On the role of Model Uncertainties in Bayesian Optimization

Probabilistic machine learning provides a framework in which it is possible to reason about uncertainty for both models and predictions (Ghahramani, 2015). It is often argued that especially in high-stakes applications (healthcare, robotics, etc.), uncertainty estimates for decisions/predictions should be a central component and that they should be well-calibrated (Kuleshov and Deshpande, 2022). The intuition behind calibration is that the uncertainty estimates should accurately reflect the reality; for example if a classification model predicts 80% probability of belonging to class A on 10 datapoints, then (on average) we would expect 8 of those 10 samples actually belong to class A. Likewise - but less intuitively - in regression, if a calibrated model generates a prediction µ and uncertainty estimate σ, we would see p percent of the data lying inside a p percentile confidence interval of µ (Busk et al., 2021). In general, uncertainty can be divided into aleatoric (irreducible inherent randomness in the data-generating process) and epistemic (lack of knowledge, i.e. it can be reduced if more data is collected) (Hüllermeier and Waegeman, 2021). However, this distinction is rarely used when evaluating uncertainty estimates for regression tasks and although this has been critiqued (Sluijterman et al., 2021), it is highly non-trivial to achieve for real-world applications because it normally requires access to the underlying true function. Uncertainty also plays a central role Bayesian Optimization (BO) (Snoek et al., 2012), which will be the focus of this paper. As a sequential design strategy for global optimization, BO has several applications with perhaps the most popular ones being general experimental design (Shahriari et al., 2015) and model selection for machine learning