Keywords: [ Theory ] [ Algorithms -> Representation Learning ] [ Algorithms -> Large Scale Learning; Applications -> Natural Language Processing; Deep Learning -> Efficient Inference Methods; ]

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