Revealing After Overwriting: An Exponential POMDP OPE Lower Bound under History-Dependent Logging

Youyu Luo, Pengzhan Zhou, Zhida Qin, Jia Wang, Zuotao Fu +2

Multi-step revealing can make off-policy evaluation tractable under memoryless logging. With history-dependent logging, state decodability and target-relevant evidence can separate. For every horizon $H\ge3$, we construct two exactly realizable POMDPs with four actions, at most four states per layer, a known logger, and a memoryless target. Action overlap, history coverage, and observation-only revealing remain bounded independently of $H$, yet the target values differ by $1/2$ and the KL divergence between the logged laws is $Θ(4^{-(H-1)})$, forcing exponential sample complexity. Logger memory makes states distinguishable, while reset erases the model-distinguishing evidence preserved by the target. A separate construction retains this barrier with common, known observation-only revealing operators. Under action and history coverage, we give a finite-class OPE guarantee using common observable value representations that remain valid at every history. The sample bound depends polynomially on their second-moment cost. In the common-operator construction, the same value direction has constant marginal decoding cost but exponential history-conditioned cost. Finally, on a fixed…