Cycles in adversarial regularized learning

International audienceRegularized learning is a fundamental technique in online optimization, machine learning, and many other fields of computer science. A natural question that arises in this context 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 Poincare 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 (zero-sum polymatrix games)