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.

X-Armed Bandits: Optimizing Quantiles and Other Risks

We propose and analyze StoROO, an algorithm for risk optimization on stochastic black-box functions derived from StoOO. Motivated by risk-averse decision making fields like agriculture, medicine, biology or finance, we do not focus on the mean payoff but on generic functionals of the return distribution, like for example quantiles. We provide a generic regret analysis of StoROO. Inspired by the bandit literature and black-box mean optimizers, StoROO relies on the possibility to construct confidence intervals for the targeted functional based on random-size samples. We explain in detail how to construct them for quantiles, providing tight bounds based on Kullback-Leibler divergence. The interest of these tight bounds is highlighted by numerical experiments that show a dramatic improvement over standard approaches.

A Review on Quantile Regression for Stochastic Computer Experiments

We report on an empirical study of the main strategies for conditional quantile estimation in the context of stochastic computer experiments. To ensure adequate diversity, six metamodels are presented, divided into three categories based on order statistics, functional approaches, and those of Bayesian inspiration. The metamodels are tested on several problems characterized by the size of the training set, the input dimension, the quantile order and the value of the probability density function in the neighborhood of the quantile. The metamodels studied reveal good contrasts in our set of 480 experiments, enabling several patterns to be extracted. Based on our results, guidelines are proposed to allow users to select the best method for a given problem.