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PreprintCC BY 4.0arXiv

Faster Learning under Relaxed Local Differential Privacy

  • Cristina Butucea
  • Huiyun Tang
  • Marie-Luce Taupin
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Abstract

We consider density estimation under the relaxed local differential privacy condition that the privatized distributions are α-close in total variation distance. We show that adding independent noise with a convenient symmetrized Gamma distribution to each sensitive observation attains the α-TV-LDP. We prove that the deconvolution estimator of r-Sobolev smooth functions attains the pointwise rate (nα)−(2r−1)/2r up to log factors which is faster than (nα2)−(2r−1)/(2r+1) under the classical α-LDP and closer to the nonprivate minimax rate n−(2r−1)/2r. Next, we use a Goldenshluger-Lepski procedure to build a free of the smoothness adaptive procedure and show optimality of our rates in the convolution model of our privatisation scheme. We illustrate the benefits of this simple privacy mechanism by implementing a neural network estimator which does not need to add more noise in the optimization steps. Numerical results show significant improvement of the estimation rate over the Laplace and the private-SGD mechanisms.

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Cite

BibTeXRIS
BibTeX
@misc{butucea2026faster,
  title = {{Faster Learning under Relaxed Local Differential Privacy}},
  author = {Cristina Butucea and Huiyun Tang and Marie-Luce Taupin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05034},
  eprint = {2609.05034},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05034},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Faster Learning under Relaxed Local Differential Privacy
AU  - Cristina Butucea
AU  - Huiyun Tang
AU  - Marie-Luce Taupin
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05034
UR  - https://arxiv.org/abs/2609.05034
AB  - We consider density estimation under the relaxed local differential privacy condition that the privatized distributions are α-close in total variation distance. We show that adding independent noise with a convenient symmetrized Gamma distribution to each sensitive observation attains the α-TV-LDP. We prove that the deconvolution estimator of r-Sobolev smooth functions attains the pointwise rate (nα)^−(2r−1)/2r up to log factors which is faster than (nα^2)^−(2r−1)/(2r+1) under the classical α-LDP and closer to the nonprivate minimax rate n^−(2r−1)/2r. Next, we use a Goldenshluger-Lepski procedure to build a free of the smoothness adaptive procedure and show optimality of our rates in the convolution model of our privatisation scheme. We illustrate the benefits of this simple privacy mechanism by implementing a neural network estimator which does not need to add more noise in the optimization steps. Numerical results show significant improvement of the estimation rate over the Laplace and the private-SGD mechanisms.
ER  -