Faster Learning under Relaxed Local Differential Privacy

2026-09-04

AI Summary

本文解决了在放松的本地差分隐私条件下进行密度估计的问题，通过向每个敏感观测值添加独立噪声，并使用去卷积估计器实现了更快的学习速率。

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 $(nα)^{-\frac{2r-1}{2r}}$ up to log factors which is faster than $(nα^2)^{-\frac{2r-1}{2r+1}}$ under the classical $α$-LDP and closer to the nonprivate minimax rate $n^{-\frac{2r-1}{2r}}$. 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.

Problem

• density estimation
• local differential privacy
• learning rate

Innovation

• relaxed local differential privacy
• symmetrized Gamma distribution
• deconvolution estimator
• Sobolev smooth functions
• adaptive procedure