arXiv:2210. 16286v2 Announce Type: replace Abstract: To understand the training dynamics of neural networks, prior studies have considered the mean-field limit of two-layer neural networks as the width tends to infinity, establishing theoretical guarantees for its convergence under gradient flow training as well as approximation and generalization capabilities.
The paper introduces a pointwise generalization theory for fully connected deep neural networks, using a pointwise Riemannian Dimension derived from eigenvalues of learned feature representations across layers. This framework provides hypothesis-dependent, representation-aware generalization bounds that are significantly tighter than traditional size- or norm-based approaches, both theoretically and experimentally. The authors analytically identify structural properties that explain deep networks’ tractability and empirically show that the pointwise Riemannian Dimension captures feature compression, over‑parameterization effects, and optimizer bias.
arXiv:2607. 23397v1 Announce Type: new Abstract: Hierarchical neural networks are widely used in artificial intelligence, yet their mathematical properties remain incompletely understood.
arXiv:2605. 10775v2 Announce Type: replace-cross Abstract: A surprising phenomenon in the training of neural networks is the ability of gradient descent to find global minimizers of the training loss despite its non-convexity.
arXiv:2606. 10324v1 Announce Type: new Abstract: The analogy between deep neural network forward passes and renormalization group (RG) flows has been repeatedly noted in the literature, but existing treatments remain qualitative: depth is described as a coarse-graining scale, attention is likened to a partition function, and representations are said to flow toward fixed points.
arXiv:2510. 24616v4 Announce Type: replace-cross Abstract: For four decades statistical physics has been providing a framework to analyse neural networks.
arXiv:2607. 03613v1 Announce Type: new Abstract: We study the implicit bias of noisy stochastic gradient descent in training wide two-layer ReLU networks for multivariate regression.
arXiv:2606. 05335v1 Announce Type: new Abstract: Theoretical studies of machine learning models commonly consider different limiting regimes in which the learning dynamics of gradient descent becomes theoretically tractable.
The paper introduces Sliding Networks, a family of residual neural network structures that aim to preserve both feature learning and feature diversity while scaling width and depth. By maintaining a limiting ratio of residual width to the product of hidden width and depth, the authors derive a joint scaling recipe that converges to a stochastic differential equation (SDE) limit with locally maximal feature learning. Experiments on residual-MLP and language models demonstrate that this scaling approach allows redistribution of computational resources between width and depth, though performance varies when parameters are regrouped at a fixed count.
arXiv:2610.30729v2 Announce Type: replace-cross Abstract: Recent work established that the infinite-width and infinite-depth limits of residual networks commute at initialization when residual branch...
arXiv:2607. 07884v1 Announce Type: new Abstract: In this short note we consider the gradient descent dynamics of deep scalar linear networks, $f(x) = \prod_{l=1}^L w_l x$, which enjoy exact time-course solutions for any integer depth.
arXiv:2507. 05164v2 Announce Type: replace-cross Abstract: In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms.