Exact and Certified Data Shapley for Weighted k-Nearest-Neighbor Regression and Soft-Label Prediction
Data Shapley answers which training points are worth what, and its nearest-neighbor specialization is the version actually deployed, shipped by toolkits such as pyDVL and OpenDataVal. Exact algorithms exist for unweighted nearest-neighbor classification and regression, and recently for weighted classification; weighted regression and soft-label prediction have resisted, the only exact method being enumeration exponential in the neighborhood size. The obstruction, in the prior authors' own words, is that the weighted regression prediction is a ratio of two coalition-dependent weighted sums: its normalization denominator blocks the additive and threshold routes, and leaves the counting route exponential in the target resolution. We close this gap with a counting dynamic program over the joint integer state of accumulated weight and weighted target, a minimal sufficient statistic for the ratio; it is exact, pseudo-polynomial, and matched exhaustive enumeration with zero mismatch. We add a certified approximation scheme for continuous weights and targets carrying a machine-checkable per-value certificate, a complexity landscape delimiting the exact problem, and a soft-label extension. We release an open-source, CPU-only library and the first exact weighted-regression ground truth. On mislabel detection our exact values are statistically equivalent to Monte-Carlo Data Shapley; exactness instead buys determinism, a certified bound, and an auditing reference, and it puts a measured price on approximation.