Competitive Mirror Descent
This work addresses constrained competitive optimization, a foundational problem in machine learning, but appears incremental as it builds on existing mirror descent and Nash equilibrium concepts.
The authors tackled the problem of constrained competitive optimization by proposing competitive mirror descent (CMD), a first-order method that avoids projection steps and accounts for global nonlinear constraints, resulting in a novel competitive multiplicative weights algorithm for problems on the positive cone.
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.