Making Coherence Out of Nothing At All: Measuring the Evolution of Gradient Alignment

We propose a new metric (m-coherence) to experimentally study the alignment of per-example gradients during training. Intuitively, given a sample of size m, m-coherence is the number of examples in the sample that benefit from a small step along the gradient of any one example on average. Using m-coherence, we study the evolution of alignment of per-example gradients in ResNet and Inception models on ImageNet and several variants with label noise, particularly from the perspective of the recently proposed Coherent Gradients (CG) theory that provides a simple, unified explanation for memorization and generalization [Chatterjee, ICLR 20]. Although we have several interesting takeaways, our most surprising result concerns memorization. Naïvely, one might expect that when training with completely random labels, each example is fitted independently, and so m-coherence should be close to 1. However, this is not the case: m-coherence reaches much higher values during training (100s), indicating that over-parameterized neural networks find common patterns even in scenarios where generalization is not possible. A detailed analysis of this phenomenon provides both a deeper confirmation of CG, but at the same point puts into sharp relief what is missing from the theory in order to provide a complete explanation of generalization in neural networks. Generalization in neural networks trained with stochastic gradient descent (SGD) is not wellunderstood. For example, the generalization gap, i.e., the difference between training and test error depends critically on the dataset and we do not understand how. This is most clearly seen when we fix all aspects of training (e.g.