Robust Active Statistical Inference: Separate Geometry from Uncertainty

Active statistical inference reduces labeling costs by combining inexpensive proxy labels with selectively queried ground-truth labels for a downstream inferential target. The sampling policy determines which labels to acquire based on the estimated error of the proxy labels. However, such policies can be sensitive to misspecification. When the error of proxy labels is severely underestimated in a region where it is in fact large, the resulting query probabilities can be small, substantially inflating the estimator's variance and confidence-interval width. To mitigate this issue, we introduce Residual-robust Active Statistical Inference (RASI), which exploits a structural decomposition of the optimal sampling policy into a target-specific loading determined by the geometry of the estimand and an unobserved residual-risk component approximated by an uncertainty proxy. RASI preserves the former and robustifies only the latter. We formulate policy design as a distributionally robust optimization problem over plausible residual risks and derive minimax sampling policies over an ambiguity set. Our analysis shows that mixing with uniform sampling, a common robustness strategy in active statistical inference, is generally not the appropriate robust fallback because it discards target-specific geometry. We extend RASI to smooth functions of parameters defined by outcome-affine estimating equations, covering linear and logistic regression coefficients and log-odds ratios. Across synthetic and real-data experiments, RASI protects against misspecification of the uncertainty proxy while retaining the efficiency gains of active sampling when the proxy is informative.