  Contrastive representation learning has been recently proved to be very
efficient for self-supervised training. These methods have been successfully
used to train encoders which perform comparably to supervised training on
downstream classification tasks. A few works have started to build a
theoretical framework around contrastive learning in which guarantees for its
performance can be proven. We provide extensions of these results to training
with multiple negative samples and for multiway classification. Furthermore, we
provide convergence guarantees for the minimization of the contrastive training
error with gradient descent of an overparametrized deep neural encoder, and
provide some numerical experiments that complement our theoretical findings
