Distributed Bayesian Estimation in Sensor Networks: Consensus on Marginal Densities (2312.01227v2)

Abstract: In this paper, we aim to design and analyze distributed Bayesian estimation algorithms for sensor networks. The challenges we address are to (i) derive a distributed provably-correct algorithm in the functional space of probability distributions over continuous variables, and (ii) leverage these results to obtain new distributed estimators restricted to subsets of variables observed by individual agents. This relates to applications such as cooperative localization and federated learning, where the data collected at any agent depends on a subset of all variables of interest. We present Bayesian density estimation algorithms using data from non-linear likelihoods at agents in centralized, distributed, and marginal distributed settings. After setting up a distributed estimation objective, we prove almost-sure convergence to the optimal set of pdfs at each agent. Then, we prove the same for a storage-aware algorithm estimating densities only over relevant variables at each agent. Finally, we present a Gaussian version of these algorithms and implement it in a mapping problem using variational inference to handle non-linear likelihood models associated with LiDAR sensing.

• The paper introduces distributed Bayesian algorithms, including SMD and DMSMD, for joint and marginal pdf estimation across sensor networks.
• It employs local agent data and neighbor consensus to enable real-time inference with reduced communication and computational overhead.
• Theoretical analysis proves almost sure convergence, validated by a distributed mapping application in robotic sensor networks.

Distributed Bayesian Estimation in Sensor Networks

Overview of Distributed Estimation

In the context of large sensor networks embedded within urban infrastructure or transportation systems, decentralized solutions to the data inference problem are increasingly desired. Centralized approaches, while potentially more accurate, are prohibitive in terms of the communication and computational overhead they introduce, especially in real-time applications. This challenge motivates the need for algorithms that can parallelize inference across nodes, reducing both communication load and vulnerability to node failures.

Distributed Estimation Algorithm

The paper presents two distributed Bayesian algorithms for estimating probability density functions (pdfs) across networked agents. Both algorithms rely on local agent data and consensus through communication with immediate neighbors. The first algorithm, named Distributed SMD (Stochastic Mirror Descent), is tasked with estimating unknown variables jointly across the network. It leverages the SMD optimization approach over functional spaces, allowing agents to integrate new observations while maintaining consensus with their neighbors.

The second algorithm, termed Distributed Marginal SMD (DMSMD), extends the capabilities of the first by focusing on the estimation of local marginal densities relevant to each agent's observations. Due to its design, DMSMD reduces memory and computational requirements significantly, enhancing the system's efficiency by allowing agents to maintain and share only subset-relevant information, rather than striving for a joint estimation across the entire network.

Convergence Results

Both algorithms' performance is analyzed rigorously. The paper provides theoretical guarantees of almost sure convergence for the estimation algorithms. Particularly, it is shown that the algorithms' iterates will converge to a pdf consistent with the true underlying data generation process that the sensor network aims to capture. These results ensure that the distributed approach does not sacrifice the reliability of the estimations compared to a centralized method.

Implementation and Use Case: Distributed Mapping

An application of the DMSMD algorithm is illustrated through a distributed mapping problem where a group of robots collects environmental data to infer a map. Employing DMSMD, robots are able to construct a map of the entire environment by sharing and updating local inferences about their immediate surroundings. The example provided utilizes Gaussian models in the estimation process, presenting a practical method for implementing these theoretical algorithms in real-world robotic applications.

Significance and Implications

The paper contributes to the field of distributed estimation by addressing the all-important question of efficient data processing in large sensor networks. By demonstrating almost sure convergence to optimal distributions while maintaining lower memory and computational costs, the work establishes a strong precedent for future exploration and application of decentralized inference in various areas. The potential impact is evident in smart city developments, autonomous vehicles, and other IoT (Internet of Things) environments where sensor data must be processed in real-time and at scale.