The paper presents convergence guarantees for Stochastic Gradient Descent (SGD) applied to the multiclass cross-entropy loss in both linear classifiers and two-layer neural networks. It shows that SGD dynamics can oscillate between an edge-of-stability regime dominated by curvature-driven oscillations and a stable regime where the expected loss decreases at a controlled rate. Despite these oscillations, the authors prove that SGD stabilizes the dynamics, ensuring that iterates return to stability within a fixed number of iterations and enabling convergence in the best-iterate sense even with large learning rates.
arXiv:2606. 30930v1 Announce Type: cross Abstract: Modern deep learning has been shown to operate at the edge of stability, routinely using learning rates far larger than those justified by classical optimization theory.
arXiv:2610.28561v1 Announce Type: new Abstract: Stochastic gradient descent (SGD) is the cornerstone of modern deep learning, yet its theoretical convergence properties remain poorly understood in no...
arXiv:2508. 03105v3 Announce Type: replace Abstract: We analyze the convergence behavior of stochastic gradient descent with momentum (SGDM) under dynamic learning-rate and batch-size schedules by introducing a novel and simpler Lyapunov function.
arXiv:2301. 06308v2 Announce Type: replace-cross Abstract: Sharpness-aware minimization (SAM) is a training method that seeks to find flat minima in deep learning, resulting in state-of-the-art performance across various domains.
arXiv:2606. 06772v2 Announce Type: replace-cross Abstract: Characterizing the optimization dynamics and statistical performance of over-parameterized deep neural networks (DNNs) remains a central challenge in understanding the remarkable success of deep learning.
arXiv:2406. 14340v2 Announce Type: replace-cross Abstract: The standard stochastic gradient descent (SGD) optimization method, as well as adaptive methods such as the Adam optimizer fail to converge if the learning rates do not converge to zero (particularly, in the situation of constant learning rates).
Deep neural networks can exhibit periodic loss spikes known as the "Slingshot Mechanism" during long‑term training. This paper shows that such spikes arise from finite‑precision floating‑point arithmetic: when the correct‑class logit becomes much larger than the others, the gradient for that class is rounded to zero while others remain nonzero, breaking the zero‑sum constraint and creating a positive feedback loop that inflates feature and classifier norms. The authors term this Numerical Feature Inflation (NFI) and demonstrate that it explains the rapid norm growth, reappearance of gradients, and loss spikes, while also showing that partial absorption can still cause hidden norm growth without visible spikes.
arXiv:2510. 03164v2 Announce Type: replace Abstract: Learning rate warm-up -- increasing the learning rate at the beginning of training -- has become a ubiquitous heuristic in modern deep learning, yet its theoretical foundations remain poorly understood.
arXiv:2606. 00340v1 Announce Type: new Abstract: We study optimal learning-rate selection in two-layer and three-layer linear neural networks trained to learn linear target functions.
The paper presents an exact, information‑theoretic analysis of stochastic gradient descent (SGD) and its variants, showing that a preconditioned SGD step corresponds to a posterior‑mean update in a Gaussian Bayes model. It decomposes one‑step regret into an intrinsic‑time cost and a change in comparator information, extending this split to an identity for the objective itself. The framework links convex convergence, saddle‑point escape, flatness‑generalization trade‑offs, learning‑rate schedules, adaptive optimizers, and various SGD variants, and it is validated on synthetic and real training runs, revealing how different optimizers achieve the same training loss through distinct step characteristics.
arXiv:2606. 06772v1 Announce Type: cross Abstract: Understanding the generalization performance of over-parameterized neural networks has become a central topic in deep learning theory.
ExpTest is an autonomous learning‑rate controller that uses the training loss curve as an online signal to perform sequential statistical tests on theoretically motivated windows, detecting convergent behavior and triggering learning‑rate reductions. It combines a covariance‑based initial learning‑rate estimate, curvature‑motivated window sizing, and a two‑phase test‑driven decay, relying on the approximately exponential decay predicted under linearized network dynamics. Experiments on regression, classification, forecasting, and natural‑language tasks across various architectures show that ExpTest achieves competitive performance compared to hand‑tuned SGD baselines and recent learning‑rate‑free methods, without requiring manual initial learning‑rate selection or predefined scheduling.