  A recent development in Bayesian optimization is the use of local
optimization strategies, which can deliver strong empirical performance on
high-dimensional problems compared to traditional global strategies. The "folk
wisdom" in the literature is that the focus on local optimization sidesteps the
curse of dimensionality; however, little is known concretely about the expected
behavior or convergence of Bayesian local optimization routines. We first study
the behavior of the local approach, and find that the statistics of individual
local solutions of Gaussian process sample paths are surprisingly good compared
to what we would expect to recover from global methods. We then present the
first rigorous analysis of such a Bayesian local optimization algorithm
recently proposed by M\"uller et al. (2021), and derive convergence rates in
both the noisy and noiseless settings.
