Prox-DBRO-VR: A Unified Analysis on Decentralized Byzantine-Resilient Composite Stochastic Optimization with Variance Reduction and Non-Asymptotic Convergence Rates

Decentralized stochastic gradient algorithms resolve efficiently large-scale finite-sum optimization problems when all agents over networks are reliable. However, most of these algorithms are not resilient to adverse conditions, such as malfunctioning agents, software bugs, and cyber attacks. This paper aims to handle a class of general composite finite-sum optimization problems over multi-agent cyber-physical systems (CPSs) in the presence of an unknown number of Byzantine agents. Based on the proximal mapping method, variance-reduced (VR) techniques, and a norm-penalized approximation strategy, we propose a decentralized Byzantine-resilient and proximal-gradient algorithmic framework, dubbed Prox-DBRO-VR,which achieves an optimization and control goal using only local computations and communications. To reduce asymptotically the variance generated by evaluating the local noisy stochastic gradients, we incorporate two localized VR techniques (SAGA and LSVRG) into Prox-DBRO-VR to design Prox-DBRO-SAGA and Prox-DBRO-LSVRG. By analyzing the contraction relationships among the gradient-learning error, robust consensus condition, and optimality gap in a unified theoretical framework, it is demonstrated that both Prox-DBRO-SAGA and Prox-DBRO-LSVRG,with a well-designed constant (resp., decaying) step-size, converge linearly (resp., sublinearly) inside an error ball around the optimal solution to the original problem under standard assumptions. The trade-off between convergence accuracy and the number of Byzantine agents in both linear and sub-linear cases is also characterized. In simulation, the effectiveness and practicability of the proposed algorithms are manifested via resolving a decentralized sparse machine-learning problem over multi-agent CPSs under various Byzantine attacks.