Researchers have found a simple, sharp answer to a theoretical puzzle in one-dimensional derivative-free optimization. They give a computationally efficient algorithm that reaches the best possible accuracy, scaling like 1 divided by the square root of the number of queries. This matches the known lower bound and removes a previous logarithmic shortfall in guarantees.

The problem is this: you want to minimize a convex function that lives on the interval [0,1], but you cannot see derivatives. You can only query the function at points and get noisy answers. The noise is assumed to be subGaussian, a standard way to say the random errors have tails no heavier than a Gaussian. The goal is to design a querying strategy that gets close to the function minimum after T noisy evaluations.

What the authors did was design a zero-order (derivative-free) algorithm that is computationally efficient and provably achieves an error of order 1/√T. That matches the earlier lower bound of order 1/√T, so there is no longer a gap between upper and lower guarantees in this one-dimensional setting. In plain terms, their method uses a careful pattern of queries and combines noisy measurements so the estimate of the minimum improves at the fastest possible rate allowed by the noise.

This result matters because derivative-free optimization is common in practice when gradients are unavailable or expensive. Having a sharp theoretical rate gives a clear benchmark: no algorithm can asymptotically beat the 1/√T scaling in this setting, and this work provides an algorithm that attains that limit. It settles a question that remained open even in the simplest one-dimensional case.

There are important caveats. The result is proved only for functions on the interval [0,1] with values in [0,1], and it assumes subGaussian noise. The paper makes no claim about higher dimensions. Also, the abstract does not report numerical experiments or give the precise constants hidden in the 1/√T notation, so practical performance and tuning are not described there. Finally, extensions beyond the specific assumptions will require further work.