# Bayesian Optimization with transformed objective function

I have a sequential learning problem where I want to rank a group of jobs in each iteration--unlike conventional Bayesian approach, I am trying to find the ordering of jobs based on g(f(x)) where GP directly models f(x)--so, maximizing the posterior mean for f(x) and ranking based on that would not help since g(f(x) does not preserve the ordering, i.e if f(x1) < f(x2) then we have no idea about the ordering of g(f(x1)) and g(f(x2))--

Note that we know the functional from of g and we know that g has parameters that are input dependent-for instance, assume g(f(x1)) = a1 * f(x1) + b1 and g(f(x2)) = a2 * f(x2) + b2. It means that a1, a2, b1, b2 are different for each input but we know them beforehand and they have nothing to do with the mapping between x and f(x)--In other words, for each train input we have (a1, b1, x1, f(x1)) where f(x1) is the observed value. Now, we want to do the prediction for item (x*, a*, b*) with unknown f(x*)--Our GP (surrogate model) tries to model the mapping between x* and f(x*) but ranking should be based on g(f(x*))

so, wanted to see whether you have an idea on how we should select jobs and how we should come up with an acquisition function? thanks for your help in advance

Goal: minimise $$h(x) = g(f(x))$$ where $$f$$ is expensive to evaluate and $$g$$ is cheap to evaluate. For maximisation we can minimise $$-h(x)$$.
• So can you choose the values of $a$ and $b$ easily? Can we not just view them as parameters of $g(x)$ i.e. $g(x \mid a, b)$? Mar 4, 2021 at 8:19