  Data-Augmentation (DA) is known to improve performance across tasks and
datasets. We propose a method to theoretically analyze the effect of DA and
study questions such as: how many augmented samples are needed to correctly
estimate the information encoded by that DA? How does the augmentation policy
impact the final parameters of a model? We derive several quantities in
close-form, such as the expectation and variance of an image, loss, and model's
output under a given DA distribution. Those derivations open new avenues to
quantify the benefits and limitations of DA. For example, we show that common
DAs require tens of thousands of samples for the loss at hand to be correctly
estimated and for the model training to converge. We show that for a training
loss to be stable under DA sampling, the model's saliency map (gradient of the
loss with respect to the model's input) must align with the smallest
eigenvector of the sample variance under the considered DA augmentation,
hinting at a possible explanation on why models tend to shift their focus from
edges to textures.
