Understanding the training dynamics of deep neural networks is challenging due to their high-dimensional nature and intricate loss landscapes. Recent studies have revealed that, along the training trajectory, the gradient approximately aligns with a low-rank top eigenspace of the training loss Hessian, referred to as the dominant subspace. Given this alignment, this paper explores whether neural networks can be trained within the dominant subspace, which, if feasible, could lead to more efficient training methods. Our primary observation is that when the SGD update is projected onto the dominant subspace, the training loss does not decrease further. This suggests that the observed alignment between the gradient and the dominant subspace is spurious. Surprisingly, projecting out the dominant subspace proves to be just as effective as the original update, despite removing the majority of the original update component. We observe similar behavior across practical setups, including the large learning rate regime (also known as Edge of Stability), Sharpness-Aware Minimization, momentum, and adaptive optimizers. We discuss the main causes and implications of this spurious alignment, shedding light on the dynamics of neural network training.

When training deep neural networks with gradient descent, sharpness often increases—a phenomenon known as progressive sharpening—before saturating at the edge of stability. Although commonly observed in practice, the underlying mechanisms behind progressive sharpening remain poorly understood. In this work, we study this phenomenon using a minimalist model: a deep linear network with a single neuron per layer. We show that this simple model effectively captures the sharpness dynamics observed in recent empirical studies, offering a simple testbed to better understand neural network training. Moreover, we theoretically analyze how dataset properties, network depth, stochasticity of optimizers, and step size affect the degree of progressive sharpening in the minimalist model. We then empirically demonstrate how these theoretical insights extend to practical scenarios. This study offers a deeper understanding of sharpness dynamics in neural network training, highlighting the interplay between depth, training data, and optimizers.

As both model and dataset sizes continue to scale rapidly, conventional pretraining strategies with fixed compute budgets—such as cosine learning rate schedules—are increasingly inadequate for large-scale training. Recent alternatives, including warmup-stable-decay (WSD) schedules and weight averaging, offer greater flexibility. However, WSD relies on explicit decay phases to track progress, while weight averaging addresses this limitation at the cost of additional memory. In search of a more principled and scalable alternative, we revisit the Schedule-Free (SF) method [Defazio et al., 2024], which has shown strong empirical performance across diverse settings. We show that SF-AdamW effectively navigates the "river" structure of the loss landscape without decay phases or auxiliary averaging, making it particularly suitable for continuously scaling training workloads. To understand this behavior, we conduct a theoretical and empirical analysis of SF dynamics, revealing that it implicitly performs weight averaging without memory overhead. Guided by this analysis, we propose a refined variant of SF that improves robustness to momentum and performs better under large batch sizes, addressing key limitations of the original method. Together, these results establish SF as a practical, scalable, and theoretically grounded approach for language model training.

Transformer training is notoriously difficult, requiring a careful design of optimizers and use of various heuristics. We make progress towards understanding the subtleties of training Transformers by carefully studying a simple yet canonical linearized *shallow* Transformer model. Specifically, we train linear Transformers to solve regression tasks, inspired by J. von Oswald et al. (ICML 2023), and K. Ahn et al. (NeurIPS 2023). Most importantly, we observe that our proposed linearized models can reproduce several prominent aspects of Transformer training dynamics. Consequently, the results obtained in this paper suggest that a simple linearized Transformer model could actually be a valuable, realistic abstraction for understanding Transformer optimization.

Cohen et al. (2021) empirically study the evolution of the largest eigenvalue of the loss Hessian, also known as sharpness, along the gradient descent (GD) trajectory and observe the Edge of Stability (EoS) phenomenon. The sharpness increases at the early phase of training (referred to as progressive sharpening), and eventually saturates close to the threshold of $2 / \text{(step size)}$. In this paper, we start by demonstrating through empirical studies that when the EoS phenomenon occurs, different GD trajectories (after a proper reparameterization) align on a specific bifurcation diagram independent of initialization. We then rigorously prove this trajectory alignment phenomenon for a two-layer fully-connected linear network and a single-neuron nonlinear network trained with a single data point. Our trajectory alignment analysis establishes both progressive sharpening and EoS phenomena, encompassing and extending recent findings in the literature.