  Recent success in training deep neural networks have prompted active
investigation into the features learned on their intermediate layers. Such
research is difficult because it requires making sense of non-linear
computations performed by millions of parameters, but valuable because it
increases our ability to understand current models and create improved versions
of them. In this paper we investigate the extent to which neural networks
exhibit what we call convergent learning, which is when the representations
learned by multiple nets converge to a set of features which are either
individually similar between networks or where subsets of features span similar
low-dimensional spaces. We propose a specific method of probing
representations: training multiple networks and then comparing and contrasting
their individual, learned representations at the level of neurons or groups of
neurons. We begin research into this question using three techniques to
approximately align different neural networks on a feature level: a bipartite
matching approach that makes one-to-one assignments between neurons, a sparse
prediction approach that finds one-to-many mappings, and a spectral clustering
approach that finds many-to-many mappings. This initial investigation reveals a
few previously unknown properties of neural networks, and we argue that future
research into the question of convergent learning will yield many more. The
insights described here include (1) that some features are learned reliably in
multiple networks, yet other features are not consistently learned; (2) that
units learn to span low-dimensional subspaces and, while these subspaces are
common to multiple networks, the specific basis vectors learned are not; (3)
that the representation codes show evidence of being a mix between a local code
and slightly, but not fully, distributed codes across multiple units.
