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Maximizing Truth Learning in a Social Network is NP-hard

Filip Úradník, Amanda Wang, Jie Gao

TL;DR

This work investigates the computational complexity of maximizing truth learning in sequential, network-constrained decision making. It develops a reduction from 3-SAT to a network learning problem under both Bayesian and majority dynamics, using variable cells and clause gadgets to couple satisfiability with higher learning rates under carefully chosen orderings. The results establish NP-hardness for deciding whether the optimal learning rate exceeds a threshold and prove APX-hardness for approximating the optimum when using Bayesian inference, with extensions to fixed private-signal accuracy. These findings reveal fundamental combinatorial barriers to designing orderings that promote network-wide truth learning in distributed settings.

Abstract

Sequential learning models situations where agents predict a ground truth in sequence, by using their private, noisy measurements, and the predictions of agents who came earlier in the sequence. We study sequential learning in a social network, where agents only see the actions of the previous agents in their own neighborhood. The fraction of agents who predict the ground truth correctly depends heavily on both the network topology and the ordering in which the predictions are made. A natural question is to find an ordering, with a given network, to maximize the (expected) number of agents who predict the ground truth correctly. In this paper, we show that it is in fact NP-hard to answer this question for a general network, with both the Bayesian learning model and a simple majority rule model. Finally, we show that even approximating the answer is hard.

Maximizing Truth Learning in a Social Network is NP-hard

TL;DR

This work investigates the computational complexity of maximizing truth learning in sequential, network-constrained decision making. It develops a reduction from 3-SAT to a network learning problem under both Bayesian and majority dynamics, using variable cells and clause gadgets to couple satisfiability with higher learning rates under carefully chosen orderings. The results establish NP-hardness for deciding whether the optimal learning rate exceeds a threshold and prove APX-hardness for approximating the optimum when using Bayesian inference, with extensions to fixed private-signal accuracy. These findings reveal fundamental combinatorial barriers to designing orderings that promote network-wide truth learning in distributed settings.

Abstract

Sequential learning models situations where agents predict a ground truth in sequence, by using their private, noisy measurements, and the predictions of agents who came earlier in the sequence. We study sequential learning in a social network, where agents only see the actions of the previous agents in their own neighborhood. The fraction of agents who predict the ground truth correctly depends heavily on both the network topology and the ordering in which the predictions are made. A natural question is to find an ordering, with a given network, to maximize the (expected) number of agents who predict the ground truth correctly. In this paper, we show that it is in fact NP-hard to answer this question for a general network, with both the Bayesian learning model and a simple majority rule model. Finally, we show that even approximating the answer is hard.

Paper Structure

This paper contains 18 sections, 28 theorems, 36 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Proof of NP-hardness for the Bayesian Model
  4. Proof Idea
  5. Graph Construction & Notation
  6. Gadget Learning Rates
  7. Optimal Ordering & Restrictions on p
  8. Picking the Epsilon
  9. Generalization for a constant range of p
  10. Proof of NP-hardness for Majority Dynamics
  11. Adapted Graph Construction
  12. Approximating the Optimal LR
  13. Conclusion
  14. Bayesian General Learning Rates
  15. Full Proof of Theorem \ref{['thm:nphardness_maj']}
  16. ...and 3 more sections

Key Result

Theorem 3.1

Network Learning with the Bayesian learning rule $\mu = \mu^B$ is NP-hard.

Figures (6)

  • Figure 1: The cell for variable $x_i$.
  • Figure 2: The graph $G_\varphi$ for $\varphi = C = \left( x \lor y \lor \lnot z \right)$.
  • Figure 3: The relationship between learning rates of $\mathbb{G}\!\left(C\right)$, depending on the number of "on" literals, as a function of $p$.
  • Figure 5: The graph $G_\varphi$ for $\varphi = C = \left( x \lor y \lor \lnot z \right)$.
  • Figure 6: The value of $\varepsilon$ w.r.t. choice of $p$ below which (additive) approximation is hard (that is, $\delta = \frac{7}{8}$Hastad2001-fg). The requirement of $\varepsilon > 0$ gives us that $p \geq \sqrt{7/8}$.
  • ...and 1 more figures

Theorems & Definitions (43)

  • Definition 2.1: Social Network
  • Definition 2.2
  • Definition 2.3: Optimal LRs
  • Definition 2.4: Opt Network Learning
  • Definition 2.5: Network Learning
  • Theorem 3.1
  • Definition 3.1: Variable cell
  • Definition 3.2: Clause gadget
  • Definition 3.2: Formula graph
  • Lemma 3.3: Bayesian Cell LR
  • ...and 33 more