  Regularized learning is a fundamental technique in online optimization,
machine learning and many other fields of computer science. A natural question
that arises in these settings is how regularized learning algorithms behave
when faced against each other. We study a natural formulation of this problem
by coupling regularized learning dynamics in zero-sum games. We show that the
system's behavior is Poincar\'e recurrent, implying that almost every
trajectory revisits any (arbitrarily small) neighborhood of its starting point
infinitely often. This cycling behavior is robust to the agents' choice of
regularization mechanism (each agent could be using a different regularizer),
to positive-affine transformations of the agents' utilities, and it also
persists in the case of networked competition, i.e., for zero-sum polymatrix
games.
