Nearly Optimal Differentially Private ReLU Regression

Meng Ding, Mingxi Lei, Shaowei Wang, Tianhang Zheng, Di Wang, Jinhui Xu
Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence, PMLR 286:1003-1038, 2025.

Abstract

In this paper, we investigate one of the most fundamental non-convex learning problems-ReLU regression-in the Differential Privacy (DP) model. Previous studies on private ReLU regression heavily rely on stringent assumptions, such as constant-bounded norms for feature vectors and labels. We relax these assumptions to a more standard setting, where data can be i.i.d. sampled from $O(1)$-sub-Gaussian distributions. We first show that when $\varepsilon = \tilde{O}(\sqrt{\frac{1}{N}})$ and there is some public data, it is possible to achieve an upper bound of $\Tilde{O}(\frac{d^2}{N^2 \varepsilon^2})$ for the excess population risk in $(\epsilon, \delta)$-DP, where $d$ is the dimension and $N$ is the number of data samples. Moreover, we relax the requirement of $\epsilon$ and public data by proposing and analyzing a one-pass mini-batch Generalized Linear Model Perceptron algorithm (DP-MBGLMtron). Additionally, using the tracing attack argument technique, we demonstrate that the minimax rate of the estimation error for $(\varepsilon, \delta)$-DP algorithms is lower bounded by $\Omega(\frac{d^2}{N^2 \varepsilon^2})$. This shows that DP-MBGLMtron achieves the optimal utility bound up to logarithmic factors. Experiments further support our theoretical results.

Cite this Paper

BibTeX
@InProceedings{pmlr-v286-ding25a, title = {Nearly Optimal Differentially Private ReLU Regression}, author = {Ding, Meng and Lei, Mingxi and Wang, Shaowei and Zheng, Tianhang and Wang, Di and Xu, Jinhui}, booktitle = {Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence}, pages = {1003--1038}, year = {2025}, editor = {Chiappa, Silvia and Magliacane, Sara}, volume = {286}, series = {Proceedings of Machine Learning Research}, month = {21--25 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v286/main/assets/ding25a/ding25a.pdf}, url = {https://proceedings.mlr.press/v286/ding25a.html}, abstract = {In this paper, we investigate one of the most fundamental non-convex learning problems-ReLU regression-in the Differential Privacy (DP) model. Previous studies on private ReLU regression heavily rely on stringent assumptions, such as constant-bounded norms for feature vectors and labels. We relax these assumptions to a more standard setting, where data can be i.i.d. sampled from $O(1)$-sub-Gaussian distributions. We first show that when $\varepsilon = \tilde{O}(\sqrt{\frac{1}{N}})$ and there is some public data, it is possible to achieve an upper bound of $\Tilde{O}(\frac{d^2}{N^2 \varepsilon^2})$ for the excess population risk in $(\epsilon, \delta)$-DP, where $d$ is the dimension and $N$ is the number of data samples. Moreover, we relax the requirement of $\epsilon$ and public data by proposing and analyzing a one-pass mini-batch Generalized Linear Model Perceptron algorithm (DP-MBGLMtron). Additionally, using the tracing attack argument technique, we demonstrate that the minimax rate of the estimation error for $(\varepsilon, \delta)$-DP algorithms is lower bounded by $\Omega(\frac{d^2}{N^2 \varepsilon^2})$. This shows that DP-MBGLMtron achieves the optimal utility bound up to logarithmic factors. Experiments further support our theoretical results.} }
Endnote
%0 Conference Paper %T Nearly Optimal Differentially Private ReLU Regression %A Meng Ding %A Mingxi Lei %A Shaowei Wang %A Tianhang Zheng %A Di Wang %A Jinhui Xu %B Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2025 %E Silvia Chiappa %E Sara Magliacane %F pmlr-v286-ding25a %I PMLR %P 1003--1038 %U https://proceedings.mlr.press/v286/ding25a.html %V 286 %X In this paper, we investigate one of the most fundamental non-convex learning problems-ReLU regression-in the Differential Privacy (DP) model. Previous studies on private ReLU regression heavily rely on stringent assumptions, such as constant-bounded norms for feature vectors and labels. We relax these assumptions to a more standard setting, where data can be i.i.d. sampled from $O(1)$-sub-Gaussian distributions. We first show that when $\varepsilon = \tilde{O}(\sqrt{\frac{1}{N}})$ and there is some public data, it is possible to achieve an upper bound of $\Tilde{O}(\frac{d^2}{N^2 \varepsilon^2})$ for the excess population risk in $(\epsilon, \delta)$-DP, where $d$ is the dimension and $N$ is the number of data samples. Moreover, we relax the requirement of $\epsilon$ and public data by proposing and analyzing a one-pass mini-batch Generalized Linear Model Perceptron algorithm (DP-MBGLMtron). Additionally, using the tracing attack argument technique, we demonstrate that the minimax rate of the estimation error for $(\varepsilon, \delta)$-DP algorithms is lower bounded by $\Omega(\frac{d^2}{N^2 \varepsilon^2})$. This shows that DP-MBGLMtron achieves the optimal utility bound up to logarithmic factors. Experiments further support our theoretical results.
APA
Ding, M., Lei, M., Wang, S., Zheng, T., Wang, D. & Xu, J.. (2025). Nearly Optimal Differentially Private ReLU Regression. Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 286:1003-1038 Available from https://proceedings.mlr.press/v286/ding25a.html.

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