  We investigate the sample complexity of learning the optimal arm for
multi-task bandit problems. Arms consist of two components: one that is shared
across tasks (that we call representation) and one that is task-specific (that
we call predictor). The objective is to learn the optimal (representation,
predictor)-pair for each task, under the assumption that the optimal
representation is common to all tasks. Within this framework, efficient
learning algorithms should transfer knowledge across tasks. We consider the
best-arm identification problem for a fixed confidence, where, in each round,
the learner actively selects both a task, and an arm, and observes the
corresponding reward. We derive instance-specific sample complexity lower
bounds satisfied by any $(\delta_G,\delta_H)$-PAC algorithm (such an algorithm
identifies the best representation with probability at least $1-\delta_G$, and
the best predictor for a task with probability at least $1-\delta_H$). We
devise an algorithm OSRL-SC whose sample complexity approaches the lower bound,
and scales at most as $H(G\log(1/\delta_G)+ X\log(1/\delta_H))$, with $X,G,H$
being, respectively, the number of tasks, representations and predictors. By
comparison, this scaling is significantly better than the classical best-arm
identification algorithm that scales as $HGX\log(1/\delta)$.
