🤖 AI Summary
Efficiently identifying the Pareto-optimal set for multi-objective black-box functions over high-dimensional continuous design spaces remains challenging due to the exponential growth of candidate solutions.
Method: This paper proposes Adaptive ε-PAL, the first algorithm integrating tree-based adaptive discretization with Gaussian process (GP) Bayesian optimization. It leverages GP surrogate modeling, Pareto front estimation, and information-theoretic analysis of compact metric spaces to dynamically partition the input space—thereby avoiding exhaustive enumeration.
Contribution/Results: We establish a provably tight upper bound on the ε-accurate sample complexity. Empirical evaluation across multiple benchmarks demonstrates substantial reductions in function evaluations compared to state-of-the-art Pareto-set identification methods. The core contribution is a theoretically grounded, adaptive multi-objective optimization framework that simultaneously ensures rigorous convergence guarantees and practical efficiency.
📝 Abstract
We consider the problem of optimizing a vector-valued objective function $oldsymbol{f}$ sampled from a Gaussian Process (GP) whose index set is a well-behaved, compact metric space $({cal X},d)$ of designs. We assume that $oldsymbol{f}$ is not known beforehand and that evaluating $oldsymbol{f}$ at design $x$ results in a noisy observation of $oldsymbol{f}(x)$. 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 $oldsymbol{epsilon}$-PAL, that exploits the smoothness of the GP-sampled function and the structure of $({cal X},d)$ to learn fast. In essence, Adaptive $oldsymbol{epsilon}$-PAL employs a tree-based adaptive discretization technique to identify an $oldsymbol{epsilon}$-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 $oldsymbol{epsilon}$-accurate Pareto set identification. We also experimentally show that our algorithm outperforms other Pareto set identification methods.
