In Euclidean space, OCO boasts a robust theoretical foundation and numerous real-world applications, such as online load balancing (Molinaro, 2017), optimal control (Li et al., 2019), revenue maximization (Lin et al., 2019), and portfolio management (Jézéquel et al., 2022).

Zhang et al. have recently shown that geodesic convex concave Riemannian problems always admit saddle-point solutions.

The primal and dual are related by the Lagrangian L (,',), L ( ',,)= E We proceed to proofs of the theorems stated in Section 4. Assumption NTK, the regularization parameter, and it may also depend indirectly on the bound R . Theorem 4.2 follows immediately from Lemmas B.1 and B.2. Theorem The following result follows from Proposition E.4 and E.5 of of Luise et al. Interestingly, the rate of estimation of the Sinkhorn plan breaks the curse of dimensionality. B.2 Log-concavity of Sinkhorn Factor The optimal entropy regularized Sinkhorn plan is given by The optimal potentials satisfy fixed point equations. Using this result, one can prove the following lemma.

However, none of these three works provided a global convergence rate analysis for their algorithms.

We establish upper bounds on the regrets of the problem with respect to time horizon, manifold curvature, and manifold dimension.

Finding the global minimum to Eq. (1) is in general NP-hard; our goal is to find an approximate second order stationary point with first order optimization methods. We are interested in first-order methods because they are extremely prevalent in machine learning, partly because computing Hessians is often too costly.

The equality is due to the fact that the parallel transport map is an isometry between tangent spaces and fixes the origin; the first inequality follows from the reverse triangle inequality, while the last follows from the upper bound on the Hessian of U in (2). U ( x,D) is the zero gradient vector field under the isometric parallel transport. Simulations pertaining to the sphere and Kendall shape space are done on a desktop computer with an Intel Xeon processor at 3.60GHz with 31.9 Simulations pertaining to symmetric positive-definite matrices were performed on the Pennsylvania State University's Institute for Computational and Data Sciences' Roar supercomputer. All simulations are done in Matlab.

It is common for data structures such as images and shapes of 2D objects to be represented as points on a manifold.