  In recent years, constrained optimization has become increasingly relevant to
the machine learning community, with applications including Neyman-Pearson
classification, robust optimization, and fair machine learning. A natural
approach to constrained optimization is to optimize the Lagrangian, but this is
not guaranteed to work in the non-convex setting, and, if using a first-order
method, cannot cope with non-differentiable constraints (e.g. constraints on
rates or proportions).
  The Lagrangian can be interpreted as a two-player game played between a
player who seeks to optimize over the model parameters, and a player who wishes
to maximize over the Lagrange multipliers. We propose a non-zero-sum variant of
the Lagrangian formulation that can cope with non-differentiable--even
discontinuous--constraints, which we call the "proxy-Lagrangian". The first
player minimizes external regret in terms of easy-to-optimize "proxy
constraints", while the second player enforces the original constraints by
minimizing swap regret.
  For this new formulation, as for the Lagrangian in the non-convex setting,
the result is a stochastic classifier. For both the proxy-Lagrangian and
Lagrangian formulations, however, we prove that this classifier, instead of
having unbounded size, can be taken to be a distribution over no more than m+1
models (where m is the number of constraints). This is a significant
improvement in practical terms.
