In Bayesian optimization, we guess the next sampling point by finding $x = \textrm{argmax}_x \alpha(x)$, where $\alpha(x)$ is the acquisition function. For simplicity, let us consider the upper confidence bound (UCB) acquisition function: $\alpha(x) = \mu(x) + \beta \sigma(x)$, where $\mu(x)$ is the usual mean and $\sigma(x)$ denotes the usual square root of the variance. Here, $\beta$ is of course the dimensionless parameter ($\beta\geq0$) that tunes the trade off between exploration (high $\beta$) and exploitation (low $\beta$).

Suppose I have a computational or experimental budget of $N$ evaluations of the objective function. Is there an algorithm or heuristic for setting $\beta$ as a function of $N$?

In Bayesian optimization, we guess the next sampling point by finding $x = \textrm{argmax}_x \alpha(x)$, where $\alpha(x)$ is the acquisition function. For simplicity, let us consider the upper confidence bound (UCB) acquisition function: $\alpha(x)=\mu(x) + \sqrt\beta \sigma(x)$, where $\mu(x)$ is the usual mean and $\sigma(x)$ denotes the usual square root of the variance. Here, $\beta$ is of course the dimensionless parameter ($\beta\geq0$) that tunes the trade off between exploration (high $\beta$) and exploitation (low $\beta$).

Suppose I have a computational or experimental budget of $N$ evaluations of the objective function. Is there an algorithm or heuristic for setting $\beta$ as a function of $N$?