Exponential Hardness of Off-Policy Evaluation under History-Dependent Logging
Researchers prove off-policy evaluation under history-dependent logging requires exponentially many episodes, resolving a hardness question for model-based POMDP evaluation.
The paper constructs POMDPs with at most two latent states per stage, three actions, and a three-memory-state logger where evaluating a known deterministic target policy to accuracy 1/8 requires Θ((3/2)^H log(1/δ)) episodes for any horizon H≥3. Coverage and outcome-revealing conditions hold with constants independent of H, yet a reset erases the unknown transition that determines the target value. The authors characterize the resulting statistical experiment exactly, derive a matching optimal estimator, and validate predictions on a two-lane gridworld. This settles the history-dependent-logging, model-based case posed by Zhang and Jiang (arXiv:2503.01134).
- Evaluating a known deterministic policy needs Θ((3/2)^H log(1/δ)) episodes even with full coverage
- A reset transition erases the unknown transition determining the target policy's value
- An optimal estimator is derived and matched to the lower bound
- Construction validated by trajectory simulations on a two-lane gridworld
Full article168 words · extracted from arxiv.org · click to collapse
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 $Θ((3/2)^H \log(1/δ))$ logged episodes at confidence $1-δ$, for $0 < δ\le 1/4$, 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.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.19135