  We identify a new phenomenon in neural network optimization which arises from
the interaction of depth and a particular heavy-tailed structure in natural
data. Our result offers intuitive explanations for several previously reported
observations about network training dynamics. In particular, it implies a
conceptually new cause for progressive sharpening and the edge of stability; we
also highlight connections to other concepts in optimization and generalization
including grokking, simplicity bias, and Sharpness-Aware Minimization.
  Experimentally, we demonstrate the significant influence of paired groups of
outliers in the training data with strong opposing signals: consistent, large
magnitude features which dominate the network output throughout training and
provide gradients which point in opposite directions. Due to these outliers,
early optimization enters a narrow valley which carefully balances the opposing
groups; subsequent sharpening causes their loss to rise rapidly, oscillating
between high on one group and then the other, until the overall loss spikes. We
describe how to identify these groups, explore what sets them apart, and
carefully study their effect on the network's optimization and behavior. We
complement these experiments with a mechanistic explanation on a toy example of
opposing signals and a theoretical analysis of a two-layer linear network on a
simple model. Our finding enables new qualitative predictions of training
behavior which we confirm experimentally. It also provides a new lens through
which to study and improve modern training practices for stochastic
optimization, which we highlight via a case study of Adam versus SGD.
