  We consider multi-objective optimization (MOO) of an unknown vector-valued
function in the non-parametric Bayesian optimization (BO) setting, with the aim
being to learn points on the Pareto front of the objectives. Most existing BO
algorithms do not model the fact that the multiple objectives, or equivalently,
tasks can share similarities, and even the few that do lack rigorous,
finite-time regret guarantees that capture explicitly inter-task structure. In
this work, we address this problem by modelling inter-task dependencies using a
multi-task kernel and develop two novel BO algorithms based on random
scalarizations of the objectives. Our algorithms employ vector-valued kernel
regression as a stepping stone and belong to the upper confidence bound class
of algorithms. Under a smoothness assumption that the unknown vector-valued
function is an element of the reproducing kernel Hilbert space associated with
the multi-task kernel, we derive worst-case regret bounds for our algorithms
that explicitly capture the similarities between tasks. We numerically
benchmark our algorithms on both synthetic and real-life MOO problems, and show
the advantages offered by learning with multi-task kernels.
