  We consider the actor-critic contextual bandit for the mobile health
(mHealth) intervention. State-of-the-art decision-making algorithms generally
ignore the outliers in the dataset. In this paper, we propose a novel robust
contextual bandit method for the mHealth. It can achieve the conflicting goal
of reducing the influence of outliers while seeking for a similar solution
compared with the state-of-the-art contextual bandit methods on the datasets
without outliers. Such performance relies on two technologies: (1) the
capped-$\ell_{2}$ norm; (2) a reliable method to set the thresholding
hyper-parameter, which is inspired by one of the most fundamental techniques in
the statistics. Although the model is non-convex and non-differentiable, we
propose an effective reweighted algorithm and provide solid theoretical
analyses. We prove that the proposed algorithm can find sufficiently decreasing
points after each iteration and finally converges after a finite number of
iterations. Extensive experiment results on two datasets demonstrate that our
method can achieve almost identical results compared with state-of-the-art
contextual bandit methods on the dataset without outliers, and significantly
outperform those state-of-the-art methods on the badly noised dataset with
outliers in a variety of parameter settings.
