STK_SAMPCRIT_AKG_EVAL computes the Approximate KG criterion CALL: AKG = stk_sampcrit_akg_eval (ZC_MEAN, ZC_STD, ZR_MEAN, ZR_STD, ZCR_COV) computes the value AKG of the Approximate KG criterion for a set of candidates points, with respect to a certain reference grid. The predictive distributions of the objective function (to be minimized) at the candidates and reference points is assumed to be jointly Gaussian, with mean ZC_MEAN and standard deviation ZC_STD for the candidate points, mean ZR_MEAN and satandard deviation ZR_STD on the reference points, and covariance matrix ZCR_COV between the candidate and reference points. The input argument must have the following sizes: * ZC_MEAN M x 1, * ZC_STD M x 1, * ZR_MEAN L x 1, * ZR_STD L x 1, * ZCR_COV M x L, where M is the number of candidate points and L the number of reference points. The output has size M x 1. NOTE ABOUT THE "KNOWLEDGE GRADIENT" CRITERION The "Knowlegde Gradient" (KG) criterion is the one-step look-ahead (a.k.a myopic) sampling criterion associated to the problem of estimating the minimizer of the objective function under the L^1 loss (equivalently, under the linear loss/utility). This sampling strategy was proposed for the first time in the work of Mockus and co-authors in the 70's (see [1] and refs therein), for the case of noiseless evaluations, but only applied to particular Brownian-like processes for which the minimum of the posterior mean coincides with the best evaluations so far (in which case the KG criterion coincides with the EI criterion introduced later by Jones et al [2]). It was later discussed for the case of a finite space with independent Gaussian priors first by Gupta and Miescke [3] and then by Frazier et al [4] who named it "knowledge gradient". It was extended to the case of correlated priors by Frazier et al [5]. NOTE ABOUT THE REFERENCE SET For the case of continuous input spaces, there is no exact expression of the KG criterion. The approximate KG criterion proposed in this function is an approximation of the KG criterion where the continuous 'min' in the expression of the criterion at the i^th candidate point are replaced by discrete mins over some reference grid *augmented* with the i^th candidate point. This type of approximation has been proposed by Scott et al [6] under the name "knowledge gradient for continuous parameters" (KGCP). In [6], the reference grid is composed of the current set of evaluation points. The implementation proposed in STK leaves this choice to the user. Note that, with the reference grid proposed in [6], the complexity of one evaluation of the AKG (KGCP) criterion increases as O(N log N), where N denotes the number of evaluation points. NOTE ABOUT THE NOISELESS CASE Simplified formulas are available for the noiseless case (see [7]) but not currenly implemented in STK. REFERENCES [1] J. Mockus, V. Tiesis and A. Zilinskas. The application of Bayesian methods for seeking the extremum. In L.C.W. Dixon and G.P. Szego, eds, Towards Global Optimization, 2:117-129, North Holland NY, 1978. [2] D. R. Jones, M. Schonlau and William J. Welch. Efficient global optimization of expensive black-box functions. Journal of Global Optimization, 13(4):455-492, 1998. [3] S. Gupta and K. Miescke, Bayesian look ahead one-stage sampling allocations for selection of the best population, J. Statist. Plann. Inference, 54:229-244, 1996. [4] P. I. Frazier, W. B. Powell, S. Dayanik, A knowledge gradient policy for sequential information collection, SIAM J. Control Optim., 47(5):2410-2439, 2008. [5] P. I. Frazier, W. B. Powell, and S. Dayanik. The Knowledge-Gradient Policy for Correlated Normal Beliefs. INFORMS Journal on Computing 21(4):599-613, 2009. [6] W. Scott, P. I. Frazier and W. B. Powell. The correlated knowledge gradient for simulation optimization of continuous parameters using Gaussian process regression. SIAM J. Optim, 21(3):996-1026, 2011. [7] J. van der Herten, I. Couckuyt, D. Deschrijver, T. Dhaene, Fast Calculation of the Knowledge Gradient for Optimization of Deterministic Engineering Simulations, arXiv preprint arXiv:1608.04550 See also: STK_SAMPCRIT_EI_EVAL
Package: stk