Jiazhao Zhang, Ying Hung, Chung-Ching Lin et al.
Choosing appropriate hyperparameters plays a crucial role in the success of neural networks as hyper-parameters directly control the behavior and performance of the training algorithms. To obtain efficient tuning, Bayesian optimization methods based on Gaussian process (GP) models are widely used. Despite numerous applications of Bayesian optimization in deep learning, the existing methodologies are developed based on a convenient but restrictive assumption that the tuning parameters are independent of each other. However, tuning parameters with conditional dependence are common in practice. In this paper, we focus on two types of them: branching and nested parameters. Nested parameters refer to those tuning parameters that exist only within a particular setting of another tuning parameter, and a parameter within which other parameters are nested is called a branching parameter. To capture the conditional dependence between branching and nested parameters, a unified Bayesian optimization framework is proposed. The sufficient conditions are rigorously derived to guarantee the validity of the kernel function, and the asymptotic convergence of the proposed optimization framework is proven under the continuum-armed-bandit setting. Based on the new GP model, which accounts for the dependent structure among input variables through a new kernel function, higher prediction accuracy and better optimization efficiency are observed in a series of synthetic simulations and real data applications of neural networks. Sensitivity analysis is also performed to provide insights into how changes in hyperparameter values affect prediction accuracy.