  We study the Linear Contextual Bandit problem in the hybrid reward setting.
In this setting every arm's reward model contains arm specific parameters in
addition to parameters shared across the reward models of all the arms. We can
reduce this setting to two closely related settings (a) Shared - no arm
specific parameters, and (b) Disjoint - only arm specific parameters, enabling
the application of two popular state of the art algorithms - $\texttt{LinUCB}$
and $\texttt{DisLinUCB}$ (Algorithm 1 in (Li et al. 2010)). When the arm
features are stochastic and satisfy a popular diversity condition, we provide
new regret analyses for both algorithms, significantly improving on the known
regret guarantees of these algorithms. Our novel analysis critically exploits
the hybrid reward structure and the diversity condition. Moreover, we introduce
a new algorithm $\texttt{HyLinUCB}$ that crucially modifies $\texttt{LinUCB}$
(using a new exploration coefficient) to account for sparsity in the hybrid
setting. Under the same diversity assumptions, we prove that
$\texttt{HyLinUCB}$ also incurs only $O(\sqrt{T})$ regret for $T$ rounds. We
perform extensive experiments on synthetic and real-world datasets
demonstrating strong empirical performance of $\texttt{HyLinUCB}$.For number of
arm specific parameters much larger than the number of shared parameters, we
observe that $\texttt{DisLinUCB}$ incurs the lowest regret. In this case,
regret of $\texttt{HyLinUCB}$ is the second best and extremely competitive to
$\texttt{DisLinUCB}$. In all other situations, including our real-world
dataset, $\texttt{HyLinUCB}$ has significantly lower regret than
$\texttt{LinUCB}$, $\texttt{DisLinUCB}$ and other SOTA baselines we considered.
We also empirically observe that the regret of $\texttt{HyLinUCB}$ grows much
slower with the number of arms compared to baselines, making it suitable even
for very large action spaces.
