Goal-oriented Tensor: Beyond Age of Information Towards Semantics-Empowered Goal-Oriented Communications

The recent advancement of the emerging 5G and beyond has spawned the proliferation of both theoretical development and application instances for Internet of Things (IoT) networks. In such networks, timely status updates are becoming increasingly crucial for enabling real-time monitoring and actuation across a plethora of applications. To address the inadequacies of traditional throughput and delay metrics in such contexts, the Age of Information (AoI) has emerged as an innovative metric to capture the data freshness perceived by the receiver [1], defined as $\mathrm{AoI}(t)=t-U(t)$, where $U(t)$ denotes the generation time of the latest received packet before time $t$. Since its inception, AoI has garnered significant research attention and has been extensively analyzed in the queuing systems [2, 3, 4, 5, 6, 7], physical-layer communications [8, 9, 10, 11, 12, 13, 14], MAC-layer communications [15, 16, 17, 18, 19], industrial IoT [20, 21], energy harvesting systems[22, 23, 24, 25], and etc. (see [26] and the references therein for more comprehensive review). These research efforts are driven by the consensus that a freshly received message typically contains more valuable information, thereby enhancing the utility of decision-making processes.

Though AoI has been proven efficient in many freshness-critical applications, it exhibits several critical shortcomings. Specifically, ($a$) AoI does not provide a direct measure of information value; ($b$) AoI does not consider the content dynamics of source data and ignores the effect of End-to-End (E2E) information mismatch on the decision-making process.

To address shortcoming (a), a typical approach is to impose a non-linear penalty on AoI[27, 28, 29, 30, 31]. In [27], the authors map the AoI to a non-linear and non-decreasing function $f(\mathrm{AoI}(t))$ to evaluate the degree of “discontent” resulting from stale information. Subsequently, the optimal sampling policy is deduced for an arbitrary non-decreasing penalty function. The authors in [28] introduce two penalty functions, namely the exponential penalty function $a^{\mathrm{AoI}(t)}-1$ and the logarithmic penalty function $\log_{a}{\left(\mathrm{AoI}(t+1)\right)}$, for evaluating the Cost of Update Delay (CoUD). In [29] and [30], the binary indicator function $\mathbbm{1}_{\left\{\mathrm{AoI(t)>d}\right\}}$ is applied to evaluate whether the most recently received message is up-to-date. Specifically, the penalty assumes a value of $1$ when the instantaneous AoI surpasses a predetermined threshold $d$; otherwise, the penalty reverts to $0$. The freshness of web crawling is evaluated through this AoI-threshold binary indicator function. In [31], an analogous binary indicator approach is implemented in caching systems to evaluate the freshness of information.

The above works tend to tailor a particular non-linear penalty function to evaluate the information value. However, the selection of penalty functions in the above research relies exclusively on empirical configurations, devoid of rigorous derivations. To this end, several information-theoretic techniques strive to explicitly derive the non-linear penalty function in terms of AoI [32, 33, 34, 35]. One such quintessential work is the auto-correlation function $\mathbb{E}\left[X_{t}^{*}X_{t-\mathrm{AoI}(t)}\right]$, which proves to be a non-linear function of AoI when the source is stationary [32]. Another methodology worth noting is the mutual information metric between the present source state $X_{t}$ and the vector consisting of an ensemble of successfully received updates $\mathbf{W}_{t}$ [33, 34]. In the context of a Markovian source, this metric can be reduced to $I(X_{t};X_{t-\mathrm{AoI}(t)})$, which demonstrates a non-linear dependency on AoI under both the Gaussian Markov source and the Binary Markov source [33]. In [34], an analogous approach is utilized to characterize the value of information (VoI) for the Ornstein-Uhlenbeck (OU) process, which likewise demonstrates a non-linear dependency on AoI. In [35], the conditional entropy $H(X_{t}|\mathbf{W}_{t})$ is further investigated to measure the uncertainty of the source for a remote estimator given the history received updates $\mathbf{W}_{t}$. When applied to a Markov Source, this conditional entropy simplifies to $H(X_{t}|{X}_{t-\mathrm{AoI}(t)})$, thus exemplifying a non-linear penalty associated with AoI.

