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In this paper, we study secure distributed optimization against arbitrarygradient attack in multi-agent networks. In distributed optimization, there isno central server to coordinate local updates, and each agent can onlycommunicate with its neighbors on a predefined network. We consider thescenario where out of $n$ networked agents, a fixed but unknown fraction $\rho$of the agents are under arbitrary gradient attack in that their stochasticgradient oracles return arbitrary information to derail the optimizationprocess, and the goal is to minimize the sum of local objective functions onunattacked agents. We propose a distributed stochastic gradient method thatcombines local variance reduction and clipping (CLIP-VRG). We show that, in aconnected network, when unattacked local objective functions are convex andsmooth, share a common minimizer, and their sum is strongly convex, CLIP-VRGleads to almost sure convergence of the iterates to the exact sum costminimizer 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 numberof the associated sum cost that guarantee exact convergence of CLIP-VRG, andcharacterize its asymptotic convergence rate. Finally, we empiricallydemonstrate the effectiveness of the proposed method under gradient attacks inboth synthetic dataset and image classification datasets.

Secure Distributed Optimization Under Gradient Attacks