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Anchoring and Mixed-Norm Contractions in Averaging-Learning Dynamics

Ionel Popescu, Jeven Syatriadi, Tushar Vaidya

TL;DR

A mixed-operator-norm framework is developed that extracts two-step contraction from the interplay between aggregate learning mass and entrywise diffusion of influence, a mechanism new to consensus literature.

Abstract

A single informed agent can draw an arbitrarily large network to the ground truth. This is the sharpest consequence of the "Averaging plus Learning" framework studied here, where agents update opinions by socially averaging neighbours while some receive private feedback at heterogeneous rates. The key is a graph-theoretic property we call condensely anchored, which implies convergence to the correct consensus on fixed networks. In the original framework of Popescu and Vaidya (2023), every agent was required to learn. Removing that requirement changes the problem fundamentally: the underlying graph must now carry the signal from a handful of anchors to everyone else. When learning rates decay to zero, a persistence condition on the rates alone suffices, with no uniform connectivity or aperiodicity assumed. The hardest case is intermittent connectivity, where no single time step contracts in any standard norm. A mixed-operator-norm framework is developed that extracts two-step contraction from the interplay between aggregate learning mass and entrywise diffusion of influence, a mechanism new to consensus literature. Finally, we demonstrate the framework's robustness: vanishing noise preserves convergence to the ground truth, whereas persistent noise drives the system to a limiting law.

Anchoring and Mixed-Norm Contractions in Averaging-Learning Dynamics

TL;DR

A mixed-operator-norm framework is developed that extracts two-step contraction from the interplay between aggregate learning mass and entrywise diffusion of influence, a mechanism new to consensus literature.

Abstract

A single informed agent can draw an arbitrarily large network to the ground truth. This is the sharpest consequence of the "Averaging plus Learning" framework studied here, where agents update opinions by socially averaging neighbours while some receive private feedback at heterogeneous rates. The key is a graph-theoretic property we call condensely anchored, which implies convergence to the correct consensus on fixed networks. In the original framework of Popescu and Vaidya (2023), every agent was required to learn. Removing that requirement changes the problem fundamentally: the underlying graph must now carry the signal from a handful of anchors to everyone else. When learning rates decay to zero, a persistence condition on the rates alone suffices, with no uniform connectivity or aperiodicity assumed. The hardest case is intermittent connectivity, where no single time step contracts in any standard norm. A mixed-operator-norm framework is developed that extracts two-step contraction from the interplay between aggregate learning mass and entrywise diffusion of influence, a mechanism new to consensus literature. Finally, we demonstrate the framework's robustness: vanishing noise preserves convergence to the ground truth, whereas persistent noise drives the system to a limiting law.
Paper Structure (23 sections, 29 theorems, 174 equations, 10 figures, 1 table)

This paper contains 23 sections, 29 theorems, 174 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Summary of contributions
  3. Notation and preliminaries
  4. Directed graphs and row-stochastic matrices
  5. Time-invariant case and the spectral radius
  6. Anchor agents and the condensely-anchored property
  7. The case where A-E is nonnegative
  8. The case where A-E can contain negative diagonal entries
  9. Impaired averaging
  10. Time-variant case
  11. Vanishing learning rates
  12. Mixed operator norms
  13. Models with noise
  14. General additive noise
  15. A stochastic variant
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $M$ be a complex square matrix. Then $M$ is zero-convergent if and only if $\rho(M)<1$.

Figures (10)

  • Figure 1: Averaging plus learning model. Solid blue edges represent averaging across a fixed influence network. Dashed green arrows represent learning pulls from the ground truth $\bar{\sigma}$.
  • Figure 2: Summary of contributions
  • Figure 3: A sink SCC has no arc leaving it.
  • Figure 4: Example of a digraph $G$ and its condensation $G^\circ$.
  • Figure 6: An example of a condensely-aperiodic digraph.
  • ...and 5 more figures

Theorems & Definitions (71)

  • Theorem 1.1: HJ85
  • Theorem 1.2: Rosenblatt57 or golub2010naive
  • Theorem 1.3
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2: popescu2023averaging
  • proof
  • Definition 2.3
  • Proposition 2.4
  • ...and 61 more