Bayesian Optimization for Categorical and Category-Specific Continuous Inputs

Many real-world functions are defined over both categorical and category-specific continuous variables and thus cannot be optimized by traditional Bayesian optimization (BO) methods. To optimize such functions, we propose a new method that formulates the problem as a multi-armed bandit problem, wherein each category corresponds to an arm with its reward distribution centered around the optimum of the objective function in continuous variables. Our goal is to identify the best arm and the maximizer of the corresponding continuous function simultaneously. Our algorithm uses a Thompson sampling scheme that helps connecting both multi-arm bandit and BO in a unified framework. We extend our method to batch BO to allow parallel optimization when multiple resources are available. We theoretically analyze our method for convergence and prove sub-linear regret bounds. We perform a variety of experiments: optimization of several benchmark functions, hyper-parameter tuning of a neural network, and automatic selection of the best machine learning model along with its optimal hyper-parameters (a.k.a automated machine learning). Comparisons with other methods demonstrate the effectiveness of our proposed method.