Renyi Differentially Private ADMM for Non-Smooth Regularized Optimization

TL;DR

This paper introduces two RDP-based differentially private ADMM algorithms for non-smooth convex optimization, demonstrating improved privacy-utility trade-offs in high privacy regimes through experiments.

Contribution

It proposes two novel stochastic ADMM algorithms under RDP for non-smooth regularized optimization, utilizing gradient and output perturbation techniques.

Findings

ssADMM reduces noise via privacy amplification.

mpADMM accurately computes sensitivity for output perturbation.

Both algorithms outperform baselines in high privacy regimes.

Abstract

In this paper we consider the problem of minimizing composite objective functions consisting of a convex differentiable loss function plus a non-smooth regularization term, such as $L_1$ norm or nuclear norm, under R\'enyi differential privacy (RDP). To solve the problem, we propose two stochastic alternating direction method of multipliers (ADMM) algorithms: ssADMM based on gradient perturbation and mpADMM based on output perturbation. Both algorithms decompose the original problem into sub-problems that have closed-form solutions. The first algorithm, ssADMM, applies the recent privacy amplification result for RDP to reduce the amount of noise to add. The second algorithm, mpADMM, numerically computes the sensitivity of ADMM variable updates and releases the updated parameter vector at the end of each epoch. We compare the performance of our algorithms with several baseline algorithms on both real and simulated datasets. Experimental results show that, in high privacy regimes (small $\epsilon$), ssADMM and mpADMM outperform other baseline algorithms in terms of classification and feature selection performance, respectively.