Zeroth-Order Decentralized Dual Averaging for Online Optimization With Privacy Consideration

Keke Zhang, Qingguo Lü, Xiaofeng Liao, Huaqing Li

Abstract: This article studies a decentralized online optimization problem with a common constraint set over time-varying and directed networks. Nodes in the network undertake local computation and communication to collaboratively solve the problem, where each node's access to the value of its local loss function is contingent upon its decision-making at each time. Besides, the persistence of information broadcasting creates a potential risk of privacy infringement for participating nodes in many of the existing decentralized algorithms designed to tackle the problem. To address these issues, we develop a novel zeroth-order decentralized dual averaging algorithm, named ZO-DDA, where we utilize a zeroth-order oracle instead of computing the true subgradient information. On the one hand, ZO-DDA via the constructed row-stochastic matrices rescales the zeroth-order oracle with an auxiliary variable to overcome the unbalancedness caused by time-varying directed networks. On the other hand, ZO-DDA has privacy protection performance, which involves the usage of perturbation with noise to preserve differential privacy. We provide rigorous theoretical analyses to illustrate that ZO-DDA achieves an expected regret with a sublinear rate under weaker initial conditions than the existing decentralized works. Also, ZO-DDA preserves differential privacy for each node's loss function. The trade-off between the privacy level and the optimization accuracy is analyzed. Numerical examples are given to demonstrate the viability and performance of ZO-DDA.