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

Influence functions provide an efficient alternative to retraining for assessing the effects of removing or reweighting training data. For groups, however, the standard first-order estimate reduces individual contributions to a sum that can conceal substantial differences among members, including opposing effects that cancel. We establish a Bregman centroid formulation of the linearized group influence objective, interpreting this aggregate as a single representative and using Bregman information (BI) to quantify the variation it omits. Extending this formulation to multiple clusters reveals distinct contributions whose sum recovers the original estimate. We further propose an estimator that jointly optimizes the weights of these cluster contributions, rather than simply summing them. Experiments uncover cluster-specific influence patterns and a positive association between target-specific BI and approximation error. The proposed estimator achieves higher $R^2$ than standard group influence in predicting target changes under the nonlinear proximal Bregman response function. Simplified clustering and correction variants reduce computational cost. The centroid view thus explains what additive group influence leaves out and provides a basis for using that information to improve estimation.