  In this paper, we consider the Byzantine-robust stochastic optimization
problem defined over decentralized static and time-varying networks, where the
agents collaboratively minimize the summation of expectations of stochastic
local cost functions, but some of the agents are unreliable due to data
corruptions, equipment failures or cyber-attacks. The unreliable agents, which
are called as Byzantine agents thereafter, can send faulty values to their
neighbors and bias the optimization process. Our key idea to handle the
Byzantine attacks is to formulate a total variation (TV) norm-penalized
approximation of the Byzantine-free problem, where the penalty term forces the
local models of regular agents to be close, but also allows the existence of
outliers from the Byzantine agents. A stochastic subgradient method is applied
to solve the penalized problem. We prove that the proposed method reaches a
neighborhood of the Byzantine-free optimal solution, and the size of
neighborhood is determined by the number of Byzantine agents and the network
topology. Numerical experiments corroborate the theoretical analysis, as well
as demonstrate the robustness of the proposed method to Byzantine attacks and
its superior performance comparing to existing methods.
