7 papers
ODE approximation for the Adam algorithm: General and overparametrized setting
Steffen Dereich, Arnulf Jentzen, Sebastian Kassing
The Adam optimizer is currently presumably the most popular optimization method in deep learning. In this article we develop an ODE based method to study the Adam optimizer in a fa…
Asymptotic stability properties and a priori bounds for Adam and other gradient descent optimization methods
Steffen Dereich, Robin Graeber, Arnulf Jentzen +1
Gradient descent (GD) based optimization methods are these days the standard tools to train deep neural networks in artificial intelligence systems. In optimization procedures in d…
SAD Neural Networks: Divergent Gradient Flows and Asymptotic Optimality via o-minimal Structures
Julian Kranz, Davide Gallon, Steffen Dereich +1
We study gradient flows for loss landscapes of fully connected feedforward neural networks with commonly used continuously differentiable activation functions such as the logistic,…
Sharp higher order convergence rates for the Adam optimizer
Steffen Dereich, Arnulf Jentzen, Adrian Riekert
Gradient descent based optimization methods are the methods of choice to train deep neural networks in machine learning. Beyond the standard gradient descent method, also suitable…
In almost all shallow analytic neural network optimization landscapes, efficient minimizers have strongly convex neighborhoods
Felix Benning, Steffen Dereich
Whether or not a local minimum of a cost function has a strongly convex neighborhood greatly influences the asymptotic convergence rate of optimizers. In this article, we rigorousl…
Mathematical analysis of the gradients in deep learning
Steffen Dereich, Thang Do, Arnulf Jentzen +1
Deep learning algorithms -- typically consisting of a class of deep artificial neural networks (ANNs) trained by a stochastic gradient descent (SGD) optimization method -- are nowa…