# The Bernstein Mechanism: Function Release under Differential Privacy

arXiv: https://arxiv.org/abs/1507.04499 · HTML: https://mlsanity.com/paper/1507.04499/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jul 2015.

## Built on by (most cited first, 13 of 13)

- [Fast Private Kernel Density Estimation via Locality Sensitive Quantization](https://mlsanity.com/paper/2307.01877/) (2023)
- [Efficiently Computing Similarities to Private Datasets](https://mlsanity.com/paper/2403.08917/) (2024)
- [Differentially Private Kernel Density Estimation](https://mlsanity.com/paper/2409.01688/) (2024)
- [Locally Private Sampling with Public Data](https://mlsanity.com/paper/2411.08791/) (2024)
- [Learning from End User Data with Shuffled Differential Privacy over Kernel Densities](https://mlsanity.com/paper/2502.14087/) (2025)
- [Differentially Private Normalizing Flows for Density Estimation, Data Synthesis, and Variational Inference with Application to Electronic Health Records](https://mlsanity.com/paper/2302.05787/) (2023)
- [Differentially Private Synthetic Data with Private Density Estimation](https://mlsanity.com/paper/2405.04554/) (2024)
- [Noise-Aware Differentially Private Regression via Meta-Learning](https://mlsanity.com/paper/2406.08569/) (2024)
- [Optimal Variance and Covariance Estimation under Differential Privacy in the Add-Remove Model and Beyond](https://mlsanity.com/paper/2509.04919/) (2025)
- [New Bounds for Kernel Sums via Fast Spherical Embeddings](https://mlsanity.com/paper/2605.01263/) (2026)
- [Pure Differential Privacy for Functional Summaries with a Laplace-like Process](https://mlsanity.com/paper/2309.00125/) (2023)
- [Gaussian Differentially Private Human Faces Under a Face Radial Curve Representation](https://mlsanity.com/paper/2409.08301/) (2024)
- [Random Fourier Features in Near-Linear Time](https://mlsanity.com/paper/2610.14751/) (2026)

## Abstract (arXiv)

We address the problem of general function release under differential privacy, by developing a functional mechanism that applies under the weak assumptions of oracle access to target function evaluation and sensitivity. These conditions permit treatment of functions described explicitly or implicitly as algorithmic black boxes. We achieve this result by leveraging the iterated Bernstein operator for polynomial approximation of the target function, and polynomial coefficient perturbation. Under weak regularity conditions, we establish fast rates on utility measured by high-probability uniform approximation. We provide a lower bound on the utility achievable for any functional mechanism that is varepsilon-differentially private. The generality of our mechanism is demonstrated by the analysis of a number of example learners, including naive Bayes, non-parametric estimators and regularized empirical risk minimization. Competitive rates are demonstrated for kernel density estimation; and varepsilon-differential privacy is achieved for a broader class of support vector machines than known previously.