A Family of Distributions on Learnable Problems without Uniform Convergence

<!DOCTYPE html> <html lang="en" dir="ltr"> <head>  <script async src="https://www.googletagmanager.com/gtag/js?id=G-P63WKM1TM1"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-P63WKM1TM1'); </script>  <script type="text/javascript" > (function(m,e,t,r,i,k,a){m[i]=m[i]||function(){(m[i].a=m[i].a||[]).push(arguments)}; m[i].l=1*new Date(); for (var j = 0; j < document.scripts.length; j++) {if (document.scripts[j].src === r) { return; }} k=e.createElement(t),a=e.getElementsByTagName(t)[0],k.async=1,k.src=r,a.parentNode.insertBefore(k,a)}) (window, document, "script", "https://mc.yandex.ru/metrika/tag.js", "ym"); ym(55165297, "init", { clickmap:false, trackLinks:true, accurateTrackBounce:true, webvisor:false }); </script> <noscript><div><img src="https://mc.yandex.ru/watch/55165297" style="position:absolute; left:-9999px;" alt="" /></div></noscript>    <title>A Family of Distributions on Learnable Problems without Uniform Convergence</title> <meta name="description" content="A Family of Distributions on Learnable Problems without Uniform Convergence"> <meta name="keywords" content="Statistical learning theory, learnability, uniform convergence, stability, regularized loss minimization."> <meta name="viewport" content="width=device-width, initial-scale=1, minimum-scale=1, maximum-scale=1, user-scalable=no"> <meta charset="utf-8"> <meta name="citation_title" content="A Family of Distributions on Learnable Problems without Uniform Convergence"> <meta name="citation_author" content="César Garza"> <meta name="citation_publication_date" content="2023/03/06"> <meta name="citation_journal_title" content="International Journal of Cognitive and Language Sciences"> <meta name="citation_volume" content="17"> <meta name="citation_issue" content="3"> <meta name="citation_firstpage" content="186"> <meta name="citation_lastpage" content="192"> <meta name="citation_pdf_url" content="https://publications.waset.org/10012987/pdf"> <link href="https://cdn.waset.org/favicon.ico" type="image/x-icon" rel="shortcut icon"> <link href="https://cdn.waset.org/static/plugins/bootstrap-4.2.1/css/bootstrap.min.css" rel="stylesheet"> <link href="https://cdn.waset.org/static/plugins/fontawesome/css/all.min.css" rel="stylesheet"> <link href="https://cdn.waset.org/static/css/site.css?v=150220211555" rel="stylesheet"> </head> <body> <header> <div class="container"> <nav class="navbar navbar-expand-lg navbar-light"> <a class="navbar-brand" href="https://waset.org"> <img src="https://cdn.waset.org/static/images/wasetc.png" alt="Open Science Research Excellence" title="Open Science Research Excellence" /> </a> <button class="d-block d-lg-none navbar-toggler ml-auto" type="button" data-toggle="collapse" data-target="#navbarMenu" aria-controls="navbarMenu" aria-expanded="false" aria-label="Toggle navigation"> <span class="navbar-toggler-icon"></span> </button> <div class="w-100"> <div class="d-none d-lg-flex flex-row-reverse"> <form method="get" action="https://waset.org/search" class="form-inline my-2 my-lg-0"> <input class="form-control mr-sm-2" type="search" placeholder="Search Conferences" value="" name="q" aria-label="Search"> <button class="btn btn-light my-2 my-sm-0" type="submit"><i class="fas fa-search"></i></button> </form> </div> <div class="collapse navbar-collapse mt-1" id="navbarMenu"> <ul class="navbar-nav ml-auto align-items-center" id="mainNavMenu"> <li class="nav-item"> <a class="nav-link" href="https://waset.org/conferences" title="Conferences in 2024/2025/2026">Conferences</a> </li> <li class="nav-item"> <a class="nav-link" href="https://waset.org/disciplines" title="Disciplines">Disciplines</a> </li> <li class="nav-item"> <a class="nav-link" href="https://waset.org/committees" rel="nofollow">Committees</a> </li> <li class="nav-item dropdown"> <a class="nav-link dropdown-toggle" href="#" id="navbarDropdownPublications" role="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false"> Publications </a> <div class="dropdown-menu" aria-labelledby="navbarDropdownPublications"> <a class="dropdown-item" href="https://publications.waset.org/abstracts">Abstracts</a> <a class="dropdown-item" href="https://publications.waset.org">Periodicals</a> <a class="dropdown-item" href="https://publications.waset.org/archive">Archive</a> </div> </li> <li class="nav-item"> <a class="nav-link" href="https://waset.org/page/support" title="Support">Support</a> </li> </ul> </div> </div> </nav> </div> </header> <main> <div