Gossip Algorithms for Distributed Signal Processing (1003.5309v1)

Abstract: Gossip algorithms are attractive for in-network processing in sensor networks because they do not require any specialized routing, there is no bottleneck or single point of failure, and they are robust to unreliable wireless network conditions. Recently, there has been a surge of activity in the computer science, control, signal processing, and information theory communities, developing faster and more robust gossip algorithms and deriving theoretical performance guarantees. This article presents an overview of recent work in the area. We describe convergence rate results, which are related to the number of transmitted messages and thus the amount of energy consumed in the network for gossiping. We discuss issues related to gossiping over wireless links, including the effects of quantization and noise, and we illustrate the use of gossip algorithms for canonical signal processing tasks including distributed estimation, source localization, and compression.

• The paper presents a theoretical framework showing how the spectral gap of gossip matrices governs convergence rates in various network topologies.
• It introduces enhanced variants like geographic and memory-based gossip to significantly improve message complexity and efficiency.
• The study demonstrates that gossip algorithms reliably handle quantization and lossy links, with applications in parameter estimation, source localization, and compression.

Gossip Algorithms for Distributed Signal Processing

In the paper "Gossip Algorithms for Distributed Signal Processing," the authors, Alexandros G. Dimakis, Soummya Kar, Jose M.F. Moura, Michael G. Rabbat, and Anna Scaglione, present an extensive review of gossip algorithms and their use in wireless sensor network (WSN) applications. The overview spans theoretical developments, practical implementations, and insights into various models and methods optimizing the efficacy of gossip algorithms, especially under constraints such as limited bandwidth and energy resources.

Introduction and Context

Gossip algorithms are fundamentally designed for in-network processing where intermediate nodes perform computations, significantly reducing the amount of raw data that needs to be transmitted to a central point. This decentralized approach avoids creating bottlenecks, single points of failure, and overhead for route maintenance, which are severe drawbacks in traditional routing schemes. This paper anchors its discussion on these benefits and extends into specific considerations for wireless sensor networks, such as robustness to unreliable wireless conditions and energy consumption optimization.

Convergence Rates and Accelerated Gossip

One of the critical aspects of gossip algorithms is understanding and optimizing their convergence rates. The convergence rate directly influences the number of iterations required and, thus, the total time and energy consumed in communication. The authors provide a robust theoretical framework, showing that the spectral gap of the gossip matrix, specifically the second largest eigenvalue, is instrumental in determining the rate of convergence.

For topologies like complete graphs, uniform pairwise gossiping converges relatively quickly, requiring $\Theta(n \log \epsilon^{-1})$ messages. Conversely, for less connected topologies like random geometric graphs and grids, traditional pairwise gossiping exhibits slower convergence, scaling quadratically (${O}(n^2)$) with the number of nodes.

To address the inefficiency in such topologies, the authors discuss several improved variants:

• Geographic Gossip: This approach introduces geographic routing to accelerate diffusion, consequently reducing the message complexity to $O(n^{1.5} \log \epsilon^{-1})$ for random geometric graphs.
• Location-Aided Distributed Averaging (LADA): This method incorporates lifting techniques from Markov chain theory, enhancing convergence while accommodating partial location information and intermittent links.
• Memory-Based and Predictive Gossip Schemes: These algorithms retain previous state values to predict future updates, significantly improving convergence rates for specific network topologies.

Wireless Considerations and Rate Limitations

The paper presents a detailed evaluation of gossip algorithms under the constraints of the wireless channel. It examines the impact of quantization and finite transmission rates on the convergence and overall performance of the algorithms. Quantization introduces distortions; however, through methods like subtractive dithered quantization, the paper shows that convergence to the true mean can still be achieved almost surely. The analysis of gossip in environments with intermittent and lossy links reveals that gossip algorithms degrade gracefully as long as the network remains connected on average.

Applications in Wireless Sensor Networks

The authors illustrate the versatility of gossip algorithms via several practical applications:

• Distributed Linear Parameter Estimation: The paper presents stochastic approximation methods that ensure convergence to the true parameter values at each sensor node despite quantized communications and noisy observations.
• Source Localization: Using simple RSS measurements and appropriately designed gossip algorithms, nodes can estimate the location of a signal-emitting source effectively.
• Distributed Compression and Field Estimation: By leveraging concepts from compressive sensing, gossip algorithms can efficiently approximate a global field measurement by communicating only a few significant coefficients, thus substantially reducing the required bandwidth.

Forward Directions and Conclusion

The paper concludes by identifying several promising areas for future research, such as extending gossip to compute non-linear functions, analyzing convergence under more complex topological changes, and mobile sensor networks. It also hints at intriguing potential applications, including social network analysis and mobile device interactions, which present new challenges and opportunities for gossip algorithms.

In summary, this paper provides an exhaustive survey of gossip algorithms, elucidating their theoretical foundations, practical implications, and applications in distributed signal processing within wireless sensor networks. It forms a vital resource for researchers embarking on exploring and exploiting gossip algorithms in various distributed systems.