We propose STARS, a randomized derivative-free algorithm for unconstrained optimization when the function evaluations are contaminated with random noise. STARS takes dynamic, noise-adjusted smoothing step-sizes that minimize the least-squares error between the true directional derivative of a noisy function and its finite difference approximation. We provide a convergence rate analysis of STARS for solving convex problems with additive or multiplicative noise. Experimental results show that (1) STARS exhibits noise-invariant behavior with respect to different levels of stochastic noise; (2) the practical performance of STARS in terms of solution accuracy and convergence rate is significantly better than that indicated by the theoretical result; and (3) STARS outperforms a selection of randomized zero-order methods on both additive and multiplicative-noisy functions.