Tight generalization guarantees for the sampling and discarding approach to scenario optimization

Abstract

We consider the scenario approach to deal with convex optimization programs affected by uncertainty, which is in turn represented by means of scenarios. An approach to deal with such programs while trading feasibility to performance is known as sampling and discarding in the scenario approach literature. Existing bounds on the probability of constraint satisfaction for such programs are not tight. In this paper we use learning theoretic concepts based on the notion of compression to show that for a particular class of convex scenario programs, namely, the so called fully-supported ones, and under a particular scenario discarding scheme, a tight bound can be obtained. We illustrate our developments by means of an example that admits an analytic solution.

The scenario approach: A tool at the service of data-driven decision making

“…Program (10) is not the only scheme to tune the intrinsic vs. the extrinsic quality. Alternatively, one can discard some of the constraints from the worstcase program (1) and the reader is referred to the papers Campi & Garatti (2011); Garatti & Campi (2013); Picallo & Dörfler (2019); Romao et al (2020) for studies in this direction.…”

On the consistency of the risk evaluation in the scenario approach

“…The design scheme in (1) is already quite general and instances of (1) have indeed found application to control system design, [4], [5], [6], [7], [8], [9], system identification, [10], [11], [12], [13], [14], and machine learning, [15], [16], [17], [18]. Moreover, design schemes alternative to (1) have been also introduced within the scenario framework, accommodating diverse design requirements, [19], [20], [21], [22], [23], [24], [25], [26], [27] -see also [28], [29], [30] for general paradigms encompassing most of the existing schemes as special cases. While in this paper we prefer to limit ourselves to (1) for the sake of simplicity, the presented results are generally applicable to other design schemes.…”