Some feedback

• You have a funny definition of the black box functions for which Bayesian optimization is applicable. It requires smooth functions with bounded derivatives. It doesn't work for functions such as 1 for irrational numbers and 0 on rational numbers. I am not sure why you think "Black-box functions are usually complex"; it is more than BO methods are particularly useful for functions that are expensive to evaluate and for which there are only noisy observations and for which it is difficult/impossible to compute the derivative (as gradient-based methods are not applicable).
• Separate what it is trying to do -- maximize a function given only noisy observations of the value some few points, while minimizing the number of observations -- from how (using a surrogate model)
• I thought that it *is* "easy to mathematically model the Neural Network's accuracy" -- it is just difficult/expensive to compute.
• You use "optimize" then do a maximize, without telling us that the aim is to maximize.
• It is okay to just explain Expected Improvement, but then it should be explained better than "considering the highest expected improvement over the current best observation"; you say formally how it defined, but we need a better intuition as to what expected improvement is meant to measure (why does the first term represent exploration and the second exploration?)
• In your pseudo-code, what is "bounds"? The two blocks of pseudo-code need to be related.
• In your end-to-end example, the function is a funny example as it doesn't have an obvious maximum in the plot. (Tell us where the maximum is!) It might be better to have an example that goes up and down multiple times.
• Your prior (that gives the variance) seems very poor. Perhaps justify how a prior can be obtained.
• Explain the qualitative difference in the left plot between iterations 7 & 8 (the plots go from rectangles to curves), and between 5 & 6 (the error bars get very big).
• The English needs fixing.
• I don't think you need more, it just needs to be explained more clearly.