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.

We theoretically study the landscape of the training error for neural networks in overparameterized cases.

An overarching goal in machine learning is to build ageneralizable model with fewsamples.

We then integrate these findings toobtain anoverallperformance guarantee forourmetalearning algorithm.

Recently there has been significant theoretical progress on understanding the convergence andgeneralization ofgradient-based methods onnonconvexlosses withoverparameterized models. Nevertheless, manyaspectsofoptimization and generalization and in particular the critical role of small random initialization are not fully understood.

Onone hand, it is widely accepted that, for two-layer neural networks, the number of parameters should growlinearlywith n(e.g.,[21,38]).