  Bayesian optimization (BO), while proved highly effective for many black-box
function optimization tasks, requires practitioners to carefully select priors
that well model their functions of interest. Rather than specifying by hand,
researchers have investigated transfer learning based methods to automatically
learn the priors, e.g. multi-task BO (Swersky et al., 2013), few-shot BO
(Wistuba and Grabocka, 2021) and HyperBO (Wang et al., 2022). However, those
prior learning methods typically assume that the input domains are the same for
all tasks, weakening their ability to use observations on functions with
different domains or generalize the learned priors to BO on different search
spaces. In this work, we present HyperBO+: a pre-training approach for
hierarchical Gaussian processes that enables the same prior to work universally
for Bayesian optimization on functions with different domains. We propose a
two-step pre-training method and analyze its appealing asymptotic properties
and benefits to BO both theoretically and empirically. On real-world
hyperparameter tuning tasks that involve multiple search spaces, we demonstrate
that HyperBO+ is able to generalize to unseen search spaces and achieves lower
regrets than competitive baselines.
