# On the Sample Size of Random Convex Programs with Structured Dependence on the Uncertainty (Extended Version)

arXiv: https://arxiv.org/abs/1502.00803 · HTML: https://mlsanity.com/paper/1502.00803/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Feb 2015.

## Built on by (most cited first, 4 of 4)

- [Compression, Generalization and Learning](https://mlsanity.com/paper/2301.12767/) (2023)
- [Margin theory for the scenario-based approach to robust optimization in high dimension](https://mlsanity.com/paper/2303.03891/) (2023)
- [Randomized algorithms and PAC bounds for inverse reinforcement learning in continuous spaces](https://mlsanity.com/paper/2405.15509/) (2024)
- [Scenario Approach with Post-Design Certification of User-Specified Properties](https://mlsanity.com/paper/2602.15568/) (2026)

## Abstract (arXiv)

The "scenario approach" provides an intuitive method to address chance constrained problems arising in control design for uncertain systems. It addresses these problems by replacing the chance constraint with a finite number of sampled constraints (scenarios). The sample size critically depends on Helly's dimension, a quantity always upper bounded by the number of decision variables. However, this standard bound can lead to computationally expensive programs whose solutions are conservative in terms of cost and violation probability. We derive improved bounds of Helly's dimension for problems where the chance constraint has certain structural properties. The improved bounds lower the number of scenarios required for these problems, leading both to improved objective value and reduced computational complexity. Our results are generally applicable to Randomized Model Predictive Control of chance constrained linear systems with additive uncertainty and affine disturbance feedback. The efficacy of the proposed bound is demonstrated on an inventory management example.