Secure Distributed Optimization Under Gradient Attacks

Abstract

In this paper, we study secure distributed optimization against arbitrary gradient attacks 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> networked agents, a fixed but unknown fraction <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\rho$</tex-math></inline-formula> of the agents are under arbitrary gradient attacks 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 ( <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CLIP-VRG</small> ). We show that, in a connected network, when the unattacked local objective functions are convex and smooth, share a common minimizer, and their sum is strongly convex, <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CLIP-VRG</small> leads to almost sure convergence of the iterates to the exact sum cost minimizer at all agents. We quantify a tight upper bound on the fraction <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\rho$</tex-math></inline-formula> of attacked agents in terms of problem parameters such as the condition number of the associated sum cost that guarantee exact convergence of <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CLIP-VRG</small> , and characterize its asymptotic convergence rate. Finally, we empirically demonstrate the effectiveness of the proposed method under gradient attacks on both synthetic and real-world image classification datasets.