# Privacy Amplification Through Synthetic Data: Insights from Linear Regression

arXiv: https://arxiv.org/abs/2506.05101 · HTML: https://mlsanity.com/paper/2506.05101/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 38 references that are in the map.
- First posted on arXiv 5 Jun 2025.

## Builds on (most cited first, 24 of 24)

- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [Renyi Differential Privacy](https://mlsanity.com/paper/1702.07476/) (2017)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [Semi-supervised Knowledge Transfer for Deep Learning from Private Training Data](https://mlsanity.com/paper/1610.05755/) (2016)
- [Privacy Amplification by Subsampling: Tight Analyses via Couplings and Divergences](https://mlsanity.com/paper/1807.01647/) (2018)
- [Differentially Private Generative Adversarial Network](https://mlsanity.com/paper/1802.06739/) (2018)
- [Amplification by Shuffling: From Local to Central Differential Privacy via Anonymity](https://mlsanity.com/paper/1811.12469/) (2018)
- [Is Model Collapse Inevitable? Breaking the Curse of Recursion by Accumulating Real and Synthetic Data](https://mlsanity.com/paper/2404.01413/) (2024)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Differentially Private Diffusion Models](https://mlsanity.com/paper/2210.09929/) (2022)
- [Deep Learning with Label Differential Privacy](https://mlsanity.com/paper/2102.06062/) (2021)
- [Winning the NIST Contest: A scalable and general approach to differentially private synthetic data](https://mlsanity.com/paper/2108.04978/) (2021)
- [Privacy Amplification by Iteration](https://mlsanity.com/paper/1808.06651/) (2018)
- [Model Collapse Demystified: The Case of Regression](https://mlsanity.com/paper/2402.07712/) (2024)
- [Graphical-model based estimation and inference for differential privacy](https://mlsanity.com/paper/1901.09136/) (2019)
- [Efficient Private ERM for Smooth Objectives](https://mlsanity.com/paper/1703.09947/) (2017)
- [Privacy of Noisy Stochastic Gradient Descent: More Iterations without More Privacy Loss](https://mlsanity.com/paper/2205.13710/) (2022)
- [SoK: Privacy-Preserving Data Synthesis](https://mlsanity.com/paper/2307.02106/) (2023)
- [Privacy Amplification by Decentralization](https://mlsanity.com/paper/2012.05326/) (2020)
- ["What do you want from theory alone?" Experimenting with Tight Auditing of Differentially Private Synthetic Data Generation](https://mlsanity.com/paper/2405.10994/) (2024)
- [Private GANs, Revisited](https://mlsanity.com/paper/2302.02936/) (2023)
- [Improving the Privacy and Practicality of Objective Perturbation for Differentially Private Linear Learners](https://mlsanity.com/paper/2401.00583/) (2023)
- [Output Perturbation for Differentially Private Convex Optimization: Faster and More General](https://mlsanity.com/paper/2102.04704/) (2021)
- [The Product of Gaussian Matrices is Close to Gaussian](https://mlsanity.com/paper/2108.09887/) (2021)

## Built on by (most cited first, 2 of 2)

- [Fully Decentralized Certified Unlearning](https://mlsanity.com/paper/2512.08443/) (2025)
- [Privacy Amplification Persists under Unlimited Synthetic Data Release](https://mlsanity.com/paper/2602.04895/) (2026)

## Abstract (arXiv)

Synthetic data inherits the differential privacy guarantees of the model used to generate it. Additionally, synthetic data may benefit from privacy amplification when the generative model is kept hidden. While empirical studies suggest this phenomenon, a rigorous theoretical understanding is still lacking. In this paper, we investigate this question through the well-understood framework of linear regression. First, we establish negative results showing that if an adversary controls the seed of the generative model, a single synthetic data point can leak as much information as releasing the model itself. Conversely, we show that when synthetic data is generated from random inputs, releasing a limited number of synthetic data points amplifies privacy beyond the model's inherent guarantees. We believe our findings in linear regression can serve as a foundation for deriving more general bounds in the future.