A Data-Augmentation Is Worth A Thousand Samples: Analytical Moments And Sampling-Free Training

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. Up to our knowledge, we obtain the first explicit regularizer that corresponds to using DA during training for non-trivial transformations such as affine transformations, color jittering, or Gaussian blur. Those derivations open new avenues to quantify the benefits and limitations of DA. For example, given a loss at hand, we find that common DAs require tens of thousands of samples for the loss to be correctly estimated and for the model training to converge. We then show that for a training loss to have reduced variance 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's covariance matrix under the considered DA augmentation; this is exactly the quantity estimated and regularized by TangentProp. Those findings also hint at a possible explanation on why models tend to shift their focus from edges to textures when specific DAs are employed.