  The standard theory of optimal stopping is based on the idealised assumption
that the underlying process is essentially known. In this paper, we drop this
restriction and study data-driven optimal stopping for a general diffusion
process, focusing on investigating the statistical performance of the proposed
estimator of the optimal stopping barrier. More specifically, we derive
non-asymptotic upper bounds on the simple regret, along with uniform and
non-asymptotic PAC bounds. Minimax optimality is verified by completing the
upper bound results with matching lower bounds on the simple regret. All
results are shown both under general conditions on the payoff functions and
under more refined assumptions that mimic the margin condition used in binary
classification, leading to an improved rate of convergence. Additionally, we
investigate how our results on the simple regret transfer to the cumulative
regret for a specific exploration-exploitation strategy, both with respect to
lower bounds and upper bounds.
