Scenario Approach with Post-Design Certification of User-Specified Properties

TL;DR

This paper extends the scenario approach by introducing a two-level framework to guarantee additional post-design properties, providing distribution-free risk bounds and practical examples in control problems.

Contribution

It introduces a novel two-level framework for post-design property certification within the scenario approach, with computable risk bounds without extra test data.

Findings

Distribution-free upper bounds on risk for post-design properties

Practical examples in H2 and pole-placement control problems

Method to infer distributional knowledge from data

Abstract

The scenario approach is an established data-driven design framework that comes equipped with a powerful theory linking design complexity to generalization properties. In this approach, data are simultaneously used both for design and for certifying the design's reliability, without resorting to a separate test dataset. This paper takes a step further by guaranteeing additional properties, useful in post-design usage but not considered during the design phase. To this end, we introduce a two-level framework of appropriateness: baseline appropriateness, which guides the design process, and post-design appropriateness, which serves as a criterion for a posteriori evaluation. We provide distribution-free upper bounds on the risk of failing to meet the post-design appropriateness; these bounds are computable without using any additional test data. Under additional assumptions, lower bounds are also derived. As part of an effort to demonstrate the usefulness of the proposed methodology, the paper presents two practical examples in H2 and pole-placement problems. Moreover, a method is provided to infer comprehensive distributional knowledge of relevant performance indexes from the available dataset.