Minimax simple‑regret rate in noisy convex zeroth‑order optimisation

Determine the minimax rate, as a function of the dimension d and the query budget n, of the simple regret f(\hat x) − min_{x \in \bar{\mathcal X}} f(x) for the sequential adaptive noisy zeroth‑order optimisation problem defined as follows: a learner selects points x_t \in \bar{\mathcal X} (a bounded convex subset of \mathbb{R}^d) and observes y_t \in [0,1] with \mathbb{E}[y_t | x_t] = f(x_t), where f: \bar{\mathcal X} \to [0,1] is convex and observations are conditionally independent; after n queries the learner outputs \hat x. The task is to establish the optimal (minimax) dependence on d and n for the achievable simple regret in this setting without imposing strong convexity or smoothness assumptions.

Background

The paper studies noisy convex zeroth‑order optimisation over a bounded convex domain with sequential, adaptive queries. Performance is measured by the simple regret f(\hat x) − inf_{x \in \bar{\mathcal X}} f(x). The authors present a conceptually simple algorithm inspired by the center‑of‑gravity method and prove a high‑probability regret bound of order d^2 / \sqrt{n} up to polylogarithmic factors, under a mild assumption that the minimiser is not too close to the boundary.

Existing results for related settings include upper bounds translating to roughly d^{2.5}/\sqrt{n} in this simple‑regret setting (from adversarial cumulative‑regret analyses) and lower bounds of order d/\sqrt{n} for linear functions, leaving a gap. The authors explicitly note that, despite their improvement, the precise minimax rate for this problem remains unknown.