PI and EI in Bayesian optimisation under gaussian noise assumption

Bayesian optimisation is a hyper-parameter tunning approach which usually adopts the gaussian process as a surrogate model. Its acquisition functions Probability of improvement(PI) and expected improvement(EI) are calculated with respect to current optima. In practice, The evaluations on loss function may have noise at sampling positions. We can take noise into account by calculating the probability of improvement and expected improvement with respect to the posterior mean and variance at the current optima found. A white kernel should be added to the Gaussian Process kernel to fit the noise. Implementations in GitHub repo and tutorial.

Below is a comparison of PI (left) and MPI+white kernel (right) optimizing on noise corrupted Goldstein-Price Function.

Objective Loss function	Searching with PI	MPI+white kernel

Publication: Huabing Wang, Modifications of PI and EI under Gaussian Noise Assumption in Current Optima.2022