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

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.