# Preserving Data-Privacy with Added Noises: Optimal Estimation and Privacy Analysis

## TL;DR

This paper develops a theoretical framework to optimize data estimation and analyze privacy in distributed systems where nodes add noise to preserve privacy, providing insights into the trade-offs between estimation accuracy and privacy guarantees.

## Contribution

It introduces a novel framework for optimal distributed estimation and privacy analysis, specifically applied to privacy-preserving average consensus algorithms.

## Key findings

- Derived the optimal estimation strategy for neighbor data.
- Analyzed the privacy guarantees under optimal estimation.
- Identified optimal noise distributions for privacy-preserving consensus.

## Abstract

Networked system often relies on distributed algorithms to achieve a global computation goal with iterative local information exchanges between neighbor nodes. To preserve data privacy, a node may add a random noise to its original data for information exchange at each iteration. Nevertheless, a neighbor node can estimate other's original data based on the information it received. The estimation accuracy and data privacy can be measured in terms of $(\epsilon, \delta)$-data-privacy, defined as the probability of $\epsilon$-accurate estimation (the difference of an estimation and the original data is within $\epsilon$) is no larger than $\delta$ (the disclosure probability). How to optimize the estimation and analyze data privacy is a critical and open issue. In this paper, a theoretical framework is developed to investigate how to optimize the estimation of neighbor's original data using the local information received, named optimal distributed estimation. Then, we study the disclosure probability under the optimal estimation for data privacy analysis. We further apply the developed framework to analyze the data privacy of the privacy-preserving average consensus algorithm and identify the optimal noises for the algorithm.