  In multi-task reinforcement learning (RL) under Markov decision processes
(MDPs), the presence of shared latent structures among multiple MDPs has been
shown to yield significant benefits to the sample efficiency compared to
single-task RL. In this paper, we investigate whether such a benefit can extend
to more general sequential decision making problems, such as partially
observable MDPs (POMDPs) and more general predictive state representations
(PSRs). The main challenge here is that the large and complex model space makes
it hard to identify what types of common latent structure of multi-task PSRs
can reduce the model complexity and improve sample efficiency. To this end, we
posit a joint model class for tasks and use the notion of $\eta$-bracketing
number to quantify its complexity; this number also serves as a general metric
to capture the similarity of tasks and thus determines the benefit of
multi-task over single-task RL. We first study upstream multi-task learning
over PSRs, in which all tasks share the same observation and action spaces. We
propose a provably efficient algorithm UMT-PSR for finding near-optimal
policies for all PSRs, and demonstrate that the advantage of multi-task
learning manifests if the joint model class of PSRs has a smaller
$\eta$-bracketing number compared to that of individual single-task learning.
We also provide several example multi-task PSRs with small $\eta$-bracketing
numbers, which reap the benefits of multi-task learning. We further investigate
downstream learning, in which the agent needs to learn a new target task that
shares some commonalities with the upstream tasks via a similarity constraint.
By exploiting the learned PSRs from the upstream, we develop a sample-efficient
algorithm that provably finds a near-optimal policy.
