# The Impact of the Geometric Properties of the Constraint Set in Safe Optimization with Bandit Feedback

arXiv: https://arxiv.org/abs/2305.00889 · HTML: https://mlsanity.com/paper/2305.00889/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 28 references that are in the map.
- First posted on arXiv 1 May 2023.

## Builds on (most cited first, 16 of 16)

- [Bayesian Optimization with Safety Constraints: Safe and Automatic Parameter Tuning in Robotics](https://mlsanity.com/paper/1602.04450/) (2016)
- [Learning-based Model Predictive Control for Safe Exploration](https://mlsanity.com/paper/1803.08287/) (2018)
- [Linear Stochastic Bandits Under Safety Constraints](https://mlsanity.com/paper/1908.05814/) (2019)
- [Stagewise Safe Bayesian Optimization with Gaussian Processes](https://mlsanity.com/paper/1806.07555/) (2018)
- [Stochastic Bandits with Linear Constraints](https://mlsanity.com/paper/2006.10185/) (2020)
- [Conservative Contextual Linear Bandits](https://mlsanity.com/paper/1611.06426/) (2016)
- [Safe Linear Thompson Sampling with Side Information](https://mlsanity.com/paper/1911.02156/) (2019)
- [Safety-Constrained Reinforcement Learning for MDPs](https://mlsanity.com/paper/1510.05880/) (2015)
- [Stage-wise Conservative Linear Bandits](https://mlsanity.com/paper/2010.00081/) (2020)
- [Best Arm Identification with Safety Constraints](https://mlsanity.com/paper/2111.12151/) (2021)
- [Safe Convex Learning under Uncertain Constraints](https://mlsanity.com/paper/1903.04626/) (2019)
- [Safe Online Convex Optimization with Unknown Linear Safety Constraints](https://mlsanity.com/paper/2111.07430/) (2021)
- [Safe Learning under Uncertain Objectives and Constraints](https://mlsanity.com/paper/2006.13326/) (2020)
- [Correlational Dueling Bandits with Application to Clinical Treatment in Large Decision Spaces](https://mlsanity.com/paper/1707.02375/) (2017)
- [Safe non-smooth black-box optimization with application to policy search](https://mlsanity.com/paper/1912.09466/) (2019)
- [Robust Model Predictive Control of Time-Delay Systems through System Level Synthesis](https://mlsanity.com/paper/2209.11841/) (2022)

## Built on by (most cited first, 3 of 3)

- [Convex Methods for Constrained Linear Bandits](https://mlsanity.com/paper/2311.04338/) (2023)
- [Constrained Linear Thompson Sampling](https://mlsanity.com/paper/2503.02043/) (2025)
- [The Safety-Privacy Tradeoff in Linear Bandits](https://mlsanity.com/paper/2504.16371/) (2025)

## Abstract (arXiv)

We consider a safe optimization problem with bandit feedback in which an agent sequentially chooses actions and observes responses from the environment, with the goal of maximizing an arbitrary function of the response while respecting stage-wise constraints. We propose an algorithm for this problem, and study how the geometric properties of the constraint set impact the regret of the algorithm. In order to do so, we introduce the notion of the sharpness of a particular constraint set, which characterizes the difficulty of performing learning within the constraint set in an uncertain setting. This concept of sharpness allows us to identify the class of constraint sets for which the proposed algorithm is guaranteed to enjoy sublinear regret. Simulation results for this algorithm support the sublinear regret bound and provide empirical evidence that the sharpness of the constraint set impacts the performance of the algorithm.