Random walk (RW)-based algorithms have long been popular in distributed systems due to low overheads and scalability, with recent growing applications in decentralized learning. However, their reliance on local interactions makes them inherently vulnerable to malicious behavior. In this work, we investigate an adversarial threat that we term the ``Pac-Man'' attack, in which a malicious node probabilistically terminates any RW that visits it. This stealthy behavior gradually eliminates active RWs from the network, effectively halting the learning process without triggering failure alarms. To counter this threat, we propose the Average Crossing (AC) algorithm--a fully decentralized mechanism for duplicating RWs to prevent RW extinction in the presence of Pac-Man. Our theoretical analysis establishes that (i) the RW population remains almost surely bounded under AC and (ii) RW-based stochastic gradient descent remains convergent under AC, even in the presence of Pac-Man, with a quantifiable deviation from the true optimum. Our extensive empirical results on both synthetic and real-world datasets corroborate our theoretical findings. Furthermore, they uncover a phase transition in the extinction probability as a function of the duplication threshold. We offer theoretical insights by analyzing a simplified variant of the AC, which sheds light on the observed phase transition.

翻译：基于随机游走（RW）的算法因其低开销和可扩展性，长期以来在分布式系统中广受欢迎，近年来在去中心化学习中的应用日益增长。然而，这些算法对局部交互的依赖使其本质上易受恶意行为的影响。在本研究中，我们探讨了一种我们称之为“吃豆人”攻击的对抗性威胁，其中恶意节点以一定概率终止任何访问它的随机游走。这种隐蔽行为会逐渐消除网络中的活跃随机游走，从而在不触发故障警报的情况下有效停止学习过程。为应对此威胁，我们提出了平均交叉（AC）算法——一种完全去中心化的机制，用于复制随机游走，以防止在存在吃豆人攻击时随机游走灭绝。我们的理论分析表明：（i）在AC机制下，随机游走数量几乎必然有界；（ii）即使在存在吃豆人攻击的情况下，基于随机游走的随机梯度下降在AC机制下仍保持收敛，且与真实最优解的偏差可量化。我们在合成数据集和真实数据集上的大量实证结果证实了我们的理论发现。此外，这些结果揭示了灭绝概率作为复制阈值函数的一个相变现象。我们通过分析AC算法的一个简化变体提供了理论见解，从而阐明了观察到的相变现象。