  Understanding what information neural networks capture is an essential
problem in deep learning, and studying whether different models capture similar
features is an initial step to achieve this goal. Previous works sought to
define metrics over the feature matrices to measure the difference between two
models. However, different metrics sometimes lead to contradictory conclusions,
and there has been no consensus on which metric is suitable to use in practice.
In this work, we propose a novel metric that goes beyond previous approaches.
Recall that one of the most practical scenarios of using the learned
representations is to apply them to downstream tasks. We argue that we should
design the metric based on a similar principle. For that, we introduce the
transferred discrepancy (TD), a new metric that defines the difference between
two representations based on their downstream-task performance. Through an
asymptotic analysis, we show how TD correlates with downstream tasks and the
necessity to define metrics in such a task-dependent fashion. In particular, we
also show that under specific conditions, the TD metric is closely related to
previous metrics. Our experiments show that TD can provide fine-grained
information for varied downstream tasks, and for the models trained from
different initializations, the learned features are not the same in terms of
downstream-task predictions. We find that TD may also be used to evaluate the
effectiveness of different training strategies. For example, we demonstrate
that the models trained with proper data augmentations that improve the
generalization capture more similar features in terms of TD, while those with
data augmentations that hurt the generalization will not. This suggests a
training strategy that leads to more robust representation also trains models
that generalize better.
