  Conformal prediction is a generic methodology for finite-sample valid
distribution-free prediction. This technique has garnered a lot of attention in
the literature partly because it can be applied with any machine learning
algorithm that provides point predictions to yield valid prediction regions. Of
course, the efficiency (width/volume) of the resulting prediction region
depends on the performance of the machine learning algorithm. In this paper, we
consider the problem of obtaining the smallest conformal prediction region
given a family of machine learning algorithms. We provide two general-purpose
selection algorithms and consider coverage as well as width properties of the
final prediction region. The first selection method yields the smallest width
prediction region among the family of conformal prediction regions for all
sample sizes, but only has an approximate coverage guarantee. The second
selection method has a finite sample coverage guarantee but only attains close
to the smallest width. The approximate optimal width property of the second
method is quantified via an oracle inequality. Asymptotic oracle inequalities
are also considered when the family of algorithms is given by ridge regression
with different penalty parameters.
