8.5. Conclusion

Details for calculating the shell bound are given in appendix section 16.3.

Shell bounds are easily calculable and can provide large improvements when we can afford to enumerate the hypotheses. Their application is less clear when it is not possible to enumerate the hypotheses because the computational burden may become too large. Via sampling techniques, it is possible to smoothly improve from earlier techniques to the best achievable shell bound result in an anytime fashion.

There remain several important open questions:

1. Can we remove the remaining division of $\delta$ by $m$? This would make shell bounds a bit more elegant and tight.
2. Is there a natural lower bound on the true error rate which uses the shell approach? This improvement is of principally theoretical interest.
3. For the continuous space shell bounds, two additional divisions by $m$ were introduced. Is it possible to remove these factors with an improved argument? This improvement would clean up the continuous shell bound.
4. Our extension to the continuous case was done in the style of PAC-Bayes bounds, but a more common technique for extending to the continuous case is via the use of covering numbers. Is there a natural way to extend Shell bounds to the continuous case using the concept of a covering number? This is another approach which might yield a result requiring less calculation.