Bayesian Optimization of Composite Functions
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Proceedings of the 36th International Conference on Machine Learning, PMLR 97:354363, 2019.
Abstract
We consider optimization of composite objective functions, i.e., of the form $f(x)=g(h(x))$, where $h$ is a blackbox derivativefree expensivetoevaluate function with vectorvalued outputs, and $g$ is a cheaptoevaluate realvalued function. While these problems can be solved with standard Bayesian optimization, we propose a novel approach that exploits the composite structure of the objective function to substantially improve sampling efficiency. Our approach models $h$ using a multioutput Gaussian process and chooses where to sample using the expected improvement evaluated on the implied nonGaussian posterior on $f$, which we call expected improvement for composite functions (EICF). Although EICF cannot be computed in closed form, we provide a novel stochastic gradient estimator that allows its efficient maximization. We also show that our approach is asymptotically consistent, i.e., that it recovers a globally optimal solution as sampling effort grows to infinity, generalizing previous convergence results for classical expected improvement. Numerical experiments show that our approach dramatically outperforms standard Bayesian optimization benchmarks, reducing simple regret by several orders of magnitude.
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