Estimating Smooth GLM in Non-interactive Local Differential Privacy Model with Public Unlabeled Data

Di Wang, Huangyu Zhang, Marco Gaboardi, Jinhui Xu
Proceedings of the 32nd International Conference on Algorithmic Learning Theory, PMLR 132:1207-1213, 2021.

Abstract

In this paper, we study the problem of estimating smooth Generalized Linear Models (GLM) in the Non-interactive Local Differential Privacy (NLDP) model. Different from its classical setting, our model allows the server to access some additional public but unlabeled data. By using Stein’s lemma and its variants, we first show that there is an $(\epsilon, \delta)$-NLDP algorithm for GLM (under some mild assumptions), if each data record is i.i.d sampled from some sub-Gaussian distribution with bounded $\ell_1$-norm. Then with high probability, the sample complexity of the public and private data, for the algorithm to achieve an $\alpha$ estimation error (in $\ell_\infty$-norm), is $O(p^2\alpha^{-2})$ and ${O}(p^2\alpha^{-2}\epsilon^{-2})$, respectively, if $\alpha$ is not too small ({\em i.e.,} $\alpha\geq \Omega(\frac{1}{\sqrt{p}})$), where $p$ is the dimensionality of the data. This is a significant improvement over the previously known exponential or quasi-polynomial in $\alpha^{-1}$, or exponential in $p$ sample complexity of GLM with no public data. We then extend our idea to the non-linear regression problem and show a similar phenomenon for it. Finally, we demonstrate the effectiveness of our algorithms through experiments on both synthetic and real world datasets. To our best knowledge, this is the first paper showing the existence of efficient and effective algorithms for GLM and non-linear regression in the NLDP model with public unlabeled data.

Cite this Paper

BibTeX
@InProceedings{pmlr-v132-wang21a, title = {Estimating Smooth {GLM} in Non-interactive Local Differential Privacy Model with Public Unlabeled Data}, author = {Wang, Di and Zhang, Huangyu and Gaboardi, Marco and Xu, Jinhui}, booktitle = {Proceedings of the 32nd International Conference on Algorithmic Learning Theory}, pages = {1207--1213}, year = {2021}, editor = {Feldman, Vitaly and Ligett, Katrina and Sabato, Sivan}, volume = {132}, series = {Proceedings of Machine Learning Research}, month = {16--19 Mar}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v132/wang21a/wang21a.pdf}, url = {https://proceedings.mlr.press/v132/wang21a.html}, abstract = {In this paper, we study the problem of estimating smooth Generalized Linear Models (GLM) in the Non-interactive Local Differential Privacy (NLDP) model. Different from its classical setting, our model allows the server to access some additional public but unlabeled data. By using Stein’s lemma and its variants, we first show that there is an $(\epsilon, \delta)$-NLDP algorithm for GLM (under some mild assumptions), if each data record is i.i.d sampled from some sub-Gaussian distribution with bounded $\ell_1$-norm. Then with high probability, the sample complexity of the public and private data, for the algorithm to achieve an $\alpha$ estimation error (in $\ell_\infty$-norm), is $O(p^2\alpha^{-2})$ and ${O}(p^2\alpha^{-2}\epsilon^{-2})$, respectively, if $\alpha$ is not too small ({\em i.e.,} $\alpha\geq \Omega(\frac{1}{\sqrt{p}})$), where $p$ is the dimensionality of the data. This is a significant improvement over the previously known exponential or quasi-polynomial in $\alpha^{-1}$, or exponential in $p$ sample complexity of GLM with no public data. We then extend our idea to the non-linear regression problem and show a similar phenomenon for it. Finally, we demonstrate the effectiveness of our algorithms through experiments on both synthetic and real world datasets. To our best knowledge, this is the first paper showing the existence of efficient and effective algorithms for GLM and non-linear regression in the NLDP model with public unlabeled data. } }
Endnote
%0 Conference Paper %T Estimating Smooth GLM in Non-interactive Local Differential Privacy Model with Public Unlabeled Data %A Di Wang %A Huangyu Zhang %A Marco Gaboardi %A Jinhui Xu %B Proceedings of the 32nd International Conference on Algorithmic Learning Theory %C Proceedings of Machine Learning Research %D 2021 %E Vitaly Feldman %E Katrina Ligett %E Sivan Sabato %F pmlr-v132-wang21a %I PMLR %P 1207--1213 %U https://proceedings.mlr.press/v132/wang21a.html %V 132 %X In this paper, we study the problem of estimating smooth Generalized Linear Models (GLM) in the Non-interactive Local Differential Privacy (NLDP) model. Different from its classical setting, our model allows the server to access some additional public but unlabeled data. By using Stein’s lemma and its variants, we first show that there is an $(\epsilon, \delta)$-NLDP algorithm for GLM (under some mild assumptions), if each data record is i.i.d sampled from some sub-Gaussian distribution with bounded $\ell_1$-norm. Then with high probability, the sample complexity of the public and private data, for the algorithm to achieve an $\alpha$ estimation error (in $\ell_\infty$-norm), is $O(p^2\alpha^{-2})$ and ${O}(p^2\alpha^{-2}\epsilon^{-2})$, respectively, if $\alpha$ is not too small ({\em i.e.,} $\alpha\geq \Omega(\frac{1}{\sqrt{p}})$), where $p$ is the dimensionality of the data. This is a significant improvement over the previously known exponential or quasi-polynomial in $\alpha^{-1}$, or exponential in $p$ sample complexity of GLM with no public data. We then extend our idea to the non-linear regression problem and show a similar phenomenon for it. Finally, we demonstrate the effectiveness of our algorithms through experiments on both synthetic and real world datasets. To our best knowledge, this is the first paper showing the existence of efficient and effective algorithms for GLM and non-linear regression in the NLDP model with public unlabeled data.
APA
Wang, D., Zhang, H., Gaboardi, M. & Xu, J.. (2021). Estimating Smooth GLM in Non-interactive Local Differential Privacy Model with Public Unlabeled Data. Proceedings of the 32nd International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 132:1207-1213 Available from https://proceedings.mlr.press/v132/wang21a.html.

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