Add to Chrome
Add to Firefox
Add to EdgeMaissam Barkeshli
Universal Sharpness Dynamics in Neural Network Training: Fixed Point Analysis, Edge of Stability, and Route to Chaos
Nov 03, 2023
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.
Phase diagram of training dynamics in deep neural networks: effect of learning rate, depth, and width
Feb 23, 2023
We systematically analyze optimization dynamics in deep neural networks (DNNs) trained with stochastic gradient descent (SGD) over long time scales and study the effect of learning rate, depth, and width of the neural network. By analyzing the maximum eigenvalue $\lambda^H_t$ of the Hessian of the loss, which is a measure of sharpness of the loss landscape, we find that the dynamics can show four distinct regimes: (i) an early time transient regime, (ii) an intermediate saturation regime, (iii) a progressive sharpening regime, and finally (iv) a late time ``edge of stability" regime. The early and intermediate regimes (i) and (ii) exhibit a rich phase diagram depending on learning rate $\eta \equiv c/\lambda^H_0$, depth $d$, and width $w$. We identify several critical values of $c$ which separate qualitatively distinct phenomena in the early time dynamics of training loss and sharpness, and extract their dependence on $d/w$. Our results have implications for how to scale the learning rate with DNN depth and width in order to remain in the same phase of learning.