New algorithm finds most influential subsets efficiently, solving combinatorial bottleneck

Identifying the most influential set (MIS) — the subset of k data points whose removal most alters a target statistic — has long been computationally infeasible due to the combinatorial explosion of checking all n-choose-k subsets. In a new paper accepted at ICML 2026, researchers Lucas D. Konrad and Nikolas Kuschnig from the statistics and econometrics community crack this bottleneck. They prove that for estimands with linear-fractional leave-set-out effects, the MIS problem reduces elegantly to a one-parameter sequence of top-k selection problems. This allows them to apply Dinkelbach's method, a classic fractional programming algorithm, yielding an iterative procedure that costs O(n) per iteration and terminates finitely. For fixed residualized inputs (e.g., the oracle-residualized partial linear model), the algorithm returns a globally optimal set. Even when nuisance functions must be estimated, they provide theoretical guarantees: uniform denominator and generated-score stability imply approximation to the first-order oracle orthogonal-score objective, and exact set recovery follows under a separation condition.

Simulations and real-data applications demonstrate that the method recovers exact MIS that were previously completely inaccessible by brute force or heuristic approximations. The implications are broad: the technique can be applied to any problem where one wants to identify which data points or features exert the most influence on a model's output, from robust regression to causal effect estimation. In econometrics, it enables precise identification of influential observations in large panels; in machine learning, it can guide data subset selection for efficient training or adversarial example detection. The paper's key insight — reformulating a combinatorial nightmare into a tractable sequence of linear-time problems — opens up a new class of influence analysis that was previously only dreamt of. The work is a significant step toward scalable, exact influence diagnostics in modern high-dimensional statistics.