Riemannian Hamiltonian Methods for Min-Max Optimization on Manifolds
研究了黎曼流形上的极小极大优化问题，通过引入黎曼哈密顿函数作为代理，提出黎曼哈密顿方法（RHMs）并分析其收敛性，适用于子空间鲁棒Wasserstein距离、神经网络鲁棒训练和生成对抗网络等应用。
.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–Łojasiewicz 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 (RHMs) and present their convergence analyses. We extend RHMs to include consensus regularization and to the stochastic setting. We illustrate the efficacy of the proposed RHMs in applications such as subspace robust Wasserstein distance, robust training of neural networks, and generative adversarial networks.KeywordsRiemannian optimizationsaddle pointconsensus optimizationHamiltonian gradient descentPolyak–Łojasiewiczgeodesic convex concaveMSC codes65K0590C3090C2290C2590C2690C2790C4658C0549M15