Groups Don't Move as One: A Bregman Clustering View of Linearized Group Influence

Influence functions approximate how removing a training example changes a quantity of interest, called the target function, such as a held-out loss. To estimate the influence of a group of examples, the standard practice is to sum the individual influences of its members. However, this sum does not capture how examples jointly affect the target: a pair of examples may be redundant or complementary, but the sum cannot distinguish these cases. We propose an interaction-aware influence function that characterizes how interactions between examples influence the target. By expanding the target to second order around the trained parameters, we obtain an estimator that augments the standard sum with a pairwise interaction term that captures the alignment between two examples' effects on the target. We empirically evaluate our estimator in two settings. First, on six dataset-model pairs spanning logistic regression, MLPs, and ResNet-9, our estimator tracks leave-group-out retraining substantially better than first-order influence across all settings. Second, when used as a greedy selection rule for instruction-tuning data on Llama-3.1-8B, it beats prior influence-based and representation-similarity baselines on five of seven downstream tasks, in a regime where standard influence-based selection underperforms random selection.

In this paper, we present a dual representation of the influence functions, whose computational complexity scales with dataset size rather than model size. Both analytically and experimentally, we show that this representation can be an efficient alternative to the original influence functions for estimating changes in parameters, model outputs and loss due to data point removal, when model size is large relative to dataset size, or when evaluating the original influence functions in parameter space is infeasible. The dual representation, however, is limited to linearizable models, which are models whose behavior can be approximated by their linearizations throughout training, and requires materializing a matrix, whose size grows with the product of model output dimension and dataset size.

Estimating the influence of training examples on model behavior is essential for data debugging, valuation, and attribution. Existing influence estimators often produce incompatible rankings, which are commonly ascribed to approximation error. We argue that a more fundamental source of disagreement is specification mismatch: influence depends on the behavior being attributed, the intervention applied to each training example, and the counterfactual training process that maps the intervention to a model response. These choices are especially important when the target behavior requires a tractable surrogate, such as query loss, a logit, or a margin. We formalize influence as a counterfactual estimand, distinguish specification mismatch across estimands from approximation error in estimating a fixed estimand, and organize representative estimators by their implied specifications. We further derive a local decomposition that exposes how behavior signals, training signals, and counterfactual parameter responses interact. Controlled experiments show that exact estimands under different specifications can induce different rankings, whereas approximation error grows as perturbations move farther from their linearization points. Experiments on noisy label detection and LLM attribution show that specification choices significantly affect attribution quality, especially for the choice of behavior surrogate. Behavior-aligned specifications can identify target-specific training examples obscured by default loss-based or similarity-based specifications. These results establish specification analysis as a necessary first step for interpreting and comparing data influence estimators.