Two-Player Games for Efficient Non-Convex Constrained Optimization

Andrew Cotter, Heinrich Jiang, Karthik Sridharan
Proceedings of the 30th International Conference on Algorithmic Learning Theory, PMLR 98:300-332, 2019.

Abstract

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 \emph{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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v98-cotter19a, title = {Two-Player Games for Efficient Non-Convex Constrained Optimization}, author = {Cotter, Andrew and Jiang, Heinrich and Sridharan, Karthik}, booktitle = {Proceedings of the 30th International Conference on Algorithmic Learning Theory}, pages = {300--332}, year = {2019}, editor = {Garivier, Aurélien and Kale, Satyen}, volume = {98}, series = {Proceedings of Machine Learning Research}, month = {22--24 Mar}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v98/cotter19a/cotter19a.pdf}, url = {https://proceedings.mlr.press/v98/cotter19a.html}, abstract = {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 \emph{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.} }
Endnote
%0 Conference Paper %T Two-Player Games for Efficient Non-Convex Constrained Optimization %A Andrew Cotter %A Heinrich Jiang %A Karthik Sridharan %B Proceedings of the 30th International Conference on Algorithmic Learning Theory %C Proceedings of Machine Learning Research %D 2019 %E Aurélien Garivier %E Satyen Kale %F pmlr-v98-cotter19a %I PMLR %P 300--332 %U https://proceedings.mlr.press/v98/cotter19a.html %V 98 %X 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 \emph{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.
APA
Cotter, A., Jiang, H. & Sridharan, K.. (2019). Two-Player Games for Efficient Non-Convex Constrained Optimization. Proceedings of the 30th International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 98:300-332 Available from https://proceedings.mlr.press/v98/cotter19a.html.

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