Thus the optimal average reward of the original MDP and modified MDP differ by O ( ϵ). To ensure Assumption 3.1 (b) is satisfied, an aperiodicity transformation can be implemented. The proof of this theorem can be found in [Sch71]. From Lemma 2.2, we thus have, ( J In order to iterate Equation (8), need to ensure the terms are non-negative. Theorem 3.3 presents an upper bound on the error in terms of the average reward.

As AI agents generate increasingly sophisticated behaviors, manually encoding human preferences to guide these agents becomes more challenging. To address this, it has been suggested that agents instead learn preferences from human choice data. This approach requires a model of choice behavior that the agent can use to interpret the data. For choices between partial trajectories of states and actions, previous models assume choice probabilities are determined by the partial return or the cumulative advantage. We consider an alternative model based instead on the bootstrapped return, which adds to the partial return an estimate of the future return. Benefits of the bootstrapped return model stem from its treatment of human beliefs. Unlike partial return, choices based on bootstrapped return reflect human beliefs about the environment. Further, while recovering the reward function from choices based on cumulative advantage requires that those beliefs are correct, doing so from choices based on bootstrapped return does not. To motivate the bootstrapped return model, we formulate axioms and prove an Alignment Theorem. This result formalizes how, for a general class of preferences, such models are able to disentangle goals from beliefs. This ensures recovery of an aligned reward function when learning from choices based on bootstrapped return. The bootstrapped return model also affords greater robustness to choice behavior. Even when choices are based on partial return, learning via a bootstrapped return model recovers an aligned reward function. The same holds with choices based on the cumulative advantage if the human and the agent both adhere to correct and consistent beliefs about the environment. On the other hand, if choices are based on bootstrapped return, learning via partial return or cumulative advantage models does not generally produce an aligned reward function.

Infinite-state Markov Decision Processes (MDPs) are essential in modeling and optimizing a wide variety of engineering problems. In the reinforcement learning (RL) context, a variety of algorithms have been developed to learn and optimize these MDPs. At the heart of many popular policy-gradient based learning algorithms, such as natural actor-critic, TRPO, and PPO, lies the Natural Policy Gradient (NPG) algorithm. Convergence results for these RL algorithms rest on convergence results for the NPG algorithm. However, all existing results on the convergence of the NPG algorithm are limited to finite-state settings. We prove the first convergence rate bound for the NPG algorithm for infinite-state average-reward MDPs, proving a $O(1/\sqrt{T})$ convergence rate, if the NPG algorithm is initialized with a good initial policy. Moreover, we show that in the context of a large class of queueing MDPs, the MaxWeight policy suffices to satisfy our initial-policy requirement and achieve a $O(1/\sqrt{T})$ convergence rate. Key to our result are state-dependent bounds on the relative value function achieved by the iterate policies of the NPG algorithm.

Many policy-based reinforcement learning (RL) algorithms can be viewed as instantiations of approximate policy iteration (PI), i.e., where policy improvement and policy evaluation are both performed approximately. In applications where the average reward objective is the meaningful performance metric, discounted reward formulations are often used with the discount factor being close to $1,$ which is equivalent to making the expected horizon very large. However, the corresponding theoretical bounds for error performance scale with the square of the horizon. Thus, even after dividing the total reward by the length of the horizon, the corresponding performance bounds for average reward problems go to infinity. Therefore, an open problem has been to obtain meaningful performance bounds for approximate PI and RL algorithms for the average-reward setting. In this paper, we solve this open problem by obtaining the first finite-time error bounds for average-reward MDPs, and show that the asymptotic error goes to zero in the limit as policy evaluation and policy improvement errors go to zero.

We study the problem of policy optimization in Markov Decision Process over infinite time horizons (Puterman, 1994). We focus on the batch (i.e., off-line) setting, where historical data of multiple trajectories has been previously collected using some behavior policy. Our goal is to learn a new policy with guaranteed performance when implemented in the future. In this work, we develop a data-efficient method to learn the policy that optimizes the long-term average reward in a pre-specified policy class from a training set composed of multiple trajectories. Furthermore, we establish a finite-sample regret guarantee, i.e., the difference between the average reward of the optimal policy in the class and the average reward of the estimated policy by our proposed method. This work is motivated by the development of justin-time adaptive intervention in mobile health (mHealth) applications (Nahum-Shani et al., 2017). Our method can be used to learn a treatment policy that maps the real-time collected information about the individual's status and context to a particular treatment at each of many decision times to support health behaviors.

With the recent advancements in wearables and sensing technology, health scientists are increasingly developing mobile health (mHealth) interventions. In mHealth interventions, mobile devices are used to deliver treatment to individuals as they go about their daily lives, generally designed to impact a near time, proximal outcome such as stress or physical activity. The mHealth intervention policies, often called Just-In-time Adaptive Interventions, are decision rules that map a user's context to a particular treatment at each of many time points. The vast majority of current mHealth interventions deploy expert-derived policies. In this paper, we provide an approach for conducting inference about the performance of one or more such policies. In particular, we estimate the performance of a mHealth policy using historical data that are collected under a possibly different policy. Our measure of performance is the average of proximal outcomes (rewards) over a long time period should the particular mHealth policy be followed. We provide a semi-parametric efficient estimator as well as the confidence intervals. This work is motivated by HeartSteps, a mobile health physical activity intervention.