This paper studies distributed online learning under Byzantine attacks. The performance of an online learning algorithm is often characterized by (adversarial) regret, which evaluates the quality of one-step-ahead decision-making when an environment provides adversarial losses, and a sublinear bound is preferred. But we prove that, even with a class of state-of-the-art robust aggregation rules, in an adversarial environment and in the presence of Byzantine participants, distributed online gradient descent can only achieve a linear adversarial regret bound, which is tight. This is the inevitable consequence of Byzantine attacks, even though we can control the constant of the linear adversarial regret to a reasonable level. Interestingly, when the environment is not fully adversarial so that the losses of the honest participants are i.i.d. (independent and identically distributed), we show that sublinear stochastic regret, in contrast to the aforementioned adversarial regret, is possible. We develop a Byzantine-robust distributed online momentum algorithm to attain such a sublinear stochastic regret bound. Extensive numerical experiments corroborate our theoretical analysis.

翻译：本文研究了拜占庭攻击下的分布式在线学习问题。在线学习算法的性能通常用（对抗性）遗憾来刻画，该指标评估环境提供对抗性损失时一步前瞻决策的质量，且次线性界是更优选择。但我们证明，即使采用一类最先进的鲁棒聚合规则，在对抗性环境及存在拜占庭参与者的情况下，分布式在线梯度下降算法仅能达到线性对抗性遗憾界，且该界是紧的。这是拜占庭攻击的必然结果，尽管我们可以将线性对抗性遗憾的常数控制在合理水平。有趣的是，当环境并非完全对抗性，使得诚实参与者的损失满足独立同分布时，我们证明与前述对抗性遗憾相对的次线性随机遗憾是可能的。为此，我们开发了一种拜占庭鲁棒的分布式在线动量算法，以实现这种次线性随机遗憾界。大量数值实验验证了我们的理论分析。