Bayesian Optimization with several acquisition functions.
Plot these behavior.
Bayesian Optimization (BO) is used in the optimization of black box functions with high observation costs. In BO, black box function is estimated by a Bayesian model and the next observation point is determined by the acquisition function. The balance between exploitation and exploration depends on the acquisition function. I use following acquisition functions.
Using the improvement probability for the current best function value.
paper: A new method of locating the maximum point of an arbitrary multipeak curve in the presence of noise. (Journal of Fluids Engineering 1964)
Using the expected improvement for the current best function value.
paper: On Bayesian methods for seeking the extremum (IFIP Technical Conference 1974)
Using lower confidence bound. In LCB, we need to set the exploitation-exploration trade-off parameter β^{1/2}. I use β^{1/2}=1,4.
In GP-UCB, β^{1/2} is determined according to iterations and in the following paper, it is shown that regret bounds are sublinear.
paper: Gaussian process optimization in the bandit setting: no regret and experimental design (ICML2010)
Using mutual information between the optimal value and function value at observation point . I sampled the optimal values using sampling from Gumbel distribution. A method of sampling the optimal value using randam feature map is also proposed in the following paper.
paper: Max-value entropy search for efficient Bayesian optimization (ICML 2017)
