High-Probability Guarantees for Random Zeroth-Order Gradient Descent on Smooth Functions

Randomized zeroth-order methods are classically analyzed in expectation, but a black-box Markov conversion can give misleading high-probability guarantees, in particular by forcing the finite-difference smoothing radius to shrink with the confidence parameter. This paper gives a direct finite-horizon high-probability analysis of a two-query Gaussian finite-difference method for deterministic objectives with Lipschitz gradients. The method uses the classical two-point estimator together with the normalized stepsize \(\eta_t=1/(4L\norm{\bu_t}^2)\). We prove that it finds an \(\varepsilon\)-suboptimal point with probability at least \(1-\delta\) using \(\cO((dL/\mu)\log(1/\varepsilon)+\log(1/\delta))\) function queries under strong convexity, subject to an explicit finite-difference smoothing-radius condition. We also establish high-probability guarantees for smooth convex objectives under a level-set distance-to-solution radius condition and a pathwise smoothing-radius condition. For lower-bounded smooth non-convex objectives, the trajectory average is certified in stationarity with \(\cO(L\Delta_0(d+\log(1/\delta))/\varepsilon)\) function queries. The proofs combine lower-tail bounds for adaptive sums of Gaussian directional projections with upper-tail bounds for accumulated finite-difference smoothing errors.