#### Fast and Robust Distributed Learning in High Dimension

##### El-Mahdi El-Mhamdi, Rachid Guerraoui, Sébastien Rouault

Could a gradient aggregation rule (GAR) for distributed machine learning be both robust and fast? This paper answers by the affirmative through multi-Bulyan. Given $n$ workers, $f$ of which are arbitrary malicious (Byzantine) and $m=n-f$ are not, we prove that multi-Bulyan can ensure a strong form of Byzantine resilience, as well as an ${\frac{m}{n}}$ slowdown, compared to averaging, the fastest (but non Byzantine resilient) rule for distributed machine learning. When $m \approx n$ (almost all workers are correct), multi-Bulyan reaches the speed of averaging. We also prove that multi-Bulyan's cost in local computation is $O(d)$ (like averaging), an important feature for ML where $d$ commonly reaches $10^9$, while robust alternatives have at least quadratic cost in $d$. Our theoretical findings are complemented with an experimental evaluation which, in addition to supporting the linear $O(d)$ complexity argument, conveys the fact that multi-Bulyan's parallelisability further adds to its efficiency.

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