Analytic Study of Families of Spurious Minima in Two-Layer ReLU Neural Networks: A Tale of Symmetry II

TL;DR

This paper analytically investigates the structure of spurious minima in two-layer ReLU neural networks, revealing detailed spectral properties and bifurcation phenomena using symmetry-based tools, without relying on limiting assumptions.

Contribution

It introduces a novel symmetry-based analytical framework for studying finite-size neural network loss landscapes and characterizes the transition from saddles to spurious minima.

Findings

Hessian spectrum concentrates near small positive constants.

Global and spurious minima Hessians are spectrally similar.

Exact fractional dimensionality of minima transition points.

Abstract

We study the optimization problem associated with fitting two-layer ReLU neural networks with respect to the squared loss, where labels are generated by a target network. We make use of the rich symmetry structure to develop a novel set of tools for studying families of spurious minima. In contrast to existing approaches which operate in limiting regimes, our technique directly addresses the nonconvex loss landscape for a finite number of inputs $d$ and neurons $k$, and provides analytic, rather than heuristic, information. In particular, we derive analytic estimates for the loss at different minima, and prove that modulo $O(d^{-1/2})$-terms the Hessian spectrum concentrates near small positive constants, with the exception of $\Theta(d)$ eigenvalues which grow linearly with~$d$. We further show that the Hessian spectrum at global and spurious minima coincide to $O(d^{-1/2})$-order, thus challenging our ability to argue about statistical generalization through local curvature. Lastly, our technique provides the exact \emph{fractional} dimensionality at which families of critical points turn from saddles into spurious minima. This makes possible the study of the creation and the annihilation of spurious minima using powerful tools from equivariant bifurcation theory.