Optimal Learning Rate Scaling Depends on Data in Deep Scalar Linear Networks

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. We show that even in this minimal model, the optimal depth-wise learning rate scaling depends on data, whereas data-agnostic scaling rules fail to transfer across depths. Under the data-dependent optimal scaling, the learning dynamics is independent of data and weakly dependent on depth, resulting in a constant linear convergence rate across all depths including infinity. We further show similar data-dependent effects in deep scalar linear networks with residual connections.

Explaining Near-Zero Hessian Eigenvalues Through Approximate Symmetries in Neural Networks

The Hessian of the training loss governs the local geometry of the loss landscape, yet despite existing explanations for its largest eigenvalues, the origin of the vast multitude of vanishingly small eigenvalues remains elusive. We argue that the bulk consists of the weakly lifted pseudo-Goldstone modes of the continuous symmetries of the network parametrization. In deep linear networks these symmetries are exact: they generate flat directions and hence exact zero modes, whose eigenvectors we construct explicitly. Introducing a ReLU nonlinearity as a perturbation, we show that it breaks these symmetries weakly and explicitly. Resolving the spectrum at the level of eigenvectors, we find that the high-curvature directions are orthogonal to the symmetry subspace, while the bulk lies almost entirely within it. We demonstrate the mechanism in a two-layer ReLU student--teacher model and in a network trained on CIFAR-10. A convolutional example demonstrates that the same diagnostic extends beyond fully connected layers. Together, these results link the Hessian bulk to weakly broken symmetries and clarify the origin of near-zero modes.

The Map Behind the Flow: Finite-Step Gradient Descent as a Dynamical System

Many phenomena of deep learning are dynamical: they concern not only which minima exist, but how gradient descent reaches, avoids, or selects among them. Edge-of-stability behavior, sharpness oscillations, catapult phases, balancing, and movement toward flatter representations are effects of the training map itself, and are poorly captured by the small-step gradient-flow limit. This paper studies fixed-step gradient descent as a discrete dynamical system in a hierarchy of exactly solvable models retaining basic structures of deep learning: depth, factorization, width, data coupling, activation, and stochasticity. The starting point is the balanced scalar reduction of a deep linear chain, giving a quartic loss and a cubic gradient map whose post-edge behavior is explicit. Under the natural large-depth scaling, this dynamics converges to a universal Ricker-type map. The edge of stability is therefore not a breakdown of optimization, but the first bifurcation of the training map. Embedding the scalar dynamics back into factored models turns these regimes into learning phenomena. Finite steps break conservation laws of gradient flow and contract factorization imbalance; residual oscillations move parameters toward flatter, more balanced representations. Wider linear networks produce a ladder of spectral edges, so the optimal learning rate can lie beyond the first edge. Data coupling, nonlinear activations, and stochastic targets preserve the same organizing principle: finite-step oscillations drive alignment, balancing, and representation selection. Thus the learning rate is not merely a numerical stability parameter. It is a structural parameter of the training dynamics, determining its attractors and shaping the representations gradient descent selects.

Deciphering Two Training Clocks in Grokking via Deep Linear Network Theory with Conditional ReLU Reduction

Grokking suggests that fitting the training data and learning a simple underlying rule may occur on different time scales. We formalize this phenomenon by separating the fast decay of the classification loss from the slower simplification of the learned representation, and we call the resulting pair of stopping times two training clocks. For deep linear networks, we show that a post-margin gap-growth or one-step tail-contraction condition reduces the cross-entropy loss to level epsilon on a logarithmic time scale. In contrast, when layerwise weight decay is present, the induced regularization on the end-to-end map can be expressed as a Schatten-type penalty; under a sharp late-time Kurdyka-Lojasiewicz tail, this structural energy closes on a polynomial time scale. The two clocks, therefore, separate fitting from representation simplification. We then explain how the same mechanism can appear in ReLU MLPs. In regions where the activation patterns on the training set remain fixed, the network reduces to a linear model in the active coordinates. In a two-layer ReLU embedding model, chain-rule estimates further show that the classifier head can receive larger effective gradients than the embedding block under controlled downstream norms. This supports a two-stage mechanism in which the classifier fits first, while the representation continues to simplify later. We use modular addition as the main experimental setting. The deep linear theory provides the rigorous core of the analysis. But the ReLU results are formulated as conditional reductions that account for empirical behavior without claiming a global proof for nonlinear training dynamics.