Semantic-Aware Remote Estimation of Multiple Markov Sources Under Constraints

This paper studies the remote estimation of multiple Markov sources over a lossy and rate-constrained channel. Unlike most existing studies that treat all source states equally, we exploit the \emph{semantics of information} and consider that the remote actuator has different tolerances for the estimation errors. We aim to find an optimal scheduling policy that minimizes the long-term $\mathit{\text{state-dependent}}$ costs of estimation errors under a transmission frequency constraint. The optimal scheduling problem is formulated as a \emph{constrained Markov decision process} (CMDP). We show that the optimal Lagrangian cost follows a piece-wise linear and concave (PWLC) function, and the optimal policy is, at most, a randomized mixture of two simple deterministic policies. By exploiting the structural results, we develop a new $\mathit{\text{intersection}}\mathit{\text{search}}$ algorithm that finds the optimal policy using only a few iterations. We further propose a reinforcement learning (RL) algorithm to compute the optimal policy without knowing $\mathit{\text{a}}\mathit{\text{priori}}$ the channel and source statistics. To avoid the ``curse of dimensionality" in MDPs, we propose an online low-complexity $\mathit{\text{drift-plus-penalty}}$ (DPP) algorithm. Numerical results show that continuous transmission is inefficient, and remarkably, our semantic-aware policies can attain the optimum by strategically utilizing fewer transmissions by exploiting the timing of the important information.