  Optimization of accelerator performance parameters is limited by numerous
trade-offs and finding the appropriate balance between optimization goals for
an unknown system is challenging to achieve. Here we show that multi-objective
Bayesian optimization can map the solution space of a laser wakefield
accelerator in a very sample-efficient way. Using a Gaussian mixture model, we
isolate contributions related to an electron bunch at a certain energy and we
observe that there exists a wide range of Pareto-optimal solutions that trade
beam energy versus charge at similar laser-to-beam efficiency. However, many
applications such as light sources require particle beams at a certain target
energy. Once such a constraint is introduced we observe a direct trade-off
between energy spread and accelerator efficiency. We furthermore demonstrate
how specific solutions can be exploited using \emph{a posteriori} scalarization
of the objectives, thereby efficiently splitting the exploration and
exploitation phases.
