论文标题
利马尼亚汉密尔顿的方法用于歧管上的最小最大优化
Riemannian Hamiltonian methods for min-max optimization on manifolds
论文作者
论文摘要
在本文中，我们研究了Riemannian歧管上的Min-Max优化问题。我们引入了Riemannian Hamiltonian功能，最小化是解决原始Min-Max问题的代理。在Riemannian polyak-dourjasiewicz关于哈密顿功能的条件下，其最小化器对应于所需的Min-Max鞍点。我们还提供满足这种情况的情况。特别是对于地球双线性优化，解决代理问题会导致正确的搜索方向朝着全球最优性方向，这在Min-Max配方中变得具有挑战性。为了最大程度地减少哈密顿函数，我们提出了Riemannian Hamiltonian方法（RHM），并介绍其收敛分析。我们将RHM扩展到包括共识正则化和随机设置。我们说明了拟议的RHM在诸如稳健的瓦斯坦距离，神经网络的鲁棒训练和生成对抗网络等应用中的功效。
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 (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.
