  Recently the surprising discovery of the Bootstrap Your Own Latent (BYOL)
method by Grill et al. shows the negative term in contrastive loss can be
removed if we add the so-called prediction head to the network. This initiated
the research of non-contrastive self-supervised learning. It is mysterious why
even when there exist trivial collapsed global optimal solutions, neural
networks trained by (stochastic) gradient descent can still learn competitive
representations. This phenomenon is a typical example of implicit bias in deep
learning and remains little understood.
  In this work, we present our empirical and theoretical discoveries on
non-contrastive self-supervised learning. Empirically, we find that when the
prediction head is initialized as an identity matrix with only its off-diagonal
entries being trainable, the network can learn competitive representations even
though the trivial optima still exist in the training objective. Theoretically,
we present a framework to understand the behavior of the trainable, but
identity-initialized prediction head. Under a simple setting, we characterized
the substitution effect and acceleration effect of the prediction head. The
substitution effect happens when learning the stronger features in some neurons
can substitute for learning these features in other neurons through updating
the prediction head. And the acceleration effect happens when the substituted
features can accelerate the learning of other weaker features to prevent them
from being ignored. These two effects enable the neural networks to learn all
the features rather than focus only on learning the stronger features, which is
likely the cause of the dimensional collapse phenomenon. To the best of our
knowledge, this is also the first end-to-end optimization guarantee for
non-contrastive methods using nonlinear neural networks with a trainable
prediction head and normalization.
