Abstract

The Expected Improvement (EI) method, proposed by Jones et al. (1998), is a widely-used Bayesian optimization method, which makes use of a fitted Gaussian process model for efficient black-box optimization. However, one key drawback of EI is that it is overly greedy in exploiting the fitted Gaussian process model for optimization, which results in suboptimal solutions even with large sample sizes. To address this, we propose a new hierarchical EI (HEI) framework, which makes use of a hierarchical Gaussian process model. HEI preserves a closed-form acquisition function, and corrects the over-greediness of EI by encouraging exploration of the optimization space. We then introduce hyperparameter estimation methods which allow HEI to mimic a fully Bayesian optimization procedure, while avoiding expensive Markov-chain Monte Carlo sampling steps. We prove the global convergence of HEI over a broad function space, and establish near-minimax convergence rates under certain prior specifications. Numerical experiments show the improvement of HEI over existing Bayesian optimization methods, for synthetic functions and a semiconductor manufacturing optimization problem.

摘要

Jones 等人提出的预期改进 (EI) 方法。(1998)，是一种广泛使用的贝叶斯优化方法，它利用拟合高斯过程模型进行有效的黑盒优化。然而，EI 的一个主要缺点是它在利用拟合的高斯过程模型进行优化时过于贪婪，即使样本量很大，也会导致次优解。为了解决这个问题，我们提出了一种新的分层 EI (HEI) 框架，它利用了分层高斯过程模型。HEI 保留了封闭形式的获取函数，并通过鼓励对优化空间的探索来纠正 EI 的过度贪婪。然后我们介绍了超参数估计方法，它允许 HEI 模拟完全贝叶斯优化过程，同时避免昂贵的马尔可夫链蒙特卡罗采样步骤。我们证明了 HEI 在广泛的函数空间上的全局收敛性，并在某些先验规范下建立了接近极小极大的收敛率。数值实验表明，HEI 对合成函数和半导体制造优化问题的现有贝叶斯优化方法有所改进。