# Optimality of the Laplace Mechanism in Differential Privacy

arXiv: https://arxiv.org/abs/1504.00065 · HTML: https://mlsanity.com/paper/1504.00065/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Mar 2015.

## Built on by (most cited first, 15 of 15)

- [The Limits of Word Level Differential Privacy](https://mlsanity.com/paper/2205.02130/) (2022)
- [Differentially Private Counterfactuals via Functional Mechanism](https://mlsanity.com/paper/2208.02878/) (2022)
- [Algorithms with More Granular Differential Privacy Guarantees](https://mlsanity.com/paper/2209.04053/) (2022)
- [Holistic Survey of Privacy and Fairness in Machine Learning](https://mlsanity.com/paper/2307.15838/) (2023)
- [Global Group Fairness in Federated Learning via Function Tracking](https://mlsanity.com/paper/2503.15163/) (2025)
- [FedXDS: Leveraging Model Attribution Methods to counteract Data Heterogeneity in Federated Learning](https://mlsanity.com/paper/2606.31742/) (2026)
- [Infinitely Divisible Noise in the Low Privacy Regime](https://mlsanity.com/paper/2110.06559/) (2021)
- [Fast, Distribution-free Predictive Inference for Neural Networks with Coverage Guarantees](https://mlsanity.com/paper/2306.06582/) (2023)
- [Differentially Private Distributed Estimation and Learning](https://mlsanity.com/paper/2306.15865/) (2023)
- [Almost linear time differentially private release of synthetic graphs](https://mlsanity.com/paper/2406.02156/) (2024)
- [Tangent differential privacy](https://mlsanity.com/paper/2406.04535/) (2024)
- [FedFlex: Federated Learning for Diverse Netflix Recommendations](https://mlsanity.com/paper/2507.21115/) (2025)
- [Power Mechanism: Private Tabular Representation Release for Model Agnostic Consumption](https://mlsanity.com/paper/2510.05581/) (2025)
- [Interpolation-Based Optimization for Enforcing lp-Norm Metric Differential Privacy in Continuous and Fine-Grained Domains](https://mlsanity.com/paper/2601.09946/) (2026)
- [Metric-Normalized Posterior Leakage (mPL): Attacker-Aligned Privacy for Joint Consumption](https://mlsanity.com/paper/2605.01137/) (2026)

## Abstract (arXiv)

In the highly interconnected realm of Internet of Things, exchange of sensitive information raises severe privacy concerns. The Laplace mechanism -- adding Laplace-distributed artificial noise to sensitive data -- is one of the widely used methods of providing privacy guarantees within the framework of differential privacy. In this work, we present Lipschitz privacy, a slightly tighter version of differential privacy. We prove that the Laplace mechanism is optimal in the sense that it minimizes the mean-squared error for identity queries which provide privacy with respect to the ℓ1-norm. In addition to the ℓ1-norm which respects individuals' participation, we focus on the use of the ℓ2-norm which provides privacy of high-dimensional data. A variation of the Laplace mechanism is proven to have the optimal mean-squared error from the identity query. Finally, the optimal mechanism for the scenario in which individuals submit their high-dimensional sensitive data is derived.