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.

中文翻译：

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竞争性镜像下降

受约束的竞争优化涉及多个代理，在受约束的情况下，试图最小化相互冲突的目标。这是一种极具表现力的建模语言，包含了大部分现代机器学习。在这项工作中，我们提出了竞争镜像下降（CMD）：一种基于可以通过自动微分获得的一阶信息来解决此类问题的通用方法。首先，通过添加拉格朗日乘子，我们获得了一个带有相关 Bregman 势的简化约束集。在每次迭代中，我们然后求解完整问题的正则化双线性近似的纳什均衡，以获得代理的移动方向。最后，我们根据 Bregman 势引起的对偶几何遵循这个方向，获得下一次迭代。通过使用对偶几何，我们获得了可行的迭代，尽管每次迭代只求解一个线性系统，消除了对投影步骤的需要，同时仍然考虑了约束集的全局非线性结构。作为一种特殊情况，我们获得了一种新颖的竞争乘法权重算法来解决正锥上的问题。