Title: Analysis of Thompson Sampling for Gaussian Process Optimization in the Bandit Setting,
Abstract: We consider the global optimization of a function over a continuous domain.
At every evaluation attempt, we can observe the function at a chosen point in
the domain and we reap the reward of the value observed. We assume that drawing
these observations are expensive and noisy. We frame it as a continuum-armed
bandit problem with a Gaussian Process prior on the function. In this regime,
most algorithms have been developed to minimize some form of regret. Contrary
to this popular norm, in this paper, we study the convergence of the sequential
point $\boldsymbol{x}^t$ to the global optimizer $\boldsymbol{x}^*$ for the
Thompson Sampling approach. Under some assumptions and regularity conditions,
we show an exponential rate of convergence to the true optimal.