class="container mt-4"> <div class="row"> <div class="col-md-9 mx-auto"> <form method="get" action="https://publications.waset.org/search"> <div id="custom-search-input"> <div class="input-group"> <i class="fas fa-search"></i> <input type="text" class="search-query" name="q" placeholder="Author, Title, Abstract, Keywords" value=""> <input type="submit" class="btn_search" value="Search"> </div> </div> </form> </div> </div> <div class="row mt-3"> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Commenced</strong> in January 2007</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Frequency:</strong> Monthly</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Edition:</strong> International</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Paper Count:</strong> 33093</div> </div> </div> </div> <div class="card publication-listing mt-3 mb-3"> <h5 class="card-header" style="font-size:.9rem">A Family of Distributions on Learnable Problems without Uniform Convergence</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/search?q=C%C3%A9sar%20Garza">César Garza</a> </p> <p class="card-text"><strong>Abstract:</strong></p> <p>In supervised binary classification and regression problems, it is well-known that learnability is equivalent to uniform convergence of the hypothesis class, and if a problem is learnable, it is learnable by empirical risk minimization. For the general learning setting of unsupervised learning tasks, there are non-trivial learning problems where uniform convergence does not hold. We present here the task of learning centers of mass with an extra feature that “activates” some of the coordinates over the unit ball in a Hilbert space. We show that the learning problem is learnable under a stable RLM rule. We introduce a family of distributions over the domain space with some mild restrictions for which the sample complexity of uniform convergence for these problems must grow logarithmically with the dimension of the Hilbert space. If we take this dimension to infinity, we obtain a learnable problem for which the uniform convergence property fails for a vast family of distributions.</p> <iframe src="https://publications.waset.org/10012987.pdf" style="width:100%; height:400px;" frameborder="0"></iframe> <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/search?q=Statistical%20learning%20theory" title="Statistical learning theory">Statistical learning theory</a>, <a href="https://publications.waset.org/search?q=learnability" title=" learnability"> learnability</a>, <a href="https://publications.waset.org/search?q=uniform%0D%0Aconvergence" title=" uniform convergence"> uniform convergence</a>, <a href="https://publications.waset.org/search?q=stability" title=" stability"> stability</a>, <a href="https://publications.waset.org/search?q=regularized%20loss%20minimization." title=" regularized loss minimization."> regularized loss minimization.</a> </p> <a href="https://publications.waset.org/10012987/a-family-of-distributions-on-learnable-problems-without-uniform-convergence" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/10012987/apa" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">APA</a> <a href="https://publications.waset.org/10012987/bibtex" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">BibTeX</a> <a href="https://publications.waset.org/10012987/chicago" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">Chicago</a> <a href="https://publications.waset.org/10012987/endnote" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">EndNote</a> <a href="https://publications.waset.org/10012987/harvard" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">Harvard</a> <a href="https://publications.waset.org/10012987/json" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">JSON</a> <a href="https://publications.waset.org/10012987/mla" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">MLA</a> <a href="https://publications.waset.org/10012987/ris" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">RIS</a> <a href="https://publications.waset.org/10012987/xml" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">XML</a> <a href="https://publications.waset.org/10012987/iso690" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">ISO 690</a> <a href="https://publications.waset.org/10012987.