# μ2-SGD: Stable Stochastic Optimization via a Double Momentum Mechanism

arXiv: https://arxiv.org/abs/2304.04172 · HTML: https://mlsanity.com/paper/2304.04172/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 34 references that are in the map.
- First posted on arXiv 9 Apr 2023.

## Builds on (most cited first, 13 of 13)

- [Google's Neural Machine Translation System: Bridging the Gap between Human and Machine Translation](https://mlsanity.com/paper/1609.08144/) (2016)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Momentum-Based Variance Reduction in Non-Convex SGD](https://mlsanity.com/paper/1905.10018/) (2019)
- [Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition](https://mlsanity.com/paper/1503.02101/) (2015)
- [Optimization, Learning, and Games with Predictable Sequences](https://mlsanity.com/paper/1311.1869/) (2013)
- [Learning-Rate-Free Learning by D-Adaptation](https://mlsanity.com/paper/2301.07733/) (2023)
- [DoG is SGD's Best Friend: A Parameter-Free Dynamic Step Size Schedule](https://mlsanity.com/paper/2302.12022/) (2023)
- [UniXGrad: A Universal, Adaptive Algorithm with Optimal Guarantees for Constrained Optimization](https://mlsanity.com/paper/1910.13857/) (2019)
- [Online Adaptive Methods, Universality and Acceleration](https://mlsanity.com/paper/1809.02864/) (2018)
- [Parameter-free Mirror Descent](https://mlsanity.com/paper/2203.00444/) (2022)
- [No-Regret Dynamics in the Fenchel Game: A Unified Framework for Algorithmic Convex Optimization](https://mlsanity.com/paper/2111.11309/) (2021)
- [At Stability's Edge: How to Adjust Hyperparameters to Preserve Minima Selection in Asynchronous Training of Neural Networks?](https://mlsanity.com/paper/1909.12340/) (2019)

## Built on by (most cited first, 4 of 4)

- [Fault Tolerant ML: Efficient Meta-Aggregation and Synchronous Training](https://mlsanity.com/paper/2405.14759/) (2024)
- [Private and Federated Stochastic Convex Optimization: Efficient Strategies for Centralized Systems](https://mlsanity.com/paper/2407.12396/) (2024)
- [Weight for Robustness: A Comprehensive Approach towards Optimal Fault-Tolerant Asynchronous ML](https://mlsanity.com/paper/2501.09621/) (2025)
- [Privacy-Preserving Federated Convex Optimization: Balancing Partial-Participation and Efficiency via Noise Cancellation](https://mlsanity.com/paper/2506.02563/) (2025)

## Abstract (arXiv)

We consider stochastic convex optimization problems where the objective is an expectation over smooth functions. For this setting we suggest a novel gradient estimate that combines two recent mechanism that are related to notion of momentum. Then, we design an SGD-style algorithm as well as an accelerated version that make use of this new estimator, and demonstrate the robustness of these new approaches to the choice of the learning rate. Concretely, we show that these approaches obtain the optimal convergence rates for both noiseless and noisy case with the same choice of fixed learning rate. Moreover, for the noisy case we show that these approaches achieve the same optimal bound for a very wide range of learning rates.