Abstract
Bayesian optimization (BO) is a powerful approach for seeking the global optimum of expensive black-box functions and has proven successful for fine tuning hyper-parameters of machine learning models. However, BO is practically limited to optimizing 10–20 parameters. To scale BO to high dimensions, we usually make structural assumptions on the decomposition of the objective and/or exploit the intrinsic lower dimensionality of the problem, e.g. by using linear projections. We could achieve a higher compression rate with nonlinear projections, but learning these nonlinear embeddings typically requires much data. This contradicts the BO objective of a relatively small evaluation budget. To address this challenge, we propose to learn a low-dimensional feature space jointly with (a) the response surface and (b) a reconstruction mapping. Our approach allows for optimization of BO’s acquisition function in the lower-dimensional subspace, which significantly simplifies the optimization problem. We reconstruct the original parameter space from the lower-dimensional subspace for evaluating the black-box function. For meaningful exploration, we solve a constrained optimization problem.
Highlights
Bayesian optimization (BO) is a useful model-based approach to global optimization of black-box functions, which are expensive to evaluate (Kushner 1964; Jones et al 1998)
The standard BO routine consists of two key steps: (1) estimating the black-box function from data through a probabilistic surrogate model, usually a Gaussian process (GP), referred to as the response surface; (2) maximizing an acquisition function that trades off exploration and exploitation according to uncertainty and optimality of the response surface
We propose a BO algorithm for high-dimensional optimization, which learns a nonlinear feature mapping ∶ RD → Rd to reduce the dimensionality of the inputs, and a reconstruction mapping ∶ Rd → RD based on GPs to evaluate the true objective function, jointly, see Fig. 1
Summary
Bayesian optimization (BO) is a useful model-based approach to global optimization of black-box functions, which are expensive to evaluate (Kushner 1964; Jones et al 1998). We propose a BO algorithm for high-dimensional optimization, which learns a nonlinear feature mapping ∶ RD → Rd to reduce the dimensionality of the inputs, and a reconstruction mapping ∶ Rd → RD based on GPs to evaluate the true objective function, jointly, see Fig. 1. This allows us to optimize the acquisition function in a lower-dimensional feature space, so that the overall BO routine scales to highdimensional problems that possess an intrinsic lower dimensionality. We use constrained maximization of the acquisition function in feature space to prevent meaningless reconstructions
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