A Posteriori Probabilistic Bounds of Convex Scenario Programs with Validation Tests
This work addresses the challenge of efficient risk assessment in decision-making under uncertainty for fields like control systems, though it is incremental as it builds on existing scenario program and validation test frameworks.
The paper tackles the problem of needing large validation samples to reliably assess risk in scenario-based decision-making under uncertainty, proposing new a posteriori probabilistic bounds that improve risk evaluation by synthesizing information from support constraints and validation tests, with case studies on aircraft lateral motion controller design validating effectiveness.
Scenario programs have established themselves as efficient tools towards decision-making under uncertainty. To assess the quality of scenario-based solutions a posteriori, validation tests based on Bernoulli trials have been widely adopted in practice. However, to reach a theoretically reliable judgement of risk, one typically needs to collect massive validation samples. In this work, we propose new a posteriori bounds for convex scenario programs with validation tests, which are dependent on both realizations of support constraints and performance on out-of-sample validation data. The proposed bounds enjoy wide generality in that many existing theoretical results can be incorporated as particular cases. To facilitate practical use, a systematic approach for parameterizing a posteriori probability bounds is also developed, which is shown to possess a variety of desirable properties allowing for easy implementations and clear interpretations. By synthesizing comprehensive information about support constraints and validation tests, improved risk evaluation can be achieved for randomized solutions in comparison with existing a posteriori bounds. Case studies on controller design of aircraft lateral motion are presented to validate the effectiveness of the proposed a posteriori bounds.