# A Bias-Accuracy-Privacy Trilemma for Statistical Estimation

arXiv: https://arxiv.org/abs/2301.13334 · HTML: https://mlsanity.com/paper/2301.13334/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 75 references that are in the map.
- First posted on arXiv 30 Jan 2023.

## Builds on (most cited first, 30 of 46)

- [Concentrated Differential Privacy: Simplifications, Extensions, and Lower Bounds](https://mlsanity.com/paper/1605.02065/) (2016)
- [Concentrated Differential Privacy](https://mlsanity.com/paper/1603.01887/) (2016)
- [Minimax Optimal Procedures for Locally Private Estimation](https://mlsanity.com/paper/1604.02390/) (2016)
- [Finite Sample Differentially Private Confidence Intervals](https://mlsanity.com/paper/1711.03908/) (2017)
- [Privately Learning High-Dimensional Distributions](https://mlsanity.com/paper/1805.00216/) (2018)
- [Algorithmic Stability for Adaptive Data Analysis](https://mlsanity.com/paper/1511.02513/) (2015)
- [Fingerprinting Codes and the Price of Approximate Differential Privacy](https://mlsanity.com/paper/1311.3158/) (2013)
- [CoinPress: Practical Private Mean and Covariance Estimation](https://mlsanity.com/paper/2006.06618/) (2020)
- [Private Mean Estimation of Heavy-Tailed Distributions](https://mlsanity.com/paper/2002.09464/) (2020)
- [Learning with User-Level Privacy](https://mlsanity.com/paper/2102.11845/) (2021)
- [Robust and Differentially Private Mean Estimation](https://mlsanity.com/paper/2102.09159/) (2021)
- [Differential privacy and robust statistics in high dimensions](https://mlsanity.com/paper/2111.06578/) (2021)
- [Private Hypothesis Selection](https://mlsanity.com/paper/1905.13229/) (2019)
- [Between Pure and Approximate Differential Privacy](https://mlsanity.com/paper/1501.06095/) (2015)
- [Average-Case Averages: Private Algorithms for Smooth Sensitivity and Mean Estimation](https://mlsanity.com/paper/1906.02830/) (2019)
- [On Differentially Private Stochastic Convex Optimization with Heavy-tailed Data](https://mlsanity.com/paper/2010.11082/) (2020)
- [Efficient Mean Estimation with Pure Differential Privacy via a Sum-of-Squares Exponential Mechanism](https://mlsanity.com/paper/2111.12981/) (2021)
- [Improved Rates for Differentially Private Stochastic Convex Optimization with Heavy-Tailed Data](https://mlsanity.com/paper/2106.01336/) (2021)
- [Robustness Implies Privacy in Statistical Estimation](https://mlsanity.com/paper/2212.05015/) (2022)
- [Covariance-Aware Private Mean Estimation Without Private Covariance Estimation](https://mlsanity.com/paper/2106.13329/) (2021)
- [Learning discrete distributions: user vs item-level privacy](https://mlsanity.com/paper/2007.13660/) (2020)
- [On the Sample Complexity of Privately Learning Unbounded High-Dimensional Gaussians](https://mlsanity.com/paper/2010.09929/) (2020)
- [Private Robust Estimation by Stabilizing Convex Relaxations](https://mlsanity.com/paper/2112.03548/) (2021)
- [Instance-optimal Mean Estimation Under Differential Privacy](https://mlsanity.com/paper/2106.00463/) (2021)
- [Preventing False Discovery in Interactive Data Analysis is Hard](https://mlsanity.com/paper/1408.1655/) (2014)
- [Tight Lower Bounds for Differentially Private Selection](https://mlsanity.com/paper/1704.03024/) (2017)
- [FriendlyCore: Practical Differentially Private Aggregation](https://mlsanity.com/paper/2110.10132/) (2021)
- [A Private and Computationally-Efficient Estimator for Unbounded Gaussians](https://mlsanity.com/paper/2111.04609/) (2021)
- [New Lower Bounds for Private Estimation and a Generalized Fingerprinting Lemma](https://mlsanity.com/paper/2205.08532/) (2022)
- [Private Estimation with Public Data](https://mlsanity.com/paper/2208.07984/) (2022)

## Built on by (most cited first, 12 of 12)

- [Differentially Private Image Classification by Learning Priors from Random Processes](https://mlsanity.com/paper/2306.06076/) (2023)
- [Mean estimation in the add-remove model of differential privacy](https://mlsanity.com/paper/2312.06658/) (2023)
- [Smooth Lower Bounds for Differentially Private Algorithms via Padding-and-Permuting Fingerprinting Codes](https://mlsanity.com/paper/2307.07604/) (2023)
- [PLAN: Variance-Aware Private Mean Estimation](https://mlsanity.com/paper/2306.08745/) (2023)
- [A Polynomial Time, Pure Differentially Private Estimator for Binary Product Distributions](https://mlsanity.com/paper/2304.06787/) (2023)
- [General Gaussian Noise Mechanisms and Their Optimality for Unbiased Mean Estimation](https://mlsanity.com/paper/2301.13850/) (2023)
- [Harnessing Synthetic Data from Generative AI for Statistical Inference](https://mlsanity.com/paper/2603.05396/) (2026)
- [High-Dimensional Differentially Private Quantile Regression: Distributed Estimation and Statistical Inference](https://mlsanity.com/paper/2508.05212/) (2025)
- [Privately Estimating Black-Box Statistics](https://mlsanity.com/paper/2510.00322/) (2025)
- [Noise-Calibrated Inference from Differentially Private Sufficient Statistics in Exponential Families](https://mlsanity.com/paper/2603.02010/) (2026)
- [Protecting and Preserving Protest Dynamics for Responsible Analysis](https://mlsanity.com/paper/2604.05256/) (2026)
- [Near-optimal node-private community estimation in polynomial-time](https://mlsanity.com/paper/2607.09441/) (2026)

## Abstract (arXiv)

Differential privacy (DP) is a rigorous notion of data privacy, used for private statistics. The canonical algorithm for differentially private mean estimation is to first clip the samples to a bounded range and then add noise to their empirical mean. Clipping controls the sensitivity and, hence, the variance of the noise that we add for privacy. But clipping also introduces statistical bias. This tradeoff is inherent: we prove that no algorithm can simultaneously have low bias, low error, and low privacy loss for arbitrary distributions. Additionally, we show that under strong notions of DP (i.e., pure or concentrated DP), unbiased mean estimation is impossible, even if we assume that the data is sampled from a Gaussian. On the positive side, we show that unbiased mean estimation is possible under a more permissive notion of differential privacy (approximate DP) if we assume that the distribution is symmetric.