I am working on optimizing a GP-based multi-fidelity surrogate model and I found this publication very helpful: 'A Tutorial on Bayesian Optimization' by Peter I. Frazier (July 10, 2018). and I have some questions regarding the conclusions on Expected Improvement (EI).
In chapter: 4 Acquisition Functions,
These alternate acquisition functions are most useful in exotic problems where an assumption made by expected improvement, that the primary benefit of sampling occurs through an improvement at the point sampled, is no longer true.
In chapter: 5 Exotic Bayesian Optimization, paragraph Multi-Fidelity and Multi-Information Source Evaluations
EI is difficult to apply directly in these problems because evaluating f(x, s) for s ≠ 0 never provides an improvement in the best objective function value seen, max { f(x_n, 0) : s_n= 0 } . ... causes EI = 0 for s ≠ 0.
Question 1: I am not sure if I understand the second statement correctly, especially "EI = 0 for s ≠ 0".
- The improvement is either a best function value so far, or 0.
- The improvement from the highest fidelity level is always larger than that from any lower fidelity level
- Hence, the acquisition function will always return the location at the highest fidelity level If the above understanding is correct, I do not understand why we can conclude that the improvement from (s=0) > improvement from (s≠0). Further, why EI = 0 for s≠0? It can be that the improvements on lower fidelity levels are not huge, but it must have some improvement, because more data points are given.
Question 2: I think I understand the statement 1 vaguely, but not fully. The main confusing part is that I do not see why (Predictive) Entropy Search can simultaneously take into account all possible improvement at all fidelity levels (while EI can only evaluate at one level)? I miss the intuition here.
Thank you for your time and help in advance!
