# Global-Entropy Driven Exploration with Distributed Models under Sparsity Constraints
## 3 Distributed Cooperative Exploration
### Exploration for Splitting-over-Examples  
Consider now an arbitrary agent \(k\) that attempts to find a next measurement position \(\mathbf{x}_{k}\in\mathcal{X}_{k}\subseteq\mathcal{X}\) by solving (28). To compute (28), we will distinguish between local quantities, which are only known to an individual agent and therefore do not require cooperation with other agents, and those that are distributed among the agents in the network. In particular, a vector \(\mathbf{\psi}_{\mathcal{A}}(\mathbf{x})\) can be evaluated locally by each agent; similarly, computing \(\varphi(\mathbf{x},\tilde{\mathbf{x}})\), as well as \(\widehat{\mathbf{\Delta}}_{\mathcal{A}}\) does not require cooperation in the SOE approach. In contrast, the quantities \(\mathbf{A}\), \(\mathbf{c}(\tilde{\mathbf{x}})\) and \(b(\tilde{\mathbf{x}})\) in (28) are only available in a distributed form. Luckily, in SOE, the distributed quantities can be recovered by means of an averaged consensus algorithm [25; 26]. Indeed, it can be recognized that:\[\mathbf{A} \triangleq\lambda\mathbf{\Psi}_{\mathcal{A}}^{T}\mathbf{\Psi}_{\mathcal{A}}+ \widehat{\mathbf{\Delta}}_{\mathcal{A}}=\sum_{k=1}^{K}\lambda\mathbf{\Psi}_{\mathcal{A} _{\mathcal{A}}}^{T}\mathbf{\Psi}_{\mathcal{A}_{\mathcal{k}}}+\widehat{\mathbf{\Delta}}_{ \mathcal{A}}, \tag{29}\] \[\mathbf{c}(\bar{\mathbf{x}}) \triangleq\lambda\mathbf{\Psi}_{\mathcal{A}}^{T}\mathbf{\varphi}(\mathbf{X}_{ M},\bar{\mathbf{x}})=\sum_{k=1}^{K}\lambda\mathbf{\Psi}_{\mathcal{A}_{\mathcal{A}}}^{T} \mathbf{\varphi}_{k}(\mathbf{X}_{M_{k}},\bar{\mathbf{x}}),\] (30) \[b(\bar{\mathbf{x}}) \triangleq\lambda\mathbf{\varphi}(\mathbf{X}_{M},\bar{\mathbf{x}})^{T}\mathbf{ \varphi}(\mathbf{X}_{M},\bar{\mathbf{x}})=\sum_{k=1}^{K}\lambda\mathbf{\varphi}_{k}(\mathbf{X }_{M_{k}},\bar{\mathbf{x}})^{T}\mathbf{\varphi}_{k}(\mathbf{X}_{M_{k}},\bar{\mathbf{x}}). \tag{31}\]  
The sums over the agents in Equations (29)-(31) can be efficiently computed using averaged consensus and thus become available at each agent. Let us stress that Equations (29)-(31) depend on an unknown feature location \(\bar{\mathbf{x}}\). However, in SOE, the features are fixed a-priori, and every agent knows \(\mathcal{X}_{\mathcal{I}}\). As such, each agent can \"scan\" \(\bar{\mathbf{x}}\in\mathcal{X}_{\mathcal{I}}\) without the participation of the other agents. This implies that solving (28) with respect to \(\bar{\mathbf{x}}\), Equations (29)-(31) can be computed for each element in \(\mathcal{X}_{\mathcal{I}}\). Furthermore, as can be seen in Equations (29)-(31), the consensus parts do not depend on \(\mathbf{x}\in\mathcal{X}\). Thus, the proposed measurement positions do not need to be distributed. Now, an agent \(k\) can evaluate (28) for any \(\mathbf{x}\in\mathcal{X}_{k}\). The whole procedure is summarized in Algorithm 1. ```
procedureExploration for SOE for all\(k=1,\dots,K\)do \(\mathbf{A}\leftarrow\)(29) \(\widehat{\mathbf{x}}_{k}=\mathbf{x}_{M_{K}}\) for all\(\bar{\mathbf{x}}\in\mathcal{X}_{\mathcal{I}}\)do \(\mathbf{c}(\bar{\mathbf{x}})\leftarrow\)(30) \(b(\bar{\mathbf{x}})\leftarrow\)(31) endfor \(\widehat{\mathbf{x}}_{k}\leftarrow\)(28), with \(\mathbf{x}\in\mathcal{X}_{k}\)  Agent \(k\) moves and makes the measurement endfor endprocedure
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**Algorithm 1** Exploration for \(K\) agents with SOE. Regarding the communication, we evaluated the amount of floats that have to be communicated. As an initial communication, the agents estimate \(\mathbf{A}\), where initially, \(\mathcal{O}(|\mathcal{A}|^{2})\) for each agent are sent to all neighbors. Due to the sparsity constraints, \(|\mathcal{A}|\) will be typically small. During the evaluation of the exploration criterion, \(N+1\) floats for each \(\bar{\mathbf{x}}\in\mathcal{X}_{\mathcal{I}}\) for each neighbor have to be communicated. In total, the communication per measurement location estimation per agent yields \(\mathcal{O}\left(\left(|\mathcal{I}|(N+1)+|\mathcal{A}|^{2}\right)(K-1)\right)\).