In decentralized machine learning, different devices communicate in a peer-to-peer manner to collaboratively learn from each other's data. Such approaches are vulnerable to misbehaving (or Byzantine) devices. We introduce $\mathrm{F}\text{-}\rm RG$, a general framework for building robust decentralized algorithms with guarantees arising from robust-sum-like aggregation rules $\mathrm{F}$. We then investigate the notion of *breakdown point*, and show an upper bound on the number of adversaries that decentralized algorithms can tolerate. We introduce a practical robust aggregation rule, coined $\rm CS_{ours}$, such that $\rm CS_{ours}\text{-}RG$ has a near-optimal breakdown. Other choices of aggregation rules lead to existing algorithms such as $\rm ClippedGossip$ or $\rm NNA$. We give experimental evidence to validate the effectiveness of $\rm CS_{ours}\text{-}RG$ and highlight the gap with $\mathrm{NNA}$, in particular against a novel attack tailored to decentralized communications.

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