论文标题
受约束最小值优化的最佳条件
Optimality Conditions for Constrained Minimax Optimization
论文作者
论文摘要
最小值优化问题既来自现代机器学习，包括生成的对抗网络，对抗性培训和多代理强化学习，以及传统研究领域，例如鞍点问题，数值部分偏微分方程和相等性的优化条件受到约束优化。对于不受限制的连续非Convex-Nonconcave情况，Jin，Netrapalli和Jordan（2019）仔细考虑了一个非常基本的问题：什么是对最小值优化问题的本地优化的适当定义，并提出了适当的局部最优定义，称为本地Minimax。我们应将局部最小点的定义扩展到约束的非Convex-Nonconcave minimax优化问题。通过分析低级最大化问题的雅各布唯一条件，以及最大化问题的Karush-Kuhn-Tucker条件的较强规律性，我们提供了必要的最佳条件和足够的最佳最佳条件，以实现约束最小值优化问题的局部最小值点。
Minimax optimization problems arises from both modern machine learning including generative adversarial networks, adversarial training and multi-agent reinforcement learning, as well as from tradition research areas such as saddle point problems, numerical partial differential equations and optimality conditions of equality constrained optimization. For the unconstrained continuous nonconvex-nonconcave situation, Jin, Netrapalli and Jordan (2019) carefully considered the very basic question: what is a proper definition of local optima of a minimax optimization problem, and proposed a proper definition of local optimality called local minimax. We shall extend the definition of local minimax point to constrained nonconvex-nonconcave minimax optimization problems. By analyzing Jacobian uniqueness conditions for the lower-level maximization problem and the strong regularity of Karush-Kuhn-Tucker conditions of the maximization problem, we provide both necessary optimality conditions and sufficient optimality conditions for the local minimax points of constrained minimax optimization problems.
