High Dimensional Bayesian Optimisation and Bandits via Additive Models

Abstract

Bayesian Optimisation (BO) is a technique used in optimising a $D$-dimensional function which is typically expensive to evaluate. While there have been many successes for BO in low dimensions, scaling it to high dimensions has been a notoriously difficult problem. Existing literature on the subject are under very restrictive settings. In this paper, we identify two key challenges in this endeavour. We tackle these challenges by assuming an additive structure for the function. This setting is substantially more expressive and contains a richer class of functions than previous work. In our theoretical analysis we prove that for additive functions the regret has only linear (as opposed to exponential) dependence on $D$ even though the function depends on all $D$ dimensions. We also demonstrate several other statistical and computational benefits in our framework. Empirically via synthetic examples, a scientific simulation and a face detection problem we demonstrate that our method outperforms naive BO on additive functions and on several examples when the function is not additive.