Bayesian Optimization for Branching and Nested Hyperparameters in Deep Learning

Hyperparameter optimization plays a crucial role in the success of neural networks as hyperparameters directly control the behavior and performance of the training algorithms. To obtain efficient tuning, Bayesian optimization based on Gaussian process is widely used. Despite numerous applications in deep learning, the existing methods rely 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 that contains other nested parameters is referred to as 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. The new model, which accounts for the dependent structure among input variables through a new kernel function, provides a higher prediction accuracy and a better optimization efficiency in simulations and real data applications in neural networks.