Differentially Private Sampling from Rashomon Sets, and the Universality of Langevin Diffusion for Convex Optimization

TL;DR

This paper introduces a Langevin diffusion-based framework for differentially private sampling from Rashomon sets, providing privacy guarantees and optimal risk bounds for convex learning problems, with applications in interpretability and fairness.

Contribution

The paper develops a novel Langevin diffusion algorithm for private sampling that is independent of convexity and offers tight stability and risk guarantees.

Findings

Provides a privacy-preserving sampler for Rashomon sets.

Achieves optimal excess risk bounds under differential privacy.

Offers a versatile framework applicable to interpretable and robust ML.

Abstract

In this paper we provide an algorithmic framework based on Langevin diffusion (LD) and its corresponding discretizations that allow us to simultaneously obtain: i) An algorithm for sampling from the exponential mechanism, whose privacy analysis does not depend on convexity and which can be stopped at anytime without compromising privacy, and ii) tight uniform stability guarantees for the exponential mechanism. As a direct consequence, we obtain optimal excess empirical and population risk guarantees for (strongly) convex losses under both pure and approximate differential privacy (DP). The framework allows us to design a DP uniform sampler from the Rashomon set. Rashomon sets are widely used in interpretable and robust machine learning, understanding variable importance, and characterizing fairness.