  We introduce the algorithm Bayesian Optimization (BO) 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 searches the original space with
alternating BO, within sub-spaces, and information exchange among sub-spaces,
to update the sub-space function evaluation. The basic idea is to distribute
the high dimensional optimization across low dimensional sub-spaces, where each
sub-space is a player in an equal interest game. At each iteration, BO produces
approximate best replies that update the players belief distribution. The
belief update and BO alternate until a stopping condition is met.
  High dimensional problems are common in real applications, and several
contributions in the BO 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.
