Gradient-tracking Based Differentially Private Distributed Optimization with Enhanced Optimization Accuracy

Privacy protection has become an increasingly pressing requirement in distributed optimization. However, equipping distributed optimization with differential privacy, the state-of-the-art privacy protection mechanism, will unavoidably compromise optimization accuracy. In this paper, we propose an algorithm to achieve rigorous $\epsilon$-differential privacy in gradient-tracking based distributed optimization with enhanced optimization accuracy. More specifically, to suppress the influence of differential-privacy noise, we propose a new robust gradient-tracking based distributed optimization algorithm that allows both stepsize and the variance of injected noise to vary with time. Then, we establish a new analyzing approach that can characterize the convergence of the gradient-tracking based algorithm under both constant and time-varying stespsizes. To our knowledge, this is the first analyzing framework that can treat gradient-tracking based distributed optimization under both constant and time-varying stepsizes in a unified manner. More importantly, the new analyzing approach gives a much less conservative analytical bound on the stepsize compared with existing proof techniques for gradient-tracking based distributed optimization. We also theoretically characterize the influence of differential-privacy design on the accuracy of distributed optimization, which reveals that inter-agent interaction has a significant impact on the final optimization accuracy. The discovery prompts us to optimize inter-agent coupling weights to minimize the optimization error induced by the differential-privacy design. Numerical simulation results confirm the theoretical predictions