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The paper discusses derivative-free optimization (DFO), which involves minimizing a function without access to gradients or directional derivatives, only function evaluations. Classical DFO methods such as Nelder-Mead and direct search have limited scalability for high-dimensional problems. Zeroth-order methods, which mimic gradient-based methods, have been gaining popularity due to the demands of large-scale machine learning applications. This paper focuses on the selection of the step size $$\alpha_k$$ in such methods. The proposed approach, called Curvature-Aware Random Search (CARS), uses first- and second-order finite difference approximations to compute a candidate $$\alpha_+$$. A safeguarding step then evaluates $$\alpha_+$$ and chooses an alternate step size in case $$\alpha_+$$ does not decrease the objective function. We prove that for strongly convex objective functions, CARS converges linearly provided that the search direction is drawn from a distribution satisfying very mild conditions. We also present a Cubic Regularized variant of CARS, named CARS-CR, which provably converges at a rate of $O(1/k)$ without the assumption of strong convexity. Numerical experiments show that CARS and CARS-CR match or exceed the state-of-the-art on benchmark problem sets.