Convergence Rates of Epsilon-Greedy Global Optimization Under Radial Basis Function Interpolation

We study a global optimization problem where the objective function can be observed exactly at individual design points with no derivative information. We suppose that the design points are determined sequentially using an epsilon-greedy algorithm, that is, by sampling uniformly on the design space with a certain probability and otherwise sampling in a local neighborhood of the current estimate of the best solution. We study the rate at which the estimate converges to the global optimum and derive two types of bounds: an asymptotic pathwise rate and a concentration inequality measuring the likelihood that the asymptotic rate has not yet gone into effect. The order of the rate becomes faster when the width of the local search neighborhood is made to shrink over time at a suitably chosen speed.