Analytic Characterization of the Hessian in Shallow ReLU Models: A Tale of Symmetry
This provides rigorous theoretical insights into optimization landscapes for neural networks, though it is incremental as it builds on prior observations of skewed spectra.
The paper analytically characterizes the Hessian spectrum at spurious minima in shallow ReLU networks, proving that for d ≥ k with Gaussian inputs, dk - O(d) eigenvalues concentrate near zero and Ω(d) grow linearly with k.
We consider the optimization problem associated with fitting two-layers ReLU networks with respect to the squared loss, where labels are generated by a target network. We leverage the rich symmetry structure to analytically characterize the Hessian at various families of spurious minima in the natural regime where the number of inputs $d$ and the number of hidden neurons $k$ is finite. In particular, we prove that for $d\ge k$ standard Gaussian inputs: (a) of the $dk$ eigenvalues of the Hessian, $dk - O(d)$ concentrate near zero, (b) $Ω(d)$ of the eigenvalues grow linearly with $k$. Although this phenomenon of extremely skewed spectrum has been observed many times before, to our knowledge, this is the first time it has been established {rigorously}. Our analytic approach uses techniques, new to the field, from symmetry breaking and representation theory, and carries important implications for our ability to argue about statistical generalization through local curvature.