Scaling Bayesian Optimization With Game Theory

Abstract

We introduce the algorithm Bayesian Optimization with Fictitious Play (BOFiP) for the optimization of high dimensional black box functions. BOFiP decomposes the original, high dimensional, space into several sub-spaces defined by non-overlapping sets of dimensions. These sets are randomly generated at the start of the algorithm, and they form a partition of the dimensions of the original space. BOFiP can search the original space through a strategic learning mechanism that alternates Bayesian optimization, to search within sub-spaces, and information exchange among sub-spaces, to update the sub-space function evaluation. The basic idea is to achieve increased efficiency by distributing the high dimensional optimization across low dimensional sub-spaces. In particular, each sub-space is interpreted as a player within an equal interest game. At each iteration, Bayesian optimization produces approximate best replies that allow the update of the players belief distribution. The belief update and Bayesian optimization continue to alternate until a stopping condition is met. High dimensional problems are often encountered in real applications, and several contributions in the Bayesian optimization literature have highlighted the difficulty in scaling to high dimensions due to the computational complexity associated to the estimation of the model hyperparameters. Such complexity is exponential in the problem dimension, resulting in substantial loss of performance for most techniques with the increase of the input dimensionality. We compare BOFiP to several state-of-the-art approaches in the field of high dimensional black box optimization. The numerical experiments show the performance over three benchmark objective functions from 20 up to 1000 dimensions. A neural network architecture design problem is tested with 42 up to 911 nodes in 6 up to 92 layers, respectively, resulting into networks with 500 up to 10,000 weights. These sets of experiments empirically show that BOFiP outperforms its competitors, showing consistent performance across different problems and increasing problem dimensionality.