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Social Learning in Community Structured Graphs

Valentina Shumovskaia, Mert Kayaalp, Ali H. Sayed

TL;DR

The paper addresses distributed hypothesis testing in heterogeneous networks where agents observe data generated from local truths. It advocates adaptive social learning (ASL) on community-structured graphs (SBMs) and derives conditions under which each community can converge to its own truth by tuning the step-size $\delta$. Theoretical results characterize the mean log-belief behavior and provide explicit thresholds (e.g., on $\delta$) for two-community SBM, with extensions to asymmetric cases, and the analysis is complemented by simulations and Twitter-based experiments. The findings demonstrate that ASL outperforms traditional SL in multi-truth, nonstationary settings and offers a practical framework for decentralized multi-task learning on networks.

Abstract

Traditional social learning frameworks consider environments with a homogeneous state, where each agent receives observations conditioned on that true state of nature. In this work, we relax this assumption and study the distributed hypothesis testing problem in a heterogeneous environment, where each agent can receive observations conditioned on their own personalized state of nature (or truth). We particularly focus on community structured networks, where each community admits their own true hypothesis. This scenario is common in various contexts, such as when sensors are spatially distributed, or when individuals in a social network have differing views or opinions. We show that the adaptive social learning strategy is a preferred choice for nonstationary environments, and allows each cluster to discover its own truth.

Social Learning in Community Structured Graphs

TL;DR

The paper addresses distributed hypothesis testing in heterogeneous networks where agents observe data generated from local truths. It advocates adaptive social learning (ASL) on community-structured graphs (SBMs) and derives conditions under which each community can converge to its own truth by tuning the step-size . Theoretical results characterize the mean log-belief behavior and provide explicit thresholds (e.g., on ) for two-community SBM, with extensions to asymmetric cases, and the analysis is complemented by simulations and Twitter-based experiments. The findings demonstrate that ASL outperforms traditional SL in multi-truth, nonstationary settings and offers a practical framework for decentralized multi-task learning on networks.

Abstract

Traditional social learning frameworks consider environments with a homogeneous state, where each agent receives observations conditioned on that true state of nature. In this work, we relax this assumption and study the distributed hypothesis testing problem in a heterogeneous environment, where each agent can receive observations conditioned on their own personalized state of nature (or truth). We particularly focus on community structured networks, where each community admits their own true hypothesis. This scenario is common in various contexts, such as when sensors are spatially distributed, or when individuals in a social network have differing views or opinions. We show that the adaptive social learning strategy is a preferred choice for nonstationary environments, and allows each cluster to discover its own truth.
Paper Structure (19 sections, 7 theorems, 88 equations, 6 figures)

This paper contains 19 sections, 7 theorems, 88 equations, 6 figures.

Table of Contents

  1. Introduction and Related Work
  2. Social Learning Model
  3. Truth Learning
  4. Traditional Social Learning
  5. Adaptive Social Learning
  6. Stochastic Block Model
  7. SBM Network Properties
  8. Truth Learning
  9. Computer Experiments
  10. Twitter data
  11. Simulated data
  12. Two communities
  13. Three communities of different sizes
  14. Sparse networks
  15. Conclusions
  16. ...and 4 more sections

Key Result

Lemma 1

Let $\Theta^{\star}\subset \Theta$ denote the solution to (eq:div2). In steady-state, as $i\rightarrow \infty$, the belief of every agent $k$ at any hypothesis $\theta\notin\Theta^{\star}$ vanishes almost surely: at the following rate of convergence: for any $\theta^\star \in \Theta^\star$.

Figures (6)

  • Figure 1: Network illustration with $n_0=20$, $n_1 = 15$, $p_0=0.8$, $p_1=0.9$, $q_0 = q_1 = 0.1$.
  • Figure 2: A Twitter network of UK parliament members.
  • Figure 3: The algorithm's performance in identifying the true state of each node, using the adaptive social learning strategy. The probabilities of error (shown inside the boxes) $\mathbb P(\widehat{\boldsymbol{\theta}}_{k,i} \neq \theta_k^\star)$ are approximated based on 500 iterations.
  • Figure 4: Evolution of log-belief ratios $\log \frac{\boldsymbol \psi_{i,k}(\theta_0)}{\boldsymbol \psi_{i,k}(\theta_1)}$ over time with different step-size $\delta$ (mean and standard deviations over 500 algorithm runs). Agent 1 belongs to the first cluster and follows $\theta_0$, while agent 21 belongs to the second cluster and follows $\theta_1$.
  • Figure 5: The algorithm's performance in identifying the true state of each node, using the adaptive social learning strategy. The probabilities of error $\mathbb P(\widehat{\boldsymbol{\theta}}_{k,i} \neq \theta_k^\star)$ are approximated based on 500 iterations. In boxes, we show an average error per cluster.
  • ...and 1 more figures

Theorems & Definitions (20)

  • Lemma 1: Convergence of traditional social learning nedic2017fast
  • Remark 1
  • Lemma 2: Log-belief ratios under adaptive social learning
  • proof
  • Remark 2
  • Lemma 3: Expected combination matrix
  • proof
  • Lemma 4: Matrix powers
  • proof
  • Theorem 1: Log-belief ratios for symmetric communities
  • ...and 10 more