Improved Compression Bounds for Scenario Decision Making

Scenario decision making offers a flexible way of making decision in an uncertain environment while obtaining probabilistic guarantees on the risk of failure of the decision. The idea of this approach is to draw samples of the uncertainty and make a decision based on the samples, called "scenarios". The probabilistic guarantees take the form of a bound on the probability of sampling a set of scenarios that will lead to a decision whose risk of failure is above a given maximum tolerance. This bound can be expressed as a function of the number of sampled scenarios, the maximum tolerated risk, and some intrinsic property of the problem called the "compression size". Several such bounds have been proposed in the literature under various assumptions on the problem. We propose new bounds that improve upon the existing ones without requiring stronger assumptions on the problem.