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

TL;DR

This paper introduces a nonparametric Bayesian method that directly models the distribution over the maximum of noisy, nonlinear functions, streamlining stochastic optimization by bypassing explicit function representation.

Contribution

It presents a novel conjugate prior based on kernel regression that directly captures uncertainty over the extrema of unknown functions.

Findings

Effective optimization of high-dimensional, noisy, non-convex functions.

Direct modeling of extrema improves efficiency over traditional two-step methods.

Demonstrated success in complex stochastic optimization tasks.

Abstract

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