Why do we use Acquisition Functions? In the context of Surrogate Modelling and Bayesian Optimization - Acquisition Functions (https://tune.tidymodels.org/articles/acquisition_functions.html) are often used as a "compass" to suggest which point to evaluate the objective function at next.
In short, the objective function that you are trying to optimize is modelled using some (simpler) surrogate model (e.g. the surface of the objective function can be modelled using a Gaussian Process), and a separate Acquisition Function "suggests" how to navigate the surface of the Gaussian Process at each iteration. This "feedback loop" is repeated iteratively until some condition is met (e.g. convergence is met or some budget constraints are exhausted). Note: In reality, the Acquisition Function itself must be optimized at each iteration using some gradient based algorithm (e.g. BFGS), but it is said that this sub-optimization problem is not as difficult as the main optimization problem.
I am trying to understand why an Acquisition Function is necessary in this above procedure. I have heard the following argument being made for why an Acquisition Function is required : In many applications where we want to use Bayesian Optimization, we only have realizations from some partially observable objective function. This means that the Gaussian Process is not very "informative" and trying to optimize the Gaussian Process directly will also not be very informative. This is why an Acquisition Function should be used - and somehow, the use of an Acquisition Function helps circumvent this "un-informativeness problem".
However, I do not fully understand this argument. If the Gaussian Process itself is uninformative, how can the use of an Acquisition Function remedy this problem?
My Question: Why is an Acquisition Function required in Bayesian Optimization - after all, why can't we directly optimize the Gaussian Process without taking into consideration the advice from the Acquisition Function?
Thanks!

In uncertain scenarios, where Bayesian optimization is employed, we are facing the exploration-exploitation tradeoff. The acquisition function is a solution for it. As you noticed, Bayesian optimization only approximates the target function. The approximation is imperfect, so we don't want to optimize it directly. Models like the Gaussian process are designed to do well at learning the uncertainties (covariances). Now recall how acquisition functions work, for example, the upper confidence bound (UCB) looks at values that are high in terms of mean $\mu$ and uncertainty $\sigma$, $\mu + \beta\sigma$, so you explore the areas where you are the most uncertain until the uncertainty is reduced enough that it doesn't play important role in the result. The same applies to other acquisition functions like expected improvement or probability of improvement. Thompson sampling overcomes this problem by sampling from the posterior distribution, so by randomizing the process according to the posterior probabilities. So acquisition function forces you to explore more rather than focusing on pure exploitation.
