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

We study how permutation symmetries in overparameterized multi-layer neural\nnetworks generate `symmetry-induced' critical points. Assuming a network with $\nL $ layers of minimal widths $ r_1^*, \\ldots, r_{L-1}^* $ reaches a zero-loss\nminimum at $ r_1^*! \\cdots r_{L-1}^*! $ isolated points that are permutations\nof one another, we show that adding one extra neuron to each layer is\nsufficient to connect all these previously discrete minima into a single\nmanifold. For a two-layer overparameterized network of width $ r^*+ h =: m $ we\nexplicitly describe the manifold of global minima: it consists of $ T(r^*, m) $\naffine subspaces of dimension at least $ h $ that are connected to one another.\nFor a network of width $m$, we identify the number $G(r,m)$ of affine subspaces\ncontaining only symmetry-induced critical points that are related to the\ncritical points of a smaller network of width $r<r^*$. Via a combinatorial\nanalysis, we derive closed-form formulas for $ T $ and $ G $ and show that the\nnumber of symmetry-induced critical subspaces dominates the number of affine\nsubspaces forming the global minima manifold in the mildly overparameterized\nregime (small $ h $) and vice versa in the vastly overparameterized regime ($h\n\\gg r^*$). Our results provide new insights into the minimization of the\nnon-convex loss function of overparameterized neural networks.\n