Random Walk Learning and the Pac-Man Attack

arXiv stat.ML / 4/16/2026

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Key Points

The paper studies a stealthy adversarial “Pac-Man” attack on random-walk (RW) based distributed and decentralized learning, where a malicious node probabilistically kills any RW that visits it and thereby silently halts learning.
It introduces a fully decentralized defense called the Average Crossing (AC) algorithm that duplicates random walks to prevent “RW extinction” under the attack.
The authors prove that under AC the RW population stays almost surely bounded and that RW-based stochastic gradient descent still converges despite Pac-Man, with a measurable deviation from the true optimum.
Extensive experiments on synthetic and real datasets confirm the theory and reveal a phase transition in extinction probability depending on the RW duplication threshold.
The work also provides additional theoretical intuition by analyzing a simplified variant of AC to explain the observed phase-transition behavior.

Abstract

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.