  When a neural network can learn multiple distinct algorithms to solve a task,
how does it "choose" between them during training? To approach this question,
we take inspiration from ecology: when multiple species coexist, they
eventually reach an equilibrium where some survive while others die out.
Analogously, we suggest that a neural network at initialization contains many
solutions (representations and algorithms), which compete with each other under
pressure from resource constraints, with the "fittest" ultimately prevailing.
To investigate this Survival of the Fittest hypothesis, we conduct a case study
on neural networks performing modular addition, and find that these networks'
multiple circular representations at different Fourier frequencies undergo such
competitive dynamics, with only a few circles surviving at the end. We find
that the frequencies with high initial signals and gradients, the "fittest,"
are more likely to survive. By increasing the embedding dimension, we also
observe more surviving frequencies. Inspired by the Lotka-Volterra equations
describing the dynamics between species, we find that the dynamics of the
circles can be nicely characterized by a set of linear differential equations.
Our results with modular addition show that it is possible to decompose
complicated representations into simpler components, along with their basic
interactions, to offer insight on the training dynamics of representations.
