We consider reinforcement learning (RL) for a class of problems with bagged decision times. A bag contains a finite sequence of consecutive decision times. The transition dynamics are non-Markovian and non-stationary within a bag. All actions within a bag jointly impact a single reward, observed at the end of the bag. For example, in mobile health, multiple activity suggestions in a day collectively affect a user’s daily commitment to being active. Our goal is to develop an online RL algorithm to maximize the discounted sum of the bag-specific rewards. To handle non-Markovian transitions within a bag, we utilize an expert-provided causal directed acyclic graph (DAG). Based on the DAG, we construct states as a dynamical Bayesian sufficient statistic of the observed history, which results in Markov state transitions within and across bags. We then formulate this problem as a periodic Markov decision process (MDP) that allows non-stationarity within a period. An online RL algorithm based on Bellman equations for stationary MDPs is generalized to handle periodic MDPs. We show that our constructed state achieves the maximal optimal value function among all state constructions for a periodic MDP. Finally, we evaluate the proposed method on testbed variants built from real data in a mobile health clinical trial.

Timely decision making is critical to the effectiveness of mobile health (mHealth) interventions. At predefined timepoints called "decision points," intelligent mHealth systems such as just-in-time adaptive interventions (JITAIs) estimate an individual's biobehavioral context from sensor or survey data and determine whether and how to intervene. For interventions targeting habitual behavior (e.g., oral hygiene), effectiveness often hinges on delivering support shortly before the target behavior is likely to occur. Current practice schedules decision points at a fixed interval (e.g., one hour) before user-provided behavior times, and the fixed interval is kept the same for all individuals. However, this one-size-fits-all approach performs poorly for individuals with irregular routines, often scheduling decision points after the target behavior has already occurred, rendering interventions ineffective. In this paper, we propose SigmaScheduling, a method to dynamically schedule decision points based on uncertainty in predicted behavior times. When behavior timing is more predictable, SigmaScheduling schedules decision points closer to the predicted behavior time; when timing is less certain, SigmaScheduling schedules decision points earlier, increasing the likelihood of timely intervention. We evaluated SigmaScheduling using real-world data from 68 participants in a 10-week trial of Oralytics, a JITAI designed to improve daily toothbrushing. SigmaScheduling increased the likelihood that decision points preceded brushing events in at least 70% of cases, preserving opportunities to intervene and impact behavior. Our results indicate that SigmaScheduling can advance precision mHealth, particularly for JITAIs targeting time-sensitive, habitual behaviors such as oral hygiene or dietary habits.

Dynamic treatment regimes or policies are a sequence of decision functions over multiple stages that are tailored to individual features. One important class of treatment policies in practice, namely multi-stage stationary treatment policies, prescribes treatment assignment probabilities using the same decision function across stages, where the decision is based on the same set of features consisting of time-evolving variables (e.g., routinely collected disease biomarkers). Although there has been extensive literature on constructing valid inference for the value function associated with dynamic treatment policies, little work has focused on the policies themselves, especially in the presence of high-dimensional features. We aim to fill the gap in this work. Specifically, we first obtain the multi-stage stationary treatment policy by minimizing the negative augmented inverse probability weighted estimator of the value function to increase asymptotic efficiency. An $L_1$ penalty is applied on the policy parameters to select important features. We then construct one-step improvements of the policy parameter estimators for valid inference. Theoretically, we show that the improved estimators are asymptotically normal, even if nuisance parameters are estimated at a slow convergence rate and the dimension of the features increases with the sample size. Our numerical studies demonstrate that the proposed method estimates a sparse policy with a near-optimal value function and conducts valid inference for the policy parameters.

An individualized treatment rule (ITR) is a decision rule that recommends treatments for patients based on their individual feature variables. In many practices, the ideal ITR for the primary outcome is also expected to cause minimal harm to other secondary outcomes. Therefore, our objective is to learn an ITR that not only maximizes the value function for the primary outcome, but also approximates the optimal rule for the secondary outcomes as closely as possible. To achieve this goal, we introduce a fusion penalty to encourage the ITRs based on different outcomes to yield similar recommendations. Two algorithms are proposed to estimate the ITR using surrogate loss functions. We prove that the agreement rate between the estimated ITR of the primary outcome and the optimal ITRs of the secondary outcomes converges to the true agreement rate faster than if the secondary outcomes are not taken into consideration. Furthermore, we derive the non-asymptotic properties of the value function and misclassification rate for the proposed method. Finally, simulation studies and a real data example are used to demonstrate the finite-sample performance of the proposed method.

Learning optimal individualized treatment rules (ITRs) has become increasingly important in the modern era of precision medicine. Many statistical and machine learning methods for learning optimal ITRs have been developed in the literature. However, most existing methods are based on data collected from traditional randomized controlled trials and thus cannot take advantage of the accumulative evidence when patients enter the trials sequentially. It is also ethically important that future patients should have a high probability to be treated optimally based on the updated knowledge so far. In this work, we propose a new design called sequentially rule-adaptive trials to learn optimal ITRs based on the contextual bandit framework, in contrast to the response-adaptive design in traditional adaptive trials. In our design, each entering patient will be allocated with a high probability to the current best treatment for this patient, which is estimated using the past data based on some machine learning algorithm (for example, outcome weighted learning in our implementation). We explore the tradeoff between training and test values of the estimated ITR in single-stage problems by proving theoretically that for a higher probability of following the estimated ITR, the training value converges to the optimal value at a faster rate, while the test value converges at a slower rate. This problem is different from traditional decision problems in the sense that the training data are generated sequentially and are dependent. We also develop a tool that combines martingale with empirical process to tackle the problem that cannot be solved by previous techniques for i.i.d. data. We show by numerical examples that without much loss of the test value, our proposed algorithm can improve the training value significantly as compared to existing methods. Finally, we use a real data study to illustrate the performance of the proposed method.