Abstract: We study the piecewise constant bandit problem where the expected reward is a piecewise constant function with one change point (discontinuity) across the action space $[0,1]$ and the learner's aim is to locate the change point. Under the assumption of a fixed exploration budget, we provide the first non-asymptotic analysis of policies designed to locate abrupt changes in the mean reward function under bandit feedback. We study the problem under a large and small budget regime, and for both settings establish lower bounds on the error probability and provide algorithms with near matching upper bounds. Interestingly, our results show a separation in the complexity of the two regimes. We then propose a regime adaptive algorithm which is near optimal for both small and large budgets simultaneously.We complement our theoretical analysis with experimental results in simulated environments to support our findings.