Action Centered Contextual Bandits
Contextual bandits have become popular as they offer a middle ground between very simple approaches based on multi-armed bandits and very complex approaches using the full power of reinforcement learning. They have demonstrated success in web applications and have a rich body of associated theoretical guarantees. Linear models are well understood theoretically and preferred by practitioners because they are not only easily interpretable but also simple to implement and debug. Furthermore, if the linear model is true, we get very strong performance guarantees. Unfortunately, in emerging applications in mobile health, the time-invariant linear model assumption is untenable. We provide an extension of the linear model for contextual bandits that has two parts: baseline reward and treatment effect. We allow the former to be complex but keep the latter simple. We argue that this model is plausible for mobile health applications. At the same time, it leads to algorithms with strong performance guarantees as in the linear model setting, while still allowing for complex nonlinear baseline modeling. Our theory is supported by experiments on data gathered in a recently concluded mobile health study.
In the context of multi-armed bandit problems for mobile health, modelling correctly the linearity or non-linearity of the rewards is crucial.
The paper tries to take advantage of the fact that a general "Do nothing" action is generally available, that corresponds to the baseline. The idea is that, while the reward for each action may not be easily captured by a linear model, it is reasonable to assume that the relative error between playing an action and playing the "Do thing" action follows a linear model. 
The article revisits bandit strategies in this context.
Comments:
1) Line 29: See also the algorithm OFUL, and more generally Kernel-UCB, GP-UCB, to cite a few other algorithms with provable regret minimization guarantees.
2) Line 35, 51, and so on: I humbly suggest you replace "mhealth" with something else, e.g. "M-health", since "mhealth" looks like a typo.
3) Question about the model: is \bar {f}_t assumed to be known? If not and if it is completely arbitrary, it seems counter-intuitive that one can prove any learning guarantee, given that a bandit setting is considered (only one action per step), and \bar {f}_t may change at any time to create maximal confusion.
4) Line 132: "might different for different"
5) Line 162: What is \bar{a}_t ? I can only see the definition for a_t and \bar{a}_t^\star.
Likewise, what is \bar{a} on line 165? It seems these are defined only later in the paper, which is confusing. Please, fix this point.
6) Question: What is the intuition for choosing \pi_t to be (up to projection) P(s_{t,\bar a}^\top \bar \theta \geq 0) ? Especially, please comment on the value 0 here.
7) Lemma 5: Should 0 be replaced with a in the probability bound?
8) Section 9: I understand that you heavily used the proof of Agrawal and Goyal, but I discourage the use of sentences such as "For any filtration such that ... is true". I suggest you rather control probability of events.
9) Proof, line 62: there is a missing reference.
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In the end the setting of the paper and definition of regret look a bit odd, but the authors provide an interesting idea to overcome the possibly nasty non-stationary term, thanks to randomization. Also deriving the weighted least-squares estimate and TS analysis in this context is interesting, even though relying heavily on previous existing analysis for TS. Finally, the idea to consider that only the relative errors follow a linear model might be generalized to other problems. 
Weak accept.