  This paper studies representation learning for multi-task linear bandits and
multi-task episodic RL with linear value function approximation. We first
consider the setting where we play $M$ linear bandits with dimension $d$
concurrently, and these bandits share a common $k$-dimensional linear
representation so that $k\ll d$ and $k \ll M$. We propose a sample-efficient
algorithm, MTLR-OFUL, which leverages the shared representation to achieve
$\tilde{O}(M\sqrt{dkT} + d\sqrt{kMT} )$ regret, with $T$ being the number of
total steps. Our regret significantly improves upon the baseline
$\tilde{O}(Md\sqrt{T})$ achieved by solving each task independently. We further
develop a lower bound that shows our regret is near-optimal when $d > M$.
Furthermore, we extend the algorithm and analysis to multi-task episodic RL
with linear value function approximation under low inherent Bellman error
\citep{zanette2020learning}. To the best of our knowledge, this is the first
theoretical result that characterizes the benefits of multi-task representation
learning for exploration in RL with function approximation.
