  We revisit the so-called sampling and discarding approach used to quantify
the probability of constraint violation of a solution to convex scenario
programs when some of the original samples are allowed to be discarded.
Motivated by two scenario programs that possess analytic solutions and the fact
that the existing bound for scenario programs with discarded constraints is not
tight, we analyze a removal scheme that consists of a cascade of optimization
problems, where at each step we remove a superset of the active constraints. By
relying on results from compression learning theory, we show that such a
removal scheme leads to less conservative bounds for the probability of
constraint violation than the existing ones. We also show that the proposed
bound is tight by characterizing a class of optimization problems that achieves
the given upper bound. The performance improvement of the proposed methodology
is illustrated by an example that involves a resource sharing linear program.
