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Asynchronous Algorithmic Alignment with Cocycles

Andrew Dudzik, Tamara von Glehn, Razvan Pascanu, Petar Veličković

TL;DR

This paper addresses the inefficiencies and learning difficulties that arise when standard GNNs execute dynamic-programming style algorithms synchronously. It introduces a node-centric, monoid/cocycle framework—where $M$ denotes messages, $S$ node states, and $A$ the generated arguments—and shows that the $1$-cocycle condition on the argument map $D: M \to [S,A]$ ensures invariance to asynchronous updates. By linking idempotence, semiring aggregations, and monoid homomorphisms, the authors derive practical, asynchronous yet faithful GNN variants, notably a PathGNN-like model with max-max aggregation and multimorphisms that can emulate Bellman-Ford and related algorithms. Empirical results on CLRS-30 demonstrate that higher levels of asynchrony invariance improve generalization and robustness, especially for algorithms with sparse meaningful updates, while preserving scalability. The work offers a principled pathway for designing neural algorithmic reasoning systems with provable push-button behavior under various asynchronous execution regimes.

Abstract

State-of-the-art neural algorithmic reasoners make use of message passing in graph neural networks (GNNs). But typical GNNs blur the distinction between the definition and invocation of the message function, forcing a node to send messages to its neighbours at every layer, synchronously. When applying GNNs to learn to execute dynamic programming algorithms, however, on most steps only a handful of the nodes would have meaningful updates to send. One, hence, runs the risk of inefficiencies by sending too much irrelevant data across the graph. But more importantly, many intermediate GNN steps have to learn the identity functions, which is a non-trivial learning problem. In this work, we explicitly separate the concepts of node state update and message function invocation. With this separation, we obtain a mathematical formulation that allows us to reason about asynchronous computation in both algorithms and neural networks. Our analysis yields several practical implementations of synchronous scalable GNN layers that are provably invariant under various forms of asynchrony.

Asynchronous Algorithmic Alignment with Cocycles

TL;DR

This paper addresses the inefficiencies and learning difficulties that arise when standard GNNs execute dynamic-programming style algorithms synchronously. It introduces a node-centric, monoid/cocycle framework—where denotes messages, node states, and the generated arguments—and shows that the -cocycle condition on the argument map ensures invariance to asynchronous updates. By linking idempotence, semiring aggregations, and monoid homomorphisms, the authors derive practical, asynchronous yet faithful GNN variants, notably a PathGNN-like model with max-max aggregation and multimorphisms that can emulate Bellman-Ford and related algorithms. Empirical results on CLRS-30 demonstrate that higher levels of asynchrony invariance improve generalization and robustness, especially for algorithms with sparse meaningful updates, while preserving scalability. The work offers a principled pathway for designing neural algorithmic reasoning systems with provable push-button behavior under various asynchronous execution regimes.

Abstract

State-of-the-art neural algorithmic reasoners make use of message passing in graph neural networks (GNNs). But typical GNNs blur the distinction between the definition and invocation of the message function, forcing a node to send messages to its neighbours at every layer, synchronously. When applying GNNs to learn to execute dynamic programming algorithms, however, on most steps only a handful of the nodes would have meaningful updates to send. One, hence, runs the risk of inefficiencies by sending too much irrelevant data across the graph. But more importantly, many intermediate GNN steps have to learn the identity functions, which is a non-trivial learning problem. In this work, we explicitly separate the concepts of node state update and message function invocation. With this separation, we obtain a mathematical formulation that allows us to reason about asynchronous computation in both algorithms and neural networks. Our analysis yields several practical implementations of synchronous scalable GNN layers that are provably invariant under various forms of asynchrony.
Paper Structure (15 sections, 7 theorems, 15 equations, 2 figures)

This paper contains 15 sections, 7 theorems, 15 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Message passing and (asynchronous) algorithms: Addition with carry
  3. Node-centric view on algorithmic alignment
  4. (A)synchrony and idempotence in GNNs
  5. Idempotent monoids
  6. Making $\phi$ a cocycle enables asynchrony
  7. The rich asynchrony of $\max$-$\max$ GNNs: Rediscovering PathGNNs
  8. Extending PathGNNs with multimorphisms: Message generation asynchrony
  9. Example: Semilattices and Bellman-Ford
  10. Evaluating asynchrony-invariant GNNs
  11. Conclusions
  12. Introduction to monoids
  13. Monoid actions as state machines
  14. Extension problems and cocycles
  15. Star-monoids and group cocycles

Key Result

Proposition 3.1

A node equipped with a rule $D$ for generating arguments is invariant to asynchrony, i.e. its output does not depend on the grouping of incoming messages, if and only if $D$ is a 1-cocycle.

Figures (2)

  • Figure 1: A possible execution trace of an asynchronous GNN. While traditional GNNs send and receive all messages synchronously, under our framework, at any step the GNN may choose to execute any number of possible operations (depicted here with a collection on the right side of the graph). Note that we do not aim to implement an asynchronous GNN---a feat concurrently explored by AMP faber2022asynchronous and Co-GNN finkelshtein2023cooperative---rather, we seek to build synchronous GNNs that are invariant---i.e., will yield identical output node embeddings---under various forms of asynchronous execution.
  • Figure 2: Test (out-of-distribution) results across all tasks in CLRS-30, for the three models described in Section \ref{['sec:eval']}, averaged across six seeds.

Theorems & Definitions (13)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • ...and 3 more