Beyond Grids: Multi-objective Bayesian Optimization With Adaptive Discretization

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