# Revisit First-order Methods for Geodesically Convex Optimization

arXiv: https://arxiv.org/abs/2504.06814 · HTML: https://mlsanity.com/paper/2504.06814/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Apr 2025.

## Built on by (most cited first, 1 of 1)

- [Mirror Descent on Riemannian Manifolds](https://mlsanity.com/paper/2603.17527/) (2026)

## Abstract (arXiv)

In a seminal work of Zhang and Sra, gradient descent methods for geodesically convex optimization were comprehensively studied. In particular, Zhang and Sra derived a comparison inequality that relates the iterative points in the optimization process. Since their seminal work, numerous follow-ups have studied different downstream usages of their comparison lemma. In this work, we introduce the concept of quasilinearization to optimization, presenting a novel framework for analyzing geodesically convex optimization. By leveraging this technique, we establish state-of-the-art convergence rates -- for both deterministic and stochastic settings -- under weaker assumptions than previously required. The technique of quasilinearization may prove valuable for other non-Euclidean optimization problems.