To address shortcoming (b), substantial research efforts have been invested in the optimization of the Mean Squared Error (MSE), denoted by $(X_{t}-\hat{X}_{t})^{2}$, with an ultimate objective of constructing a real-time reconstruction remote estimation system [36, 37, 38, 39]. In [36], a metric termed effective age is proposed to minimize the MSE for the remote estimation of a Markov source. In [37] and [38], two Markov sources of interest, the Wiener process and the OU process are investigated to deduce the MSE-optimal sampling policy. Intriguingly, these policies were found to be threshold-based, activating sampling only when the instantaneous MSE exceeds a predefined threshold, otherwise maintaining a state of idleness. The authors in [39] explored the trade-off between MSE and quantization over a noisy channel, and derived the MSE-optimal sampling strategy for the OU process.

Complementary to the above MSE-centered research, variants of AoI that address shortcoming $(b)$ have also been conceptualized [40, 41, 42, 43]. In [40], Age of Changed Information (AoCI) is proposed to address the ignorance of content dynamics of AoI. In this regard, unchanged statuses do not necessarily provide new information and thus are not prioritized for transmission. In [41], the authors introduce the context-aware weighting coefficient to propose the Urgency of Information (UoI), a metric capable of measuring the weighted MSE in diverse urgency contexts. In [42], the authors propose a novel age penalty named Age of Synchronization (AoS), a novel metric quantifying the time since the most recent end-to-end synchronization. Moreover, considering that an E2E status mismatch may exert a detrimental effect on the overall system’s performance over time, the authors of [43] propose a novel metric called Age of Incorrect Information (AoII). This metric quantifies the adverse effect stemming from the duration of the E2E mismatch, revealing that both the degree and duration of E2E semantic mismatches lead to a utility reduction for subsequent decision-making.

The above studies focused on the open-loop generation-to-reception process within a transmitter-receiver pair, neglecting the closed-loop perception-actuation timeliness. A notable development addressing this issue is the extension from Up/Down-Link (UL/DL) AoI to a closed-loop AoI metric, referred to as the Age of Loop (AoL) [44]. Unlike the traditional open-loop AoI, which diminishes upon successful reception of a unidirectional update, the AoL decreases solely when both the UL status and the DL command are successfully received. Another advanced metric in [45], called Age of Actuation (AoA), also encapsulates the actuation timeliness, which proves pertinent when the receiver employs the received update to execute timely actuation.

Notwithstanding the above advancements, the question on how the E2E mismatch affects the closed-loop utility of decision-making has yet to be addressed. To address this issue, [46, 47, 48, 49] introduce a metric termed Cost of Actuation Error to delve deeper into the cost resulting from the error actuation due to imprecise real-time estimations. Specifically, the Cost of Actuation Error is denoted by an asymmetric zero diagonal matrix $\mathbf{C}$, with each value $C_{X_{t},\hat{X}_{t}}$ representing the instant cost under the E2E mismatch status $(X_{t},\hat{X}_{t})_{X_{t}\neq\hat{X}_{t}}$. In this regard, the authors reveal that the utility of decision-making bears a close relation to the E2E semantic mismatch category, as opposed to the mismatch duration (AoII) or mismatch degree (MSE). For example, an E2E semantic mismatch category that “Fire” is estimated as “No Fire” will result in high cost; while in the opposite scenario, the cost is low. Nonetheless, we notice that $i$) the method to obtain a Cost of Actuation Error remains unclear, which implicitly necessitates a pre-established decision-making policy; $ii$) Cost of Actuation Error does not consider the context-varying factors, which may also affect the decision-making utility; $iii$) the zero diagonal property of the matrix implies the supposition that if $X_{t}=\hat{X}_{t}$, then $C_{X_{t},\hat{X}_{t}}=0$, thereby signifying that error-less actuation necessitates no energy expenditure. Fig. 1 provides an overview of the existing metrics.

Against this background, the present authors have recently proposed a new metric referred to as GoT in [50], which, compared to Cost of Actuation Error, introduces new dimensions of the context $\Phi_{t}$ and the decision-making policy $\pi_{A}$ to describe the true utility of decision-making. Remarkably, we find that GoT offers a versatile degeneration to established metrics such as AoI, MSE, UoI, AoII, and Cost of Actuation Error. Additionally, it provides a balanced evaluation of the cost trade-off between the sampling and decision-making. The contributions of this work could be summarized as follows:

$\bullet$ We focus on the decision utility issue directly by employing the GoT. A controlled Markov source is observed, wherein the transition of the source is dependent on both the decision-making at the receiver and the contextual situation it is situated. In this case, the decision-making will lead to three aspects in utility: $i$) the future evolution of the source; $ii$) the instant cost at the source; $iii$) the energy and resources consumed by actuation.

