  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 multiple context directions -- in contrast to the
conventional estimator that only updates the 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 exploration
samples 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.
