Document Zbl 1539.62116

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Stat.</a> <a href="/?q=in%3A499793" title="Articles in this Issue">51, No. 2, 738-761 (2023)</a>. </div> <div class="abstract">Summary: We study the covariate shift problem in the context of nonparametric regression over a reproducing kernel Hilbert space (RKHS). We focus on two natural families of covariate shift problems defined using the likelihood ratios between the source and target distributions. When the likelihood ratios are uniformly bounded, we prove that the kernel ridge regression (KRR) estimator with a carefully chosen regularization parameter is minimax rate-optimal (up to a log factor) for a large family of RKHSs with regular kernel eigenvalues. Interestingly, KRR does not require full knowledge of the likelihood ratio apart from an upper bound on it. In striking contrast to the standard statistical setting without covariate shift, we also demonstrate that a naïve estimator, which minimizes the empirical risk over the function class, is strictly suboptimal under covariate shift as compared to KRR. We then address the larger class of covariate shift problems where likelihood ratio is possibly unbounded yet has a finite second moment. Here, we propose a reweighted KRR estimator that weights samples based on a careful truncation of the likelihood ratios. Again, we are able to show that this estimator is minimax optimal, up to logarithmic factors.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7714179">Cited in <strong>4</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A62G08" title="MSC2020">62G08</a> </td> <td class="space"> Nonparametric regression and quantile regression </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62C20" title="MSC2020">62C20</a> </td> <td class="space"> Minimax procedures in statistical decision theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62H30" title="MSC2020">62H30</a> </td> <td class="space"> Classification and discrimination; cluster analysis (statistical aspects) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62J07" title="MSC2020">62J07</a> </td> <td class="space"> Ridge regression; shrinkage estimators (Lasso) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68T05" title="MSC2020">68T05</a> </td> <td class="space"> Learning and adaptive systems in artificial intelligence </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acovariate+shift">covariate shift</a>; <a href="/?q=ut%3Akernel+ridge+regression">kernel ridge regression</a>; <a href="/?q=ut%3Anonparametric+regression">nonparametric regression</a>; <a href="/?q=ut%3Areproducing+kernel+Hilbert+spaces">reproducing kernel Hilbert spaces</a>; <a href="/?q=ut%3Atransfer+learning">transfer learning</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1539.62116" data-ciurl="/ci/07714179" data-biburl="/bibtex/07714179.bib" data-amsurl="/amsrefs/07714179.bib" data-xmlurl="/xml/07714179.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07714179.pdf" title="Zbl 1539.62116 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1214/23-AOS2268" aria-label="DOI for “Optimally tackling covariate shift in RKHS-based nonparametric regression”" title="10.1214/23-AOS2268">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2205.02986"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Cai, T. 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