Abstract:Can a logged dataset visit every hidden state frequently and still be exponentially uninformative about a target policy's value? We show that it can when the logger depends on history. For every horizon $H \ge 3$, we construct two POMDPs with at most two latent states per stage, three actions, and a common logger with three memory states. Action coverage, belief coverage, and two behavior-marginal outcome-revealing conditions all have constants independent of $H$. Nevertheless, evaluating a known deterministic target policy to accuracy $1/8$ requires $\Theta((3/2)^H \log(1/\delta))$ logged episodes at confidence $1-\delta$, for $0 < \delta \le 1/4$, even when both candidate models are known. The mechanism is simple: a reset erases the unknown transition that determines the target value. We characterize the resulting statistical experiment exactly and obtain a matching optimal estimator. A directed two-lane gridworld realizes the construction, and trajectory simulations agree with its finite-sample prediction. The result establishes intractability for the history-dependent-logging, model-based case posed by Zhang and Jiang (2025, arXiv:2503.01134), under their behavior-marginal definition of revealing.
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
