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

2026-07-14

AI Summary

This work addresses zeroth-order stochastic convex optimization in one dimension under sub-Gaussian noise, proposing a computationally efficient algorithm that achieves the optimal $O(1/\sqrt{T})$ convergence rate using only noisy function-value feedback. By employing a refined sampling and function estimation strategy, the method matches the known information-theoretic lower bound of $\Omega(1/\sqrt{T})$, thereby closing the longstanding logarithmic gap that has persisted in this setting. This result resolves a notable theoretical gap in the literature on one-dimensional zeroth-order stochastic convex optimization, establishing for the first time an algorithm that attains the minimax-optimal rate without extraneous logarithmic factors.

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 $Ω(1/\sqrt{T})$ 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.

Problem

• zero-order optimization
• stochastic convex optimization
• derivative-free
• convergence rate
• one dimension

Innovation

• zero-order optimization
• stochastic convex optimization
• optimal convergence rate
• derivative-free
• one-dimensional