I have gone through the GP-UCB N. Srinivas et.al where the acquisition function attempts to strike balance between exploration and exploitation. The regression model is $y=f(x)+\varepsilon$ following all the conditions mentioned in the original manuscript. The Gaussian process is given as $y(x) \sim \mathcal{N}(\mu(x),\sigma^2(x))$. The GP-UCB acquisition function is:
$$x_{t+1}=\arg max_x=\mu_t(x)+\sqrt{\beta_{t+1}}\sigma_t(x)$$
This leads to the result where the probability of the difference between mean estimate $\mu(x)$ and true (but unknown) function $f(x)$ is bounded as, with $\delta \in (0,1)$ $$\mathbb{P}\bigg \{|f(x)-\mu_T(x)|\leq \sqrt{\beta_{t+1}}\sigma_t(x) \bigg \} \geq 1-\delta $$
My question is that if i modify the acquisition function to perform pure exploration as $x_{t+1}=\arg max_x=\sqrt{\beta_{t+1}}\sigma_t(x)$, does the result on probabilisitc bound hold? Or I need to modify the bound as well.
My understanding is that if the function is sampled from a Gaussian process, then the bound should hold irrespective of the acquation function. Something similar to the Hoeffding’s Inequality based UCB in Given in this blog. How can I construct argument to show that, if true.