New Algorithm Provides Exact Data Shapley for Weighted KNN Regression

Data Shapley is a crucial method for determining the value of individual data points in a training set, particularly its k-nearest-neighbor (KNN) variant used in practical toolkits. While exact algorithms existed for unweighted KNN and weighted KNN classification, weighted KNN regression and soft-label prediction remained computationally challenging, previously requiring an exponential brute-force approach. This difficulty stemmed from the complex, coalition-dependent ratio involved in weighted regression predictions, which broke the additive structures leveraged by prior polynomial algorithms. A new breakthrough introduces the first pseudo-polynomial-time exact algorithm for weighted KNN-regression Data Shapley. This method, based on a counting dynamic program, has been rigorously verified against exhaustive enumeration. Additionally, the research provides a certified Fully Polynomial-Time Approximation Scheme (FPTAS) for continuous weights and targets, complete with machine-checkable error certificates. The findings also include a complexity landscape analysis and a weighted soft-label multi-class extension. An open-source, CPU-only library has been released, offering the first exact weighted-regression Data Shapley ground truth. While Monte-Carlo Data Shapley showed statistical equivalence for downstream mislabel detection, the exact method provides determinism and a certified error bound, proving invaluable for auditing and ensuring precise data point ranking.