The paper investigates the relationship between discrete gradient descent (GD) and its continuous-time gradient-flow counterpart in the context of ReLU neural networks. It shows that while GD states converge over a finite horizon, the exact discrete derivatives obtained via automatic differentiation do not necessarily match the derivative of the limiting flow, due to singular curvature at activation events. The authors provide a Stieltjes representation that separates continuous regional Hessians from atomic interface curvature, revealing rank-one discrepancies at activation jumps and demonstrating that even globally strongly convex residual-ReLU losses can exhibit large sensitivity ratios on certain initialization sets.
By Xiaoyang Li, Runni Zhou
Population gradient flow is a common tool for reasoning about how neural networks adapt, including after pretraining. We show that it can mispredict finite-batch stochastic gradient descent (SGD) qualitatively, and we trace the discrepancy to a specific mechanism.
arXiv:2606. 04476v1 Announce Type: new Abstract: In this paper, we study the gradient descent dynamics for jointly training both layers of a one-hidden-layer ReLU network to fit a linear target function.
By Berk Tinaz, Changzhi Xie, Mahdi Soltanolkotabi
arXiv:2610. 11475v1 Announce Type: new Abstract: Population gradient flow is a common tool for reasoning about how neural networks adapt, including after pretraining.
By Ruoyu Zhao, Mingxuan Zhang, Jianbo Dai, Jiaqi Wu, Chenyu Zhu, Tong Che
arXiv:2606. 18080v1 Announce Type: new Abstract: Gradient descent in deep learning may operate at the edge of stability (EoS), a regime in which the largest eigenvalue of the loss Hessian hovers near the stability threshold $2/\eta$, where $\eta$ is the learning rate.
By Pierre Marion
arXiv:2606. 27759v1 Announce Type: new Abstract: Training binary neural networks (BNNs) from scratch is dominated by the straight-through estimator (STE), whose forward/backward mismatch produces severe accuracy degradation as networks deepen.
By Evan Gibson Smith, Bashima Islam
We develop a finite-width geometric framework describing how learned feature geometries are organized, transported, and selectively aligned in deep neural networks. Incompatibility among weight-genera...
The paper analyzes Gradient Descent (GD) on a minimal binary classification problem with a two‑neuron ReLU network trained on two instances. It proves that GD converges to an optimal robustness margin—maximizing the distance between the decision boundary and training points—but does so at a very slow rate, specifically θ(1/ln(t)). Empirical simulations confirm this slow convergence across various initializations, and the authors provide a rigorous analysis of GD trajectories across different activation patterns to establish the first explicit lower bound on robustness convergence in a non‑linear model.
By Guy Smorodinsky, Sveta Gimpleson, Itay Safran
arXiv:2608.25631v1 Announce Type: cross
Abstract: Continuous-time Markov chains (CTMCs) provide the backbone for modeling discrete stochastic dynamics across applied, physical, and biological science...
By Jose M. G. Vilar, Leonor Saiz
The paper introduces the "lift" technique for training input‑convex neural networks, replacing the traditional non‑negative weight constraint enforced by projected gradient descent or a softplus map. By adding a learnable slack variable and an unconstrained network that processes a permutation‑invariant batch summary, the lift couples batch‑dependent latent weights to the gradient, increasing update variance and enabling faster escape from the softplus shoulder. Experiments show that when the softplus method stalls at the shoulder, the lift achieves tighter fits and reconstructs targets roughly three times faster, while both methods agree when the shoulder is rarely reached.
By Ali Siahkoohi
The paper introduces a finite‑width geometric framework that explains how learned feature geometries are organized, transported, and selectively aligned in deep neural networks. It quantifies incompatibilities among weight‑generated covariance, gates, and backward sensitivities using three families of commutators, and provides exact layerwise identities that decompose these commutators into sources such as downstream transport, adjacent‑layer imbalance, and nonlinear gate‑covariance interactions. The study demonstrates that spectral alignment is a layer‑ and scale‑dependent compatibility phenomenon governed by transport, interaction, cancellation, and possible damping, rather than a universal consequence of training.
By Kaj Nystr\"om
The paper investigates how hard‑ReLU training behaves when perturbations have a finite radius. It shows that the usual infinitesimal sensitivities are insufficient to predict the response at a chosen radius, and it characterizes the intermediate regime where the perturbation radius scales with the gradient‑descent step. The authors derive crossing indices, a uniform endpoint expansion for separated transverse events, and provide explicit remainder terms in contractive affine regions to certify finite candidate comparisons, supported by experiments on nonlinear networks.
By Xiaoyang Li, Runni Zhou, Xinghao Yan