  In gradient descent dynamics of neural networks, the top eigenvalue of the
Hessian of the loss (sharpness) displays a variety of robust phenomena
throughout training. This includes early time regimes where the sharpness may
decrease during early periods of training (sharpness reduction), and later time
behavior such as progressive sharpening and edge of stability. We demonstrate
that a simple $2$-layer linear network (UV model) trained on a single training
example exhibits all of the essential sharpness phenomenology observed in
real-world scenarios. By analyzing the structure of dynamical fixed points in
function space and the vector field of function updates, we uncover the
underlying mechanisms behind these sharpness trends. Our analysis reveals (i)
the mechanism behind early sharpness reduction and progressive sharpening, (ii)
the required conditions for edge of stability, and (iii) a period-doubling
route to chaos on the edge of stability manifold as learning rate is increased.
Finally, we demonstrate that various predictions from this simplified model
generalize to real-world scenarios and discuss its limitations.
