In this work we introduce PHOENICS, a probabilistic global optimization algorithm combining ideas from Bayesian optimization with concepts from Bayesian kernel density estimation. We propose an inexpensive acquisition function balancing the explorative and exploitative behavior of the algorithm. This acquisition function enables intuitive sampling strategies for an efficient parallel search of global minima. The performance of PHOENICS is assessed via an exhaustive benchmark study on a set of 15 discrete, quasi-discrete and continuous multidimensional functions. Unlike optimization methods based on Gaussian processes (GP) and random forests (RF), we show that PHOENICS is less sensitive to the nature of the co-domain, and outperforms GP and RF optimizations. We illustrate the performance of PHOENICS on the Oregonator, a difficult case-study describing a complex chemical reaction network. We demonstrate that only PHOENICS was able to reproduce qualitatively and quantitatively the target dynamic behavior of this nonlinear reaction dynamics. We recommend PHOENICS for rapid optimization of scalar, possibly non-convex, black-box unknown objective functions.
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.
Student-$t$ processes have recently been proposed as an appealing alternative non-parameteric function prior. They feature enhanced flexibility and predictive variance. In this work the use of Student-$t$ processes are explored for multi-objective Bayesian optimization. In particular, an analytical expression for the hypervolume-based probability of improvement is developed for independent Student-$t$ process priors of the objectives. Its effectiveness is shown on a multi-objective optimization problem which is known to be difficult with traditional Gaussian processes.
The PC algorithm is a popular method for learning the structure of Gaussian Bayesian networks. It carries out statistical tests to determine absent edges in the network. It is hence governed by two parameters: (i) The type of test, and (ii) its significance level. These parameters are usually set to values recommended by an expert. Nevertheless, such an approach can suffer from human bias, leading to suboptimal reconstruction results. In this paper we consider a more principled approach for choosing these parameters in an automatic way. For this we optimize a reconstruction score evaluated on a set of different Gaussian Bayesian networks. This objective is expensive to evaluate and lacks a closed-form expression, which means that Bayesian optimization (BO) is a natural choice. BO methods use a model to guide the search and are hence able to exploit smoothness properties of the objective surface. We show that the parameters found by a BO method outperform those found by a random search strategy and the expert recommendation. Importantly, we have found that an often overlooked statistical test provides the best over-all reconstruction results.
Multi-objective optimization aims at finding trade-off solutions to conflicting objectives. These constitute the Pareto optimal set. In the context of expensive-to-evaluate functions, it is impossible and often non-informative to look for the entire set. As an end-user would typically prefer a certain part of the objective space, we modify the Bayesian multi-objective optimization algorithm which uses Gaussian Processes to maximize the Expected Hypervolume Improvement, to focus the search in the preferred region. The cumulated effects of the Gaussian Processes and the targeting strategy lead to a particularly efficient convergence to the desired part of the Pareto set. To take advantage of parallel computing, a multi-point extension of the targeting criterion is proposed and analyzed.