Bayesian optimisation is a popular method for efficient optimisation of expensive black-box functions. Traditionally, BO assumes that the search space is known. However, in many problems, this assumption does not hold. To this end, we propose a novel BO algorithm which expands (and shifts) the search space over iterations based on controlling the expansion rate thought a \emph{hyperharmonic series}. Further, we propose another variant of our algorithm that scales to high dimensions. We show theoretically that for both our algorithms, the cumulative regret grows at sub-linear rates. Our experiments with synthetic and real-world optimisation tasks demonstrate the superiority of our algorithms over the current state-of-the-art methods for Bayesian optimisation in unknown search space.

Additional Feedback: Algorithm 2. X_t is never defined. I assumed that X_t is defined by Equation 2 like Algorithm 1. Authors mentioned the same computational budget for acquisition function optimization. What is the optimizer though? Constrained optimization of the acquisition function inside H_t (Equation 3) does not seem trivial. It isn't mentioned anywhere how the acquisition funciton was optimized.

The paper has been discussed after the rebuttal that the reviewers found useful and actionable (e.g., concerns about the confidence bound). The paper is recommended for acceptance. All reviewers have acknowledged that the paper is well motivated, well written and establishes a nice interplay between theory and a practical problem of interest.

Bayesian optimisation is a popular method for efficient optimisation of expensive black-box functions. Traditionally, BO assumes that the search space is known. However, in many problems, this assumption does not hold. To this end, we propose a novel BO algorithm which expands (and shifts) the search space over iterations based on controlling the expansion rate thought a \emph{hyperharmonic series}. Further, we propose another variant of our algorithm that scales to high dimensions. We show theoretically that for both our algorithms, the cumulative regret grows at sub-linear rates.