Gilles Blanchard and François Fleuret
In: 20th conference on Learning Theory (COLT 2007), 13-15 June 2007, San Diego, CA, USA.
We establish a generic theoretical tool to construct probabilistic
bounds for algorithms where the output is a subset of objects from
an initial pool of candidates (or more generally, a probability
distribution on said pool). This general device, dubbed ``Occam's
hammer'', acts as a meta layer when a probabilistic bound is
already known on the objects of the pool taken individually,
and aims at controlling the proportion of the objects in the set
output not satisfying their individual bound.
In this regard, it can be seen as a non-trivial generalization of
the ``union bound with a prior'' (``Occam's razor''), a familiar
tool in learning theory. We give applications of
this principle to randomized classifiers (providing
an interesting alternative approach to PAC-Bayes bounds) and multiple testing
(where it allows to retrieve exactly and extend the so-called
Benjamini-Yekutieli testing procedure).