Geometry of the Loss Landscape in Overparameterized Neural Networks: Symmetries and Invariances

• URL: http://arxiv.org/abs/2105.12221v1
• Date: Tue, 25 May 2021 21:19:07 GMT
• Title: Geometry of the Loss Landscape in Overparameterized Neural Networks: Symmetries and Invariances
• Authors: Berfin \c{S}im\c{s}ek, Fran\c{c}ois Ged, Arthur Jacot, Francesco Spadaro, Cl\'ement Hongler, Wulfram Gerstner, Johanni Brea
• Abstract summary: We show that adding one extra neuron to each is sufficient to connect all previously discrete minima into a single manifold. We show that the number of symmetry-induced critical subspaces dominates the number of affine subspaces forming the global minima manifold.
• Score: 9.390008801320024
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $) and vice versa in the vastly overparameterized regime ($h \gg r^*$). Our results provide new insights into the minimization of the non-convex loss function of overparameterized neural networks.

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