Symmetry & critical points for a model shallow neural network

Abstract

We consider the optimization problem associated with fitting two-layer ReLU networks with $k$ hidden neurons, where labels are assumed to be generated by a (teacher) neural network. We leverage the rich symmetry exhibited by such models to identify various families of critical points and express them as power series in $k^{-\frac{1}{2}}$. These expressions are then used to derive estimates for several related quantities which imply that not all spurious minima are alike. In particular, we show that while the loss function at certain types of spurious minima decays to zero like $k^{-1}$, in other cases the loss converges to a strictly positive constant. The methods used depend on symmetry, the geometry of group actions, bifurcation, and Artin's implicit function theorem.

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Cited by in corpus (4)

• Dynamic Neural Diversification: Path to Computationally Sustainable Neural Networks
• On the Stability Properties and the Optimization Landscape of Training Problems with Squared Loss for Neural Networks and General Nonlinear Conic Approximation Schemes
• Analytic Study of Families of Spurious Minima in Two-Layer ReLU Neural Networks: A Tale of Symmetry II
• Equivariant bifurcation, quadratic equivariants, and symmetry breaking for the standard representation of $S_n$