# EF-BV: A Unified Theory of Error Feedback and Variance Reduction Mechanisms for Biased and Unbiased Compression in Distributed Optimization

arXiv: https://arxiv.org/abs/2205.04180 · HTML: https://mlsanity.com/paper/2205.04180/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 60 references that are in the map.
- First posted on arXiv 9 May 2022.

## Builds on (most cited first, 30 of 36)

- [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)
- [Federated Learning: Challenges, Methods, and Future Directions](https://mlsanity.com/paper/1908.07873/) (2019)
- [Federated Learning: Strategies for Improving Communication Efficiency](https://mlsanity.com/paper/1610.05492/) (2016)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Robust and Communication-Efficient Federated Learning from Non-IID Data](https://mlsanity.com/paper/1903.02891/) (2019)
- [TernGrad: Ternary Gradients to Reduce Communication in Distributed Deep Learning](https://mlsanity.com/paper/1705.07878/) (2017)
- [The Convergence of Sparsified Gradient Methods](https://mlsanity.com/paper/1809.10505/) (2018)
- [Gradient Sparsification for Communication-Efficient Distributed Optimization](https://mlsanity.com/paper/1710.09854/) (2017)
- [A Survey on Distributed Machine Learning](https://mlsanity.com/paper/1912.09789/) (2019)
- [Qsparse-local-SGD: Distributed SGD with Quantization, Sparsification, and Local Computations](https://mlsanity.com/paper/1906.02367/) (2019)
- [On Biased Compression for Distributed Learning](https://mlsanity.com/paper/2002.12410/) (2020)
- [EF21: A New, Simpler, Theoretically Better, and Practically Faster Error Feedback](https://mlsanity.com/paper/2106.05203/) (2021)
- [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)
- [Variance-Reduced Methods for Machine Learning](https://mlsanity.com/paper/2010.00892/) (2020)
- [Stochastic Distributed Learning with Gradient Quantization and Variance Reduction](https://mlsanity.com/paper/1904.05115/) (2019)
- [Linearly Converging Error Compensated SGD](https://mlsanity.com/paper/2010.12292/) (2020)
- [Bidirectional compression in heterogeneous settings for distributed or federated learning with partial participation: tight convergence guarantees](https://mlsanity.com/paper/2006.14591/) (2020)
- [Uncertainty Principle for Communication Compression in Distributed and Federated Learning and the Search for an Optimal Compressor](https://mlsanity.com/paper/2002.08958/) (2020)
- [Optimal Gradient Compression for Distributed and Federated Learning](https://mlsanity.com/paper/2010.03246/) (2020)
- [Unified Analysis of Stochastic Gradient Methods for Composite Convex and Smooth Optimization](https://mlsanity.com/paper/2006.11573/) (2020)
- [Stochastic Quasi-Gradient Methods: Variance Reduction via Jacobian Sketching](https://mlsanity.com/paper/1805.02632/) (2018)
- [On the Discrepancy between the Theoretical Analysis and Practical Implementations of Compressed Communication for Distributed Deep Learning](https://mlsanity.com/paper/1911.08250/) (2019)
- [Permutation Compressors for Provably Faster Distributed Nonconvex Optimization](https://mlsanity.com/paper/2110.03300/) (2021)
- [rTop-k: A Statistical Estimation Approach to Distributed SGD](https://mlsanity.com/paper/2005.10761/) (2020)
- [RATQ: A Universal Fixed-Length Quantizer for Stochastic Optimization](https://mlsanity.com/paper/1908.08200/) (2019)
- [Proximal Splitting Algorithms for Convex Optimization: A Tour of Recent Advances, with New Twists](https://mlsanity.com/paper/1912.00137/) (2019)
- [MURANA: A Generic Framework for Stochastic Variance-Reduced Optimization](https://mlsanity.com/paper/2106.03056/) (2021)

