We consider Pareto front identification~(PFI) for linear bandits (PFILin), i.e., the goal is to identify a set of arms with undominated mean reward vectors when the mean reward vector is a linear function of the context. PFILin includes the best arm identification problem and multi-objective active learning as special cases. The sample complexity of our proposed algorithm is optimal up to a logarithmic factor. In addition, the regret incurred by our algorithm during the estimation is within a logarithmic factor of the optimal regret among all algorithms that identify the Pareto front.Our key contribution is a new estimator that in every round updates the estimate for the unknown parameter along \emph{multiple} context directions -- in contrast to the conventional estimator that only updatesthe parameter estimate along the chosen context. This allows us to use low-regret arms to collect information about Pareto optimal arms. Our key innovation is to reuse the explorationsamples multiple times; in contrast to conventional estimators that use each sample only once. Numerical experiments demonstrate that the proposed algorithm successfully identifies the Pareto front while controlling the regret.