Beyond Grids: Multi-objective Bayesian Optimization With Adaptive Discretization
This work addresses the challenge of multi-objective Bayesian optimization in large design spaces, offering an incremental improvement in efficiency for applications like engineering design or hyperparameter tuning.
The paper tackles the problem of optimizing a vector-valued objective function sampled from a Gaussian Process over a large design space, proposing an algorithm called Adaptive ε-PAL that uses adaptive discretization to identify an ε-accurate Pareto set with fewer evaluations, achieving sample complexity bounds and outperforming other methods in experiments.
We consider the problem of optimizing a vector-valued objective function $\boldsymbol{f}$ sampled from a Gaussian Process (GP) whose index set is a well-behaved, compact metric space $({\cal X},d)$ of designs. We assume that $\boldsymbol{f}$ is not known beforehand and that evaluating $\boldsymbol{f}$ at design $x$ results in a noisy observation of $\boldsymbol{f}(x)$. 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.