  We investigate the role of projection heads, also known as projectors, within
the encoder-projector framework (e.g., SimCLR) used in contrastive learning. We
aim to demystify the observed phenomenon where representations learned before
projectors outperform those learned after -- measured using the downstream
linear classification accuracy, even when the projectors themselves are linear.
  In this paper, we make two significant contributions towards this aim.
Firstly, through empirical and theoretical analysis, we identify two crucial
effects -- expansion and shrinkage -- induced by the contrastive loss on the
projectors. In essence, contrastive loss either expands or shrinks the signal
direction in the representations learned by an encoder, depending on factors
such as the augmentation strength, the temperature used in contrastive loss,
etc. Secondly, drawing inspiration from the expansion and shrinkage phenomenon,
we propose a family of linear transformations to accurately model the
projector's behavior. This enables us to precisely characterize the downstream
linear classification accuracy in the high-dimensional asymptotic limit. Our
findings reveal that linear projectors operating in the shrinkage (or
expansion) regime hinder (or improve) the downstream classification accuracy.
This provides the first theoretical explanation as to why (linear) projectors
impact the downstream performance of learned representations. Our theoretical
findings are further corroborated by extensive experiments on both synthetic
data and real image data.
