Bayesian Quantile and Expectile Optimisation

Bayesian optimisation is widely used to optimise stochastic black box functions. While most strategies are focused on optimising conditional expectations, a large variety of applications require risk-averse decisions and alternative criteria accounting for the distribution tails need to be considered. In this paper, we propose new variational models for Bayesian quantile and expectile regression that are well-suited for heteroscedastic settings. Our models consist of two latent Gaussian processes accounting respectively for the conditional quantile (or expectile) and variance that are chained through asymmetric likelihood functions. Furthermore, we propose two Bayesian optimisation strategies, either derived from a GP-UCB or Thompson sampling, that are tailored to such models and that can accommodate large batches of points. As illustrated in the experimental section, the proposed approach clearly outperforms the state of the art.