Recently, decentralized learning has emerged as a popular peer-to-peer signal and information processing paradigm that enables model training across geographically distributed agents in a scalable manner, without the presence of any central server. When some of the agents are malicious (also termed as Byzantine), resilient decentralized learning algorithms are able to limit the impact of these Byzantine agents without knowing their number and identities, and have guaranteed optimization errors. However, analysis of the generalization errors, which are critical to implementations of the trained models, is still lacking. In this paper, we provide the first analysis of the generalization errors for a class of popular Byzantine-resilient decentralized stochastic gradient descent (DSGD) algorithms. Our theoretical results reveal that the generalization errors cannot be entirely eliminated because of the presence of the Byzantine agents, even if the number of training samples are infinitely large. Numerical experiments are conducted to confirm our theoretical results.

翻译：近年来，分布式学习作为一种流行的点对点信号与信息处理范式，能够在无中心服务器的情况下，以可扩展的方式实现地理分布式智能体间的模型训练。当部分智能体存在恶意行为（亦称为拜占庭节点）时，鲁棒的分布式学习算法能够在不知晓恶意节点数量与身份的条件下限制其影响，并保证优化误差的收敛性。然而，对于训练模型部署至关重要的泛化误差分析仍属空白。本文首次对一类主流的拜占庭鲁棒分布式随机梯度下降（DSGD）算法进行了泛化误差分析。理论结果表明，由于拜占庭节点的存在，即使训练样本数量趋于无穷，泛化误差也无法完全消除。数值实验验证了理论分析的正确性。