Regret-Optimal and Stability-Enhanced Online Sampling of the Wiener Process for Remote Estimation over an Unreliable Channel with Unknown Statistics

We study the online sampling problem of a Wiener process, where the samples are transmitted to the remote estimator via an unreliable channel. The objective is to minimize the long-term average mean squared error (MSE) of the remote estimator subject to a sampling frequency constraint. To derive the online sampling strategy when the channel statistics is unknown, we reformulate the problem as an optimal stopping problem and review the sufficient condition of the offline MSE-optimal sampling threshold. Utilizing the sufficient condition, we then propose a stochastic approximation algorithm that adaptively approximate the optimum threshold. Our theoretical analysis shows the approximated threshold converges to the optimum sampling threshold almost surely. Moreover, let k be the number of successfully received samples, we proved that: the MSE gap between our online algorithm and the optimum offline algorithm decays with rate O(1/k), while for any online sampling algorithm that does not know the channel statistics in advance, the minimum MSE gap under the worst case delay distribution decays with rate more than Ω(1/k). To improve the speed of the adaptive learning algorithm, we integrate the Polyak–Ruppert (PR) averaging algorithm when updating the sampling threshold, achieving a faster convergence speed both theoretically and in simulations.