Differentially Private Sampling from Rashomon Sets, and the Universality of Langevin Diffusion for Convex Optimization
This work addresses privacy concerns in interpretable and robust machine learning, such as fairness and variable importance analysis, by offering a novel method for differentially private sampling from Rashomon sets.
The paper tackles the problem of sampling from Rashomon sets under differential privacy by introducing an algorithmic framework based on Langevin diffusion, which provides optimal excess risk guarantees for convex losses and enables differentially private uniform sampling from these sets.
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.