Convergence Rate Analysis of Distributed Gossip (Linear Parameter) Estimation: Fundamental Limits and Tradeoffs (1011.1677v1)

Abstract: The paper considers gossip distributed estimation of a (static) distributed random field (a.k.a., large scale unknown parameter vector) observed by sparsely interconnected sensors, each of which only observes a small fraction of the field. We consider linear distributed estimators whose structure combines the information \emph{flow} among sensors (the \emph{consensus} term resulting from the local gossiping exchange among sensors when they are able to communicate) and the information \emph{gathering} measured by the sensors (the \emph{sensing} or \emph{innovations} term.) This leads to mixed time scale algorithms--one time scale associated with the consensus and the other with the innovations. The paper establishes a distributed observability condition (global observability plus mean connectedness) under which the distributed estimates are consistent and asymptotically normal. We introduce the distributed notion equivalent to the (centralized) Fisher information rate, which is a bound on the mean square error reduction rate of any distributed estimator; we show that under the appropriate modeling and structural network communication conditions (gossip protocol) the distributed gossip estimator attains this distributed Fisher information rate, asymptotically achieving the performance of the optimal centralized estimator. Finally, we study the behavior of the distributed gossip estimator when the measurements fade (noise variance grows) with time; in particular, we consider the maximum rate at which the noise variance can grow and still the distributed estimator being consistent, by showing that, as long as the centralized estimator is consistent, the distributed estimator remains consistent.

• The paper introduces a framework for distributed gossip estimation that combines consensus and innovation to achieve robust parameter convergence.
• It employs mixed time-scale algorithms to manage sensor communications and measurements, mimicking centralized estimation performance.
• The authors demonstrate that with proper network observability and gain tuning, the estimator attains consistency and asymptotic normality even amid communication failures.

Convergence Rate Analysis of Distributed Gossip Estimation

The paper "Convergence Rate Analysis of Distributed Gossip (Linear Parameter) Estimation: Fundamental Limits and Tradeoffs" by Soummya Kar and Jose M. F. Moura presents a comprehensive paper on distributed estimation processes in sensor networks, focusing on gossip protocols for communication and parameter estimation. The research addresses the problem of estimating a static, distributed random field using sparsely interconnected sensors where the sensors only capture fragments of the field. The authors propose a class of linear distributed estimators that combine consensus and innovations to achieve robust estimation across the network.

Key Contributions

1. Estimation under Gossip Protocols: The paper introduces a framework for distributed estimation using gossip protocols, where sensors communicate and share information intermittently due to network constraints and possible link failures. The framework addresses both the communication gaps and variations in observation quality due to noise.
2. Mixed Time Scale Algorithms: A significant contribution is the development of algorithms that operate on mixed time scales—consensus for information flow and innovations for parameter measurement. This dual time scale is pivotal in ensuring that despite network limitations, the distributed estimation aligns with performance expectations of centralized systems.
3. Distributed Observability and Fisher Information Rate: The authors propose a distributed observability condition and extend the concept of Fisher information rate to the distributed setting. They demonstrate that under specific conditions, distributed estimators can achieve convergence rates similar to centralized estimators.
4. Consistency and Convergence Analysis: The paper rigorously analyzes the conditions under which distributed estimations remain consistent, even when the noise variance increases over time. Highlighting the importance of structural network properties and gain tuning, it establishes the consistency and asymptotic normality of the estimators.
5. Impact of Network Dynamics: Analysis extends to quantify the impact of network changes such as link failures, including conditions where distributed estimators can still converge to true parameter values effectively.

Implications and Future Directions

The insights offered in this paper have profound implications for the design and implementation of sensor networks and distributed systems. Practically, this means that even in the presence of significant network limitations and environmental noise, sensor networks can still perform estimations robustly with suitable algorithmic adjustments. The theoretical underpinning extends the scope of distributed consensus and estimation algorithms and contributes to fields like wireless sensor networks, cyber-physical systems, and adaptive control systems.

Future developments could focus on extending these concepts to nonlinear estimators and dynamic fields, accommodating more complex network topologies and communication uncertainties. Additionally, leveraging these insights for designing energy-efficient, robust, and fault-tolerant networks would be a logical advancement.

In summary, Kar and Moura's paper provides valuable methodologies for bridging the gap between distributed and centralized estimation performance through innovative use of gossip protocols and time scale management. It sets a foundation for future explorations into more resilient and adaptive distributed estimation paradigms in sensor networks.