Built First, Kept: Task Symmetry Selects What a Recurrent Network Represents

Several different sets of features can solve one task, and a trained network holds one of them. We ask what selects that set, and whether it is kept. We study the word problem of a finite abelian group, written as modular counters, with one bit of supervision per position. A recurrent network solves it, yet in most runs one counter is nearly missing from its features. A symmetry is a relabelling of the symbols that leaves the task unchanged. We prove that updates treating alike the symbols a symmetry exchanges can raise only a feature that cannot tell them apart. At counters of equal size, a feature that ignores which symbol advanced it is built first in every run that built a feature before its symmetry broke. Rewriting the task in a new alphabet, with the counters fixed, moves the symmetry and the first feature with it. Making that feature cost more to learn does not move the choice to cost while the symmetry lasts. Whatever feature the network builds first it keeps, in almost every run that solves, so the start of training settles the choice. We prove why under an idealisation of the network. At counters of equal size, the kept feature is one the task could do without, and it takes one counter’s place. Thus, what a trained network holds depends on how its task was written down.