Can a One-Point Feedback Zeroth-order Algorithm Achieve Linear Dimension Dependent Sample Complexity?

We revisit the one-point feedback zeroth-order (ZO) optimization problem, a classical setting in derivative-free optimization where only a single noisy function evaluation is available per query. Compared to their two-point counterparts, existing one-point feedback ZO algorithms typically suffer from poor dimension dependence in their sample complexities – often quadratic or worse – even for convex problems. This gap has led to the open question of whether one-point feedback ZO algorithms can match the optimal \emph{linear} dimension dependence achieved by two-point methods. In this work, we answer this question \emph{affirmatively}.