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arXiv · 2607.12938

Sharp Optimal Algorithm for Derivative-Free Stochastic Convex Optimization in One Dimension

Alexandra Carpentier·Chloé Rouyer·Alexandre Tsybakov·Arya Akhavan
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Abstract

Stochastic convex optimization is a classical problem with well-understood guarantees under first-order feedback. In contrast, for zero-order optimization with noisy function evaluations, a logarithmic gap has persisted between known upper bounds and the $\Omega(1/\sqrt{T})$ lower bound, even in the one-dimensional case. In this work, we study the problem of minimizing a convex function $f : [0,1] \to [0,1]$ using a zero-order oracle with subGaussian noise. We propose a computationally efficient algorithm that achieves the optimal $O(1/\sqrt{T})$ convergence rate, matching the lower bound. The result closes the existing gap in one dimension, providing the first sharp rate guarantee in this setting.

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Alexandra Carpentier, Chloé Rouyer, Alexandre Tsybakov, Arya Akhavan. 2026-07-14. Sharp Optimal Algorithm for Derivative-Free Stochastic Convex Optimization in One Dimension. https://arxiv.org/abs/2607.12938

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