Bayesian optimisation is a powerful method for non-convex black-box
optimization in low data regimes. However, the question of establishing tight
upper bounds for common algorithms in the noiseless setting remains a largely
open question. In this paper, we establish new and tightest bounds for two
algorithms, namely GP-UCB and Thompson sampling, under the assumption that the
objective function is smooth in terms of having a bounded norm in a Mat\'ern
RKHS. Importantly, unlike several related works, we do not consider perfect
knowledge of the kernel of the Gaussian process emulator used within the
Bayesian optimization loop. This allows us to provide results for practical
algorithms that sequentially estimate the Gaussian process kernel parameters
from the available data