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.
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