12 papers
On MUON optimization: From non-convergence to an error analysis with Polar Express and the Newton-Schulz polynomial from implementations
Thang Do, Steffen Dereich, Arnulf Jentzen
Stochastic gradient descent (SGD) optimization methods are the standard instruments for the training of deep neural networks (DNNs). In many relevant artificial intelligence (AI) s…
Learning rate adaptive stochastic gradient descent optimization methods: numerical simulations for deep learning methods for partial differential equations and convergence analyses
Steffen Dereich, Arnulf Jentzen, Adrian Riekert
The standard stochastic gradient descent (SGD) optimization method, as well as adaptive methods such as the Adam optimizer fail to converge if the learning rates do not converge to…
Adam symmetry theorem: characterization of the convergence of the stochastic Adam optimizer
Steffen Dereich, Thang Do, Arnulf Jentzen +1
Beside the standard stochastic gradient descent (SGD) method, the Adam optimizer due to Kingma & Ba (2014) is currently probably the best-known optimization method for the training…
Central limit theorem for the averaged Adam optimizer
Steffen Dereich, Arnulf Jentzen
In this article, we analyse convergence of the averaged Adam optimizer to an attracting zero of the Adam vector field. We provide a central limit theorem that, in particular, quant…
Uniform a priori bounds and error analysis for the Adam stochastic gradient descent optimization method
Steffen Dereich, Thang Do, Arnulf Jentzen
The adaptive moment estimation (Adam) optimizer proposed by Kingma & Ba (2014) is presumably the most popular stochastic gradient descent (SGD) optimization method for the training…
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,…