  Bayesian optimisation (BO) is widely used to optimise stochastic black box
functions. While most BO approaches focus on optimising conditional
expectations, many applications require risk-averse strategies 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 noise settings. Our models
consist of two latent Gaussian processes accounting respectively for the
conditional quantile (or expectile) and the scale parameter of an asymmetric
likelihood functions. Furthermore, we propose two BO strategies based on
max-value entropy search and Thompson sampling, that are tailored to such
models and that can accommodate large batches of points. Contrary to existing
BO approaches for risk-averse optimisation, our strategies can directly
optimise for the quantile and expectile, without requiring replicating
observations or assuming a parametric form for the noise. As illustrated in the
experimental section, the proposed approach clearly outperforms the state of
the art in the heteroscedastic, non-Gaussian case.
