A Nonparametric Conjugate Prior Distribution for the Maximizing Argument\n of a Noisy Function

We propose a novel Bayesian approach to solve stochastic optimization\nproblems that involve finding extrema of noisy, nonlinear functions. Previous\nwork has focused on representing possible functions explicitly, which leads to\na two-step procedure of first, doing inference over the function space and\nsecond, finding the extrema of these functions. Here we skip the representation\nstep and directly model the distribution over extrema. To this end, we devise a\nnon-parametric conjugate prior based on a kernel regressor. The resulting\nposterior distribution directly captures the uncertainty over the maximum of\nthe unknown function. We illustrate the effectiveness of our model by\noptimizing a noisy, high-dimensional, non-convex objective function.\n