  In this paper we consider the contextual multi-armed bandit problem for
linear payoffs under a risk-averse criterion. At each round, contexts are
revealed for each arm, and the decision maker chooses one arm to pull and
receives the corresponding reward. In particular, we consider mean-variance as
the risk criterion, and the best arm is the one with the largest mean-variance
reward. We apply the Thompson Sampling algorithm for the disjoint model, and
provide a comprehensive regret analysis for a variant of the proposed
algorithm. For $T$ rounds, $K$ actions, and $d$-dimensional feature vectors, we
prove a regret bound of $O((1+\rho+\frac{1}{\rho}) d\ln T \ln
\frac{K}{\delta}\sqrt{d K T^{1+2\epsilon} \ln \frac{K}{\delta}
\frac{1}{\epsilon}})$ that holds with probability $1-\delta$ under the
mean-variance criterion with risk tolerance $\rho$, for any
$0<\epsilon<\frac{1}{2}$, $0<\delta<1$. The empirical performance of our
proposed algorithms is demonstrated via a portfolio selection problem.
