Modern machine learning is distributed and the work of several machines is typically aggregated by \emph{averaging} which is the optimal rule in terms of speed, offering a speedup of $n$ (with respect to using a single machine) when $n$ processes are learning together. Distributing data and models poses however fundamental vulnerabilities, be they to software bugs, asynchrony, or worse, to malicious attackers controlling some machines or injecting misleading data in the network. Such behavior is best modeled as Byzantine failures, and averaging does not tolerate a single one from a worker. Krum, the first provably Byzantine resilient aggregation rule for SGD only uses one worker per step, which hampers its speed of convergence, especially in best case conditions when none of the workers is actually Byzantine. An idea, coined multi-Krum, of using $m$ different workers per step was mentioned, without however any proof neither on its Byzantine resilience nor on its slowdown. More recently, it was shown that in high dimensional machine learning, guaranteeing convergence is not a sufficient condition for \emph{strong} Byzantine resilience. A improvement on Krum, coined Bulyan, was proposed and proved to guarantee stronger resilience. However, Bulyan suffers from the same weakness of Krum: using only one worker per step. This adds up to the aforementioned open problem and leaves the crucial need for both fast and strong Byzantine resilience unfulfilled. The present paper proposes using Bulyan over Multi-Krum (we call it Multi-Bulyan), a combination for which we provide proofs of strong Byzantine resilience, as well as an ${\frac{m}{n}}$ slowdown, compared to averaging, the fastest (but non Byzantine resilient) rule for distributed machine learning, finally we prove that Multi-Bulyan inherits the $O(d)$ merits of both multi-Krum and Bulyan.

证明multi-Bulyan 在分布式机器学习中是鲁棒且快速的梯度聚合规则，并能够实现拜占庭容错，其本地计算成本与平均计算法相同，且并行化加快了其效率。

高维度快速稳健分布式学习