Fisher information and the geometry of memorization in neural networks

Daniel T. Bernstein, Chase Goddard, Luca Di Carlo, Vudtiwat Ngampruetikorn, David J. Schwab

A fruitful approach towards understanding generalization in machine learning involves characterizing how models separate generalizable structure from idiosyncratic or even corrupted features of training data. Fisher information has emerged as a diagnostic tool for this separation, leveraging loss geometry to distinguish generalization from memorization. Here we investigate this connection in vision models trained on CIFAR-10 with controlled noise that can be learned only through memorization. We show that projecting model weights onto the top Fisher eigenvectors decouples general task performance from noisy sample memorization, and we investigate how this decoupling presents differentially across layers. We show that the same phenomena are present in small transformers trained to solve a synthetic in-context learning task, and are reproduced in closed form in a simple, analytically solvable linear model. Our results reveal that Fisher information captures aspects of the geometric structure underlying how neural networks allocate capacity between generalization and memorization.