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.

This paper studies a natural generalization of the problem of minimizing a univariate convex function $f$ by querying its values sequentially. At each time-step $t$, the optimizer can invest a budget $b_t$ in a query point $X_t$ of their choice to obtain a fuzzy evaluation of $f$ at $X_t$ whose accuracy depends on the amount of budget invested in $X_t$ across times. This setting is motivated by the minimization of objectives whose values can only be determined approximately through lengthy or expensive computations. We design an any-time parameter-free algorithm called Dyadic Search, for which we prove near-optimal optimization error guarantees. As a byproduct of our analysis, we show that the classical dependence on the global Lipschitz constant in the error bounds is an artifact of the granularity of the budget. Finally, we illustrate our theoretical findings with numerical simulations.