  Learning features from data is one of the defining characteristics of deep
learning, but our theoretical understanding of the role features play in deep
learning is still rudimentary. To address this gap, we introduce a new tool,
the interaction tensor, for empirically analyzing the interaction between data
and model through features. With the interaction tensor, we make several key
observations about how features are distributed in data and how models with
different random seeds learn different features. Based on these observations,
we propose a conceptual framework for feature learning. Under this framework,
the expected accuracy for a single hypothesis and agreement for a pair of
hypotheses can both be derived in closed-form. We demonstrate that the proposed
framework can explain empirically observed phenomena, including the recently
discovered Generalization Disagreement Equality (GDE) that allows for
estimating the generalization error with only unlabeled data. Further, our
theory also provides explicit construction of natural data distributions that
break the GDE. Thus, we believe this work provides valuable new insight into
our understanding of feature learning.
