Thats Good Advice From Bandit

That's Good Advice From Bandit

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IF I become a mercenary for one kingdom and my contract ends can I then go to a kingdom I've gotten a bad rep with and become a mercenary for them as well? I know that sounds like a stupid question lol but I'd like to know how it works.

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Annoying both neighbouring powers and the local populace at the same time never went well in the long run for anyone. Unless we're talking grand strategy type games, and you are like, at least 2 tech levels above everyone else Eventually, everyone decides you're better off dead, and does that. Recruit mercenaries from towns. Mercs are expensive, so that's something to note as well.

Capture enemy troops, then you can see if any of them are willing to work for you. Morale will take a hit, and some might run off in the night. Take in the prisoners that the enemy forces have taken. 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: 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.

Unlike bandit algorithms, which cannot use any side-information or context, contextual bandit algorithms can learn to map the context into appropriate actions. However, contextual bandits do not consider the impact of actions on the evolution of future contexts. An influential thread within the contextual bandit literature models the expected reward for any action in a given context using a linear mapping from a d -dimensional context vector to a real-valued reward.

Algorithms using this assumption include LinUCB and Thompson Sampling, for both of which regret bounds have been derived. These analyses often allow the context sequence to be chosen adversarially, but require the linear model, which links rewards to contexts, to be time-invariant. There has been little effort to extend these algorithms and analyses when the data follow an unknown nonlinear or time-varying model. In this paper, we consider a particular type of non-stationarity and non-linearity that is motivated by problems arising in mobile health mHealth.

Action Centered Contextual Bandits

Mobile health is a fast developing field that uses mobile and wearable devices for health care delivery. These devices provide us with a real-time stream of dynamically evolving contextual information about the user location, calendar, weather, physical activity, internet activity, etc. Contextual bandit algorithms can learn to map this contextual information to a set of available intervention options e.

However, human behavior is hard to model using stationary, linear models. We make a fundamental assumption in this paper that is quite plausible in the mHealth setting. The expected reward for this action is the baseline reward and it can change in a very non-stationary, non-linear fashion. However, the treatment effect of a non-zero action, i.

We show, both theoretically and empirically, that the performance of an appropriately designed action-centered contextual bandit algorithm is agnostic to the high model complexity of the baseline reward. Instead, we get the same level of performance as expected in a stationary, linear model setting.

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Note that it might be tempting to make the entire model non-linear and non-stationary. However, the sample complexity of learning very general non-stationary, non-linear models is likely to be so high that they will not be useful in mHealth where data is often noisy, missing, or collected only over a few hundred decision points.

We connect our algorithm design and theoretical analysis to the real world of mHealth by using data from a pilot study of HeartSteps, an Android-based walking intervention. HeartSteps encourages walking by sending individuals contextually-tailored suggestions to be active. HeartSteps contains two types of suggestions: While the initial pilot study of HeartSteps micro-randomized the delivery of activity suggestions Klasnja et al. We introduce a variant of the standard linear contextual bandit model that allows the baseline reward model to be quite complex while keeping the treatment effect model simple.

We then introduce the idea of using action centering in contextual bandits as a way to decouple the estimation of the above two parts of the model. We show that action centering is effective in dealing with time-varying and non-linear behavior in our model, leading to regret bounds that scale as nicely as previous bounds for linear contextual bandits. Finally, we use data gathered in the recently conducted HeartSteps study to validate our model and theory. Contextual bandits have been the focus of considerable interest in recent years. Works such as Seldin et al. Methods for reducing the regret under complex reward functions include the nonparametric approach of May et al.

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Each of these approaches has regret that scales with the complexity of the overall reward model including the baseline, and requires the reward function to remain constant over time. Consider a contextual bandit with a baseline zero action and N non-baseline arms actions or treatments. This form is used to achieve maximum generality, as it allows for infinite possible actions so long as the reward can be modeled using a d -dimensional s t,a.

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Note that the optimal action depends in no way on g t , which merely confounds the observation of regret. We hypothesize that the regret bounds for such a contextual bandit asymptotically depend only on the complexity of f , not of g t. We emphasize that we do not require any assumptions about or bounds on the complexity or smoothness of g t , allowing g t to be arbitrarily nonlinear and to change abruptly in time.

