Missingness-MDPs: Bridging the Theory of Missing Data and POMDPs

Partially Observable Markov Decision Processes (POMDPs) are systems in which one agent interacts with a stochastic environment, and receives only partial information about the current state. In a multi-environment POMDP (MEPOMDP), the initial state is unknown, and assumed to be adversarially chosen. In this work we focus on computing the optimal value and policy in MEPOMDPs with finite-horizon objectives. That problem is known to be PSPACE-complete in POMDPs. Our main results are as follows: (1) we establish that it is also PSPACE-complete in the more general setting of MEPOMDPs; (2) we present a practical algorithm and evaluate it on classical benchmarks, significantly outperforming the only previously known algorithm.

In offline Reinforcement Learning, immediate rewards in logged batch data are often unobserved due to sparse or irregular record-keeping, or censored beyond certain reward values. This issue arises in practical settings, including health care and marketing. We investigate off-policy evaluation (OPE) in finite-horizon Markov decision processes when rewards are missing not at random (MNAR), which breaks ignorability and induces selection bias even after conditioning on states and actions. To address this, we formalize a reward-dependent propensity model and use future states as shadow variables to identify the full-data conditional mean reward. We further introduce a bridge function that recovers the conditional mean reward without explicitly modeling the MNAR mechanism, and estimate it via a min-max procedure to avoid double sampling. Building upon these identification results, we propose an Fitted-Q-Evaluation-style estimator that propagates the recovered rewards while allowing target policies to depend on past missingness indicators. Finally, we establish consistency and finite-sample error bounds for our OPE estimator, and show through experiments the strong performance of our method compared to existing methods on simulated and MIMIC-III Sepsis data.

We are interested in enabling autonomous agents to learn and reason about systems with hidden states, such as locking mechanisms. We cast this problem as learning the parameters of a discrete Partially Observable Markov Decision Process (POMDP). The agent begins with knowledge of the POMDP's actions and observation spaces, but not its state space, transitions, or observation models. These properties must be constructed from a sequence of actions and observations. Spectral approaches to learning models of partially observable domains, such as Predictive State Representations (PSRs), learn representations of state that are sufficient to predict future outcomes. PSR models, however, do not have explicit transition and observation system models that can be used with different reward functions to solve different planning problems. Under a mild set of rankness assumptions on the products of transition and observation matrices, we show how PSRs learn POMDP matrices up to a similarity transform, and this transform may be estimated via tensor decomposition methods. Our method learns observation matrices and transition matrices up to a partition of states, where the states in a single partition have the same observation distributions corresponding to actions whose transition matrices are full-rank. Our numerical experiments suggest that explicit observation and transition likelihoods can be leveraged to generate new plans for different goals and reward functions after the model has been learned. We also show that learning a POMDP beyond a partition of states is impossible from sequential data by constructing two POMDPs that agree on all observation distributions but differ in their transition dynamics.