The Price of Hidden Assignments: Calibration and Minimax Rates in Pooled CATE Learning

How accurately can the conditional average treatment effect (CATE) be estimated from noisy pooled treatment reports? We study randomized experiments with $G$ fixed-size pools, linked individual covariates and outcomes, hidden treatment assignments, known assignment probabilities, and a common unknown nondifferential reporting channel. Validation of $m$ uniformly selected pools reveals their true OR states: whether any member received treatment. Under fixed overlap and channel bounds, pooling with a known channel preserves the usual CATE minimax rate. For an unknown channel, assuming smoothness only of the CATE, we establish matching classwise bounds for expected integrated squared risk, of order $G^{-\alpha }+\{G(1+m+G{\sigma }_0){\}}^{-1/2}$, where $\alpha =2\beta /(2\beta +d)$, $\beta \in (0,1]$ is CATE Hölder smoothness, $d$ is covariate dimension, and ${\sigma }_0$ is the class-specified propensity-variance lower bound.The two terms quantify the regression error and the additional calibration cost.Validation and propensity heterogeneity jointly determine the calibration information scale $1+m+G{\sigma }_0$, yielding distinct validation scales for improving on the no-validation minimax rate and recovering the known-channel rate. The lower bound retains treatment-effect information in complete validation records. An upper-bound construction nevertheless attains the same minimax rate using only covariates, true OR states, and noisy reports from a reserved validation subset, without its outcomes, and estimates CATE from the remaining pools. Finite-budget diagnostics show that qualification checks and clipping can suppress calibration gains in this construction. Across all 90 prespecified settings, its mean risk exceeded that of the zero-effect estimator.