# ECO: Quantized Training without Full-Precision Master Weights

arXiv: https://arxiv.org/abs/2601.22101 · HTML: https://mlsanity.com/paper/2601.22101/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in MoE large language models, one of 12 directions in Mixture-of-experts.
- Builds on 39 of its 44 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## Placed in

- Mixture-of-experts / [MoE large language models](https://mlsanity.com/moe/moe-large-language-models/)

## Builds on (most cited first, 30 of 39)

- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [SentencePiece: A simple and language independent subword tokenizer and detokenizer for Neural Text Processing](https://mlsanity.com/paper/1808.06226/) (2018)
- [GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers](https://mlsanity.com/paper/2210.17323/) (2022)
- [Gemma 3 Technical Report](https://mlsanity.com/paper/2503.19786/) (2025)
- [AWQ: Activation-aware Weight Quantization for LLM Compression and Acceleration](https://mlsanity.com/paper/2306.00978/) (2023)
- [Mixed Precision Training](https://mlsanity.com/paper/1710.03740/) (2017)
- [DeepSeekMoE: Towards Ultimate Expert Specialization in Mixture-of-Experts Language Models](https://mlsanity.com/paper/2401.06066/) (2024)
- [QuaRot: Outlier-Free 4-Bit Inference in Rotated LLMs](https://mlsanity.com/paper/2404.00456/) (2024)
- [Learned Step Size Quantization](https://mlsanity.com/paper/1902.08153/) (2019)
- [DoReFa-Net: Training Low Bitwidth Convolutional Neural Networks with Low Bitwidth Gradients](https://mlsanity.com/paper/1606.06160/) (2016)
- [Kimi K2: Open Agentic Intelligence](https://mlsanity.com/paper/2507.20534/) (2025)
- [SpinQuant: LLM quantization with learned rotations](https://mlsanity.com/paper/2405.16406/) (2024)
- [PACT: Parameterized Clipping Activation for Quantized Neural Networks](https://mlsanity.com/paper/1805.06085/) (2018)
- [8-bit Optimizers via Block-wise Quantization](https://mlsanity.com/paper/2110.02861/) (2021)
- [LLM-QAT: Data-Free Quantization Aware Training for Large Language Models](https://mlsanity.com/paper/2305.17888/) (2023)
- [One Billion Word Benchmark for Measuring Progress in Statistical Language Modeling](https://mlsanity.com/paper/1312.3005/) (2013)
- [BitNet: Scaling 1-bit Transformers for Large Language Models](https://mlsanity.com/paper/2310.11453/) (2023)
- [Learning to Quantize Deep Networks by Optimizing Quantization Intervals with Task Loss](https://mlsanity.com/paper/1808.05779/) (2018)
- [On-Device Training Under 256KB Memory](https://mlsanity.com/paper/2206.15472/) (2022)
- [Microscaling Data Formats for Deep Learning](https://mlsanity.com/paper/2310.10537/) (2023)
- [EF21: A New, Simpler, Theoretically Better, and Practically Faster Error Feedback](https://mlsanity.com/paper/2106.05203/) (2021)
- [FlatQuant: Flatness Matters for LLM Quantization](https://mlsanity.com/paper/2410.09426/) (2024)
- [EfficientQAT: Efficient Quantization-Aware Training for Large Language Models](https://mlsanity.com/paper/2407.11062/) (2024)
- [FP8-LM: Training FP8 Large Language Models](https://mlsanity.com/paper/2310.18313/) (2023)
- [1-bit Adam: Communication Efficient Large-Scale Training with Adam's Convergence Speed](https://mlsanity.com/paper/2102.02888/) (2021)
- [Scaling FP8 training to trillion-token LLMs](https://mlsanity.com/paper/2409.12517/) (2024)
- [Memory Efficient Optimizers with 4-bit States](https://mlsanity.com/paper/2309.01507/) (2023)
- [ZeRO++: Extremely Efficient Collective Communication for Giant Model Training](https://mlsanity.com/paper/2306.10209/) (2023)

## Built on by (most cited first, 5 of 5)

- [Demystifying Manifold Constraints in LLM Pre-training](https://mlsanity.com/paper/2605.04418/) (2026)
- [When Can Depth Replace Precision? A Resource Theory of Quantized Neural Computation](https://mlsanity.com/paper/2607.23390/) (2026)
- [Auditing Invisible Weight Updates with Reference Traces](https://mlsanity.com/paper/2607.09800/) (2026)
- [Blog: Survey of Optimizers](https://mlsanity.com/paper/2608.28557/) (2026)
- [Lost in the bf16 Cast: Exporting Ternary Language Models Can Revert Most Low-Learning-Rate Code Changes](https://mlsanity.com/paper/2610.07853/) (2026)

## Abstract (arXiv)

Quantization has significantly improved the compute and memory efficiency of Large Language Model (LLM) training. However, existing approaches still rely on accumulating their updates in high-precision: concretely, gradient updates must be applied to a high-precision weight buffer, known as textitmaster weights. This buffer introduces substantial memory overhead, particularly for Sparse Mixture of Experts (SMoE) models, where model parameters and optimizer states dominate memory usage. To address this, we introduce the Error-Compensating Optimizer (ECO), which eliminates master weights by applying updates directly to quantized parameters. ECO quantizes weights after each step and carefully injects the resulting quantization error into the optimizer momentum, forming an error-feedback loop with no additional memory. We prove that, under standard assumptions and a decaying learning rate, ECO converges to a constant-radius neighborhood of the optimum, while naive master-weight removal can incur an error that is inversely proportional to the learning rate. We show empirical results for pretraining small Transformers (30-800M), a Gemma-3 1B model, and a 2.1B parameter Sparse MoE model with FP8 quantization, and fine-tuning DeepSeek-MoE-16B in INT4 precision. Throughout, ECO matches baselines with master weights up to near-lossless accuracy, significantly shifting the static memory vs validation loss Pareto frontier.