# On Polynomial Stochastic Barrier Functions: Bernstein Versus Sum-of-Squares

arXiv: https://arxiv.org/abs/2506.09164 · HTML: https://mlsanity.com/paper/2506.09164/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2025.

## Built on by (most cited first, 1 of 1)

- [Universal Learning of Stochastic Dynamics for Exact Belief Propagation using Bernstein Normalizing Flows](https://mlsanity.com/paper/2509.15533/) (2025)

## Abstract (arXiv)

Stochastic Barrier Functions (SBFs) certify the safety of stochastic systems by formulating a functional optimization problem, which state-of-the-art methods solve using Sum-of-Squares (SoS) polynomials. This work focuses on polynomial SBFs and introduces a new formulation based on Bernstein polynomials and provides a comparative analysis of its theoretical and empirical performance against SoS methods. We show that the Bernstein formulation leads to a linear program (LP), in contrast to the semi-definite program (SDP) required for SoS, and that its relaxations exhibit favorable theoretical convergence properties. However, our empirical results reveal that the Bernstein approach struggles to match SoS in practical performance, exposing an intriguing gap between theoretical advantages and real-world feasibility.