Different gradient-based methods for optimizing overparameterized models can all achieve zero training error yet converge to distinctly different solutions inducing different generalization properties. We provide the first complete characterization of implicit optimization bias for p-norm normalized steepest descent (NSD) and momentum steepest descent (NMD) algorithms in multi-class linear classification with cross-entropy loss. Our key theoretical contribution is proving that these algorithms converge to solutions maximizing the margin with respect to the classifier matrix's p-norm, with established convergence rates. These results encompass important special cases including Spectral Descent and Muon, which we show converge to max-margin solutions with respect to the spectral norm. A key insight of our contribution is that the analysis of general entry-wise and Schatten p-norms can be reduced to the analysis of NSD/NMD with max-norm by exploiting a natural ordering property between all p-norms relative to the max-norm and its dual sumnorm. For the specific case of descent with respect to the max-norm, we further extend our analysis to include preconditioning, showing that Adam converges to the matrix's max-norm solution. Our results demonstrate that the multi-class linear setting, which is inherently richer than the binary counterpart, provides the most transparent framework for studying implicit biases of matrix-parameter optimization algorithms.

Adam has become one of the most favored optimizers in deep learning problems.

The properties of complexes with transuranium elements have long been the object of research in various fields of chemistry. However, their experimental study is complicated by their rarity, high cost and special conditions necessary for working with such elements, and the complexity of quantum chemical calculations does not allow their use for large systems. To overcome these problems, we used modern machine learning methods to create a novel neural network architecture that allows to use available experimental data on a number of elements and thus significantly improve the quality of the resulting models. We also described the applicability domain of the presented model and identified the molecular fragments that most influence the stability of the complexes.

The solution of linear inverse problems arising, for example, in signal and image processing is a challenging problem since the ill-conditioning amplifies the noise on the data. Recently introduced algorithms based on deep learning overwhelm the more traditional model-based approaches, but they typically suffer from instability with respect to data perturbation. In this paper, we theoretically analyze the trade-off between neural networks stability and accuracy in the solution of linear inverse problems. Moreover, we propose different supervised and unsupervised solutions to increase network stability that maintains good accuracy by inheriting, in the network training, regularization from a model-based iterative scheme. Extensive numerical experiments on image deblurring confirm the theoretical results and the effectiveness of the proposed deep learning-based solutions to stably solve noisy inverse problems.