Average number of mistakes in sequential risk-averse scenario decision-making

Data-driven methods aim to design predictors and controllers that adapt to the environment by utilizing information sourced from data. Due to their reliance on a finite amount of data, these designs are inevitably subject to a degree of imprecision, which can result in mistakes when they are applied to new cases. In this contribution, we introduce a sequential decision scheme in which the user is provided at each step with both a design and an assessment of the associated risk of making mistakes. The user decides whether to apply the design based on a threshold on the acceptable risk level. Novel results are presented to evaluate the average number of mistakes in this sequential data-driven risk-averse decision making framework. This requires in-depth analyses because, as we will see, naive evaluations based on common sense may lead to misleading results. Many are the potential applications, including the optimization of control actions over shifting windows (as in MPC), investments with recourse, and sequential prediction approaches.