  Consider the sequential optimization of an expensive to evaluate and possibly
non-convex objective function $f$ from noisy feedback, that can be considered
as a continuum-armed bandit problem. Upper bounds on the regret performance of
several learning algorithms (GP-UCB, GP-TS, and their variants) are known under
both a Bayesian (when $f$ is a sample from a Gaussian process (GP)) and a
frequentist (when $f$ lives in a reproducing kernel Hilbert space) setting. The
regret bounds often rely on the maximal information gain $\gamma_T$ between $T$
observations and the underlying GP (surrogate) model. We provide general bounds
on $\gamma_T$ based on the decay rate of the eigenvalues of the GP kernel,
whose specialisation for commonly used kernels, improves the existing bounds on
$\gamma_T$, and subsequently the regret bounds relying on $\gamma_T$ under
numerous settings. For the Mat\'ern family of kernels, where the lower bounds
on $\gamma_T$, and regret under the frequentist setting, are known, our results
close a huge polynomial in $T$ gap between the upper and lower bounds (up to
logarithmic in $T$ factors).
