Self-Influence Governs Generalization: A von Mises Expansion Approach

We study generalization through the lens of self-influence: how strongly a training example affects the learned predictor and its associated losses. Using a second-order von Mises expansion of the loss functional, we derive leading-order influence-based estimators for the expected generalization gap. For i.i.d. samples, the expected gap is governed by average self-influence. For hierarchical sampling, additional same-parent cross-influence terms appear, providing a mechanism by which augmentations and correlated views affect generalization. Empirically, the resulting estimators accurately track generalization in well-generalizing regimes and become conservative near memorization, providing a diagnostic signal for the breakdown of the leading-order approximation.