This study investigates the contextual best arm identification (BAI) problem, aiming to design an adaptive experiment to identify the best treatment arm conditioned on contextual information (covariates). We consider a decision-maker who assigns treatment arms to experimental units during an experiment and recommends the estimated best treatment arm based on the contexts at the end of the experiment. The decision-maker uses a policy for recommendations, which is a function that provides the estimated best treatment arm given the contexts. In our evaluation, we focus on the worst-case \emph{expected regret}, a relative measure between the expected outcomes of an optimal policy and our proposed policy. We derive a lower bound for the expected simple regret and then propose a strategy called \emph{Adaptive Sampling-Policy Learning} (PLAS). We prove that this strategy is minimax rate-optimal in the sense that its leading factor in the regret upper bound matches the lower bound as the number of experimental units increases.