Document Zbl 07983606

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Comput. Sci.</a> <a href="/?q=in%3A523765" title="Articles in this Issue">1030, Article ID 115071, 13 p. (2025)</a>. </div> <div class="abstract">Summary: We study the problem of Differentially Private Stochastic Convex Optimization (DPSCO) with heavy-tailed data. Specifically, we focus on the problem of Least Absolute Deviations, i.e., $ \ell_1$-norm linear regression, in the $\varepsilon $-DP model. While most previous work focuses on the case where the loss function is Lipschitz, in this paper we only need to assume the variates have bounded moments. Firstly, we study the case where the $\ell_2$ norm of data has a bounded second-order moment. We propose an algorithm that is based on the exponential mechanism and show that it is possible to achieve an upper bound of $\widetilde{O}(\sqrt{ \frac{ d}{ n \epsilon}})$ (with high probability). Next, we relax the assumption to bounded $\theta $-th order moment with some $\theta \in(1, 2)$ and show that it is possible to achieve an upper bound of $\widetilde{O}( ( \frac{ d}{ n \epsilon} )^{\frac{ \theta - 1}{ \theta}})$. Our algorithms can also be extended to more relaxed cases where only each coordinate of the data has bounded moments, and we can get an upper bound of $\widetilde{O}(\frac{ d}{ \sqrt{ n \epsilon}})$ and $\widetilde{O}(\frac{ d}{ ( n \epsilon )^{\frac{ \theta - 1}{ \theta}}})$ in the second and $\theta $-th moment case respectively.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A68P27" title="MSC2020">68P27</a> </td> <td class="space"> Privacy of data </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62J05" title="MSC2020">62J05</a> </td> <td class="space"> Linear regression; mixed models </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C15" title="MSC2020">90C15</a> </td> <td class="space"> Stochastic programming </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C25" title="MSC2020">90C25</a> </td> <td class="space"> Convex programming </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C90" title="MSC2020">90C90</a> </td> <td class="space"> Applications of mathematical programming </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Adifferential+privacy">differential privacy</a>; <a href="/?q=ut%3Arobust+estimation">robust estimation</a>; <a href="/?q=ut%3Aregression">regression</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 07983606" data-ciurl="/ci/07983606" data-biburl="/bibtex/07983606.bib" data-amsurl="/amsrefs/07983606.bib" data-xmlurl="/xml/07983606.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07983606.pdf" title="Zbl 07983606 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/10.1016/j.tcs.2025.115071" aria-label="DOI for “Private least absolute deviations with heavy-tailed data”" title="10.1016/j.tcs.2025.115071">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Wang, D.; Xu, J., Differentially private $\ell_1$-norm linear regression with heavy-tailed data, (IEEE International Symposium on Information Theory. 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