On the Price of Decentralization in Decentralized Detection

2024-09-01

IEEE Transactions on Information Theory

AI Summary

This paper investigates the fundamental limits of error probability for decentralized detection algorithms based on social learning over directed graphs. We first analyze the distributed Bayesian rule proposed by Lalitha et al., showing that while it achieves asymptotic consistency, its error probability decays exponentially at a rate strictly slower than the centralized optimal rate—due to imbalanced information diffusion caused by local weighting. To bridge this gap, we propose a corrected learning rule that restructures neighbor information fusion; remarkably, each node achieves the exact same error exponent as the centralized oracle, using only knowledge of its own observation distribution. This is the first result proving that, in the large-sample limit, decentralized detection can eliminate the “first-order penalty” in error exponent decay, fully closing the theoretical performance gap with centralized detection. Our analysis leverages large deviations theory, graph-based propagation modeling, and distributed Bayesian updating.

Technology Category

• Machine Learning: Bayesian Learning
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

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.

Problem

• Investigates error limits in decentralized detection algorithms
• Proposes modified rule to close error exponent gap
• Shows no first-order penalty from decentralization

Innovation

• Decentralized detection algorithms analysis
• Modified learning rule proposal
• Error exponent gap closure