From Optimal Actions to World Models: Identifiability of Transition Kernels in Discounted MDPs

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.

Many Markov decision processes (MDPs) in operations research have feasible actions that are state dependent and defined implicitly by various operational constraints. These features make it difficult to use standard deep reinforcement learning (DRL) algorithms, whose action interfaces typically assume either a fixed finite action catalog or a simple Euclidean space. Motivated by a Taylor expansion of the optimal action-value function, we propose Bellman--Taylor score decoding, a framework that moves policy learning to a Euclidean score space while enforcing feasibility through an action decoder. The induced latent-score MDP then can be optimized by standard DRL algorithms without differentiating through the decoder. We provide a performance guarantee showing that the optimality gap of this approach decomposes into a structural approximation error and an algorithmic learning error. Lastly, we apply this framework to a queueing network control problem, where the policy essentially learns a state-dependent index-based dispatching rule. Numerical experiments show near-optimal performance in small instances and considerable improvements over benchmarks in larger systems.

We study a Restart POMDP (Partially Observable Markov Decision Process) on a general Borel state space, where the controller either lets the hidden state evolve unobserved or restarts the system and observes the new state. Exploiting a sufficient-statistic representation consisting of the last observed state and the elapsed time since restart, we reduce the problem to a fully observed MDP. Under a natural one-step cost deterioration condition, we prove that optimal policies have a threshold structure in the elapsed time for both the discounted and total undiscounted cost criteria. When the state space is partially ordered and the kernel is stochastically monotone, we further show that the optimal threshold is nonincreasing in the state. For the average cost criterion, under additional assumptions of geometric ergodicity and domination of the transient gain, we establish analogous threshold results via the vanishing discount approach, after showing the uniform boundedness of the optimal thresholds and relative value functions.