This paper focuses on Bayesian Optimization (BO) for objectives on combinatorial search spaces, including ordinal and categorical variables. Despite the abundance of potential applications of Combinatorial BO, including chipset configuration search and neural architecture search, only a handful of methods have been proposed. We introduce COMBO, a new Gaussian Process (GP) BO.

Applying BO with GPs on combinatorial spaces is, therefore, not straightforward.

This manuscript proposes a system for combinatorial Bayesian optimization called COMBO, aimed at problems with large numbers of categorical and/or ordinal features. The main contribution is an effective kernel for this setting based on applying a graph kernel to the graph Cartesian product of each of the features, which can be computed efficiently by exploiting structure. This kernel can be further enhanced using an ARD extension and a horseshoe prior to encourage sparse feature selection. The COMBO system then creates a GP with this kernel and does random local search to maximize an acquisition function such as EI in the combinatorial space. A series of experiments demonstrate COMBO performing better on real and synthetic tasks than alternatives such as systems using one-hot encodings.

This paper focuses on Bayesian Optimization (BO) for objectives on combinatorial search spaces, including ordinal and categorical variables. Despite the abundance of potential applications of Combinatorial BO, including chipset configuration search and neural architecture search, only a handful of methods have been pro- posed. We introduce COMBO, a new Gaussian Process (GP) BO. The vertex set of the combinatorial graph consists of all possible joint assignments of the variables, while edges are constructed using the graph Cartesian product of the sub-graphs that represent the individual variables. On this combinatorial graph, we propose an ARD diffusion kernel with which the GP is able to model high-order interactions between variables leading to better performance.

This paper focuses on Bayesian Optimization (BO) for objectives on combinatorial search spaces, including ordinal and categorical variables. Despite the abundance of potential applications of Combinatorial BO, including chipset configuration search and neural architecture search, only a handful of methods have been pro- posed. We introduce COMBO, a new Gaussian Process (GP) BO. The vertex set of the combinatorial graph consists of all possible joint assignments of the variables, while edges are constructed using the graph Cartesian product of the sub-graphs that represent the individual variables. On this combinatorial graph, we propose an ARD diffusion kernel with which the GP is able to model high-order interactions between variables leading to better performance.

This paper focuses on Bayesian Optimization - typically considered with continuous inputs - for discrete search input spaces, including integer, categorical or graph structured input variables. In Gaussian process-based Bayesian Optimization a problem arises, as it is not straightforward to define a proper kernel on discrete input structures, where no natural notion of smoothness or similarity could be provided. We propose COMBO, a method that represents values of discrete variables as vertices of a graph and then use the diffusion kernel on that graph. As the graph size explodes with the number of categorical variables and categories, we propose the graph Cartesian product to decompose the graph into smaller sub-graphs, enabling kernel computation in linear time with respect to the number of input variables. Moreover, in our formulation we learn a scale parameter per subgraph. In empirical studies on four discrete optimization problems we demonstrate that our method is on par or outperforms the state-of-the-art in discrete Bayesian optimization.