[Paper Review] Locally Private Mean Estimation: Z-test and Tight Confidence Intervals

Marco Gaboardi, Ryan Rogers+1 more|arXiv (Cornell University)|Oct 18, 2018

Privacy-Preserving Technologies in Data18 citations

TL;DR

This paper presents optimal locally differentially private algorithms for mean estimation under Gaussian assumptions, achieving tight confidence intervals of length $\widetilde{O}\left(\frac{\sigma\sqrt{\log(1/\beta)}}{\epsilon\sqrt{n}}\right)$ using a combination of bit-flipping mechanisms and binary search for quantile estimation. It establishes matching upper and lower bounds (up to poly-log factors), proving the optimality of the interval length in the local model.

ABSTRACT

This work provides tight upper- and lower-bounds for the problem of mean estimation under $ε$-differential privacy in the local model, when the input is composed of $n$ i.i.d. drawn samples from a normal distribution with variance $σ$. Our algorithms result in a $(1-β)$-confidence interval for the underlying distribution's mean $μ$ of length $ ilde O\left( \frac{σ\sqrt{\log(\frac 1 β)}}{ε\sqrt n} ight)$. In addition, our algorithms leverage binary search using local differential privacy for quantile estimation, a result which may be of separate interest. Moreover, we prove a matching lower-bound (up to poly-log factors), showing that any one-shot (each individual is presented with a single query) local differentially private algorithm must return an interval of length $Ω\left( \frac{σ\sqrt{\log(1/β)}}{ε\sqrt{n}} ight)$.

Motivation & Objective

• To close the gap between upper and lower bounds for locally differentially private mean estimation under Gaussian data.
• To design efficient, one-shot LDP algorithms that produce confidence intervals of near-minimal length.
• To provide a private analog of the classical Z-test in the local model, enabling hypothesis testing under privacy constraints.
• To prove that the derived confidence interval length is optimal up to poly-logarithmic factors.
• To develop a novel application of local differential privacy for private quantile estimation, which may be of independent interest.

Proposed method

• Uses bit-flipping mechanisms to identify a narrow interval containing the true mean with high probability, leveraging the fact that the mean lies within $2\sigma$ of the most frequent interval of length $\sigma$.
• Employs binary search over a discretized range $[-R, R]$ to estimate quantiles privately, using a carefully constructed distribution family to simulate quantile queries.
• Applies a private averaging mechanism with calibrated Gaussian noise $\mathcal{N}\left(0, \frac{2|I|^2 \log(2/\delta)}{\epsilon^2}\right)$ on clipped and projected data points to estimate the mean.
• Constructs a family of discrete distributions $\mathcal{P}_i$ with three points to prove lower bounds via total variation distance and privacy amplification.
• Uses a hybrid privacy analysis combining $\epsilon$-LDP and $\delta$-relaxations to derive tight bounds on sample complexity and error.
• Applies concentration inequalities and privacy loss accounting to bound the probability of error in quantile estimation and interval recovery.

Experimental results

Research questions

• RQ1 Can we design a locally differentially private algorithm that achieves confidence intervals for the mean of length $\widetilde{O}\left(\frac{\sigma\sqrt{\log(1/\beta)}}{\epsilon\sqrt{n}}\right)$?
• RQ2 Is this interval length optimal up to poly-logarithmic factors in the local model?
• RQ3 Can private quantile estimation via binary search be used as a building block for private mean estimation?
• RQ4 What are the fundamental limits of one-shot locally private mean estimation in terms of confidence interval length?
• RQ5 How does the known vs. unknown variance case affect the utility and privacy trade-offs in private statistical inference?

Key findings

• The proposed algorithm achieves a confidence interval of length $O\left(\frac{\sigma\sqrt{\log(n)}}{\epsilon\sqrt{n}}\right)$ for the known variance case, under the condition $n = \Omega\left(\frac{\log(R/\sigma)}{\epsilon^2}\right)$.
• For the unknown variance case, the algorithm produces a valid confidence interval of similar length assuming a lower bound on $\sigma$ is available.
• A matching lower bound of $\Omega\left(\frac{\sigma\sqrt{\log(1/\beta)}}{\epsilon\sqrt{n}}\right)$ is proven for any one-shot $\epsilon$-LDP algorithm, showing optimality up to poly-log factors.
• The private Z-test is empirically validated, demonstrating practical utility and tightness of the confidence intervals.
• The binary search-based quantile estimation technique is shown to be effective and may be of independent interest for other LDP applications.
• The proof of the lower bound demonstrates the necessity of both $\alpha_{\rm quant}$ and $\alpha_{\rm dist}$ parameters in the construction, as omitting either leads to unbounded sample complexity.