  Bayesian optimization is a broadly applied methodology to optimize the
expensive black-box function. Despite its success, it still faces the challenge
from the high-dimensional search space. To alleviate this problem, we propose a
novel Bayesian optimization framework (termed SILBO), which finds a
low-dimensional space to perform Bayesian optimization iteratively through
semi-supervised dimension reduction. SILBO incorporates both labeled points and
unlabeled points acquired from the acquisition function to guide the embedding
space learning. To accelerate the learning procedure, we present a randomized
method for generating the projection matrix. Furthermore, to map from the
low-dimensional space to the high-dimensional original space, we propose two
mapping strategies: $\text{SILBO}_{FZ}$ and $\text{SILBO}_{FX}$ according to
the evaluation overhead of the objective function. Experimental results on both
synthetic function and hyperparameter optimization tasks demonstrate that SILBO
outperforms the existing state-of-the-art high-dimensional Bayesian
optimization methods.
