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# A Nonmyopic Approach to Cost-Constrained Bayesian Optimization

*Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence*, PMLR 161:568-577, 2021.

#### Abstract

Bayesian optimization (BO) is a popular method for optimizing expensive-to-evaluate black-box functions. BO budgets are typically given in iterations, which implicitly assumes each evaluation has the same cost. In fact, in many BO applications, evaluation costs vary significantly in different regions of the search space. In hyperparameter optimization, the time spent on neural network training increases with layer size; in clinical trials, the monetary cost of drug compounds vary; and in optimal control, control actions have differing complexities. Cost-constrained BO measures convergence with alternative cost metrics such as time, money, or energy, for which the sample efficiency of standard BO methods is ill-suited. For cost-constrained BO, cost efficiency is far more important than sample efficiency. In this paper, we formulate cost-constrained BO as a constrained Markov decision process (CMDP), and develop an efficient rollout approximation to the optimal CMDP policy that takes both the cost and future iterations into account. We validate our method on a collection of hyperparameter optimization problems as well as a sensor set selection application.