Sampling of the Wiener Process for Remote Estimation Over a Channel With Unknown Delay Statistics

• Published in: IEEE/ACM transactions on networking Vol. 32; no. 3; pp. 1920 - 1935
• Main Authors: Tang, Haoyue, Sun, Yin, Tassiulas, Leandros
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.06.2024; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Age of Information; Algorithms; Delays; Network interfaces; online learning; Optimization; Optimized production technology; Receivers; Sampling; Sensors; Signal processing algorithms; stochastic approximation
• ISSN: 1063-6692; 1558-2566
• DOI: 10.1109/TNET.2023.3331266

In this paper, we study an online sampling problem of the Wiener process. The goal is to minimize the mean squared error (MSE) of the remote estimator under a sampling frequency constraint when the transmission delay distribution is unknown. The sampling problem is reformulated into an optional stopping problem, and we propose an online sampling algorithm that can adaptively learn the optimal stopping threshold through stochastic approximation. We prove that the cumulative MSE regret grows with rate <inline-formula> <tex-math notation="LaTeX">\mathcal {O}(\ln k) </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> is the number of samples. Through Le Cam's two point method, we show that the worst-case cumulative MSE regret of any online sampling algorithm is lower bounded by <inline-formula> <tex-math notation="LaTeX">\Omega (\ln k) </tex-math></inline-formula>. Hence, the proposed online sampling algorithm is minimax order-optimal. Finally, we validate the performance of the proposed algorithm via numerical simulations.