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

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)(*), ..., r(L-1)* reaches a zero-loss minimum at r(1)(*)! ... 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 >> r*). Our results provide new insights into the minimization of the non-convex loss function of overparameterized neural networks.