# A Posteriori Probabilistic Bounds of Convex Scenario Programs with Validation Tests

## TL;DR

This paper introduces new probabilistic bounds for convex scenario programs that integrate support constraints and validation data, enabling more reliable risk assessment with fewer samples.

## Contribution

It proposes a general framework for a posteriori bounds that incorporate validation tests and support constraints, improving risk evaluation in scenario optimization.

## Key findings

- Bounds depend on support constraints and validation data
- Framework encompasses many existing theoretical results
- Case studies demonstrate improved risk assessment

## Abstract

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.