1 Introduction

These conditions create a partially agnostic setting where we have a simple model for the interaction but the baseline cannot be modeled with a simple linear function. In this paper, we consider the linear model for the reward difference at time t:. As is common in the literature, we assume that both the baseline and interaction rewards are bounded by a constant for all t. In the mHealth setting, a contextual bandit must choose at each time point whether to deliver to the user a behavior-change intervention, and if so, what type of intervention to deliver.

Whether or not an intervention, such as an activity suggestion or a medication reminder, is sent is a critical aspect of the user experience. We are thus motivated to introduce a constraint on the size of the probabilities of delivering an intervention.

Conceptually, we can view the bandit as pulling two arms at each time t: While these probability constraints are motivated by domain science, these constraints also enable our proposed action-centering algorithm to effectively orthogonalize the baseline and interaction term rewards, achieving sublinear regret in complex scenarios that often occur in mobile health and other applications and for which existing approaches have large regret.

Under this probability constraint, we can now derive the optimal policy with which to compare the bandit. The policy that maximizes the expected reward 2 will play the optimal action. The remainder of the probability is assigned as follows. If the optimal action is nonzero, the optimal policy will then play the zero action with the remaining probability which is the minimum allowed probability of playing the zero action.

If the optimal action is zero, the optimal policy will play the nonzero action with the highest expected reward. Since the observed reward always contains the sum of the baseline reward and the differential reward we are estimating, and the baseline reward is arbitrarily complex, the main challenge is to isolate the differential reward at each time step. This corresponds to the Bayesian estimator when the reward is Gaussian.

As the Thompson sampling approach generates probabilities of taking an action, rather than selecting an action, Thompson sampling is particularly suited to our regression approach. We thus introduce a two-step hierarchical procedure. This probability is easily computed using the normal CDF. Our action-centered Thompson sampling algorithm is summarized in Algorithm 1. The probability constraint implies that the optimal policy 3 plays the optimal arm with a probability bounded away from 0 and 1, hence definition 7 is no longer meaningful.

In the following theorem we show that with high probability, the probability-constrained Thompson sampler has low regret relative to the optimal probability-constrained policy. Under this regime, the total regret at time T for the action-centered Thompson sampling contextual bandit Algorithm 1 satisfies. The constant C is in the proof. Observe that this regret bound does not depend on the number of actions N , is sublinear in T , and scales only with the complexity d of the interaction term, not the complexity of the baseline reward g. Furthermore, when the baseline reward is time-varying, the worst case regret of the standard Thompson sampling approach is O T , while the regret of our method remains O d 2 T.

We will first bound the regret 8 at time t. We bound I using concentration inequalities, and bound II using arguments paralleling those for standard Thompson sampling. Suppose that the conditions of Theorem 1 apply. Then term II can be bounded as. The proofs are contained in Sections D and E in the supplement respectively. Combining Lemmas 1 and 2 via the union bound gives Theorem 1.

In each experiment, we choose a true reward generative model r t s, a inspired by data from the HeartSteps study for details see Section A. We consider both nonlinear and nonstationary baselines, while keeping the treatment effect models the same. The bandit under evaluation iterates through the T time points, at each choosing an action and receiving a reward generated according to the chosen model. At each time step, the reward under the optimal policy is calculated and compared to the reward received by the bandit to form the regret regret t.

We can then plot the cumulative regret cumulative regret. In the first experiment, the baseline reward is nonlinear.

This simulates the quite likely scenario that for a given individual the baseline reward is higher for small absolute deviations from the mean of the first context feature, i. The results are shown in Figure 1 , demonstrating linear growth of the benchmark Thompson sampling algorithm and significantly lower, sublinear regret for our proposed method.

Nonlinear baseline reward g , in scenario with 2 nonzero actions and reward function based on collected HeartSteps data.

Cumulative regret shown for proposed Action-Centered approach, compared to baseline contextual bandit, median computed over random trials. The cumulative regret is shown in Figure 2 , again demonstrating linear regret for the baseline approach and significantly lower sublinear regret for our proposed action-centering algorithm as expected. Nonstationary baseline reward g , in scenario with 2 nonzero actions and reward function based on collected HeartSteps data.

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