Bayesian optimization (BO) is a sample-efficient approach to optimizing costly-to-evaluate black-box functions. Most BO methods ignore how evaluation costs may vary over the optimization domain. However, these costs can be highly heterogeneous and are often unknown in advance. This occurs in many practical settings, such as hyperparameter tuning of machine learning algorithms or physics-based simulation optimization. Moreover, those few existing methods that acknowledge cost heterogeneity do not naturally accommodate a budget constraint on the total evaluation cost. This combination of unknown costs and a budget constraint introduces a new dimension to the exploration-exploitation trade-off, where learning about the cost incurs the cost itself. Existing methods do not reason about the various trade-offs of this problem in a principled way, leading often to poor performance. We formalize this claim by proving that the expected improvement and the expected improvement per unit of cost, arguably the two most widely used acquisition functions in practice, can be arbitrarily inferior with respect to the optimal non-myopic policy. To overcome the shortcomings of existing approaches, we propose the budgeted multi-step expected improvement, a non-myopic acquisition function that generalizes classical expected improvement to the setting of heterogeneous and unknown evaluation costs. Finally, we show that our acquisition function outperforms existing methods in a variety of synthetic and real problems.
翻译:贝叶斯优化(BO)是一种优化成本到评估黑盒功能的抽样高效方法。大多数BO方法忽略了评估成本在优化领域之间可能发生的差异。然而,这些费用可能差异很大,而且往往事先不为人所知。这在许多实际环境中发生,例如机器学习算法的超参数调或基于物理的模拟优化。此外,承认成本差异性的少数现有方法自然不能满足总评价成本的预算限制。这种未知成本和预算限制的结合给勘探-开发交易带来了新的层面,因为在那里,了解成本本身就会产生成本。现有方法并不说明这一问题的各种权衡,往往导致业绩不佳。我们通过证明预期的改进和预期的单位成本改进,即实际中最常用的两种购置功能,可能任意低于最佳的非微观政策。为了克服现有方法的缺陷,我们建议预算的多步骤预期改进,一种非微观的购置功能,以原则性方式说明这一问题的各种交易,往往导致业绩不佳。我们正式确认这一主张,通过证明预期的改进和预期的单位成本的改进,即实际的购置方法的改进,最终显示我们无法想象的合成方法的改进。