proposed by <a href="/wiki/Dana_Angluin" title="Dana Angluin">Dana Angluin</a><sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2017)">citation needed</span></a></i>]</sup>;</li> <li><a href="/wiki/Probably_approximately_correct_learning" title="Probably approximately correct learning">Probably approximately correct learning</a> (PAC learning), proposed by <a href="/wiki/Leslie_Valiant" title="Leslie Valiant">Leslie Valiant</a>;<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/VC_theory" class="mw-redirect" title="VC theory">VC theory</a>, proposed by <a href="/wiki/Vladimir_Vapnik" title="Vladimir Vapnik">Vladimir Vapnik</a> and <a href="/wiki/Alexey_Chervonenkis" title="Alexey Chervonenkis">Alexey Chervonenkis</a>;<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Solomonoff%27s_theory_of_inductive_inference" title="Solomonoff's theory of inductive inference">Inductive inference</a> as developed by <a href="/wiki/Ray_Solomonoff" title="Ray Solomonoff">Ray Solomonoff</a>;<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Algorithmic_learning_theory" title="Algorithmic learning theory">Algorithmic learning theory</a>, from the work of <a href="/wiki/E._Mark_Gold" title="E. Mark Gold">E. Mark Gold</a>;<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Online_machine_learning" title="Online machine learning">Online machine learning</a>, from the work of Nick Littlestone<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2017)">citation needed</span></a></i>]</sup>.</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="/wiki/Boosting_(meta-algorithm)" class="mw-redirect" title="Boosting (meta-algorithm)">boosting</a>, VC theory led to <a href="/wiki/Support_vector_machine" title="Support vector machine">support vector machines</a>, and Bayesian inference led to <a href="/wiki/Belief_networks" class="mw-redirect" title="Belief networks">belief networks</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_learning_theory&action=edit&section=2" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Error_tolerance_(PAC_learning)" title="Error tolerance (PAC learning)">Error tolerance (PAC learning)</a></li> <li><a href="/wiki/Grammar_induction" title="Grammar induction">Grammar induction</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Occam_learning" title="Occam learning">Occam learning</a></li> <li><a href="/wiki/Stability_(learning_theory)" title="Stability (learning theory)">Stability (learning theory)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_learning_theory&action=edit&section=3" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-ACL-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-ACL_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.learningtheory.org/">"ACL - Association for Computational Learning"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=ACL+-+Association+for+Computational+Learning&rft_id=http%3A%2F%2Fwww.learningtheory.org%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+learning+theory" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKearnsVazirani1994" class="citation book cs1">Kearns, Michael; Vazirani, Umesh (August 15, 1994). <i>An Introduction to Computational Learning Theory</i>. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0262111935" title="Special:BookSources/978-0262111935"><bdi>978-0262111935</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Computational+Learning+Theory&rft.pub=MIT+Press&rft.date=1994-08-15&rft.isbn=978-0262111935&rft.aulast=Kearns&rft.aufirst=Michael&rft.au=Vazirani%2C+Umesh&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+learning+theory" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: date and year (<a href="/wiki/Category:CS1_maint:_date_and_year" title="Category:CS1 maint: date and year">link</a>)</span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="ValTotL" class="citation journal cs1">Valiant, Leslie (1984). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190517235548/http://www.montefiore.ulg.ac.be/~geurts/Cours/AML/Readings/Valiant.pdf">"A Theory of the Learnable"</a> <span class="cs1-format">(PDF)</span>. <i>Communications of the ACM</i>. <b>27</b> (11): <span class="nowrap">1134–</span>1142. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1968.1972">10.1145/1968.1972</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:12837541">12837541</a>. Archived from <a rel="nofollow" class="external text" href="https://www.montefiore.ulg.ac.be/~geurts/Cours/AML/Readings/Valiant.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2019-05-17<span class="reference-accessdate">. Retrieved <span class="nowrap">2022-11-24</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=A+Theory+of+the+Learnable&rft.volume=27&rft.issue=11&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1134-%3C%2Fspan%3E1142&rft.date=1984&rft_id=info%3Adoi%2F10.1145%2F1968.1972&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A12837541%23id-name%3DS2CID&rft.aulast=Valiant&rft.aufirst=Leslie&rft_id=https%3A%2F%2Fwww.montefiore.ulg.ac.be%2F~geurts%2FCours%2FAML%2FReadings%2FValiant.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+learning+theory" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="VCdim" class="citation journal cs1">Vapnik, V.; Chervonenkis, A. (1971). <a rel="nofollow" class="external text" href="https://courses.engr.illinois.edu/ece544na/fa2014/vapnik71.pdf">"On the uniform convergence of relative frequencies of events to their probabilities"</a> <span class="cs1-format">(PDF)</span>. <i>Theory of Probability and Its Applications</i>. <b>16</b> (2): <span class="nowrap">264–</span>280. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1116025">10.1137/1116025</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theory+of+Probability+and+Its+Applications&rft.atitle=On+the+uniform+convergence+of+relative+frequencies+of+events+to+their+probabilities&rft.volume=16&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E264-%3C%2Fspan%3E280&rft.date=1971&rft_id=info%3Adoi%2F10.1137%2F1116025&rft.aulast=Vapnik&rft.aufirst=V.&rft.au=Chervonenkis%2C+A.&rft_id=https%3A%2F%2Fcourses.engr.illinois.edu%2Fece544na%2Ffa2014%2Fvapnik71.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+learning+theory" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSolomonoff1964" class="citation journal cs1">Solomonoff, Ray (March 1964). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0019-9958%2864%2990223-2">"A Formal Theory of Inductive Inference Part 1"</a>. <i>Information and Control</i>. <b>7</b> (1): <span class="nowrap">1–</span>22. