# Minimizing Dynamic Regret on Geodesic Metric Spaces

arXiv: https://arxiv.org/abs/2302.08652 · HTML: https://mlsanity.com/paper/2302.08652/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 55 references that are in the map.
- First posted on arXiv 17 Feb 2023.

## Builds on (most cited first, 26 of 26)

- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Non-stationary Stochastic Optimization](https://mlsanity.com/paper/1307.5449/) (2013)
- [Fast Convergence of Regularized Learning in Games](https://mlsanity.com/paper/1507.00407/) (2015)
- [Adaptive Online Learning in Dynamic Environments](https://mlsanity.com/paper/1810.10815/) (2018)
- [Online Optimization : Competing with Dynamic Comparators](https://mlsanity.com/paper/1501.06225/) (2015)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Dynamic Regret of Convex and Smooth Functions](https://mlsanity.com/paper/2007.03479/) (2020)
- [Differentiating through the Fr'echet Mean](https://mlsanity.com/paper/2003.00335/) (2020)
- [MetaGrad: Multiple Learning Rates in Online Learning](https://mlsanity.com/paper/1604.08740/) (2016)
- [Improved Dynamic Regret for Non-degenerate Functions](https://mlsanity.com/paper/1608.03933/) (2016)
- [Optimal Dynamic Regret in Exp-Concave Online Learning](https://mlsanity.com/paper/2104.11824/) (2021)
- [Escaping from saddle points on Riemannian manifolds](https://mlsanity.com/paper/1906.07355/) (2019)
- [Improved Analysis for Dynamic Regret of Strongly Convex and Smooth Functions](https://mlsanity.com/paper/2006.05876/) (2020)
- [A Continuous-time Perspective for Modeling Acceleration in Riemannian Optimization](https://mlsanity.com/paper/1910.10782/) (2019)
- [From Nesterov's Estimate Sequence to Riemannian Acceleration](https://mlsanity.com/paper/2001.08876/) (2020)
- [An accelerated first-order method for non-convex optimization on manifolds](https://mlsanity.com/paper/2008.02252/) (2020)
- [Geodesic Convex Optimization: Differentiation on Manifolds, Geodesics, and Convexity](https://mlsanity.com/paper/1806.06373/) (2018)
- [Tracking and regret bounds for online zeroth-order Euclidean and Riemannian optimisation](https://mlsanity.com/paper/2010.00211/) (2020)
- [Negative curvature obstructs acceleration for strongly geodesically convex optimization, even with exact first-order oracles](https://mlsanity.com/paper/2111.13263/) (2021)
- [Online Improper Learning with an Approximation Oracle](https://mlsanity.com/paper/1804.07837/) (2018)
- [Global Riemannian Acceleration in Hyperbolic and Spherical Spaces](https://mlsanity.com/paper/2012.03618/) (2020)
- [On Geodesically Convex Formulations for the Brascamp-Lieb Constant](https://mlsanity.com/paper/1804.04051/) (2018)
- [Accelerated Riemannian Optimization: Handling Constraints with a Prox to Bound Geometric Penalties](https://mlsanity.com/paper/2211.14645/) (2022)
- [Elements of Convex Geometry in Hadamard Manifolds with Application to Equilibrium Problems](https://mlsanity.com/paper/2107.02223/) (2021)
- [Riemannian stochastic approximation algorithms](https://mlsanity.com/paper/2206.06795/) (2022)

## Built on by (most cited first, 6 of 6)

- [Data-Dependent Bounds for Online Portfolio Selection Without Lipschitzness and Smoothness](https://mlsanity.com/paper/2305.13946/) (2023)
- [Riemannian Projection-free Online Learning](https://mlsanity.com/paper/2305.19349/) (2023)
- [Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds](https://mlsanity.com/paper/2509.07779/) (2025)
- [Dynamic Regret via Discounted-to-Dynamic Reduction with Applications to Curved Losses and Adam Optimizer](https://mlsanity.com/paper/2602.08372/) (2026)
- [Extragradient Type Methods for Riemannian Variational Inequality Problems](https://mlsanity.com/paper/2309.14155/) (2023)
- [Finite sample bounds for barycenter estimation in geodesic spaces](https://mlsanity.com/paper/2502.14069/) (2025)

## Abstract (arXiv)

In this paper, we consider the sequential decision problem where the goal is to minimize the general dynamic regret on a complete Riemannian manifold. The task of offline optimization on such a domain, also known as a geodesic metric space, has recently received significant attention. The online setting has received significantly less attention, and it has remained an open question whether the body of results that hold in the Euclidean setting can be transplanted into the land of Riemannian manifolds where new challenges (e.g., curvature) come into play. In this paper, we show how to get optimistic regret bound on manifolds with non-positive curvature whenever improper learning is allowed and propose an array of adaptive no-regret algorithms. To the best of our knowledge, this is the first work that considers general dynamic regret and develops "optimistic" online learning algorithms which can be employed on geodesic metric spaces.