Alternative Acquisition Functions of Bayesian Optimization in terms of Noisy Observation

In this paper, we introduce a variety of acquisition functions currently used in Bayesian optimization. Besides the traditional acquisition functions like Probability Improvement (PI), Expected Improvement (EI) and Gaussian Process-Upper Confidence Bound (GP-UCB), we also present some modified or improved EI and PI methods, Knowledge Gradient (KG) and Predictive Entropy Search (PES) methods to explore ways to reduce the impact of observational noise. In experimental part, we choose a benchmark function and use Bayesian optimization algorithm to find its global minimum. We add different scales of noise in particular following the Gaussian distribution to the benchmark function, to compare the performance of BO algorithm using different acquisition functions. Combined with the experimental results, we also present a discussion of the pros and cons of using those acquisition functions. Hope this can provide some experience and suggestions for choosing acquisition functions in terms of noisy observation.