  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.
