  We consider the problem of \emph{blocked} collaborative bandits where there
are multiple users, each with an associated multi-armed bandit problem. These
users are grouped into \emph{latent} clusters such that the mean reward vectors
of users within the same cluster are identical. Our goal is to design
algorithms that maximize the cumulative reward accrued by all the users over
time, under the \emph{constraint} that no arm of a user is pulled more than
$\mathsf{B}$ times. This problem has been originally considered by
\cite{Bresler:2014}, and designing regret-optimal algorithms for it has since
remained an open problem. In this work, we propose an algorithm called
\texttt{B-LATTICE} (Blocked Latent bAndiTs via maTrIx ComplEtion) that
collaborates across users, while simultaneously satisfying the budget
constraints, to maximize their cumulative rewards. Theoretically, under certain
reasonable assumptions on the latent structure, with $\mathsf{M}$ users,
$\mathsf{N}$ arms, $\mathsf{T}$ rounds per user, and $\mathsf{C}=O(1)$ latent
clusters, \texttt{B-LATTICE} achieves a per-user regret of
$\widetilde{O}(\sqrt{\mathsf{T}(1 + \mathsf{N}\mathsf{M}^{-1})}$ under a budget
constraint of $\mathsf{B}=\Theta(\log \mathsf{T})$. These are the first
sub-linear regret bounds for this problem, and match the minimax regret bounds
when $\mathsf{B}=\mathsf{T}$. Empirically, we demonstrate that our algorithm
has superior performance over baselines even when $\mathsf{B}=1$.
\texttt{B-LATTICE} runs in phases where in each phase it clusters users into
groups and collaborates across users within a group to quickly learn their
reward models.
