# Tight analyses of first-order methods with error feedback

arXiv: https://arxiv.org/abs/2506.05271 · HTML: https://mlsanity.com/paper/2506.05271/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 68 references that are in the map.
- First posted on arXiv 5 Jun 2025.

## Builds on (most cited first, 30 of 48)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Advances and Open Problems in Federated Learning](https://mlsanity.com/paper/1912.04977/) (2019)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [Robust and Communication-Efficient Federated Learning from Non-IID Data](https://mlsanity.com/paper/1903.02891/) (2019)
- [Sparsified SGD with Memory](https://mlsanity.com/paper/1809.07599/) (2018)
- [FedPAQ: A Communication-Efficient Federated Learning Method with Periodic Averaging and Quantization](https://mlsanity.com/paper/1909.13014/) (2019)
- [Error Feedback Fixes SignSGD and other Gradient Compression Schemes](https://mlsanity.com/paper/1901.09847/) (2019)
- [The Convergence of Sparsified Gradient Methods](https://mlsanity.com/paper/1809.10505/) (2018)
- [Decentralized Stochastic Optimization and Gossip Algorithms with Compressed Communication](https://mlsanity.com/paper/1902.00340/) (2019)
- [Federated Learning with Compression: Unified Analysis and Sharp Guarantees](https://mlsanity.com/paper/2007.01154/) (2020)
- [On Biased Compression for Distributed Learning](https://mlsanity.com/paper/2002.12410/) (2020)
- [Dissecting Adam: The Sign, Magnitude and Variance of Stochastic Gradients](https://mlsanity.com/paper/1705.07774/) (2017)
- [EF21: A New, Simpler, Theoretically Better, and Practically Faster Error Feedback](https://mlsanity.com/paper/2106.05203/) (2021)
- [Dynamic Model Pruning with Feedback](https://mlsanity.com/paper/2006.07253/) (2020)
- [Distributed Learning with Compressed Gradient Differences](https://mlsanity.com/paper/1901.09269/) (2019)
- [DoubleSqueeze: Parallel Stochastic Gradient Descent with Double-Pass Error-Compensated Compression](https://mlsanity.com/paper/1905.05957/) (2019)
- [Natural Compression for Distributed Deep Learning](https://mlsanity.com/paper/1905.10988/) (2019)
- [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)
- [Communication-efficient distributed SGD with Sketching](https://mlsanity.com/paper/1903.04488/) (2019)
- [Error Compensated Quantized SGD and its Applications to Large-scale Distributed Optimization](https://mlsanity.com/paper/1806.08054/) (2018)
- [Linearly Converging Error Compensated SGD](https://mlsanity.com/paper/2010.12292/) (2020)
- [Distributed learning with compressed gradients](https://mlsanity.com/paper/1806.06573/) (2018)
- [Communication-Efficient Distributed Blockwise Momentum SGD with Error-Feedback](https://mlsanity.com/paper/1905.10936/) (2019)
- [Bidirectional compression in heterogeneous settings for distributed or federated learning with partial participation: tight convergence guarantees](https://mlsanity.com/paper/2006.14591/) (2020)
- [BEER: Fast O(1/T) Rate for Decentralized Nonconvex Optimization with Communication Compression](https://mlsanity.com/paper/2201.13320/) (2022)
- [Smooth Strongly Convex Interpolation and Exact Worst-case Performance of First-order Methods](https://mlsanity.com/paper/1502.05666/) (2015)
- [A Double Residual Compression Algorithm for Efficient Distributed Learning](https://mlsanity.com/paper/1910.07561/) (2019)
- [Momentum Provably Improves Error Feedback!](https://mlsanity.com/paper/2305.15155/) (2023)
- [EF21-P and Friends: Improved Theoretical Communication Complexity for Distributed Optimization with Bidirectional Compression](https://mlsanity.com/paper/2209.15218/) (2022)

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

- [A Tight Theory of Error Feedback Algorithms in Distributed Optimization](https://mlsanity.com/paper/2605.31594/) (2026)

## Abstract (arXiv)

Communication between agents often constitutes a major computational bottleneck in distributed learning. One of the most common mitigation strategies is to compress the information exchanged, thereby reducing communication overhead. To counteract the degradation in convergence associated with compressed communication, error feedback schemes -- most notably EF and EF21 -- were introduced. In this work, we provide a tight analysis of both of these methods. Specifically, we find the Lyapunov function that yields the best possible convergence rate for each method -- with matching lower bounds. This principled approach yields sharp performance guarantees and enables a rigorous, apples-to-apples comparison between EF, EF21, and compressed gradient descent. Our analysis is carried out in the simplified single-agent setting, which allows for clean theoretical insights and fair comparison of the underlying mechanisms.