Abstract: We systematically analyze optimization dynamics in deep neural networks (DNNs) trained with stochastic gradient descent (SGD) and study the effect of learning rate $\eta$, depth $d$, and width $w$ 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 (iv) a late time "edge of stability" regime. The early and intermediate regimes (i) and (ii) exhibit a rich phase diagram depending on $\eta \equiv c / \lambda_0^H$, $d$, and $w$. We identify several critical values of $c$, which separate qualitatively distinct phenomena in the early time dynamics of training loss and sharpness. Notably, we discover the opening up of a "sharpness reduction" phase, where sharpness decreases at early times, as $d$ and $1/w$ are increased.