Safe Online Convex Optimization with Multi-Point Feedback
This addresses safety-critical applications where constraints must not be violated, though it is incremental as it builds on existing safe optimization methods.
The paper tackles safe online convex optimization with zero constraint violation using only zero-order feedback, achieving O(d√T) regret under smooth and strongly convex constraints.
Motivated by the stringent safety requirements that are often present in real-world applications, we study a safe online convex optimization setting where the player needs to simultaneously achieve sublinear regret and zero constraint violation while only using zero-order information. In particular, we consider a multi-point feedback setting, where the player chooses $d + 1$ points in each round (where $d$ is the problem dimension) and then receives the value of the constraint function and cost function at each of these points. To address this problem, we propose an algorithm that leverages forward-difference gradient estimation as well as optimistic and pessimistic action sets to achieve $\mathcal{O}(d \sqrt{T})$ regret and zero constraint violation under the assumption that the constraint function is smooth and strongly convex. We then perform a numerical study to investigate the impacts of the unknown constraint and zero-order feedback on empirical performance.