$\bullet$ We accomplish the goal-oriented sampler & decision-maker co-design, which, to the best of our knowledge, represents the first work that addresses the trade-off between semantics-aware sampling and goal-oriented decision-making. Specifically, we formulate this problem as a two-agent infinite-horizon Decentralized Partially Observable Markov Decision Process (Dec-POMDP) problem, with one agent embodying the semantics and context-aware sampler, and the other representing the goal-oriented decision-maker. Note that the optimal solution of even a finite-horizon Dec-POMDP is known to be NEXP-hard [51], we develop the RVI-Brute-Force-Search Algorithm. This algorithm seeks to derive optimal deterministic joint policies for both sampling and decision-making. A thorough discussion on the optimality of our algorithm is also presented.

$\bullet$ To further mitigate the “curse of dimensionality” intricately linked with the execution of the optimal RVI-Brute-Force-Search, we introduce a low-complexity yet efficient algorithm to solve the problem. The algorithm is designed by decoupling the problem into two distinct components: a Markov Decision Process (MDP) problem and a Partially Observable Markov Decision Process (POMDP) issue. Following this separation, the algorithm endeavors to search for the joint Nash Equilibrium between the sampler and the decision-maker, providing a sub-optimal solution to this goal-oriented communication-decision-making co-design.

We consider a time-slotted perception-actuation loop where both the perceived semantics $X_{t}\in\mathcal{S}=\left\{s_{1},\cdots,s_{|\mathcal{S}|}\right\}$ and context $\Phi_{t}\in\mathcal{V}=\left\{v_{1},\cdots,v_{|\mathcal{V}|}\right\}$ are input into a semantic sampler, tasked with determining the significance of the present status $X_{t}$ and subsequently deciding if it warrants transmission via an unreliable channel. The semantics and context are extracted and assumed to perfectly describe the status of the observed process. The binary indicator, $a_{S}(t)=\pi_{S}(X_{t},\Phi_{t},\hat{X}_{t})\in\left\{0,1\right\}$, signifies the sampling and transmission action at time slot $t$, with the value $1$ representing the execution of sampling and transmission, and the value $0$ indicating the idleness of the sampler. $\pi_{S}$ here represents the sampling policy. We consider a perfect and delay-free feedback channel [46, 47, 48, 49], with ACK representing a successful transmission and NACK representing the otherwise. The decision-maker at the receiver will make decisions $a_{A}(t)\in\mathcal{A}_{A}=\left\{a_{1},\cdots,a_{|\mathcal{A}_{A}|}\right\}$ based on the estimate $\hat{X}_{t}$³³3We consider a general abstract decision-making set $\mathcal{A}_{A}$ that exhibits adaptability to diverse applications. Notably, this decision-making set can be customized and tailored to suit specific requirements., which will ultimately affect the utility of the system. An illustration of our considered model is shown in Fig. 4.

Traditionally, the development of sampling strategies has been designed separately from the decision-making process. An archetypal illustration of this two-stage methodology involves first determining the optimal sampling policy based on AoI, MSE, and their derivatives, such as AoII, and subsequently accomplishing goal-oriented decision-making. This two-stage separate design arises from the inherent limitation of existing metrics that they fail to capture the closed-loop decision utility. Nevertheless, the metric GoT empowers us to undertake a co-design of sampling and decision-making.

We adopt the team decision theory, wherein two agents, one embodying the sampler and the other the decision-maker, collaborate to achieve a shared goal. We aim at determining a joint deterministic policy $\boldsymbol{\pi}_{C}=(\pi_{S},\pi_{A})$ that minimizes the long-term average cost of the system. It is considered that the sampling and transmission of an update also consumes energy, incurring a $C_{S}$ cost. In this case, the instant cost of the system could be clarified by $\mathrm{GoT}^{\pi_{A}}(t)+C_{S}\cdot a_{S}(t)$, and the problem is characterized as:

where $\boldsymbol{\pi}_{C}=(\pi_{S},\pi_{A})$ denotes the joint sampling and decision-making policy, comprising $\pi_{S}=(a_{S}(0),a_{S}(1),\cdots)$ and $\pi_{A}=(a_{A}(0),a_{A}(1),\cdots)$, which correspond to the sampling action sequence and actuation sequence, respectively. Note that $\mathrm{GoT}^{\pi_{A}}(t)$ is characterized by (9).