Reproduce “Privacy Amplification Through Synthetic Data: Insights from Linear Regression” on OpenResearch

About this paper

Clément Pierquin, Aurélien Bellet, Marc Tommasi, Matthieu Boussard

Researchers from Craft AI, Université de Lille, Inria, and Université de Montpellier provide the first rigorous theoretical framework for quantifying privacy amplification in synthetic data release from differentially private linear regression models, proving that when adversaries control the input "seed" for generation, releasing even a single synthetic data point leaks as much information as releasing the entire private model (establishing the post-processing property's tightness), while conversely demonstrating that synthetic data generated from random Gaussian inputs achieves O(1/d) privacy amplification for single points and O(d^(-1/2)) for multiple points when input dimension d exceeds output dimension and number of synthetic samples, leveraging advanced probabilistic tools including multivariate Central Limit Theorems and convergence results for products of Gaussian matrices to show that the original "mean shift" between adjacent datasets transforms into a more privacy-preserving "variance shift" in synthetic outputs, with numerical estimations confirming theoretical bounds and revealing that privacy loss scales as nl/(d-n) for releasing nl synthetic points, thereby establishing conditions under which differentially private generative models can provide both post-processing guarantees for large releases and significantly tighter amplified bounds for controlled small releases.

Key insights

• Privacy amplification only occurs when the randomness in synthetic data generation is controlled by a trusted party and hidden from adversaries - adversarial control of inputs eliminates amplification benefits entirely
• The mathematical mechanism behind amplification involves transforming "mean shifts" between models trained on adjacent datasets into "variance shifts" in synthetic outputs, which are inherently more privacy-preserving
• Amplification effectiveness depends critically on dimensional relationships: input dimension d must exceed both output dimension n and number of synthetic samples l for convergence guarantees to hold

Summary by alphaXiv.