Papers › Curvature-Aware Derivative-Free Optimization
Curvature-Aware Derivative-Free Optimization
Bumsu Kim, HanQin Cai, Daniel Mckenzie, Wotao Yin
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, which mimic gradient-based methods, such as Nelder-Mead and direct search have limited scalability for high-dimensional problems. Zeroth-order methods have been gaining popularity due to the demands of large-scale machine learning applications, and the paper focuses on the selection of the step size αₖ in these methods. The proposed approach, called Curvature-Aware Random Search (CARS), uses first- and second-order finite difference approximations to compute a candidate α₊. 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 converges in a rate of 𝒪(k⁻¹) without the assumption of strong convexity. Numerical experiments show that CARS and CARS-CR match or exceed the state-of-the-arts on benchmark problem sets.
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