12.02.2024 15:00 Alexandru Craciun (LMU):
On the Stability of Gradient Descent for Large Learning RateMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

There currently is a significant interest in understanding the Edge of Stability (EoS) phenomenon, which has been observed in neural networks training, characterized by a non-monotonic decrease of the loss function over epochs, while the sharpness of the loss (spectral norm of the Hessian) progressively approaches and stabilizes around 2/(learning rate). Reasons for the existence of EoS when training using gradient descent have recently been proposed—a lack of flat minima near the gradient descent trajectory together with the presence of compact forward-invariant sets. In this paper, we show that linear neural networks with a quadratic loss function satisfy the first assumption and also a necessary condition for the second assumption. More precisely, we prove that the gradient descent map is non-singular, the set of global minimizers of the loss function forms a smooth manifold, and the stable minima form a bounded subset in parameter space. Additionally, we prove that if the step-size is too big, then the set of initializations from which gradient descent converges to a critical point has measure zero.