In this paper, we study min-max optimization problems on Riemannian manifolds. We introduce a Riemannian Hamiltonian function, minimization of which serves as a proxy for solving the original min-max problems. Under the Riemannian Polyak--{\L}ojasiewicz condition on the Hamiltonian function, its minimizer corresponds to the desired min-max saddle point. We also provide cases where this condition is satisfied. For geodesic-bilinear optimization in particular, solving the proxy problem leads to the correct search direction towards global optimality, which becomes challenging with the min-max formulation. To minimize the Hamiltonian function, we propose Riemannian Hamiltonian methods (RHM) and present their convergence analyses. We extend RHM to include consensus regularization and to the stochastic setting. We illustrate the efficacy of the proposed RHM in applications such as subspace robust Wasserstein distance, robust training of neural networks, and generative adversarial networks.

翻译：本文研究了黎曼流形上的极小化极大优化问题。我们引入了一种黎曼哈密顿函数，通过最小化该函数来求解原始极小化极大问题。在哈密顿函数满足黎曼Polyak--{\L}ojasiewicz条件的情况下，其最小值点对应于所需的极小化极大鞍点。我们还给出了该条件成立的情形。特别地，对于测地线双线性优化问题，求解代理问题能引导出通向全局最优的正确搜索方向，而使用极小化极大公式则会在此过程中产生困难。为最小化哈密顿函数，我们提出了黎曼哈密顿方法（RHM）并给出了其收敛性分析。我们将RHM扩展至包含一致性正则化的情形以及随机设定中。我们通过子空间鲁棒Wasserstein距离、神经网络的鲁棒训练和生成对抗网络等应用展示了所提出的RHM的有效性。