A Data-Augmentation Is Worth A Thousand Samples: Exact Quantification From Analytical Augmented Sample Moments

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.