Signal-Comparison-Based Distributed Estimation Under Decaying Average Data Rate Communications

TL;DR

The paper tackles distributed parameter estimation over networks with decaying communication frequency by integrating a signal-comparison (SC) consensus protocol with a stochastic event-triggered mechanism and dithering-based binary quantization. It proves that the proposed SC-based estimator achieves almost sure and mean-square convergence to the true parameter $\theta$, with a polynomial rate in the almost-sure sense. The analysis also shows that both local and global average data rates decay polynomially, enabling significantly reduced communication without sacrificing convergence, and it characterizes the trade-off between convergence speed and communication cost via event-triggering exponents. Simulation demonstrates faster convergence and lower communication relative to benchmarks, highlighting practical viability for low-rate networks. The framework offers avenues for extension to more complex topologies and online distributed learning under bandwidth constraints.

Abstract

The paper investigates the distributed estimation problem under low bit rate communications. Based on the signal-comparison (SC) consensus protocol under binary-valued communications, a new consensus+innovations type distributed estimation algorithm is proposed. Firstly, the high-dimensional estimates are compressed into binary-valued messages by using a periodic compressive strategy, dithered noises and a sign function. Next, based on the dithered noises and expanding triggering thresholds, a new stochastic event-triggered mechanism is proposed to reduce the communication frequency. Then, a modified SC consensus protocol is applied to fuse the neighborhood information. Finally, a stochastic approximation estimation algorithm is used to process innovations. The proposed SC-based algorithm has the advantages of high effectiveness and low communication cost. For the effectiveness, the estimates of the SC-based algorithm converge to the true value in the almost sure and mean square sense. A polynomial almost sure convergence rate is also obtained. For the communication cost, the local and global average bit rates for communications decay to zero at a polynomial rate. The trade-off between the convergence rate and the communication cost is established through event-triggered coefficients. A better convergence rate can be achieved by decreasing event-triggered coefficients, while lower communication cost can be achieved by increasing event-triggered coefficients. A simulation example is given to demonstrate the theoretical results.

Figures (6)

• Figure 1: Communication topology.
• Figure 2: The trajectory of $\frac{1}{N} \sum_{i=1}^{N} \lVert \tilde{\uptheta}_{i,k} \rVert^2$
• Figure 3: Convergence rates with different $\nu$
• Figure 4: Average data rates with different $\nu$
• Figure 5: The trajectories of $\ln \left( \frac{1}{50N} \sum_{t=1}^{50} \sum_{i=1}^{N} \lVert \tilde{\uptheta}_{i,k}^t \rVert^2 \right)$ or different algorithms

Theorems & Definitions (45)

• Remark 2.2
• Remark 2.4
• Definition 2.6
• Remark 2.7
• Remark 2.8
• Theorem 3.1: Theorem 1 of Ke2023Signal
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Remark 3.6