# Rennala MVR: Improved Time Complexity for Parallel Stochastic Optimization via Momentum-Based Variance Reduction

arXiv: https://arxiv.org/abs/2605.08871 · HTML: https://mlsanity.com/paper/2605.08871/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 37 references that are in the map.
- First posted on arXiv 9 May 2026.

## Builds on (most cited first, 29 of 29)

- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [SAGA: A Fast Incremental Gradient Method With Support for Non-Strongly Convex Composite Objectives](https://mlsanity.com/paper/1407.0202/) (2014)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (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)
- [SARAH: A Novel Method for Machine Learning Problems Using Stochastic Recursive Gradient](https://mlsanity.com/paper/1703.00102/) (2017)
- [PAGE: A Simple and Optimal Probabilistic Gradient Estimator for Nonconvex Optimization](https://mlsanity.com/paper/2008.10898/) (2020)
- [Sharper Convergence Guarantees for Asynchronous SGD for Distributed and Federated Learning](https://mlsanity.com/paper/2206.08307/) (2022)
- [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)
- [MARS: Unleashing the Power of Variance Reduction for Training Large Models](https://mlsanity.com/paper/2411.10438/) (2024)
- [Stochastic Nested Variance Reduction for Nonconvex Optimization](https://mlsanity.com/paper/1806.07811/) (2018)
- [Asynchronous SGD Beats Minibatch SGD Under Arbitrary Delays](https://mlsanity.com/paper/2206.07638/) (2022)
- [On the Ineffectiveness of Variance Reduced Optimization for Deep Learning](https://mlsanity.com/paper/1812.04529/) (2018)
- [Advances in Asynchronous Parallel and Distributed Optimization](https://mlsanity.com/paper/2006.13838/) (2020)
- [Improved SVRG for Non-Strongly-Convex or Sum-of-Non-Convex Objectives](https://mlsanity.com/paper/1506.01972/) (2015)
- [Stochastic Quasi-Gradient Methods: Variance Reduction via Jacobian Sketching](https://mlsanity.com/paper/1805.02632/) (2018)
- [Permutation Compressors for Provably Faster Distributed Nonconvex Optimization](https://mlsanity.com/paper/2110.03300/) (2021)
- [Optimal Time Complexities of Parallel Stochastic Optimization Methods Under a Fixed Computation Model](https://mlsanity.com/paper/2305.12387/) (2023)
- [Ringmaster ASGD: The First Asynchronous SGD with Optimal Time Complexity](https://mlsanity.com/paper/2501.16168/) (2025)
- [Semi-Stochastic Gradient Descent Methods](https://mlsanity.com/paper/1312.1666/) (2013)
- [Shadowheart SGD: Distributed Asynchronous SGD with Optimal Time Complexity Under Arbitrary Computation and Communication Heterogeneity](https://mlsanity.com/paper/2402.04785/) (2024)
- [Tight Time Complexities in Parallel Stochastic Optimization with Arbitrary Computation Dynamics](https://mlsanity.com/paper/2408.04929/) (2024)
- [On the Optimal Time Complexities in Decentralized Stochastic Asynchronous Optimization](https://mlsanity.com/paper/2405.16218/) (2024)
- [Freya PAGE: First Optimal Time Complexity for Large-Scale Nonconvex Finite-Sum Optimization with Heterogeneous Asynchronous Computations](https://mlsanity.com/paper/2405.15545/) (2024)
- [ATA: Adaptive Task Allocation for Efficient Resource Management in Distributed Machine Learning](https://mlsanity.com/paper/2502.00775/) (2025)
- [Ringleader ASGD: The First Asynchronous SGD with Optimal Time Complexity under Data Heterogeneity](https://mlsanity.com/paper/2509.22860/) (2025)
- [First Provably Optimal Asynchronous SGD for Homogeneous and Heterogeneous Data](https://mlsanity.com/paper/2601.02523/) (2026)
- [Optimizing Asynchronous Federated Learning: A Delicate Trade-Off Between Model-Parameter Staleness and Update Frequency](https://mlsanity.com/paper/2502.08206/) (2025)
- [Optimal Asynchronous Stochastic Nonconvex Optimization under Heavy-Tailed Noise](https://mlsanity.com/paper/2601.19379/) (2026)

## Abstract (arXiv)

Large-scale machine learning models are trained on clusters of machines that exhibit heterogeneous performance due to hardware variability, network delays, and system-level instabilities. In such environments, time complexity rather than iteration complexity becomes the relevant performance metric for optimization algorithms. Recent work by Tyurin and Richt'arik (2023) established the first time complexity analysis for parallel first-order stochastic optimization, proposing Rennala SGD as a time-optimal method for smooth nonconvex optimization. However, Rennala SGD is fundamentally a modification of SGD, and variance reduction techniques are known to improve the iteration complexity of SGD. In this work, we investigate whether variance reduction can also improve time complexity in heterogeneous systems. We show that, under a mean-squared smoothness assumption, variance reduction can improve time complexity in relevant parameter regimes. To this end, we propose Rennala MVR, a variance-reduced extension of Rennala SGD based on momentum-based variance reduction, and analyze its oracle and time complexity. We establish lower bounds for time complexity under these assumptions. On a stochastic quadratic benchmark, experiments with the exact method support the theory, while neural-network experiments with a practical inexact variant show similar empirical gains over Rennala SGD.