  While multitask representation learning has become a popular approach in
reinforcement learning (RL), theoretical understanding of why and when it works
remains limited. This paper presents analyses for the statistical benefit of
multitask representation learning in linear Markov Decision Process (MDP) under
a generative model. In this paper, we consider an agent to learn a
representation function $\phi$ out of a function class $\Phi$ from $T$ source
tasks with $N$ data per task, and then use the learned $\hat{\phi}$ to reduce
the required number of sample for a new task. We first discover a
\emph{Least-Activated-Feature-Abundance} (LAFA) criterion, denoted as $\kappa$,
with which we prove that a straightforward least-square algorithm learns a
policy which is $\tilde{O}(H^2\sqrt{\frac{\mathcal{C}(\Phi)^2 \kappa
d}{NT}+\frac{\kappa d}{n}})$ sub-optimal. Here $H$ is the planning horizon,
$\mathcal{C}(\Phi)$ is $\Phi$'s complexity measure, $d$ is the dimension of the
representation (usually $d\ll \mathcal{C}(\Phi)$) and $n$ is the number of
samples for the new task. Thus the required $n$ is $O(\kappa d H^4)$ for the
sub-optimality to be close to zero, which is much smaller than
$O(\mathcal{C}(\Phi)^2\kappa d H^4)$ in the setting without multitask
representation learning, whose sub-optimality gap is
$\tilde{O}(H^2\sqrt{\frac{\kappa \mathcal{C}(\Phi)^2d}{n}})$. This
theoretically explains the power of multitask representation learning in
reducing sample complexity. Further, we note that to ensure high sample
efficiency, the LAFA criterion $\kappa$ should be small. In fact, $\kappa$
varies widely in magnitude depending on the different sampling distribution for
new task. This indicates adaptive sampling technique is important to make
$\kappa$ solely depend on $d$. Finally, we provide empirical results of a noisy
grid-world environment to corroborate our theoretical findings.
