The Hidden Vulnerability of Distributed Learning in Byzantium

TLDR

Machine learning’s reliance on stochastic gradient descent has raised concerns about its vulnerability to Byzantine workers, prompting recent methods to make distributed SGD robust. The paper demonstrates that mere convergence is insufficient and introduces Bulyan to tighten Byzantine resilience. We construct a high‑dimensional attack that exploits loss‑function non‑convexity to force SGD into ineffective models, and show its effectiveness on CIFAR‑10 and MNIST. We find that while some Byzantine‑resilient schemes guarantee convergence, they still allow a poisoning margin that grows at least as √d, and Bulyan reduces this margin to O(1/√d) while empirically achieving robust convergence comparable to non‑Byzantine updates.

Abstract

While machine learning is going through an era of celebrated success, concerns have been raised about the vulnerability of its backbone: stochastic gradient descent (SGD). Recent approaches have been proposed to ensure the robustness of distributed SGD against adversarial (Byzantine) workers sending poisoned gradients during the training phase. Some of these approaches have been proven Byzantine-resilient: they ensure the convergence of SGD despite the presence of a minority of adversarial workers. We show in this paper that convergence is not enough. In high dimension $d \gg 1$, an adver\-sary can build on the loss function's non-convexity to make SGD converge to ineffective models. More precisely, we bring to light that existing Byzantine-resilient schemes leave a margin of poisoning of $Ω\left(f(d)\right)$, where $f(d)$ increases at least like $\sqrt{d~}$. Based on this leeway, we build a simple attack, and experimentally show its strong to utmost effectivity on CIFAR-10 and MNIST. We introduce Bulyan, and prove it significantly reduces the attackers leeway to a narrow $O( \frac{1}{\sqrt{d~}})$ bound. We empirically show that Bulyan does not suffer the fragility of existing aggregation rules and, at a reasonable cost in terms of required batch size, achieves convergence as if only non-Byzantine gradients had been used to update the model.