## 9 Discussion and Conclusions
Our RRF theorems offer some technical insights to help explain why it is difficult for DQL methods to find optimal RRFs for a given dataset. These theorems should by no means be construed as stating that the RRFs are, in a general sense, worse than the SCNs, insomuch as they say nothing about different reward strategies or different LRFs. Nor do they apply to HQL methods. Theorems 1 and 2 present a probabilistic argument that when the number of optimal RRFs for a given dataset is low, the probability of not finding any of them with DQL is high, and as the number of optimal RFs tends to 0, the probability of not finding any of them approaches 1. Theorem 3 shows that there is a specific threshold for the number of optimal RRFs that makes the probability of finding at least one optimal RRF less than 50% with DQL. Our research has three practical implications. The first implication is that RRFs may present a reasonable alternative to ConvNets on domain-specific datasets with smaller numbers of categories and smaller numbers of examples per category. The second implication is that LRFs are important for RRFs and formal investigations of possible relationships between LRFs and datasets may be warranted. The third implication is that it is important for embedded systems (e.g., EBM systems) to seek energy-efficient models with limited memory and power resources. We remain optimistic about the prospects of RRFs, especially for domain-specific datasets, because it may well be possible with innovative reward strategies and domain-specific LRFs to find near-optimal (if not optimal) RRFs that perform on par with SCNs. In our future work, we plan to investigate different reward strategies, improve domain-specific LRFs for RRFs, and explore relative energy efficiencies of SCNs and RRFs. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **RFs** & **RRFs** & **SCNs** \\ \hline Parameters & 10,000 & 91,920 & 4,195,328 \\ Operations & 20,404 & 839,604 & 15,991,560 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Numbers of parameters and operations in RFs, RRFs, and SCNs involved in classifying an image into two classes. **Author Contributions:** Conceptualization: V.K. ; methodology: V.K. ; software: N.G., V.K. ; validation, N.G., V.K., A.T.; formal analysis: N.G., A.T., V.K. ; investigation: V.K., N.G., A.T.; resources: V.K. ; data curation: N.G., V.K. ; writing--original draft preparation: V.K., N.G. ; writing--review and editing: V.K., N.G., A.T.; supervision: V.K. ; project administration: V.K. ; funding acquisition: V.K. All authors have read and agreed to the published version of the manuscript. **Funding:** This research is funded, in part, by two Kickstarter research fundraisers in 2017 and 2019. **Institutional Review Board Statement:** No applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Interested partis can contact the first author about the availability of datasets. **Acknowledgments:** The first author would like to thank all Kickstarter backers of the BeePi project and especially the Angel Backers (in alphabetical order): [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON]. The author expresses his gratitude to [PERSON], who backed both fundraisers and donated hardware to the BeePi project. The author is grateful to [PERSON], [PERSON], and [PERSON] for letting him use their private property in northern Utah for longitudinal CEBM tests. The first author would like to thank his students [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON] for their pro bono software, hardware, and data curation contributions to the BeePi project. The first author expresses his gratitude to [PERSON] of the Department of Mathematics and Statistics of Utah State University for reading a draft of this article and making valuable suggestions and comments. **Conflicts of Interest:** The authors declare no conflict of interest.