Distributed Consensus Algorithms in Sensor Networks: Quantized Data and Random Link Failures (0712.1609v3)

Abstract: The paper studies the problem of distributed average consensus in sensor networks with quantized data and random link failures. To achieve consensus, dither (small noise) is added to the sensor states before quantization. When the quantizer range is unbounded (countable number of quantizer levels), stochastic approximation shows that consensus is asymptotically achieved with probability one and in mean square to a finite random variable. We show that the meansquared error (m.s.e.) can be made arbitrarily small by tuning the link weight sequence, at a cost of the convergence rate of the algorithm. To study dithered consensus with random links when the range of the quantizer is bounded, we establish uniform boundedness of the sample paths of the unbounded quantizer. This requires characterization of the statistical properties of the supremum taken over the sample paths of the state of the quantizer. This is accomplished by splitting the state vector of the quantizer in two components: one along the consensus subspace and the other along the subspace orthogonal to the consensus subspace. The proofs use maximal inequalities for submartingale and supermartingale sequences. From these, we derive probability bounds on the excursions of the two subsequences, from which probability bounds on the excursions of the quantizer state vector follow. The paper shows how to use these probability bounds to design the quantizer parameters and to explore tradeoffs among the number of quantizer levels, the size of the quantization steps, the desired probability of saturation, and the desired level of accuracy $\epsilon$ away from consensus. Finally, the paper illustrates the quantizer design with a numerical study.

• The paper introduces a dithering-based stochastic approximation approach to mitigate quantization errors under random link failures.
• The Quantized Consensus (QC) algorithm is shown to converge almost surely to a random variable centered on the true sensor average, balancing speed and accuracy.
• The study provides quantizer design guidelines that ensure bounded state excursions and high-probability ε-consensus in practical sensor networks.

Insightful Overview of "Distributed Consensus Algorithms in Sensor Networks: Quantized Data and Random Link Failures"

The paper conducted by Soummya Kar and Jose M. F. Moura tackles significant challenges in distributed consensus algorithms within sensor networks operating under constraints of quantized data and random link failures. In modern sensor networks, achieving consensus is integral for applications such as distributed inference, data fusion, and multi-agent coordination, where agents (sensors) must reliably compute a collective average of their initial states through local information exchanges.

Problem Characterization and Methodology

The paper examines how quantized data transmission and the inherent unreliability of wireless links—manifested as random packet losses—affect consensus algorithms. It is known that traditional consensus protocols assume perfect and continuous communication channels, yet this assumption falters in real-world scenarios marked by finite bandwidth and environmental interference.

To mitigate quantization-induced errors, the authors incorporate dithering—a controlled injection of noise before quantization. This approach randomizes quantization noise, ensuring the errors are independent across communication rounds. The core innovation lies in applying stochastic approximation theories to tackle the consensus problem within such imperfect communication models.

Algorithm Analysis: QC and QCF

Two primary algorithms are discussed:

1. Quantized Consensus (QC): When sensors propagate their states using unbounded quantizers, the authors establish that consensus can be reached asymptotically almost surely. Their stochastic approximation framework proves that sensor states converge to a random variable whose expectation is anchored to the true average of initial sensor values.
2. Quantized Consensus with Finite Quantizer (QCF): Considering bounded initial conditions, the authors delve into consensus with finite-range quantizers. They rigorously analyze the sample paths of the QC algorithm to ensure bounded excursions, facilitating finite quantizer designs that achieve ε-consensus with high probability.

Key Findings and Theoretical Contributions

The paper introduces several pivotal concepts:

• Convergence Analysis: Through stochastic approximation, the paper formalizes conditions under which the mean squared error (MSE) between the achieved and true consensus can be tightly bounded. Interestingly, by appropriately tuning the link weight sequence, one can trade-off convergence speed with precision.
• Uniform Sample Path Bounds: For QCF, the authors demonstrate that, even with limited information accuracy due to finite quantization, the algorithm can maintain stability of sensor states within prescribed error margins.
• Quantizer Design: By deriving probability bounds on excursions and formulating optimization criteria, the authors provide quantizer design guidelines—balancing the quantization step size and range to meet desired performance objectives.

Practical Implications and Future Directions

The practical implications of this research extend to several fields:

• Wireless Sensor Networks (WSNs): The ability to operate under quantization and network-induced imperfections is crucial for energy-constrained WSNs, where finite bandwidth and power are limiting factors.
• Optimization of Communication Protocols: The insights allow network designers to tailor consensus protocols that adapt to the statistical properties of noise and data losses.

Future research could expand upon optimizing these methods further for dynamic, time-varying networks, and investigate applications in distributed neural networks where communication constraints play a significant role. Additionally, exploring the impact of different network topologies and their interplay with quantization-induced errors would enrich the theory's applicability.

In conclusion, this paper offers valuable contributions to the field of distributed computing and consensus in sensor networks, providing analytical tools and theoretical insights that balance robustness with efficiency in a quantized communication landscape.