Adaptive Step Sizes for Preconditioned Stochastic Gradient Descent

This paper proposes a novel approach to adaptive step sizes in stochastic gradient descent (SGD) by utilizing quantities that we have identified as numerically traceable -- the Lipschitz constant for gradients and a concept of the local variance in search directions. Our findings yield a nearly hyperparameter-free algorithm for stochastic optimization, which has provable convergence properties when applied to quadratic problems and exhibits truly problem adaptive behavior on classical image classification tasks. Our framework enables the potential inclusion of a preconditioner, thereby enabling the implementation of adaptive step sizes for stochastic second-order optimization methods.

Sensitivity-Based Layer Insertion for Residual and Feedforward Neural Networks

The training of neural networks requires tedious and often manual tuning of the network architecture. We propose a systematic method to insert new layers during the training process, which eliminates the need to choose a fixed network size before training. Our technique borrows techniques from constrained optimization and is based on first-order sensitivity information of the objective with respect to the virtual parameters that additional layers, if inserted, would offer. We consider fully connected feedforward networks with selected activation functions as well as residual neural networks. In numerical experiments, the proposed sensitivity-based layer insertion technique exhibits improved training decay, compared to not inserting the layer. Furthermore, the computational effort is reduced in comparison to inserting the layer from the beginning. The code is available at \url{https://github.com/LeonieKreis/layer_insertion_sensitivity_based}.

Frobenius-Type Norms and Inner Products of Matrices and Linear Maps with Applications to Neural Network Training

The Frobenius norm is a frequent choice of norm for matrices. In particular, the underlying Frobenius inner product is typically used to evaluate the gradient of an objective with respect to matrix variable, such as those occuring in the training of neural networks. We provide a broader view on the Frobenius norm and inner product for linear maps or matrices, and establish their dependence on inner products in the domain and co-domain spaces. This shows that the classical Frobenius norm is merely one special element of a family of more general Frobenius-type norms. The significant extra freedom furnished by this realization can be used, among other things, to precondition neural network training.

Introducing Thermodynamics-Informed Symbolic Regression -- A Tool for Thermodynamic Equations of State Development

Thermodynamic equations of state (EOS) are essential for many industries as well as in academia. Even leaving aside the expensive and extensive measurement campaigns required for the data acquisition, the development of EOS is an intensely time-consuming process, which does often still heavily rely on expert knowledge and iterative fine-tuning. To improve upon and accelerate the EOS development process, we introduce thermodynamics-informed symbolic regression (TiSR), a symbolic regression (SR) tool aimed at thermodynamic EOS modeling. TiSR is already a capable SR tool, which was used in the research of https://doi.org/10.1007/s10765-023-03197-z. It aims to combine an SR base with the extensions required to work with often strongly scattered experimental data, different residual pre- and post-processing options, and additional features required to consider thermodynamic EOS development. Although TiSR is not ready for end users yet, this paper is intended to report on its current state, showcase the progress, and discuss (distant and not so distant) future directions. TiSR is available at https://github.com/scoop-group/TiSR and can be cited as https://doi.org/10.5281/zenodo.8317547.

Time Regularization in Optimal Time Variable Learning

Recently, optimal time variable learning in deep neural networks (DNNs) was introduced in arXiv:2204.08528. In this manuscript we extend the concept by introducing a regularization term that directly relates to the time horizon in discrete dynamical systems. Furthermore, we propose an adaptive pruning approach for Residual Neural Networks (ResNets), which reduces network complexity without compromising expressiveness, while simultaneously decreasing training time. The results are illustrated by applying the proposed concepts to classification tasks on the well known MNIST and Fashion MNIST data sets. Our PyTorch code is available on https://github.com/frederikkoehne/time_variable_learning.