Label-Efficient Group Risk Control for Selective Prediction

Certifying selective-prediction risk separately for each group requires labeled evidence from every group. Under a fixed label budget, proportional sampling can leave a small group with an overly conservative certified threshold, forcing abstention even when more of its predictions are safe to accept. We call this coverage loss certification disparity and cast it as a sequential label-allocation problem. Our method, , combines prediction-powered loss corrections with simultaneous betting confidence sequences, allowing labels to follow groups that still need evidence. We prove simultaneous, anytime-valid group-wise risk control under predictable allocation, data-dependent stopping and bounded importance sampling. For a fixed-configuration empirical-Bernstein variant, the certification cost separates into residual-variance and bounded-range terms, and passive and active lower bounds identify the corresponding variance orders. For the EB variant, round-robin stopping certifies targets within the sum of group requirements with high probability under the stated conditions; fixed allocation is bottlenecked by the largest requirement per share. On a controlled four-group benchmark, round-robin allocation raises worst-group coverage from $1.6\%$ to $61.7\%$ over proportional allocation while also increasing population coverage, and auxiliary information adds $2.54$ percentage points at matched egalitarian allocation. With stopping, round-robin allocation reduces the median labels needed to reach common coverage targets by $37.1\%$ relative to fixed uniform allocation.