  We consider the problem of model selection for two popular stochastic linear
bandit settings, and propose algorithms that adapts to the unknown problem
complexity. In the first setting, we consider the $K$ armed mixture bandits,
where the mean reward of arm $i \in [K]$, is $\mu_i+ \langle
\alpha_{i,t},\theta^* \rangle $, with $\alpha_{i,t} \in \mathbb{R}^d$ being the
known context vector and $\mu_i \in [-1,1]$ and $\theta^*$ are unknown
parameters. We define $\|\theta^*\|$ as the problem complexity and consider a
sequence of nested hypothesis classes, each positing a different upper bound on
$\|\theta^*\|$. Exploiting this, we propose Adaptive Linear Bandit (ALB), a
novel phase based algorithm that adapts to the true problem complexity,
$\|\theta^*\|$. We show that ALB achieves regret scaling of
$O(\|\theta^*\|\sqrt{T})$, where $\|\theta^*\|$ is apriori unknown. As a
corollary, when $\theta^*=0$, ALB recovers the minimax regret for the simple
bandit algorithm without such knowledge of $\theta^*$. ALB is the first
algorithm that uses parameter norm as model section criteria for linear
bandits. Prior state of art algorithms \cite{osom} achieve a regret of
$O(L\sqrt{T})$, where $L$ is the upper bound on $\|\theta^*\|$, fed as an input
to the problem. In the second setting, we consider the standard linear bandit
problem (with possibly an infinite number of arms) where the sparsity of
$\theta^*$, denoted by $d^* \leq d$, is unknown to the algorithm. Defining
$d^*$ as the problem complexity, we show that ALB achieves $O(d^*\sqrt{T})$
regret, matching that of an oracle who knew the true sparsity level. This
methodology is then extended to the case of finitely many arms and similar
results are proven. This is the first algorithm that achieves such model
selection guarantees. We further verify our results via synthetic and real-data
experiments.
