  Thompson Sampling is one of the oldest heuristics for multi-armed bandit
problems. It is a randomized algorithm based on Bayesian ideas, and has
recently generated significant interest after several studies demonstrated it
to have better empirical performance compared to the state-of-the-art methods.
However, many questions regarding its theoretical performance remained open. In
this paper, we design and analyze a generalization of Thompson Sampling
algorithm for the stochastic contextual multi-armed bandit problem with linear
payoff functions, when the contexts are provided by an adaptive adversary. This
is among the most important and widely studied versions of the contextual
bandits problem. We provide the first theoretical guarantees for the contextual
version of Thompson Sampling. We prove a high probability regret bound of
$\tilde{O}(d^{3/2}\sqrt{T})$ (or $\tilde{O}(d\sqrt{T \log(N)})$), which is the
best regret bound achieved by any computationally efficient algorithm available
for this problem in the current literature, and is within a factor of
$\sqrt{d}$ (or $\sqrt{\log(N)}$) of the information-theoretic lower bound for
this problem.