## Built on by (most cited first, 17 of 17)

- [A Guide Through the Zoo of Biased SGD](https://mlsanity.com/paper/2305.16296/) (2023)
- [Communication Acceleration of Local Gradient Methods via an Accelerated Primal-Dual Algorithm with Inexact Prox](https://mlsanity.com/paper/2207.03957/) (2022)
- [GradSkip: Communication-Accelerated Local Gradient Methods with Better Computational Complexity](https://mlsanity.com/paper/2210.16402/) (2022)
- [z-SignFedAvg: A Unified Stochastic Sign-based Compression for Federated Learning](https://mlsanity.com/paper/2302.02589/) (2023)
- [CompressedScaffnew: The First Theoretical Double Acceleration of Communication from Local Training and Compression in Distributed Optimization](https://mlsanity.com/paper/2210.13277/) (2022)
- [LoCoDL: Communication-Efficient Distributed Learning with Local Training and Compression](https://mlsanity.com/paper/2403.04348/) (2024)
- [QUIC-FL: Quick Unbiased Compression for Federated Learning](https://mlsanity.com/paper/2205.13341/) (2022)
- [Shifted Compression Framework: Generalizations and Improvements](https://mlsanity.com/paper/2206.10452/) (2022)
- [Unbiased and Sign Compression in Distributed Learning: Comparing Noise Resilience via SDEs](https://mlsanity.com/paper/2502.17009/) (2025)
- [TAMUNA: Doubly Accelerated Distributed Optimization under Partial Participation](https://mlsanity.com/paper/2302.09832/) (2023)
- [Problem-dependent convergence bounds for randomized linear gradient compression](https://mlsanity.com/paper/2411.12898/) (2024)
- [Mask-Encoded Sparsification: Mitigating Biased Gradients in Communication-Efficient Split Learning](https://mlsanity.com/paper/2408.13787/) (2024)
- [BiCoLoR: Communication-Efficient Optimization with Bidirectional Compression and Local Training](https://mlsanity.com/paper/2601.12400/) (2026)
- [Unified analysis of SGD-type methods](https://mlsanity.com/paper/2303.16502/) (2023)
- [Beyond Communication Overhead: A Multilevel Monte Carlo Approach for Mitigating Compression Bias in Distributed Learning](https://mlsanity.com/paper/2507.05508/) (2025)
- [Byzantine-Robust and Communication-Efficient Distributed Training: Compressive and Cyclic Gradient Coding](https://mlsanity.com/paper/2603.28780/) (2026)
- [A Tight Theory of Error Feedback Algorithms in Distributed Optimization](https://mlsanity.com/paper/2605.31594/) (2026)

## Abstract (arXiv)

In distributed or federated optimization and learning, communication between the different computing units is often the bottleneck and gradient compression is widely used to reduce the number of bits sent within each communication round of iterative methods. There are two classes of compression operators and separate algorithms making use of them. In the case of unbiased random compressors with bounded variance (e.g., rand-k), the DIANA algorithm of Mishchenko et al. (2019), which implements a variance reduction technique for handling the variance introduced by compression, is the current state of the art. In the case of biased and contractive compressors (e.g., top-k), the EF21 algorithm of Richt'arik et al. (2021), which instead implements an error-feedback mechanism, is the current state of the art. These two classes of compression schemes and algorithms are distinct, with different analyses and proof techniques. In this paper, we unify them into a single framework and propose a new algorithm, recovering DIANA and EF21 as particular cases. Our general approach works with a new, larger class of compressors, which has two parameters, the bias and the variance, and includes unbiased and biased compressors as particular cases. This allows us to inherit the best of the two worlds: like EF21 and unlike DIANA, biased compressors, like top-k, whose good performance in practice is recognized, can be used. And like DIANA and unlike EF21, independent randomness at the compressors allows to mitigate the effects of compression, with the convergence rate improving when the number of parallel workers is large. This is the first time that an algorithm with all these features is proposed. We prove its linear convergence under certain conditions. Our approach takes a step towards better understanding of two so-far distinct worlds of communication-efficient distributed learning.