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Geometry of the Loss Landscape in Overparameterized Neural Networks: Symmetries and Invariances by Şimşek, Berfin; Ged, François; Jacot, Arthur; Spadaro, Francesco; Hongler, Clément; Gerstner, Wulfram; Brea, Johanni is a scholarly article available to read on EtoBox.

What is Geometry of the Loss Landscape in Overparameterized Neural Networks: Symmetries and Invariances about?

We study how permutation symmetries in overparameterized multi-layer neural networks generate `symmetry-induced' critical points. Assuming a network with $ L $ layers of minimal widths $ r_1^*, \ldots, r_{L-1}^* $ reaches a zero-loss minimum at $ r_1^*! \cdots r_{L-1}^*! $ isolated points that are permutations of one another, we show that adding one extra neuron to each layer is sufficient to connect all these previously discrete minima into a single manifold. For a two-layer overparameterized network of width $ r^*+ h =: m $ we explicitly describe the manifold of global minima: it consists of $ T(r^*, m) $ affine subspaces of dimension at least $ h $ that are connected to one another. For a network of width $m$, we identify the number $G(r,m)$ of affine subspaces containing only symmetry-induced critical points that are related to the critical points of a smaller network of width $r<r^*$. Via a combinatorial analysis, we derive closed-form formulas for $ T $ and $ G $ and show that the number of symmetry-induced critical subspaces dominates the number of affine subspaces forming the global minima manifold in the mildly overparameterized regime (small $ h $) and vice versa in the v

Author

Şimşek, Berfin; Ged, François; Jacot, Arthur; Spadaro, Francesco; Hongler, Clément; Gerstner, Wulfram; Brea, Johanni

Published

2021

Language

EN