Compressed Differentially Private Distributed Optimization with Linear Convergence

This paper addresses the problem of differentially private distributed optimization under limited communication, where each agent aims to keep their cost function private while minimizing the sum of all agents' cost functions. In response, we propose a novel Compressed differentially Private distributed Gradient Tracking algorithm (CPGT). We demonstrate that CPGT achieves linear convergence for smooth and strongly convex cost functions, even with a class of biased but contractive compressors, and achieves the same accuracy as the idealized communication algorithm. Additionally, we rigorously prove that CPGT ensures differential privacy. Simulations are provided to validate the effectiveness of the proposed algorithm.