When approximating a black-box function, sampling with active learning
focussing on regions with non-linear responses tends to improve accuracy. We
present the FLOLA-Voronoi method introduced previously for deterministic
responses, and theoretically derive the impact of output uncertainty. The
algorithm automatically puts more emphasis on exploration to provide more
information to the models

Couckuyt, Ivo

Deschrijver, Dirk

Dhaene, Tom

van der Herten, Joachim

English

arXiv

Ghent University Academic Bibliography

Active Learning for Approximation of Expensive Functionswith Normal Distributed Output UncertaintyJoachim van der Herten JOACHIM.VANDERHERTEN@INTEC.UGENT.BEIvo Couckuyt IVO.COUCKUYT@INTEC.UGENT.BEDirk Deschrijver DIRK.DESCHRIJVER@INTEC.UGENT.BETom Dhaene TOM.DHAENE@INTEC.UGENT.BEGhent University, Technologiepark 15, B-9052 Ghent, BELGIUMAbstractWhen approximating a black-box function, sam-pling with active learning focussing on regionswith non-linear responses tends to improve accu-racy. We present the FLOLA-Voronoi method in-troduced previously for deterministic responses,and theoretically derive the impact of output un-certainty. The algorithm automatically puts moreemphasis on exploration to provide more infor-mation to the models.1. IntroductionConsider an unknown multivariate black-box function f :Rd → R. Our aim is obtaining an approximation of theresponse f˜ using machine learning methodology (regres-sion) by evaluating f on distinct points pi ∈ Rd. The cor-responding responses are subject to normal distributed un-certainty λ: yi ∼ N (f(pi), λ) . However, assuming evalu-ation of f involves a significant cost, only a limited amountof N evaluations can be performed. Intelligent choice ofthese evaluations (experimental design) can result in sig-nificantly better approximations.Traditional Design of Experiments for computer experi-ments select all N data points prior to evaluation. Thesespace-filling methodologies do not allow exploiting infor-mation obtained by each evaluation. In contrast, we presentan active learning approach developed in the context ofsurrogate modeling known as FLOLA-Voronoi (van derHerten et al., 2015). Starting from a small set of initialevaluations, iteratively new evaluation(s) are chosen basedon an analysis of the previous responses. As a result, thealgorithm increases the information density in non-linearareas that are more difficult to approximate. OriginallyProceedings of the 33 rd International Conference on MachineLearning, New York, NY, USA, 2016. JMLR: W&CP volume48. Copyright 2016 by the author(s).the algorithm was formulated for deterministic responses,we analyze the impact of uncertain outputs theoretically inSection 2, and illustrate the effect in Section 3.2. MethodWe present the FLOLA-Voronoi method, an improved ver-sion of the LOLA-Voronoi method (Crombecq et al., 2010)which has been applied successfully previously (Deschri-jver et al., 2011; Rosenbaum & Schulz, 2012; Deschrijveret al., 2012). The method can operate without intermedi-ate model training, balances exploration and exploitationand remains efficient for high-dimensional functions. Twoscores (computed by two algorithms) are aggregated foreach previously evaluated point p. The aggregated scoresare ranked, and the highest ranked points are chosen to se-lect a new point for evaluation local to the chosen point,using for instance the maximin criterion.2.1. ExplorationThe Voronoi exploration algorithm (Crombecq et al., 2009)estimates the size of the voronoi cells whose centroids cor-responds to earlier evaluated data points: each point is as-signed a score proportional to the relative size of the cellwith respect to the other cells. As only a ranking of cellsizes is required, a Monte-Carlo estimate is sufficiently ac-curate and no full Voronoi tesselation needs to be com-puted. This algorithm only considers the d-dimensional in-put space and is insensitive to λ.2.2. ExploitationFor a previously evaluated point pr, neighbouring pointsare selected to compute a gradient estimate g by solvinga linear system of equations. Denote the set N(pr) asthe set containing the indices of the neighbouring pointsof pr. The original crosspolytope selection criterion pre-sented in (Crombecq et al., 2010) does not scale well toActive Learning for Approximation of Expensive Functions with Normal Distributed Output Uncertaintyx-5 0 5y-505(a) λ = 0x-5 0 5y-505(b) λ = 1Figure 1. FLOLA-Voronoi applied to the Peaks problem, with and without noise. Clearly, the noiseless case is more focussed on thenon-linear area whereas the noisy case results in more exploration.high-dimensional input spaces, however this can be re-solved by using a faster selection scheme (van der Hertenet al., 2015). The non-linearity error E sums the differ-ences between the linear prediction and the response ob-tained from the simulator. This score depends on the out-put and therefore is sensitive to λ. We now show how muchthe uncertainty impacts E, under the assumption of normaldistributed noise: yi = f(pi) + i with i ∼ N (0, λ). Byexpanding the terms and applying the triangular inequality:E(pr) ≤∑i∈N(pr)|f(pi)− (f(pr) + g.