Beyond Grids: Multi-objective Bayesian Optimization With Adaptive Discretization

We consider the problem of optimizing a vector-valued objective function boldsymbolf sampled from a Gaussian Process (GP) whose index set is a well-behaved, compact metric space (cal X,d) of designs. We assume that boldsymbolf is not known beforehand and that evaluating boldsymbolf at design x results in a noisy observation of boldsymbolf(x). Since identifying the Pareto optimal designs via exhaustive search is infeasible when the cardinality of cal X is large, we propose an algorithm, called Adaptive boldsymbolε-PAL, that exploits the smoothness of the GP-sampled function and the structure of (cal X,d) to learn fast. In essence, Adaptive boldsymbolε-PAL employs a tree-based adaptive discretization technique to identify an boldsymbolε-accurate Pareto set of designs in as few evaluations as possible. We provide both information-type and metric dimension-type bounds on the sample complexity of boldsymbolε-accurate Pareto set identification. We also experimentally show that our algorithm outperforms other Pareto set identification methods.