Regret Analysis of Dyadic Search

We analyze the cumulative regret of the Dyadic Search algorithm of Bachoc et al. [2022]. In this section, we introduce the formal setting for our budget convex optimization problem. Given a bounded intervalI R, our goal is to minimize an unknown convex functionf I R picked by a possibly adversarial and adaptive environment by only requesting fuzzy evaluations of f. The interactions between the optimizer and the environment are described in Optimization Protocol 1. Optimization Protocol 1 input: A non-empty bounded interval I R (the domain of the unknown objective f) We stress that the environment is adaptive. The idea is that the more budget is invested, the more accurate approximation of the objectivef can be determined, in a quantifiable way.