  In this paper, we study secure distributed optimization against arbitrary
gradient attack in multi-agent networks. In distributed optimization, there is
no central server to coordinate local updates, and each agent can only
communicate with its neighbors on a predefined network. We consider the
scenario where out of $n$ networked agents, a fixed but unknown fraction $\rho$
of the agents are under arbitrary gradient attack in that their stochastic
gradient oracles return arbitrary information to derail the optimization
process, and the goal is to minimize the sum of local objective functions on
unattacked agents. We propose a distributed stochastic gradient method that
combines local variance reduction and clipping (CLIP-VRG). We show that, in a
connected network, when unattacked local objective functions are convex and
smooth, share a common minimizer, and their sum is strongly convex, CLIP-VRG
leads to almost sure convergence of the iterates to the exact sum cost
minimizer at all agents. We quantify a tight upper bound of the fraction $\rho$
of attacked agents in terms of problem parameters such as the condition number
of the associated sum cost that guarantee exact convergence of CLIP-VRG, and
characterize its asymptotic convergence rate. Finally, we empirically
demonstrate the effectiveness of the proposed method under gradient attacks in
both synthetic dataset and image classification datasets.
