Only Learn What Changes the Decision: Exact Quotient Geometry for Active Acquisition

Active learning for decision making often estimates more information than is actually needed: it may try to recover a latent parameter even when the final decision depends only on a few observable directions. We study when and how this unnecessary identification can be avoided. Our key insight is to quotient out parameter directions that cannot affect any decision comparison, yielding a decision-observability quotient that contains exactly the information needed for decision making. We show that this reduction is valid if and only if the query space spans all decision-relevant differences, and that under this condition it preserves regret, acceptable answers, and the information-theoretic complexity of the problem. For Gaussian observations, we characterize the exact fixed-confidence sample complexity through a quotient characteristic time and give a practical adaptive algorithm that attains this rate under explicit regularity conditions. We further show that identifying the full parameter can be substantially more expensive: on an explicit family, coordinate-wise identification requires exactly $n$ times the characteristic time of decision identification, with stopping-time ratios between $n$ and $4n$ under the self-normalized ellipsoid boundary. Experiments reveal a stark practical gap. On synthetic instances, stopping-time ratios follow the predicted $n$-to-$4n$ separation, while on 18 public rating panels the coordinate-wise test fails to stop in all $64,800$ runs. On the same runs and confidence events, the quotient test stops $18,993$ times and makes no incorrect recommendation on held-out raters.