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We introduce the algorithm Bayesian Optimization (BO) with Fictitious Play(BOFiP) for the optimization of high dimensional black box functions. BOFiPdecomposes the original, high dimensional, space into several sub-spacesdefined by non-overlapping sets of dimensions. These sets are randomlygenerated at the start of the algorithm, and they form a partition of thedimensions of the original space. BOFiP searches the original space withalternating BO, within sub-spaces, and information exchange among sub-spaces,to update the sub-space function evaluation. The basic idea is to distributethe high dimensional optimization across low dimensional sub-spaces, where eachsub-space is a player in an equal interest game. At each iteration, BO producesapproximate best replies that update the players belief distribution. Thebelief update and BO alternate until a stopping condition is met.  High dimensional problems are common in real applications, and severalcontributions in the BO literature have highlighted the difficulty in scalingto high dimensions due to the computational complexity associated to theestimation of the model hyperparameters. Such complexity is exponential in theproblem dimension, resulting in substantial loss of performance for mosttechniques with the increase of the input dimensionality.  We compare BOFiP to several state-of-the-art approaches in the field of highdimensional black box optimization. The numerical experiments show theperformance over three benchmark objective functions from 20 up to 1000dimensions. A neural network architecture design problem is tested with 42 upto 911 nodes in 6 up to 92 layers, respectively, resulting into networks with500 up to 10,000 weights. These sets of experiments empirically show that BOFiPoutperforms its competitors, showing consistent performance across differentproblems and increasing problem dimensionality.

Scaling Bayesian Optimization With Game Theory