Differentially Private Stochastic Convex Optimization under a Quantile Loss Function

Du Chen, Geoffrey A. Chua
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:4435-4461, 2023.

Abstract

We study $(\varepsilon,\delta)$-differentially private (DP) stochastic convex optimization under an $r$-th quantile loss function taking the form $c(u) = ru^+ + (1-r)(-u)^+$. The function is non-smooth, and we propose to approximate it with a smooth function obtained by convolution smoothing, which enjoys both structure and bandwidth flexibility and can address outliers. This leads to a better approximation than those obtained from existing methods such as Moreau Envelope. We then design private algorithms based on DP stochastic gradient descent and objective perturbation, and show that both algorithms achieve (near) optimal excess generalization risk $O(\max\{\frac{1}{\sqrt{n}}, \frac{\sqrt{d\ln(1/\delta)}}{n\varepsilon}\})$. Through objective perturbation, we further derive an upper bound $O(\max\{\sqrt{\frac{d}{n}}, \sqrt{\frac{d\ln(1/\delta)}{n\varepsilon}}\})$ on the parameter estimation error under mild assumptions on data generating processes. Some applications in private quantile regression and private inventory control will be discussed.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-chen23d, title = {Differentially Private Stochastic Convex Optimization under a Quantile Loss Function}, author = {Chen, Du and Chua, Geoffrey A.}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {4435--4461}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/chen23d/chen23d.pdf}, url = {https://proceedings.mlr.press/v202/chen23d.html}, abstract = {We study $(\varepsilon,\delta)$-differentially private (DP) stochastic convex optimization under an $r$-th quantile loss function taking the form $c(u) = ru^+ + (1-r)(-u)^+$. The function is non-smooth, and we propose to approximate it with a smooth function obtained by convolution smoothing, which enjoys both structure and bandwidth flexibility and can address outliers. This leads to a better approximation than those obtained from existing methods such as Moreau Envelope. We then design private algorithms based on DP stochastic gradient descent and objective perturbation, and show that both algorithms achieve (near) optimal excess generalization risk $O(\max\{\frac{1}{\sqrt{n}}, \frac{\sqrt{d\ln(1/\delta)}}{n\varepsilon}\})$. Through objective perturbation, we further derive an upper bound $O(\max\{\sqrt{\frac{d}{n}}, \sqrt{\frac{d\ln(1/\delta)}{n\varepsilon}}\})$ on the parameter estimation error under mild assumptions on data generating processes. Some applications in private quantile regression and private inventory control will be discussed.} }
Endnote
%0 Conference Paper %T Differentially Private Stochastic Convex Optimization under a Quantile Loss Function %A Du Chen %A Geoffrey A. Chua %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-chen23d %I PMLR %P 4435--4461 %U https://proceedings.mlr.press/v202/chen23d.html %V 202 %X We study $(\varepsilon,\delta)$-differentially private (DP) stochastic convex optimization under an $r$-th quantile loss function taking the form $c(u) = ru^+ + (1-r)(-u)^+$. The function is non-smooth, and we propose to approximate it with a smooth function obtained by convolution smoothing, which enjoys both structure and bandwidth flexibility and can address outliers. This leads to a better approximation than those obtained from existing methods such as Moreau Envelope. We then design private algorithms based on DP stochastic gradient descent and objective perturbation, and show that both algorithms achieve (near) optimal excess generalization risk $O(\max\{\frac{1}{\sqrt{n}}, \frac{\sqrt{d\ln(1/\delta)}}{n\varepsilon}\})$. Through objective perturbation, we further derive an upper bound $O(\max\{\sqrt{\frac{d}{n}}, \sqrt{\frac{d\ln(1/\delta)}{n\varepsilon}}\})$ on the parameter estimation error under mild assumptions on data generating processes. Some applications in private quantile regression and private inventory control will be discussed.
APA
Chen, D. & Chua, G.A.. (2023). Differentially Private Stochastic Convex Optimization under a Quantile Loss Function. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:4435-4461 Available from https://proceedings.mlr.press/v202/chen23d.html.

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