Exponential Hardness of Off-Policy Evaluation under History-Dependent Logging

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.

Machine Learning (cs.LG)

arXiv:2609.19135 [cs.LG]

(or arXiv:2609.19135v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.19135 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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