Convergence of Byzantine-Resilient Gradient Tracking via Probabilistic Edge Dropout

TL;DR

This paper introduces a stochastic distributed optimization method, GT-PD, resilient to Byzantine agents, combining message clipping and probabilistic edge dropout to ensure convergence under adversarial conditions.

Contribution

It proposes a novel gradient tracking algorithm with probabilistic edge dropout and leaky integration, enhancing Byzantine resilience while maintaining convergence guarantees.

Findings

GT-PD converges linearly to a neighborhood under complete Byzantine isolation.

GT-PD-L controls error accumulation with leaky integrator, achieving convergence under partial isolation.

Experiments show GT-PD-L outperforms trimmed mean by up to 4.3% under stealth attacks.

Abstract

We study distributed optimization over networks with Byzantine agents that may send arbitrary adversarial messages. We propose \emph{Gradient Tracking with Probabilistic Edge Dropout} (GT-PD), a stochastic gradient tracking method that preserves the convergence properties of gradient tracking under adversarial communication. GT-PD combines two complementary defense layers: a universal self-centered projection that clips each incoming message to a ball of radius $\tau$ around the receiving agent, and a fully decentralized probabilistic dropout rule driven by a dual-metric trust score in the decision and tracking channels. This design bounds adversarial perturbations while preserving the doubly stochastic mixing structure, a property often lost under robust aggregation in decentralized settings. Under complete Byzantine isolation ($p_b=0$), GT-PD converges linearly to a neighborhood determined solely by stochastic gradient variance. For partial isolation ($p_b>0$), we introduce \emph{Gradient Tracking with Probabilistic Edge Dropout and Leaky Integration} (GT-PD-L), which uses a leaky integrator to control the accumulation of tracking errors caused by persistent perturbations and achieves linear convergence to a bounded neighborhood determined by the stochastic variance and the clipping-to-leak ratio. We further show that under two-tier dropout with $p_h=1$, isolating Byzantine agents introduces no additional variance into the honest consensus dynamics. Experiments on MNIST under Sign Flip, ALIE, and Inner Product Manipulation attacks show that GT-PD-L outperforms coordinate-wise trimmed mean by up to 4.3 percentage points under stealth attacks.