Summary: The task of finding the optimum of some function $f(x)$ is commonly accomplished by generating and testing sample solutions iteratively, choosing each new sample x heuristically on the basis of results to date. We use Gaussian processes to represent predictions and uncertainty about the true function, and describe how to use these predictions to choose where to take each new sample in an optimal way. By doing this we were able to solve a difficult optimization problem ‒ finding weights in a neural network controller to simultaneously balance two vertical poles ‒ using an order of magnitude fewer samples than reported elsewhere.