  We propose a practical Bayesian optimization method over sets, to minimize a
black-box function that takes a set as a single input. Because set inputs are
permutation-invariant, traditional Gaussian process-based Bayesian optimization
strategies which assume vector inputs can fall short. To address this, we
develop a Bayesian optimization method with \emph{set kernel} that is used to
build surrogate functions. This kernel accumulates similarity over set elements
to enforce permutation-invariance, but this comes at a greater computational
cost. To reduce this burden, we propose two key components: (i) a more
efficient approximate set kernel which is still positive-definite and is an
unbiased estimator of the true set kernel with upper-bounded variance in terms
of the number of subsamples, (ii) a constrained acquisition function
optimization over sets, which uses symmetry of the feasible region that defines
a set input. Finally, we present several numerical experiments which
demonstrate that our method outperforms other methods.
