Differential Privacy in Distributed Optimization With Gradient Tracking

Optimization with gradient tracking is particularly notable for its superior convergence results among the various distributed algorithms, especially in the context of directed graphs. However, privacy concerns arise when gradient information is transmitted directly which would induce more information leakage. Surprisingly, literature has not adequately addressed the associated privacy issues. In response to the gap, our article proposes a privacy-preserving distributed optimization algorithm with gradient tracking by adding noises to transmitted messages, namely, the decision variables and the estimate of the aggregated gradient. We prove two dilemmas for this kind of algorithm. In the first dilemma, we reveal that this distributed optimization algorithm with gradient tracking cannot achieve epsilon-differential privacy (DP) and exact convergence simultaneously. Building on this, we subsequently highlight that the algorithm fails to achieve epsilon-DP when employing nonsummable stepsizes in the presence of Laplace noises. It is crucial to emphasize that these findings hold true regardless of the size of the privacy metric epsilon. After that, we rigorously analyze the convergence performance and privacy level given summable stepsize sequences under the Laplace distribution since it is only with summable stepsizes that is meaningful for us to study. We derive sufficient conditions that allow for the simultaneous stochastically bounded accuracy and epsilon-DP. Recognizing that several options can meet these conditions, we further derive an upper bound of the mean error's variance and specify the mathematical expression of epsilon under such conditions. Numerical simulations are provided to demonstrate the effectiveness of our proposed algorithm.