Differentially Private Gradient-Tracking-Based Distributed Stochastic Optimization Over Directed Graphs

J. Chen

doi:10.1109/tac.2026.3678834

Differentially Private Gradient-Tracking-Based Distributed Stochastic Optimization Over Directed Graphs

2026-03-30

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DOI：10.1109/tac.2026.3678834

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This article proposes a differentially private gradient-tracking-based distributed stochastic optimization algorithm over directed graphs. In particular, privacy noises are incorporated into each agent’s state and tracking variable to mitigate information leakage, after which the perturbed states and tracking variables are transmitted to neighbors. We design two novel schemes for the step-sizes and the sampling number within the algorithm. The sampling parameter-controlled subsampling method employed by both schemes enhances the differential privacy level and ensures a finite cumulative privacy budget even over infinite iterations. The algorithm achieves both almost sure and mean square convergence for nonconvex objectives. Furthermore, when nonconvex objectives satisfy the Polyak–Łojasiewicz condition, Scheme (S1) achieves a polynomial mean square convergence rate, and Scheme (S2) achieves an exponential mean square convergence rate. The tradeoff between privacy and convergence is presented. The effectiveness of the algorithm and its superior performance compared to existing works are illustrated through numerical examples of distributed training on the benchmark datasets “MNIST” and “CIFAR-10.”

Keywords:

Differential privacy

directed graphs

distributed stochastic optimization

exponential mean square convergence rate

gradient tracking

Differentially Private Gradient-Tracking-Based Distributed Stochastic Optimization Over Directed Graphs

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