# Error Feedback Can Accurately Compress Preconditioners

arXiv: https://arxiv.org/abs/2306.06098 · HTML: https://mlsanity.com/paper/2306.06098/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 40 references that are in the map.
- First posted on arXiv 9 Jun 2023.

## Builds on (most cited first, 20 of 20)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [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)
- [PowerSGD: Practical Low-Rank Gradient Compression for Distributed Optimization](https://mlsanity.com/paper/1905.13727/) (2019)
- [The Convergence of Sparsified Gradient Methods](https://mlsanity.com/paper/1809.10505/) (2018)
- [ADAHESSIAN: An Adaptive Second Order Optimizer for Machine Learning](https://mlsanity.com/paper/2006.00719/) (2020)
- [A Kronecker-factored approximate Fisher matrix for convolution layers](https://mlsanity.com/paper/1602.01407/) (2016)
- [Three Mechanisms of Weight Decay Regularization](https://mlsanity.com/paper/1810.12281/) (2018)
- [EigenDamage: Structured Pruning in the Kronecker-Factored Eigenbasis](https://mlsanity.com/paper/1905.05934/) (2019)
- [M-FAC: Efficient Matrix-Free Approximations of Second-Order Information](https://mlsanity.com/paper/2107.03356/) (2021)
- [Large-Scale Distributed Second-Order Optimization Using Kronecker-Factored Approximate Curvature for Deep Convolutional Neural Networks](https://mlsanity.com/paper/1811.12019/) (2018)
- [On Distributed Adaptive Optimization with Gradient Compression](https://mlsanity.com/paper/2205.05632/) (2022)
- [Neumann Optimizer: A Practical Optimization Algorithm for Deep Neural Networks](https://mlsanity.com/paper/1712.03298/) (2017)
- [Elastic Consistency: A General Consistency Model for Distributed Stochastic Gradient Descent](https://mlsanity.com/paper/2001.05918/) (2020)

## Built on by (most cited first, 6 of 6)

- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [MicroAdam: Accurate Adaptive Optimization with Low Space Overhead and Provable Convergence](https://mlsanity.com/paper/2405.15593/) (2024)
- [SUMO: Subspace-Aware Moment-Orthogonalization for Accelerating Memory-Efficient LLM Training](https://mlsanity.com/paper/2505.24749/) (2025)
- [AdaRankGrad: Adaptive Gradient-Rank and Moments for Memory-Efficient LLMs Training and Fine-Tuning](https://mlsanity.com/paper/2410.17881/) (2024)
- [LORENZA: Enhancing Generalization in Low-Rank Gradient LLM Training via Efficient Zeroth-Order Adaptive SAM](https://mlsanity.com/paper/2502.19571/) (2025)
- [DASH: Faster Shampoo via Batched Block Preconditioning and Efficient Inverse-Root Solvers](https://mlsanity.com/paper/2602.02016/) (2026)

## Abstract (arXiv)

Leveraging second-order information about the loss at the scale of deep networks is one of the main lines of approach for improving the performance of current optimizers for deep learning. Yet, existing approaches for accurate full-matrix preconditioning, such as Full-Matrix Adagrad (GGT) or Matrix-Free Approximate Curvature (M-FAC) suffer from massive storage costs when applied even to small-scale models, as they must store a sliding window of gradients, whose memory requirements are multiplicative in the model dimension. In this paper, we address this issue via a novel and efficient error-feedback technique that can be applied to compress preconditioners by up to two orders of magnitude in practice, without loss of convergence. Specifically, our approach compresses the gradient information via sparsification or low-rank compression before it is fed into the preconditioner, feeding the compression error back into future iterations. Experiments on deep neural networks show that this approach can compress full-matrix preconditioners to up to 99% sparsity without accuracy loss, effectively removing the memory overhead of full-matrix preconditioners such as GGT and M-FAC. Our code is available at urlhttps://github.com/IST-DASLab/EFCP.