Pareto-frontier Entropy Search with Variational Lower Bound Maximization

This study considers multi-objective Bayesian optimization (MOBO) through the information gain of the Pareto-frontier. To calculate the information gain, a predictive distribution conditioned on the Pareto-frontier plays a key role, which is defined as a distribution truncated by the Pareto-frontier. However, it is usually impossible to obtain the entire Pareto-frontier in a continuous domain, and therefore, the complete truncation cannot be known. We consider an approximation of the truncated distribution by using a mixture distribution consisting of two possible approximate truncations obtainable from a subset of the Pareto-frontier, which we call over- and under-truncation. Since the optimal balance of the mixture is unknown beforehand, we propose optimizing the balancing coefficient through the variational lower bound maximization framework, by which the approximation error of the information gain can be minimized. Our empirical evaluation demonstrates the effectiveness of the proposed method particularly when the number of objective functions is large.

In many real-world scenarios, we need to find the best balance between multiple goals such as maximizing performance while minimizing cost. This is known as the multi-objective optimization, and it is especially challenging when we do not know how these goals trade-off with each other. Our research addresses this challenge by using an approach called information-theoretic Bayesian optimization, which helps efficiently find a set of the best trade-off solutions. A key idea of this method involves evaluating how much information we can gain when new data is observed, but the calculation of this information gain is computationally difficult particularly when the candidate space is continuous. To overcome this difficulty, we developed a novel approach to approximating the information gain, by combining two approximation schemes of the so-called ``Pareto-frontier''. Further, we improved approximation accuracy by automatically adjusting this combination using a framework called variational inference. Our technique improved performance of information-theoretic Bayesian optimization especially when optimizing many objectives. This research can help improve decision-making in fields like engineering design, materials science, and hyper-parameter optimization of machine leaning.