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We consider the problem of global optimization of a function over acontinuous domain. In our setup, we can evaluate the function sequentially atpoints of our choice and the evaluations are noisy. We frame it as acontinuum-armed bandit problem with a Gaussian Process prior on the function.In this regime, most algorithms have been developed to minimize some form ofregret. In this paper, we study the convergence of the sequential point $x^t$to the global optimizer $x^*$ for the Thompson Sampling approach. Under someassumptions and regularity conditions, we prove concentration bounds for $x^t$where the probability that $x^t$ is bounded away from $x^*$ decaysexponentially fast in $t$. Moreover, the result allows us to derive adaptiveconvergence rates depending on the function structure.

Adaptive Rate of Convergence of Thompson Sampling for Gaussian Process Optimization