# A new integral loss function for Bayesian optimization

arXiv: https://arxiv.org/abs/1408.4622 · HTML: https://mlsanity.com/paper/1408.4622/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Aug 2014.

## Built on by (most cited first, 1 of 1)

- [Relaxed Gaussian process interpolation: a goal-oriented approach to Bayesian optimization](https://mlsanity.com/paper/2206.03034/) (2022)

## Abstract (arXiv)

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 vZilinskas in the 70's, the problem of optimization is usually formulated by considering the loss function max f - Mn (where Mn 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 (mathbbX, λ), we propose to consider instead the integral loss function int_mathbbX (f - Mn)+, 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 EI2 (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.