Optimal Sampling and Scheduling for Remote Estimation of Multiple Gauss-Markov Processes over Parallel Channels

Core Concepts

The paper develops a Whittle index policy to optimally schedule the sampling and transmission of multiple continuous-time Gauss-Markov processes over parallel channels with the objective of minimizing the weighted sum of the time-average expected estimation errors.

Abstract

The paper studies a remote estimation system with N source-estimator pairs and L channels, where each source is a continuous-time Gauss-Markov process. A centralized scheduler determines when to take samples from the sources and transmit them over the channels to the remote estimators. The objective is to minimize the weighted sum of the time-average expected estimation errors of the Gauss-Markov sources.
The key highlights and insights are:

The problem is formulated as a continuous-time Restless Multi-armed Bandit (RMAB) problem with a continuous state space, which is challenging to solve.

The authors prove the indexability of the RMAB problem and derive an exact expression for the Whittle index, which is the first such result for multi-source remote estimation of Gauss-Markov processes.

The paper also investigates signal-agnostic remote estimation, where the scheduling problem becomes a multi-source Age of Information (AoI) minimization problem over parallel channels with i.i.d. random transmission times. The authors establish the indexability and derive the Whittle index for this case as well.

The results unify two important theoretical frameworks for remote estimation and AoI minimization: threshold-based sampling and Whittle index-based scheduling. In the single-source, single-channel scenario, the authors show that the optimal solution can be expressed using both approaches.

The numerical results demonstrate that the proposed Whittle index policy outperforms existing policies, especially when some of the Gauss-Markov processes are highly unstable.

Stats

The following sentences contain key metrics or important figures:
The objective is to minimize the weighted sum of the time-average expected estimation errors of the N Gauss-Markov sources.
The sample transmission times Yn,i's are i.i.d. across samples and channels with mean 0 < E[Yn,i] < ∞.
The Gauss-Markov processes Xn,t are defined by the stochastic differential equation dXn,t = θn(μn - Xn,t)dt + σndWn,t, where θn, μn, and σn > 0 are the parameters.

Quotes

"To the extent of our knowledge, this is the first Whittle index policy for multi-source signal-aware remote estimation of Gauss-Markov processes."
"Notably, the Whittle index is equal to zero if and only if two conditions are satisfied: (i) the channel must be idle, and (ii) the threshold condition is precisely met at time t."

Key Insights Distilled From

Remote Estimation of Gauss-Markov Processes over Multiple Channels: A Whittle Index Policy

by Tasmeen Zama... at arxiv.org 04-17-2024

https://arxiv.org/pdf/2305.04809.pdf

Deeper Inquiries

How can the proposed Whittle index policy be extended to handle correlated Gauss-Markov processes or non-Gaussian processes

The proposed Whittle index policy can be extended to handle correlated Gauss-Markov processes or non-Gaussian processes by incorporating the correlation structure or non-Gaussian characteristics into the formulation of the problem. For correlated Gauss-Markov processes, the covariance matrix of the processes can be taken into account in the estimation error calculations and the Whittle index computation. By considering the interdependencies between the processes, the scheduling policy can be adjusted to optimize the estimation performance while taking into consideration the correlation structure.
When dealing with non-Gaussian processes, such as processes with heavy-tailed distributions or asymmetric distributions, the estimation error metrics and the Whittle index calculations may need to be adapted to account for the non-Gaussian nature of the processes. Techniques from robust estimation theory or non-parametric statistics can be utilized to handle the non-Gaussian processes effectively. By incorporating the appropriate statistical models and estimation methods, the Whittle index policy can be extended to accommodate a wider range of process types, including non-Gaussian processes.

What are the potential limitations of the threshold-based sampling and Whittle index-based scheduling approaches, and how can they be addressed

The potential limitations of the threshold-based sampling and Whittle index-based scheduling approaches include:

Sensitivity to model assumptions: Both approaches rely on specific assumptions about the underlying processes, such as stationarity, linearity, and known parameters. Deviations from these assumptions can lead to suboptimal performance.
Computational complexity: The calculation of the Whittle index and the determination of threshold values can be computationally intensive, especially for complex systems with multiple sources and channels.
Lack of adaptability: The fixed thresholds and Whittle index values may not adapt well to changing system dynamics or environmental conditions, leading to suboptimal performance in dynamic scenarios.
Limited scalability: The approaches may face challenges when scaling up to systems with a large number of sources and channels, as the complexity of optimization and decision-making increases.

These limitations can be addressed by:

Robustness analysis: Conducting sensitivity analysis to assess the robustness of the approaches to deviations from model assumptions and exploring robust optimization techniques.
Algorithm optimization: Developing efficient algorithms for calculating the Whittle index and determining thresholds to reduce computational complexity and improve scalability.
Adaptive strategies: Incorporating adaptive mechanisms that allow the thresholds and Whittle index values to be updated based on real-time feedback and system conditions for improved adaptability.
Hybrid approaches: Integrating the threshold-based sampling and Whittle index-based scheduling with machine learning or reinforcement learning techniques to enhance performance and scalability in complex systems.

How can the joint optimization of the sampler and estimator design be incorporated into the framework to further improve the remote estimation performance

The joint optimization of the sampler and estimator design can be incorporated into the framework to further improve remote estimation performance by considering the interaction between the sampling strategy and the estimation algorithm. This joint optimization aims to jointly optimize the sampling policy, transmission scheduling, and estimation algorithm to minimize the overall estimation error and maximize the information gain.
Some approaches to incorporate joint optimization include:

Co-design framework: Develop a co-design framework that jointly optimizes the sampling strategy (including sampling rates, transmission scheduling, and channel allocation) and the estimation algorithm (such as the MMSE estimator) to achieve the best trade-off between sampling cost and estimation accuracy.
Reinforcement learning: Utilize reinforcement learning techniques to learn an adaptive sampling and scheduling policy that maximizes the long-term estimation performance based on feedback from the estimation process.
Dynamic programming: Apply dynamic programming methods to solve the joint optimization problem by considering the sequential decision-making process of sampling, transmission, and estimation in a unified framework.
Online learning: Implement online learning algorithms that continuously update the sampling and scheduling decisions based on real-time data and feedback, allowing the system to adapt to changing conditions and optimize performance over time.

By integrating the sampler and estimator design into a unified optimization framework, the remote estimation system can achieve enhanced performance, adaptability, and efficiency in capturing and processing information from multiple sources.