# Weight-Space Symmetry in Deep Networks Gives Rise to Permutation Saddles, Connected by Equal-Loss Valleys Across the Loss Landscape

Johanni Brea*, Berfin Şimşek*, Bernd Illing, Wulfram Gerstner

July 2019### Abstract

The permutation symmetry of neurons in each layer of a deep neural network gives rise not only to multiple equivalent global minima of the loss function, but also to first-order saddle points located on the path between the global minima. In a network of $d-1$ hidden layers with $n_k$ neurons in layers $k = 1,\ldots, d$, we construct smooth paths between equivalent global minima that lead through a ‘permutation point’ where the input and output weight vectors of two neurons in the same hidden layer $k$ collide and interchange. We show that such permutation points are critical points with at least $n_k+1$ vanishing eigenvalues of the Hessian matrix of second derivatives indicating a local plateau of the loss function. We find that a permutation point for the exchange of neurons $i$ and $j$ transits into a flat valley (or generally, an extended plateau of $n_k+1$ flat dimensions) that enables all $n_k!$ permutations of neurons in a given layer $k$ at the same loss value. Moreover, we introduce high-order permutation points by exploiting the recursive structure in neural network functions, and find that the number of $K$-th order permutation points is at least by a factor $\sum_{k=1}^{d-1}\frac{1}{2!^K} {n_k -K \choose K}$ larger than the (already huge) number of equivalent global minima. In two tasks, we illustrate numerically that some of the permutation points correspond to first-order saddles (`permutation saddles’): first, in a toy network with a single hidden layer on a function approximation task and, second, in a multilayer network on the MNIST task. Our geometric approach yields a lower bound on the number of critical points generated by weight-space symmetries and provides a simple intuitive link between previous mathematical results and numerical observations.

Publication

*arXiv:1907.02911 [cs, stat]*