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">355</span> </span> <p class="card-text"><strong>References:</strong></p> <br>[1] V. N. Vapnik and A. Y. Chervonenkis, “On the uniform convergence of relative frequencies of events to their probabilities,” in Measures of complexity, pp. 11–30, Springer, Cham, 2015. Reprint of Theor. Probability Appl. 16 (1971), 264–280. <br>[2] M. J. Kearns, R. E. Schapire, and L. M. Sellie, “Toward efficient agnostic learning,” Machine Learning, vol. 17, pp. 115–141, 1994. <br>[3] B. K. Natarajan, “On learning sets and functions,” Machine Learning, vol. 4, pp. 67–97, 2004. <br>[4] S. Shalev-Shwartz, O. Shamir, N. Srebro, and K. Sridharan, “Learnability, stability and uniform convergence,” J. Mach. Learn. Res., vol. 11, pp. 2635–2670, 2010. <br>[5] S. Shalev-Shwartz and S. Ben-David, Understanding Machine Learning - From Theory to Algorithms. Cambridge University Press, 2014. <br>[6] N. Alon, S. Ben-David, N. Cesa-Bianchi, and D. Haussler, “Scale-sensitive dimensions, uniform convergence, and learnability,” J. ACM, vol. 44, no. 4, pp. 615–631, 1997. </div> </div> </div> </main> <footer> <div id="infolinks" class="pt-3 pb-2"> <div class="container"> <div style="background-color:#f5f5f5;" class="p-3"> <div class="row"> <div class="col-md-2"> <ul class="list-unstyled"> About <li><a href="https://waset.org/page/support">About Us</a></li> <li><a href="https://waset.org/page/support#legal-information">Legal</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/WASET-16th-foundational-anniversary.pdf">WASET celebrates its 16th foundational anniversary</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Account <li><a href="https://waset.org/profile">My Account</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Explore <li><a href="https://waset.org/disciplines">Disciplines</a></li> <li><a href="https://waset.org/conferences">Conferences</a></li> <li><a href="https://waset.org/conference-programs">Conference Program</a></li> <li><a href="https://waset.org/committees">Committees</a></li> <li><a href="https://publications.waset.org">Publications</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Research <li><a href="https://publications.waset.org/abstracts">Abstracts</a></li> <li><a href="https://publications.waset.org">Periodicals</a></li> <li><a href="https://publications.waset.org/archive">Archive</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Open Science <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Open-Science-Philosophy.pdf">Open Science Philosophy</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Open-Science-Award.pdf">Open Science Award</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Open-Society-Open-Science-and-Open-Innovation.pdf">Open Innovation</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Postdoctoral-Fellowship-Award.pdf">Postdoctoral Fellowship Award</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Scholarly-Research-Review.pdf">Scholarly Research Review</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Support <li><a href="https://waset.org/page/support">Support</a></li> <li><a href="https://waset.org/profile/messages/create">Contact Us</a></li> <li><a href="https://waset.org/profile/messages/create">Report Abuse</a></li> </ul> </div> </div> </div> </div> </div> <div class="container text-center"> <hr style="margin-top:0;margin-bottom:.3rem;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" class="text-muted small">Creative Commons Attribution 4.0 International License</a> <div id="copy" class="mt-2">© 2024 World Academy of Science, Engineering and Technology</div> </div> </footer> <a href="javascript:" id="return-to-top"><i class="fas fa-arrow-up"></i></a> <div class="modal" id="modal-template"> <div class="modal-dialog"> <div class="modal-content"> <div class="row m-0 mt-1"> <div class="col-md-12"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> </div> </div> <div class="modal-body"></div> </div> </div> </div> <script src="https://cdn.waset.org/static/plugins/jquery-3.3.1.min.js"></script> <script src="https://cdn.waset.org/static/plugins/bootstrap-4.2.1/js/bootstrap.bundle.min.js"></script> <script src="https://cdn.waset.org/static/js/site.js?v=150220211556"></script> <script> jQuery(document).ready(function() { /*jQuery.get("https://publications.waset.org/xhr/user-menu", function (response) { jQuery('#mainNavMenu').append(response); });*/ jQuery.get({ url: "https://publications.waset.org/xhr/user-menu", cache: false }).then(function(response){ jQuery('#mainNavMenu').append(response); }); }); </script> </body> </html>

CINXE.COM

A Family of Distributions on Learnable Problems without Uniform Convergence