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0019-9958%2864%2990223-2">10.1016/S0019-9958(64)90223-2</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Information+and+Control&rft.atitle=A+Formal+Theory+of+Inductive+Inference+Part+1&rft.volume=7&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E22&rft.date=1964-03&rft_id=info%3Adoi%2F10.1016%2FS0019-9958%2864%2990223-2&rft.aulast=Solomonoff&rft.aufirst=Ray&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252FS0019-9958%252864%252990223-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+learning+theory" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSolomonoff1964" class="citation journal cs1">Solomonoff, Ray (1964). "A Formal Theory of Inductive Inference Part 2". <i>Information and Control</i>. <b>7</b> (2): <span class="nowrap">224–</span>254. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0019-9958%2864%2990131-7">10.1016/S0019-9958(64)90131-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Information+and+Control&rft.atitle=A+Formal+Theory+of+Inductive+Inference+Part+2&rft.volume=7&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E224-%3C%2Fspan%3E254&rft.date=1964&rft_id=info%3Adoi%2F10.1016%2FS0019-9958%2864%2990131-7&rft.aulast=Solomonoff&rft.aufirst=Ray&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+learning+theory" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGold1967" class="citation journal cs1">Gold, E. Mark (1967). <a rel="nofollow" class="external text" href="http://web.mit.edu/~6.863/www/spring2009/readings/gold67limit.pdf">"Language identification in the limit"</a> <span class="cs1-format">(PDF)</span>. <i>Information and Control</i>. <b>10</b> (5): <span class="nowrap">447–</span>474. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0019-9958%2867%2991165-5">10.1016/S0019-9958(67)91165-5</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Information+and+Control&rft.atitle=Language+identification+in+the+limit&rft.volume=10&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E447-%3C%2Fspan%3E474&rft.date=1967&rft_id=info%3Adoi%2F10.1016%2FS0019-9958%2867%2991165-5&rft.aulast=Gold&rft.aufirst=E.+Mark&rft_id=http%3A%2F%2Fweb.mit.edu%2F~6.863%2Fwww%2Fspring2009%2Freadings%2Fgold67limit.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+learning+theory" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_learning_theory&action=edit&section=4" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A description of some of these publications is given at important publications in machine learning. </p> <div class="mw-heading mw-heading3"><h3 id="Surveys">Surveys</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_learning_theory&action=edit&section=5" title="Edit section: Surveys"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 rel="nofollow" class="external free" 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 rel="nofollow" class="external free" href="http://citeseer.ist.psu.edu/haussler90probably.html">http://citeseer.ist.psu.edu/haussler90probably.html</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Feature_selection">Feature selection</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_learning_theory&action=edit&section=6" title="Edit section: Feature selection"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 rel="nofollow" class="external free" href="http://citeseer.ist.psu.edu/dhagat94pac.html">http://citeseer.ist.psu.edu/dhagat94pac.html</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Optimal_O_notation_learning">Optimal O notation learning</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_learning_theory&action=edit&section=7" title="Edit section: Optimal O notation learning"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Oded_Goldreich" title="Oded Goldreich">Oded Goldreich</a>, <a href="/wiki/Dana_Ron" title="Dana Ron">Dana Ron</a>. <i><a rel="nofollow" class="external text" href="http://www.wisdom.weizmann.ac.il/~oded/PS/ul.ps">On universal learning algorithms</a></i>. <a rel="nofollow" class="external free" 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> <div class="mw-heading mw-heading3"><h3 id="Negative_results">Negative results</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_learning_theory&action=edit&section=8" title="Edit section: Negative results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>M. Kearns and <a href="/wiki/Leslie_Valiant" title="Leslie Valiant">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 rel="nofollow" class="external free" href="http://citeseer.ist.psu.edu/kearns89cryptographic.html">http://citeseer.ist.psu.edu/kearns89cryptographic.html</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged August 2024">dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Error_tolerance">Error tolerance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_learning_theory&action=edit&section=9" title="Edit section: Error tolerance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 rel="nofollow" class="external free" 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 rel="nofollow" class="external free" href="http://citeseer.ist.psu.edu/kearns93efficient.html">http://citeseer.ist.psu.edu/kearns93efficient.html</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Equivalence">Equivalence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_learning_theory&action=edit&section=10" title="Edit section: Equivalence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>D.Haussler, M.Kearns, N.Littlestone and <a href="/wiki/Manfred_K._Warmuth" title="Manfred K. Warmuth">M. Warmuth</a>, Equivalence of models for polynomial learnability, Proc. 1st ACM Workshop on Computational Learning Theory, (1988) 42-55.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPittWarmuth1990" class="citation journal cs1">Pitt, L.; Warmuth, M. K. (1990). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-0000%2890%2990028-J">"Prediction-Preserving Reducibility"</a>. <i>Journal of Computer and System Sciences</i>. <b>41</b> (3): <span class="nowrap">430–</span>467. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-0000%2890%2990028-J">10.1016/0022-0000(90)90028-J</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computer+and+System+Sciences&rft.atitle=Prediction-Preserving+Reducibility&rft.volume=41&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E430-%3C%2Fspan%3E467&rft.date=1990&rft_id=info%3Adoi%2F10.1016%2F0022-0000%2890%2990028-J&rft.aulast=Pitt&rft.aufirst=L.&rft.au=Warmuth%2C+M.+K.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0022-0000%252890%252990028-J&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+learning+theory" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_learning_theory&action=edit&section=11" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" 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