Bayesian Optimization and Data Science by Francesco Archetti & Antonio Candelieri

Author:Francesco Archetti & Antonio Candelieri
Language: eng
Format: epub
ISBN: 9783030244941
Publisher: Springer International Publishing

The following figure shows another case where TS is, instead, more explorative than LCB (Fig. 4.5).

Fig. 4.5Next point to evaluate according to TS and LCB: the sample from GP posterior implies a more explorative choice than LCB

Theoretical analysis was provided for the classical multi-armed bandit problem and later extended to the analysis of the continuous case in Russo et al. (2018) drawing on an analogy between TS and UCB.

4.2.4 Entropy-Based Acquisition Functions

The traditional acquisition functions, presented in Sect. 4.1, are based on probabilistic measures of improvement in the f domain. Exploitation and exploration are represented, respectively, by the mean value and the “uncertainty bonus” represented by the variance. Recent interest has been focused on querying at points that can help to learn most about the location of the unknown minimum, leading to information-based acquisition functions. This informational approach was originally proposed in Villemonteix et al. (2009), and developer into the Entropy Search (Henning and Schuler (2012)), Predictive Entropy Search (Hernández-Lobato et al. 2014). In both Entropy Search (ES) and Predictive Entropy Search (PES), the basic idea is to maximize the information about the global optimizer.

The current information about the global optimizer can be computed as the negative differential entropy of and the next point to evaluate is given by the maximization of the expected reduction in this quantity. The ES and PES acquisition functions have the following equations, respectively:

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