Privacy-Preserving Distributed Online Optimization over Unbalanced Digraphs via Subgradient Rescaling

In this article, we investigate a distributed online constrained optimization problem with differential privacy where the network is modeled by an unbalanced digraph with a row-stochastic adjacency matrix. To address such a problem, a distributed differentially private algorithm without introducing a trusted third-party is proposed to preserve the privacy of the participating nodes. Under mild conditions, we show that the proposed algorithm attains an OlogT expected regret bound for strongly convex local cost functions, where T is the time horizon. Moreover, we remove the need for knowing the time horizon T in advance by adopting doubling trick scheme, and derive an OT expected regret bound for general convex local cost functions. Our results coincide with the best theoretical regrets that can be achieved in the state-of-the-art algorithms. Finally, simulation results are conducted to validate the efficiency of our proposed algorithm.