X-Armed Bandits: Optimizing Quantiles, CVaR and Other Risks

We propose and analyze StoROO, an algorithm for risk optimization on stochastic blackbox 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. We provide a generic regret analysis of StoROO and illustrate its applicability with two examples: the optimization of quantiles and CVaR. 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 randomsize samples. We detail their construction in the case of quantiles, providing tight bounds based on Kullback-Leibler divergence. We finally present numerical experiments that show a dramatic impact of tight bounds for the optimization of quantiles and CVaR.