# Safe Learning under Uncertain Objectives and Constraints

arXiv: https://arxiv.org/abs/2006.13326 · HTML: https://mlsanity.com/paper/2006.13326/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jun 2020.

## Built on by (most cited first, 7 of 7)

- [Safe Online Convex Optimization with Unknown Linear Safety Constraints](https://mlsanity.com/paper/2111.07430/) (2021)
- [Log Barriers for Safe Black-box Optimization with Application to Safe Reinforcement Learning](https://mlsanity.com/paper/2207.10415/) (2022)
- [Directional Optimism for Safe Linear Bandits](https://mlsanity.com/paper/2308.15006/) (2023)
- [Dynamic Regret Analysis of Safe Distributed Online Optimization for Convex and Non-convex Problems](https://mlsanity.com/paper/2302.12320/) (2023)
- [The Impact of the Geometric Properties of the Constraint Set in Safe Optimization with Bandit Feedback](https://mlsanity.com/paper/2305.00889/) (2023)
- [Online Learning with Unknown Constraints](https://mlsanity.com/paper/2403.04033/) (2024)
- [Optimistic Safety for Online Convex Optimization with Unknown Linear Constraints](https://mlsanity.com/paper/2403.05786/) (2024)

## Abstract (arXiv)

In this paper, we consider non-convex optimization problems under textitunknown yet safety-critical constraints. Such problems naturally arise in a variety of domains including robotics, manufacturing, and medical procedures, where it is infeasible to know or identify all the constraints. Therefore, the parameter space should be explored in a conservative way to ensure that none of the constraints are violated during the optimization process once we start from a safe initialization point. To this end, we develop an algorithm called Reliable Frank-Wolfe (Reliable-FW). Given a general non-convex function and an unknown polytope constraint, Reliable-FW simultaneously learns the landscape of the objective function and the boundary of the safety polytope. More precisely, by assuming that Reliable-FW has access to a (stochastic) gradient oracle of the objective function and a noisy feasibility oracle of the safety polytope, it finds an ε-approximate first-order stationary point with the optimal O(1/ε2) gradient oracle complexity (resp. tildeO(1/ε3) (also optimal) in the stochastic gradient setting), while ensuring the safety of all the iterates. Rather surprisingly, Reliable-FW only makes tildeO((d2/ε2)log 1/δ) queries to the noisy feasibility oracle (resp. tildeO((d2/ε4)log 1/δ) in the stochastic gradient setting) where d is the dimension and δ is the reliability parameter, tightening the existing bounds even for safe minimization of convex functions. We further specialize our results to the case that the objective function is convex. A crucial component of our analysis is to introduce and apply a technique called geometric shrinkage in the context of safe optimization.