Contour location via entropy reduction leveraging multiple information sources
We introduce an algorithm to locate contours of functions that are expensive to evaluate. The problem of locating contours arises in many applications, including classification, constrained optimization, and performance analysis of mechanical and dynamical systems (reliability, probability of failure, stability, etc.). Our algorithm locates contours using information from multiple sources, which are available in the form of relatively inexpensive, biased, and possibly noisy approximations to the original function. Considering multiple information sources can lead to significant cost savings. We also introduce the concept of contour entropy, a formal measure of uncertainty about the location of the zero contour of a function approximated by a statistical surrogate model. Our algorithm locates contours efficiently by maximizing the reduction of contour entropy per unit cost.
This paper introduces an active learning strategy for estimating some contour line of an expensive to evaluate function in the case where lower fidelity  evaluations of the objective function can be performed at lower cost. The proposed worklow relies on a multi-information source Gaussian Process model  following notably the approach of Poloczek et al. (on "Multi-information source optimization") presented at NIPS 2017. Here however the acquisition  function substantially differs from the one used in the former reference, and focuses specifically on contour line estimation rather than optimization.  The proposed acquisition function builds upon the entropy of discrete distributions indicating at each point whether the response is in a prescribed  interval around the contour level, below the interval lower bound or above the interval upper bound. At any given point, the entropy of this distribution can be written as a sum of three terms written in closed form (up to the Gaussian CDF) as a function of the predictive GP mean and standard deviation.  Then this entropy is integrated over the input space (up to a normalization by the domain volume) and the considered sampling criterion is obtained at any  given candidate evaluation point and fidelity level as the ratio between the expected decrease in this integrated entropy if a corresponding evaluation is  performed and the (known) cost of performing it. While the integrand is not calculated in closed form, a tight approximation is provided for which a closed  form expression is presened in the paper. An algorithm is then devised that consists, from a starting design, in performing at every iteration a new  evaluation at selected point and fidelity level so as to achieve optimal resource allocation in the sense of the proposed criterion, until budget exhaustion.  The contour line of the GP predictive mean at the final state is finally returned. The algorithm is tested and illustrated on two examples, one 2-dimensional  analytical one and one 1-dimensional test case stemming from a chemical engineering problem.   As motivated at the beginning of the paper, contour line estimation is an important topic arising in a number of applications and methodological contexts.  Also, integrating several information sources withing a GP model in order to study expensive to evaluate functions under a smart resource allocation scheme  is a timely and useful approach with promising benefits in a variety of settings. Adressing contour line estimation using a multi-information source GP and  a dedicated acquisition function thus appears as a legitimate and worthwhile development. From that perspective, I very much welcome the present contribution. This being said, the paper also has some weaknesses, and while I recommend acceptance, I think that it deserves to be substantially improved before publication.  One of the main concerns is that part of the relevant literature has been ignored, and also importantly that the proposed approach has not really been  extensively compared to potential competitors (that might need to be adapted to the multi-source framework; not e also that single-fidelity experiments could be  run in order to better understand how the proposed acquisition function compares to others from the literature). Another main concern in connection with the  previous one is that the presented examples remain relatively simple, one testcase being an analytical function and the other one a one-dimensional mototonic  function. While I am not necessarily requesting a gigantic benchmark or a list of complicated high-dimensional real-world test cases, the paper would significantly  benefit from a more informative application section. Ideally, the two aspects of improving the representativity of numerical test cases and of better benchmarking  against competitor strategies could be combined. As of missing approaches from the literature, some entry points follow:   * Adaptive Designs of Experiments for Accurate Approximation of a Target Region (Picheny et al. 2010) http://mechanicaldesign.asmedigitalcollection.asme.org/article.aspx?articleid=1450081  *Fast kriging-based stepwise uncertainty reduction with application to the identification of an excursion set (Chevalier et al. 2014a) http://amstat.tandfonline.com/doi/full/10.1080/00401706.2013.860918   * NB: approaches from the two articles above and more (some cited in the submitted paper but not benchmarked against) are coded for  instance in the R package "KrigInv". The following article gives an overview of some of the package's functionalities:  KrigInv: An efficient and user-friendly implementation of batch-sequential inversion strategies based on kriging (Chevalier et al. 2014b)  * In the following paper, an entropy-based approach (but not in the same fashion as the one proposed in the submitted paper) is used, in  a closely related reliability framework:  Gaussian process surrogates for failure detection: A Bayesian experimental design approach (Wang et al. 2016)  https://www.sciencedirect.com/science/article/pii/S002199911600125X  * For an overall discussion on estimating and quantifying uncertainty on sets under GP priors (with an example in contour line estimation),  see Quantifying Uncertainties on Excursion Sets Under a Gaussian Random Field Prior (Azzimonti et al. 2016) https://epubs.siam.org/doi/abs/10.1137/141000749 NB: the presented approaches to quantify uncertainties on sets under GP priors could also be useful here to return a more complete output  (than just the contour line of the predictive GP mean) in the CLoVER algorithm.   * Coming to the multi-fidelity framework and sequential design for learning quantities (e.g. probabilities of threshold exceedance) of  interest, see notably Assessing Fire Safety using Complex Numerical Models with a Bayesian Multi-fidelity Approach (Stroh et al. 2017) https://www.sciencedirect.com/science/article/pii/S0379711217301297?via%3Dihub    Some further points  * Somehow confusing to read "relatively inexpensive" in the abstract and then "expensive to evaluate" in the first line of the introduction! * L 55 "A contour" should be "A contour line"?  * L88: what does "f(l,x) being the normal distribution..." mean?    * It would be nice to have a point-by-point derivation of equation (10) in the supplementary material (that would among others help readers  including referees proofchecking the calculation). * About the integral appearing in the criterion, some more detail on how its computation is dealt with could be worth.  ##### added after the rebuttal  I updated my overall grade from 7 to 8 as I found the response to the point and it made me confident that the final paper would be improved (by suitably accounting for my remarks and those of the other referees) upon acceptance.  Let me add a comment about related work by Rémi Stroh. The authors are right, the Fire Safety paper is of relevance but does actually not address the design of GP-based multi-fidelity acquisition functions (in the BO fashion). However, this point has been further developed by Stroh et al; see "Sequential design of experiments to estimate a probability of exceeding a threshold in a multi-fidelity stochastic simulator" in conference contributions listed in  http://www.l2s.centralesupelec.fr/en/perso/remi.stroh/publications