Takeno, Shion

Multi-objective Bayesian Optimization using Pareto-frontier Entropy

arXiv.org Machine Learning

We propose Pareto-frontier entropy search (PFES) for multi-objective Bayesian optimization (MBO). Unlike the existing entropy search for MBO which considers the entropy of the input space, we define the entropy of Pareto-frontier in the output space. By using a sampled Pareto-frontier from the current model, PFES provides a simple formula for directly evaluating the entropy. Besides the usual MBO setting, in which all the objectives are simultaneously observed, we also consider the "decoupled" setting, in which the objective functions can be observed separately. PFES can easily derive an acquisition function for the decoupled setting through the entropy of the marginal density for each output variable. For the both settings, by conditioning on the sampled Pareto-frontier, dependence among different objectives arises in the entropy evaluation. PFES can incorporate this dependency into the acquisition function, while the existing information-based MBO employs an independent Gaussian approximation. Our numerical experiments show effectiveness of PFES through synthetic functions and real-world datasets from materials science.

Multi-fidelity Bayesian Optimization with Max-value Entropy Search

arXiv.org Machine Learning

Bayesian optimization (BO) is an effective tool for black-box optimization in which objective function evaluation is usually quite expensive. In practice, lower fidelity approximations of the objective function are often available. Recently, multi-fidelity Bayesian optimization (MFBO) has attracted considerable attention because it can dramatically accelerate the optimization process by using those cheaper observations. We propose a novel information theoretic approach to MFBO. Information-based approaches are popular and empirically successful in BO, but existing studies for information-based MFBO are plagued by difficulty for accurately estimating the information gain. Our approach is based on a variant of information-based BO called max-value entropy search (MES), which greatly facilitates evaluation of the information gain in MFBO. In fact, computations of our acquisition function is written analytically except for one dimensional integral and sampling, which can be calculated efficiently and accurately. We demonstrate effectiveness of our approach by using synthetic and benchmark datasets, and further we show a real-world application to materials science data.