A new integral loss function for Bayesian optimization
This work addresses the limitation of focusing only on the maximum value in Bayesian optimization, offering a more balanced approach for applications requiring accurate maximizer location, though it is incremental as it builds on existing Stepwise Uncertainty Reduction strategies.
The authors tackled the problem of maximizing a continuous function using Bayesian optimization by proposing a new integral loss function, which leads to a sampling criterion called EI^2, and showed in experiments that it reduces errors on both the value and location of the maximizer faster than the classical Expected Improvement method.
We consider the problem of maximizing a real-valued continuous function $f$ using a Bayesian approach. Since the early work of Jonas Mockus and Antanas Žilinskas in the 70's, the problem of optimization is usually formulated by considering the loss function $\max f - M_n$ (where $M_n$ denotes the best function value observed after $n$ evaluations of $f$). This loss function puts emphasis on the value of the maximum, at the expense of the location of the maximizer. In the special case of a one-step Bayes-optimal strategy, it leads to the classical Expected Improvement (EI) sampling criterion. This is a special case of a Stepwise Uncertainty Reduction (SUR) strategy, where the risk associated to a certain uncertainty measure (here, the expected loss) on the quantity of interest is minimized at each step of the algorithm. In this article, assuming that $f$ is defined over a measure space $(\mathbb{X}, λ)$, we propose to consider instead the integral loss function $\int_{\mathbb{X}} (f - M_n)_{+}\, dλ$, and we show that this leads, in the case of a Gaussian process prior, to a new numerically tractable sampling criterion that we call $\rm EI^2$ (for Expected Integrated Expected Improvement). A numerical experiment illustrates that a SUR strategy based on this new sampling criterion reduces the error on both the value and the location of the maximizer faster than the EI-based strategy.