# B'ezier Flow: a Surface-wise Gradient Descent Method for Multi-objective Optimization

arXiv: https://arxiv.org/abs/2205.11099 · HTML: https://mlsanity.com/paper/2205.11099/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 33 references that are in the map.
- First posted on arXiv 23 May 2022.

## Builds on (most cited first, 9 of 9)

- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [Predictive Entropy Search for Multi-objective Bayesian Optimization](https://mlsanity.com/paper/1511.05467/) (2015)
- [jMetalPy: a Python Framework for Multi-Objective Optimization with Metaheuristics](https://mlsanity.com/paper/1903.02915/) (2019)
- [Bezier Simplex Fitting: Describing Pareto Fronts of Simplicial Problems with Small Samples in Multi-objective Optimization](https://mlsanity.com/paper/1812.05222/) (2018)
- [Topology of Pareto sets of strongly convex problems](https://mlsanity.com/paper/1904.03615/) (2019)
- [Asymptotic Risk of Bezier Simplex Fitting](https://mlsanity.com/paper/1906.06924/) (2019)
- [Simpliciality of strongly convex problems](https://mlsanity.com/paper/1912.09328/) (2019)
- [Ensuring smoothly navigable approximation sets by Bezier curve parameterizations in evolutionary bi-objective optimization -- applied to brachytherapy treatment planning for prostate cancer](https://mlsanity.com/paper/2006.06449/) (2020)
- [All unconstrained strongly convex problems are weakly simplicial](https://mlsanity.com/paper/2106.12704/) (2021)

## Abstract (arXiv)

In this paper, we propose a strategy to construct a multi-objective optimization algorithm from a single-objective optimization algorithm by using the B'ezier simplex model. Also, we extend the stability of optimization algorithms in the sense of Probability Approximately Correct (PAC) learning and define the PAC stability. We prove that it leads to an upper bound on the generalization with high probability. Furthermore, we show that multi-objective optimization algorithms derived from a gradient descent-based single-objective optimization algorithm are PAC stable. We conducted numerical experiments and demonstrated that our method achieved lower generalization errors than the existing multi-objective optimization algorithm.