We consider an inexact variant of the popular Riemannian trust-region algorithm for structured big-data minimization problems. The proposed algorithm approximates the gradient and the Hessian in addition to the solution of a trust-region sub-problem. Addressing large-scale finite-sum problems, we specifically propose sub-sampled algorithms with a fixed bound on sub-sampled Hessian and gradient sizes, where the gradient and Hessian are computed by a random sampling technique. Numerical evaluations demonstrate that the proposed algorithms outperform state-of-the-art Riemannian deterministic and stochastic gradient algorithms across different applications.

Let (x0,y0) in M beanaugmentedg.

Recurrent neural networks (RNNs) notoriously struggle to learn long-term memories, primarily due to vanishing and exploding gradients.

SAM can improve generalization in deep learning. Seemingly similar methods like weight noise and gradient penalties often fail to provide such benefits. We investigate this inconsistency and reveal its connection to the the structure of the Hessian of the loss.

CNNs, MLPs, and quadratic problems, and find that SGD can perform on par with Adam on problems without block heterogeneity, but performs worse than Adam when the heterogeneity exists.

Recently, there has been growing evidence that if the width and depth of a neural network are scaled toward the so-called rich feature learning limit ( µ P and its depth extension), then some hyperparameters -- such as the learning rate -- exhibit transfer from small to very large models. From an optimization perspective, this phenomenon is puzzling, as it implies that the loss landscape is consistently similar across very different model sizes. In this work, we study the landscape through the lens of the loss Hessian, with a focus on its largest eigenvalue (i.e. the sharpness), and find that certain spectral properties under µ P are largely independent of the size of the network, and remain consistent as training progresses. We name this property Super Consistency of the landscape. On the other hand, we show that in the Neural Tangent Kernel (NTK) and other scaling regimes, the sharpness exhibits very different dynamics at different scales.