  We investigate the connections between compression learning and scenario
based optimization. We first show how to strengthen, or relax the consistency
assumption at the basis of compression learning and study the learning and
generalization properties of the algorithm involved. We then consider different
constrained optimization problems affected by uncertainty represented by means
of scenarios. We show that the issue of providing guarantees on the probability
of constraint violation reduces to a learning problem for an appropriately
chosen algorithm that enjoys compression learning properties. The compression
learning perspective provides a unifying framework for scenario based
optimization and allows us to revisit the scenario approach and the
probabilistically robust design, a recently developed technique based on a
mixture of randomized and robust optimization, and to extend the guarantees on
the probability of constraint violation to cascading optimization problems.
