On the geometry of generalization and memorization in deep neural networks

This part of the gradient behaves similarly for permuted and unpermuted examples. In Eq. 25 we see that the contribution to the label dependent part of the gradient from permuted examples vanishes for large datasets, while the contribution from unpermuted examples does not provided the cross correlation between input features and labels is nonzero. This suggests that with small weight initialization, the gradient descent dynamics initially ignores the labels of permuted examples. Figure A.1 shows a breakdown of how the two components of the gradient computed on both unpermuted and permuted examples evolve over the course of training for the different layers of the VGG16 model trained on CIFAR-100. We see that the label dependent part behaves qualitatively differently for the unpermuted examples than for the permuted examples, as the permuted examples give close to zero contribution early in training in agreement with Eq. 25. The label independent part of the gradient shows similar trends between unpermuted and permuted examples, though in the final epochs, the unpermuted examples have a slightly larger label independent gradient indicating slightly greater model confidence on these examples. As the label dependent and label independent parts of the gradient have differing signs, they compete with each other and cancel when the loss is minimized, but are not independently zero and in fact grow during training. The slightly larger label independent gradient for unpermuted examples is balanced by a corresponding slightly larger label dependent gradient at the end of training.