# Streamlining in the Riemannian Realm: Efficient Riemannian Optimization with Loopless Variance Reduction

arXiv: https://arxiv.org/abs/2403.06677 · HTML: https://mlsanity.com/paper/2403.06677/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 74 of its 114 references that are in the map.
- First posted on arXiv 11 Mar 2024.

## Builds on (most cited first, 30 of 74)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [SAGA: A Fast Incremental Gradient Method With Support for Non-Strongly Convex Composite Objectives](https://mlsanity.com/paper/1407.0202/) (2014)
- [Sparsified SGD with Memory](https://mlsanity.com/paper/1809.07599/) (2018)
- [Error Feedback Fixes SignSGD and other Gradient Compression Schemes](https://mlsanity.com/paper/1901.09847/) (2019)
- [Momentum-Based Variance Reduction in Non-Convex SGD](https://mlsanity.com/paper/1905.10018/) (2019)
- [SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator](https://mlsanity.com/paper/1807.01695/) (2018)
- [The Convergence of Sparsified Gradient Methods](https://mlsanity.com/paper/1809.10505/) (2018)
- [SGD: General Analysis and Improved Rates](https://mlsanity.com/paper/1901.09401/) (2019)
- [Gradient Sparsification for Communication-Efficient Distributed Optimization](https://mlsanity.com/paper/1710.09854/) (2017)
- [Minimizing Finite Sums with the Stochastic Average Gradient](https://mlsanity.com/paper/1309.2388/) (2013)
- [Better Theory for SGD in the Nonconvex World](https://mlsanity.com/paper/2002.03329/) (2020)
- [Stochastic Variance Reduction for Nonconvex Optimization](https://mlsanity.com/paper/1603.06160/) (2016)
- [Katyusha: The First Direct Acceleration of Stochastic Gradient Methods](https://mlsanity.com/paper/1603.05953/) (2016)
- [EF21: A New, Simpler, Theoretically Better, and Practically Faster Error Feedback](https://mlsanity.com/paper/2106.05203/) (2021)
- [PAGE: A Simple and Optimal Probabilistic Gradient Estimator for Nonconvex Optimization](https://mlsanity.com/paper/2008.10898/) (2020)
- [Distributed Learning with Compressed Gradient Differences](https://mlsanity.com/paper/1901.09269/) (2019)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Acceleration for Compressed Gradient Descent in Distributed and Federated Optimization](https://mlsanity.com/paper/2002.11364/) (2020)
- [A Unified Theory of SGD: Variance Reduction, Sampling, Quantization and Coordinate Descent](https://mlsanity.com/paper/1905.11261/) (2019)
- [On the Convergence of SGD with Biased Gradients](https://mlsanity.com/paper/2008.00051/) (2020)
- [Communication-efficient distributed SGD with Sketching](https://mlsanity.com/paper/1903.04488/) (2019)
- [Variance-Reduced Methods for Machine Learning](https://mlsanity.com/paper/2010.00892/) (2020)
- [MARINA: Faster Non-Convex Distributed Learning with Compression](https://mlsanity.com/paper/2102.07845/) (2021)
- [Variance Reduction for Faster Non-Convex Optimization](https://mlsanity.com/paper/1603.05643/) (2016)
- [Don't Jump Through Hoops and Remove Those Loops: SVRG and Katyusha are Better Without the Outer Loop](https://mlsanity.com/paper/1901.08689/) (2019)
- [Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifolds](https://mlsanity.com/paper/1605.07147/) (2016)
- [Global rates of convergence for nonconvex optimization on manifolds](https://mlsanity.com/paper/1605.08101/) (2016)
- [Sparse Binary Compression: Towards Distributed Deep Learning with minimal Communication](https://mlsanity.com/paper/1805.08768/) (2018)

## Built on by (most cited first, 2 of 2)

- [Provable Model-Parallel Distributed Principal Component Analysis with Parallel Deflation](https://mlsanity.com/paper/2502.17615/) (2025)
- [Theoretical Foundations of Communication-Efficient, Robust, and Practical Distributed and Federated Optimization](https://mlsanity.com/paper/2608.06563/) (2026)

## Abstract (arXiv)

In this study, we investigate stochastic optimization on Riemannian manifolds, focusing on the crucial variance reduction mechanism used in both Euclidean and Riemannian settings. Riemannian variance-reduced methods usually involve a double-loop structure, computing a full gradient at the start of each loop. Determining the optimal inner loop length is challenging in practice, as it depends on strong convexity or smoothness constants, which are often unknown or hard to estimate. Motivated by Euclidean methods, we introduce the Riemannian Loopless SVRG (R-LSVRG) and PAGE (R-PAGE) methods. These methods replace the outer loop with probabilistic gradient computation triggered by a coin flip in each iteration, ensuring simpler proofs, efficient hyperparameter selection, and sharp convergence guarantees. Using R-PAGE as a framework for non-convex Riemannian optimization, we demonstrate its applicability to various important settings. For example, we derive Riemannian MARINA (R-MARINA) for distributed settings with communication compression, providing the best theoretical communication complexity guarantees for non-convex distributed optimization over Riemannian manifolds. Experimental results support our theoretical findings.