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On the Price of Decentralization in Decentralized Detection

Bruce, Huang, I-Hsiang Wang

TL;DR

It is shown that while the original social learning rule of Lalitha et al., 2018 achieves asymptotically vanishing error probabilities as the number of samples tends to infinity, it suffers a gap in the achievable error exponent compared to the centralized case.

Abstract

Fundamental limits on the error probabilities of a family of decentralized detection algorithms (eg., the social learning rule proposed by Lalitha et al. over directed graphs are investigated. In decentralized detection, a network of nodes locally exchanging information about the samples they observe with their neighbors to collectively infer the underlying unknown hypothesis. Each node in the network weighs the messages received from its neighbors to form its private belief and only requires knowledge of the data generating distribution of its observation. In this work, it is first shown that while the original social learning rule of Lalitha et al. achieves asymptotically vanishing error probabilities as the number of samples tends to infinity, it suffers a gap in the achievable error exponent compared to the centralized case. The gap is due to the network imbalance caused by the local weights that each node chooses to weigh the messages received from its neighbors. To close this gap, a modified learning rule is proposed and shown to achieve error exponents as large as those in the centralized setup. This implies that there is essentially no first-order penalty caused by decentralization in the exponentially decaying rate of error probabilities.

On the Price of Decentralization in Decentralized Detection

TL;DR

It is shown that while the original social learning rule of Lalitha et al., 2018 achieves asymptotically vanishing error probabilities as the number of samples tends to infinity, it suffers a gap in the achievable error exponent compared to the centralized case.

Abstract

Fundamental limits on the error probabilities of a family of decentralized detection algorithms (eg., the social learning rule proposed by Lalitha et al. over directed graphs are investigated. In decentralized detection, a network of nodes locally exchanging information about the samples they observe with their neighbors to collectively infer the underlying unknown hypothesis. Each node in the network weighs the messages received from its neighbors to form its private belief and only requires knowledge of the data generating distribution of its observation. In this work, it is first shown that while the original social learning rule of Lalitha et al. achieves asymptotically vanishing error probabilities as the number of samples tends to infinity, it suffers a gap in the achievable error exponent compared to the centralized case. The gap is due to the network imbalance caused by the local weights that each node chooses to weigh the messages received from its neighbors. To close this gap, a modified learning rule is proposed and shown to achieve error exponents as large as those in the centralized setup. This implies that there is essentially no first-order penalty caused by decentralization in the exponentially decaying rate of error probabilities.
Paper Structure (32 sections, 10 theorems, 131 equations, 10 figures)

This paper contains 32 sections, 10 theorems, 131 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Contribution
  3. Related Work
  4. Paper Organization
  5. Problem Formulation and Preliminaries
  6. Problem Formulation
  7. Social Learning Rule
  8. Main Results
  9. Modified Learning Rule
  10. First-Order Results
  11. Higher-Order Asymptotics
  12. Obtaining the Geometric Weights
  13. Estimating the Optimal Geometric Weights in a Decentralized Way
  14. Combining the Learning Rule with the Estimation of the Optimal Geometric Weights
  15. Simulations
  16. ...and 17 more sections

Key Result

Theorem 1

Suppose that Assumptions assu:graph_strongly_connected and assum:irreducible_aperiodic hold. For the Neyman-Pearson problem, the type-II error exponent for each node $i$ is characterized as shown on the top of the next page.

Figures (10)

  • Figure 1: Effect of network imbalance.
  • Figure 2: A node (colored red) with small $\pi_i$.
  • Figure 3: Convergence of the error exponent.
  • Figure 4: Convergence of the error exponent in log scale.
  • Figure 5: Impact of network imbalance.
  • ...and 5 more figures

Theorems & Definitions (15)

  • Definition 1: Log-Belief Ratio
  • Definition 2: Log-Belief Ratio Test
  • Remark 1: Equivalence to geometrically weighting the likelihood function
  • Definition 3: Probability of Error
  • Theorem 1: Neyman-Pearson Error Exponent for General Geometric Weights
  • Theorem 2: Social Learning is Almost as Good as Centralized Detection in Neyman-Pearson Problem
  • Theorem 3: Social Learning is Almost as Good as Centralized Detection under the Bayes Setting
  • Theorem 4: The Effect of Decentralization in Neyman-Pearson Problem
  • Corollary 1: Viewing the Price of Decentralization as a Constant Time Delay for Decentralized Testing
  • proof
  • ...and 5 more