  Bayesian optimization (BO) has shown impressive results in a variety of
applications within low-to-moderate dimensional Euclidean spaces. However,
extending BO to high-dimensional settings remains a significant challenge. We
address this challenge by proposing a two-step optimization framework.
Initially, we identify the effective dimension reduction (EDR) subspace for the
objective function using the minimum average variance estimation (MAVE) method.
Subsequently, we construct a Gaussian process model within this EDR subspace
and optimize it using the expected improvement criterion. Our algorithm offers
the flexibility to operate these steps either concurrently or in sequence. In
the sequential approach, we meticulously balance the exploration-exploitation
trade-off by distributing the sampling budget between subspace estimation and
function optimization, and the convergence rate of our algorithm in
high-dimensional contexts has been established. Numerical experiments validate
the efficacy of our method in challenging scenarios.
