  Constrained competitive optimization involves multiple agents trying to
minimize conflicting objectives, subject to constraints. This is a highly
expressive modeling language that subsumes most of modern machine learning. In
this work we propose competitive mirror descent (CMD): a general method for
solving such problems based on first order information that can be obtained by
automatic differentiation. First, by adding Lagrange multipliers, we obtain a
simplified constraint set with an associated Bregman potential. At each
iteration, we then solve for the Nash equilibrium of a regularized bilinear
approximation of the full problem to obtain a direction of movement of the
agents. Finally, we obtain the next iterate by following this direction
according to the dual geometry induced by the Bregman potential. By using the
dual geometry we obtain feasible iterates despite only solving a linear system
at each iteration, eliminating the need for projection steps while still
accounting for the global nonlinear structure of the constraint set. As a
special case we obtain a novel competitive multiplicative weights algorithm for
problems on the positive cone.
