提出新型黎曼乐观在线学习法，解决流形上极小极大问题。
Implicit Riemannian Optimism with Applications to Min-Max Problems
- 基于不精确隐式更新，支持流形内约束
- 达到欧氏空间最优后悔界，无几何常数依赖
- 首次逼近欧氏下界梯度复杂度，适合流形优化
我们提出一种基于不精确隐式更新的黎曼乐观在线学习算法，适用于哈达玛流形。与以往工作不同，该方法可处理流形内约束，并在无几何常数（如最小曲率）依赖的前提下，达到欧氏情形下的最优后悔界。基于此，我们设计了用于哈达玛流形上g-凸、g-凹光滑极小极大问题的算法。其中一种方法首次几乎逼近欧氏问题的梯度预言机复杂度下界。
原文摘要 · Abstract (English)
We introduce a Riemannian optimistic online learning algorithm for Hadamard manifolds based on inexact implicit updates. Unlike prior work, our method can handle in-manifold constraints, and matches the best known regret bounds in the Euclidean setting with no dependence on geometric constants, like the minimum curvature. Building on this, we develop algorithms for g-convex, g-concave smooth min-max problems on Hadamard manifolds. Notably, one method nearly matches the gradient oracle complexity of the lower bound for Euclidean problems, for the first time.
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