Bayesian Optimization (BO) has become a core method for solving expensive black-box optimization problems. While much research focussed on the choice of the acquisition function, we focus on online length-scale adaption and the choice of kernel function. Instead of choosing hyperparameters in view of maximum likelihood on past data, we propose to use the acquisition function to decide on hyperparameter adaptation more robustly and in view of the future optimization progress. Further, we propose a particular kernel function that includes non-stationarity and local anisotropy and thereby implicitly integrates the efficiency of local convex optimization with global Bayesian optimization. Comparisons to state-of-the art BO methods underline the efficiency of these mechanisms on global optimization benchmarks.
Optimising black-box functions is important in many disciplines, such as tuning machine learning models, robotics, finance and mining exploration. Bayesian optimisation is a state-of-the-art technique for the global optimisation of black-box functions which are expensive to evaluate. At the core of this approach is a Gaussian process prior that captures our belief about the distribution over functions. However, in many cases a single Gaussian process is not flexible enough to capture non-stationarity in the objective function. Consequently, heteroscedasticity negatively affects performance of traditional Bayesian methods. In this paper, we propose a novel prior model with hierarchical parameter learning that tackles the problem of non-stationarity in Bayesian optimisation. Our results demonstrate substantial improvements in a wide range of applications, including automatic machine learning and mining exploration.  This paper introduces a new method of calculating the expected improvement infill criterion, which does not rely on accurate model parameter estimation. The parameter estimation is embedded within the search of the infill criterion, wherein parameter changes are assessed using likelihood ratio tests. Unlike the traditional expected improvement, a new formulation we present cannot be fooled' by unlucky sampling or deceptive functions. The new method is introduced both mathematically and illustratively using a one-variable test function. It is then shown to outperform traditional expected improvement when optimizing the geometry of a passive vibration isolating truss. This paper presents a taxonomy of existing approaches for using response surfaces for global optimization. Each method is illustrated with a simple numerical example that brings out its advantages and disadvantages. The central theme is that methods that seem quite reasonable often have non-obvious failure modes. Understanding these failure modes is essential for the development of practical algorithms that fulfill the intuitive promise of the response surface approach. The Maximum Likelihood (ML) and Cross Validation (CV) methods for estimating covariance hyper-parameters are compared, in the context of Kriging with a misspecified covariance structure. A two-step approach is used. First, the case of the estimation of a single variance hyper-parameter is addressed, for which the fixed correlation function is misspecified. A predictive variance based quality criterion is introduced and a closed-form expression of this criterion is derived. It is shown that when the correlation function is misspecified, the CV does better compared to ML, while ML is optimal when the model is well-specified. In the second step, the results of the first step are extended to the case when the hyper-parameters of the correlation function are also estimated from data.