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：本文研究黎曼流形上的极小极大优化问题。我们引入黎曼哈密顿函数，其最小化可作为求解原极小极大问题的代理。当哈密顿函数满足黎曼Polyak-Łojasiewicz条件时，其极小化器对应所需的极小极大鞍点。我们还给出了该条件成立的情形。特别地，对于测地双线性优化，求解代理问题可导向全局最优的正确搜索方向，而这在极小极大形式下难以实现。为最小化哈密顿函数，我们提出黎曼哈密顿方法(RHM)并给出其收敛性分析。我们将RHM扩展至包含共识正则化与随机场景。通过子空间鲁棒Wasserstein距离、神经网络鲁棒训练及生成对抗网络等应用，验证了所提RHM的有效性。