Goal-oriented Tensor: Beyond AoI Towards Semantics-Empowered Goal-oriented Communications

The intricate interplay of source dynamics, unreliable channels, and staleness of information has long been recognized as a significant impediment for the receiver to achieve accurate, timely, and most importantly, goal-oriented decision making. Thus, a plethora of promising metrics, such as Age of Information, Value of Information, and Mean Square Error, have emerged to quantify these underlying adverse factors. Following this avenue, optimizing these metrics has indirectly improved the utility of goal-oriented decision making. Nevertheless, no metric has hitherto been expressly devised to evaluate the utility of a goal-oriented decision-making process. To this end, this paper investigates a novel performance metric, the Goal-oriented Tensor (GoT), to directly quantify the impact of semantic mismatches on the goal-oriented decision making. Based on the GoT, we consider a sampler-decision maker pair that work collaboratively and distributively to achieve a shared goal of communications. We formulate an infinite-horizon Decentralized Partially Observable Markov Decision Process (Dec-POMDP) to conjointly deduce the optimal deterministic sampling policy and decision-making policy. The simulation results reveal that the sampler-decision maker co-design surpasses beyond the current literature on AoI and its variants in terms of both goal achievement utility and sparse sampling rate, signifying a notable accomplishment for a sparse sampler and goal-oriented decision maker co-design.