Bayesian Optimization Using Simulation-Based Multiple Information Sources over Combinatorial Structures

Bayesian optimization due to its flexibility and sample efficiency has become a standard approach for simulation optimization. To reduce this problem, one can resort to cheaper surrogates of the objective function. Examples are ubiquitous, from protein engineering or material science to tuning machine learning algorithms, where one could use a subset of the full training set or even a smaller related dataset. Cheap information sources in the optimization scheme have been studied in the literature as the multi-fidelity optimization problem. Of course, cheaper sources may hold some promise toward tractability, but cheaper models offer an incomplete model inducing unknown bias and epistemic uncertainty. In this manuscript, we are concerned with the discrete case, where (Formula presented.) is the value of the performance measure associated with the environmental condition (Formula presented.) and (Formula presented.) represents the relevance of the condition (Formula presented.) (i.e., the probability of occurrence or the fraction of time this condition occurs). The main contribution of this paper is the proposal of a Gaussian-based framework, called augmented Gaussian process (AGP), based on sparsification, originally proposed for continuous functions and its generalization in this paper to stochastic optimization using different risk profiles for combinatorial optimization. The AGP leverages sample and cost-efficient Bayesian optimization (BO) of multiple information sources and supports a new acquisition function to select the new source–location pair considering the cost of the source and the (location-dependent) model discrepancy. An extensive set of computational results supports risk-aware optimization based on CVaR (conditional value-at-risk). Computational experiments confirm the actual performance of the MISO-AGP method and the hyperparameter optimization on benchmark functions and real-world problems.