Functional Ambiguity Reduction: Minimax Acquisition under Observational Equivalence

Finite observations can leave distinct latent populations observationally indistinguishable while those populations imply different values of a target functional. We study how this target-relevant ambiguity limits recovery and how it should guide subsequent acquisition. We first characterize recoverability under a finite response design: the largest functional ambiguity is exactly twice the best uniform approximation error of the target by the observed response coordinates, and this approximation error equals the minimax recovery error of any estimator based only on the observed response vector. This characterization yields Functional Ambiguity Reduction (FAR), which selects queries according to the target disagreement that can remain among observationally compatible populations. FAR is one-step minimax-optimal for worst-case absolute target recovery under exact responses and extends to finite response batches and learned response models. Across five public datasets, FAR shows its clearest gains on CIFAR-10H and Duolingo-FR, reducing MAE on Duolingo-FR from \(0.1330\) for the strongest baseline to \(0.1263\). We further apply FAR to budgeted LLM evaluation, where five additional queries reduce score-estimation MAE by $7.0\%$ on Qwen3-4B and $25.7\%$ on Phi-3.5-mini relative to the strongest baseline for each model. These results connect finite-design identifiability directly to adaptive acquisition for population-level functional recovery. Code is available at https://anonymous.4open.science/r/FAR-D5B5/.