  Many black-box optimization tasks arising in high-stakes applications require
risk-averse decisions. The standard Bayesian optimization (BO) paradigm,
however, optimizes the expected value only. We generalize BO to trade mean and
input-dependent variance of the objective, both of which we assume to be
unknown a priori. In particular, we propose a novel risk-averse heteroscedastic
Bayesian optimization algorithm (RAHBO) that aims to identify a solution with
high return and low noise variance, while learning the noise distribution on
the fly. To this end, we model both expectation and variance as (unknown) RKHS
functions, and propose a novel risk-aware acquisition function. We bound the
regret for our approach and provide a robust rule to report the final decision
point for applications where only a single solution must be identified. We
demonstrate the effectiveness of RAHBO on synthetic benchmark functions and
hyperparameter tuning tasks.
