# Global-Entropy Driven Exploration with Distributed Models under Sparsity Constraints
## 5 Conclusions  
In this paper, we discussed distributed learning and exploration algorithms for two different data distribution paradigms, namely SOE and SOF. A general linear model was used to describe a static spatial process that is explored with a set of multiple agents. In both, SOE and SOF, cases, we used the ADMM algorithm to estimate model parameters in a distributed fashion over the communication network between the agents. The used model also explicitly included an \(\ell_{1}\) penalty term that allowed computing a sparse parameter vector estimate. This penalty, on the one hand, regularizes the numerical optimization problem when only a few measurements are available; on the other hand, it leads to a sparse signal estimate, which allows reducing the model complexity at each agent. The latter allows reducing the communication load on the network (as in the SOE case) or reducing the local computational load per agent (as in the SOF case). As an exploration objective, we proposed to use a D-optimality criterion from the theory of experiment design. This criterion is designed to select new measurement positions, which reduce the estimator's covariance. Yet, due to the use of sparsity constraints, the evaluation of the D-optimality criterion is difficult due to the non-smoothness of the resulting objective function. To circumvent this difficulty, we proposed a modification of the objective function that allows approximating the estimator's covariance. We then proposed algorithms that permit optimizing the resulting approximate D-optimality criterion in a distributed fashion using averaged consensus for both SOE and SOF data distribution paradigms. Finally, we performed numerical simulations where the exploration criterion for each paradigm was tested against benchmark algorithms. These benchmark algorithms cover the types of systematic and random exploration strategies. Simulations have shown that the proposed exploration strategy outperforms the corresponding benchmarks with respect to the NMSE. We found that the SOF method has more flexibility with selecting feature positions and thus yields a lower NMSE compared to the SOE approach. It is possible that this effect can be attributed to the ratio between measurements \(M\) and features \(N\). This ratio is very low for SOE in the beginning of the exploration, because the number of features is predefined and rather large, thus the difference in the NMSE performance. The SOF can, in contrast, build up the number of features in accordance with the number of measurements and can thus better fit the data. Regarding the number of measurements, we can also conclude that the proposed exploration strategy permits collecting relevant data more efficiently, as compared to the benchmark schemes, where the agents are not driven to more informative locations. Both approaches seem very interesting for developing real distributed systems with efficient navigation algorithms and are next to be tested in real experiments. SOE, with a fixed number of features and a splitting in measurements, is useful when many measurements have to be observed, i.e., when we expect that \(M\gg N\). For few features, this approach leads to fast computations. Furthermore, the communication with neighbors for computing the exploration criterion is reduced, since it is dependent on the number of features. The SOF scheme is computationally more efficient when the number of features is higher than the number of measurements, i.e., \(N\gg M\). Therefore, it can be seen as an alternative to SOE in this case. Formal analysis, C.M. and D.S. Investigation, C.M. Software, C.M. Supervision, D.S. Writing, original draft, C.M. Writing, review and editing, C.M. and D.S. The authors declare no conflict of interest. The following abbreviations are used in this manuscript: \begin{tabular}{l l}
**NMSE** & normalized mean square error \\
**SAR** & synthetic aperture radar \\
**LASSO** & least absolute shrinkage and selection operator \\
**ADMM** & alternating direction method of multipliers \\ \end{tabular}