Escape dynamics and implicit bias of one-pass SGD in overparameterized quadratic networks

We analyze the one-pass stochastic gradient descent dynamics of a two-layer neural network with quadratic activations in a teacher--student framework. In the high-dimensional regime, where the input dimension $N$ and the number of samples $M$ diverge at fixed ratio $α= M/N$, and for finite hidden widths $(p,p^*)$ of the student and teacher, respectively, we study the low-dimensional ordinary differential equations that govern the evolution of the student--teacher and student--student overlap matrices. We show that overparameterization ($p>p^*$) only modestly accelerates escape from a plateau of poor generalization by modifying the prefactor of the exponential decay of the loss. We then examine how unconstrained weight norms introduce a continuous rotational symmetry that results in a nontrivial manifold of zero-loss solutions for $p>1$. From this manifold the dynamics consistently selects the closest solution to the random initialization, as enforced by a conserved quantity in the ODEs governing the evolution of the overlaps. Finally, a Hessian analysis of the population-loss landscape confirms that the plateau and the solution manifold correspond to saddles with at least one negative eigenvalue and to marginal minima in the population-loss geometry, respectively.

Implicit bias produces neural scaling laws in learning curves, from perceptrons to deep networks

Scaling laws in deep learning - empirical power-law relationships linking model performance to resource growth - have emerged as simple yet striking regularities across architectures, datasets, and tasks. These laws are particularly impactful in guiding the design of state-of-the-art models, since they quantify the benefits of increasing data or model size, and hint at the foundations of interpretability in machine learning. However, most studies focus on asymptotic behavior at the end of training or on the optimal training time given the model size. In this work, we uncover a richer picture by analyzing the entire training dynamics through the lens of spectral complexity norms. We identify two novel dynamical scaling laws that govern how performance evolves during training. These laws together recover the well-known test error scaling at convergence, offering a mechanistic explanation of generalization emergence. Our findings are consistent across CNNs, ResNets, and Vision Transformers trained on MNIST, CIFAR-10 and CIFAR-100. Furthermore, we provide analytical support using a solvable model: a single-layer perceptron trained with binary cross-entropy. In this setting, we show that the growth of spectral complexity driven by the implicit bias mirrors the generalization behavior observed at fixed norm, allowing us to connect the performance dynamics to classical learning rules in the perceptron.