(pi − pr))|+∑i∈N(pr)ζi. (1)The upper bound of the score corresponds to the score fordeterministic responses, complemented with a sum of noiseterms ζi = |i − r|. We denote the sum of the noise termsas random variable X . All  are normally distributed withzero mean, so the difference of both variance terms is alsonormally distributed with zero mean and variance 2λ. Thismeans each ζi is distributed according to a folded normaldistribution:ζi ∼ FN(2√λpi, 2λ(1− 2pi)).The distribution of X depends on T = |N(pr)|, and theshape of its probability density function resembles the log-normal distribution, its explicit formulation can be calcu-lated but is a lengthy expression. Defining u(t) = (t −1)2 + 1 and v(t) = t2 − 2t + 2, the expectancy and vari-ance are given by following expressions:E [X] =2pi√λ(pi − 2)(T + (T − 1)√u(T )),Var [X] =2λ(pi − 2) (4(T − 1) + 5piv(T ))pi2− E [X] .The variance of X perturbs the non-linearity score most.From the equation it can be derived that higher λ and ahigher T cause more variance on the distribution of X .Using the neighbourhood selection mechanism of FLOLA-Voronoi the latter can be countered by including only themost significant neighbours in Equation 1 (instead of all).3. IllustrationThe perturbations caused by Var [X] impact the ranking ofFLOLA-Voronoi causing more exploration, as the exploita-tion score becomes less decisive. This is illustrated byselecting 120 data points from the same function (Peaks)twice, once with output noise and once without for a con-stant T . The sample distributions together with a con-tour plot of a fitted Kriging model are shown in Figure 1.Clearly, the perturbed non-linearity scores cause more ex-ploration which is a desirable property.4. ConclusionWe presented the FLOLA-Voronoi algorithm for effi-cient model-independent sequential sampling of expensiveblack-box functions, and analysed the impact of uncertainresponse values on the algorithm. We have shown how theexploitation component of the algorithm is affected, andillustrated the resulting enhanced exploration. This is a de-sirable property in the presence of output uncertainty, as themodel typically requires more information for linear areas.AcknowledgementsThis research has (partially) been funded by the Inter uni-versity Attraction Poles Programme BESTCOM initiatedby the Belgian Science Policy Office. Ivo Couckuyt is apost-doctoral research fellow of the Research FoundationFlanders (FWO).Active Learning for Approximation of Expensive Functions with Normal Distributed Output UncertaintyReferencesCrombecq, K., Couckuyt, I., Gorissen, D., and Dhaene, T.Space-filling Sequential Design Strategies for AdaptiveSurrogate Modelling. In Soft Computing Technology inCivil, Structural and Environmental Engineering (CSC2009), volume 92, pp. Paper–50, 2009.Crombecq, K., Gorissen, D., Deschrijver, D., and Dhaene,T. A Novel Hybrid Sequential Design Strategy forGlobal Surrogate Modelling of Computer Experiments.SIAM Journal of Scientific Computing, 33(4):1948–1974, 2010.Deschrijver, Dirk, Crombecq, Karel, Nguyen, Huu Minh,and Dhaene, Tom. Adaptive Sampling Algorithm forMacromodeling of Parameterized-Parameter Responses.Microwave Theory and Techniques, IEEE Transactionson, 59(1):39–45, 2011.Deschrijver, Dirk, Vanhee, Filip, Pissoort, Davy, andDhaene, Tom. Automated near-field scanning algorithmfor the EMC analysis of electronic devices. IEEE Trans-actions on Electromagnetic Compatibility, 54(3):502–510, 2012.Rosenbaum, Benjamin and Schulz, Volker. Comparingsampling strategies for aerodynamic Kriging surrogatemodels. ZAMM-Journal of Applied Mathematics andMechanics/Zeitschrift fu¨r Angewandte Mathematik undMechanik, 92(11-12):852–868, 2012.van der Herten, J, Couckuyt, I, Deschrijver, D, and Dhaene,T. A Fuzzy Hybrid Sequential Design Strategy forGlobal Surrogate Modeling of High-Dimensional Com-puter Experiments. SIAM Journal on Scientific Comput-ing, 37(2):A1020–A1039, 2015.

Active learning for approximation of expensive functions with normal distributed output uncertainty

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