Jointly Optimal Policies for Remote Estimation of Autoregressive Markov Processes over Time-Correlated Fading Channel

Abstract: We study a remote estimation setup with an autoregressive (AR) Markov process, a sensor, and a remote estimator. The sensor observes the process and sends encoded observations to the estimator as packets over an unreliable communication channel modeled as the Gilbert-Elliot (GE) channel. We assume that the sensor gets to observe the channel state by the ACK/NACK feedback mechanism only when it attempts a transmission while it does not observe the channel state when no transmission attempt is made. The objective is to design a transmission scheduling strategy for the sensor, and an estimation strategy for the estimator that are jointly optimal, i.e., they minimize the expected value of an infinite-horizon cumulative discounted cost defined as the sum of squared estimation error over time and the sensor's transmission power. Since the sensor and the estimator have access to different information sets, this constitutes a decentralized stochastic control problem. We formulate this problem as a partially observed Markov decision process (POMDP) and show the existence of jointly optimal transmission and estimation strategies that have a simple structure. More specifically, an optimal transmission strategy exhibits a threshold structure, i.e., the sensor attempts a transmission only when its belief about the channel being in a good state exceeds a threshold that depends on a certain error. Moreover, an optimal estimation strategy follows a `Kalman-like' update rule. When the channel parameters are unknown, we exploit this structure to design an actor-critic reinforcement learning algorithm that converges to a locally optimal policy. Simulations show the learned policy performs close to a globally optimal one, with about a 5.5% average relative gap across evaluated parameters.