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Random Walk Learning and the Pac-Man Attack

Xingran Chen, Parimal Parag, Rohit Bhagat, Zonghong Liu, Salim El Rouayheb

TL;DR

This work tackles the vulnerability of random-walk based decentralized learning to a stealthy Pac-Man attack that terminates incoming random walks with probability $\zeta$. It proposes the Average Crossing (AC) algorithm to locally duplicate RWs, ensuring a bounded RW population and enabling continued learning via RW-SGD. Theoretical results show almost-sure boundedness of RW population, a phase transition in extinction probability tied to the duplication threshold, and convergence of the chain of RWs to a surrogate optimization problem with bounded bias from the true optimum. Empirical results on synthetic and real datasets—including MNIST—validate the theory, reveal a soft phase transition, and demonstrate competitive convergence and robustness against Pac-Man, highlighting the practical viability of fully decentralized defenses in RW-based learning systems.

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.

Random Walk Learning and the Pac-Man Attack

TL;DR

This work tackles the vulnerability of random-walk based decentralized learning to a stealthy Pac-Man attack that terminates incoming random walks with probability . It proposes the Average Crossing (AC) algorithm to locally duplicate RWs, ensuring a bounded RW population and enabling continued learning via RW-SGD. Theoretical results show almost-sure boundedness of RW population, a phase transition in extinction probability tied to the duplication threshold, and convergence of the chain of RWs to a surrogate optimization problem with bounded bias from the true optimum. Empirical results on synthetic and real datasets—including MNIST—validate the theory, reveal a soft phase transition, and demonstrate competitive convergence and robustness against Pac-Man, highlighting the practical viability of fully decentralized defenses in RW-based learning systems.

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.

Paper Structure

This paper contains 40 sections, 10 theorems, 108 equations, 30 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Graph and Random Walks
  4. Threat Model: The Pac-Man Attack
  5. RW-based Stochastic Gradient Descent
  6. Designable Properties of the Decentralized Mechanism
  7. No Blowup
  8. Low Probability of Extinction
  9. Convergence
  10. Average Crossing Algorithm
  11. Fundamental Analysis
  12. Population Boundedness
  13. Phase Transition in Extinction Behavior
  14. Convergence
  15. Simulations
  16. ...and 25 more sections

Key Result

Theorem 1

On any finite graph $\mathcal{G}^\prime = (\mathcal{V}^\prime, E^\prime)$ of Definition defn:PacMan in Appendix Appe:assumptions, with $z_0\geqslant 1$, $A_u \geqslant 1$, $q\leqslant 1$, and $\zeta \in (0,1]$, the AC algorithm ensures that $\limsup_{t\to\infty}Z_t < \infty$ almost surely.

Figures (30)

  • Figure 1: Illustration of a chain of RWs. Node $u$ duplicates a new RW (green) by copying the purple RW. Node $v$ duplicates a new RW (orange) by copying the green RW. The blue trajectory shows how these RWs are connected over time, forming a chain.
  • Figure 4: Loss v.s. learning steps on a complete graph: comparison between AC, DecAFork and gossip-based SGD.
  • Figure : (a) bounded population
  • Figure : synthetic dataset
  • Figure : (a) random regular graph
  • ...and 25 more figures

Theorems & Definitions (36)

  • Theorem 1: Boundedness
  • proof
  • Proposition 1: Soft Phase Transition
  • proof
  • Definition 1: Chain of RWs
  • Theorem 2
  • proof
  • Proposition 2: Shift of Optima
  • proof
  • Definition 2: Communication topologies and RWs
  • ...and 26 more