On the consistency of the risk evaluation in the scenario approach

The scenario approach is a well-established methodology that allows one to generate solutions from a sample of observations (data-driven decision making). In the recent wait-and-judge paradigm to the scenario approach, the risk (i.e., the probability with which a scenario solution does not satisfy new, out-of-sample, constraints) is estimated from an observable called the complexity and this result is used to compute intervals that contain with high confidence the value of the risk. In this paper, we establish a new analytical expression for these confidence intervals and we show that they are centered around the complexity divided by the sample size N while their width uniformly (in the complexity) shrinks to zero for increasing N at the rate $O(\ln (N)/\sqrt N )$ (which is close to the convergence rate of the central limit theorem). This result bears profound implications: (i) it proves the asymptotic consistency of the evaluation of the risk; (ii) as a corollary, it shows that the complexity is an observable that carries the fundamental information on the risk (a quantity that is not directly accessible); (iii) it extends the result that the empirical mean tends to the true probability of an event to the case when the event is chosen based on observations via a scenario decision scheme.