  A number of machine learning tasks entail a high degree of invariance: the
data distribution does not change if we act on the data with a certain group of
transformations. For instance, labels of images are invariant under
translations of the images. Certain neural network architectures -- for
instance, convolutional networks -- are believed to owe their success to the
fact that they exploit such invariance properties. With the objective of
quantifying the gain achieved by invariant architectures, we introduce two
classes of models: invariant random features and invariant kernel methods. The
latter includes, as a special case, the neural tangent kernel for convolutional
networks with global average pooling. We consider uniform covariates
distributions on the sphere and hypercube and a general invariant target
function. We characterize the test error of invariant methods in a
high-dimensional regime in which the sample size and number of hidden units
scale as polynomials in the dimension, for a class of groups that we call
`degeneracy $\alpha$', with $\alpha \leq 1$. We show that exploiting invariance
in the architecture saves a $d^\alpha$ factor ($d$ stands for the dimension) in
sample size and number of hidden units to achieve the same test error as for
unstructured architectures.
  Finally, we show that output symmetrization of an unstructured kernel
estimator does not give a significant statistical improvement; on the other
hand, data augmentation with an unstructured kernel estimator is equivalent to
an invariant kernel estimator and enjoys the same improvement in statistical
efficiency.
