# QUASAR: Lowering the Loss Floor of Quantization-Aware Training with Loss-Aware Reconstruction

arXiv: https://arxiv.org/abs/2608.13966 · HTML: https://mlsanity.com/paper/2608.13966/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 63 of its 80 references that are in the map.
- First posted on arXiv 14 Aug 2026.

## Builds on (most cited first, 30 of 63)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [LiveCodeBench: Holistic and Contamination Free Evaluation of Large Language Models for Code](https://mlsanity.com/paper/2403.07974/) (2024)
- [GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers](https://mlsanity.com/paper/2210.17323/) (2022)
- [MMLU-Pro: A More Robust and Challenging Multi-Task Language Understanding Benchmark](https://mlsanity.com/paper/2406.01574/) (2024)
- [RULER: What's the Real Context Size of Your Long-Context Language Models?](https://mlsanity.com/paper/2404.06654/) (2024)
- [AWQ: Activation-aware Weight Quantization for LLM Compression and Acceleration](https://mlsanity.com/paper/2306.00978/) (2023)
- [Instruction-Following Evaluation for Large Language Models](https://mlsanity.com/paper/2311.07911/) (2023)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [A Survey of Quantization Methods for Efficient Neural Network Inference](https://mlsanity.com/paper/2103.13630/) (2021)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Learned Step Size Quantization](https://mlsanity.com/paper/1902.08153/) (2019)
- [Up or Down? Adaptive Rounding for Post-Training Quantization](https://mlsanity.com/paper/2004.10568/) (2020)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [BRECQ: Pushing the Limit of Post-Training Quantization by Block Reconstruction](https://mlsanity.com/paper/2102.05426/) (2021)
- [QuIP: 2-Bit Quantization of Large Language Models With Guarantees](https://mlsanity.com/paper/2307.13304/) (2023)
- [LongBench v2: Towards Deeper Understanding and Reasoning on Realistic Long-context Multitasks](https://mlsanity.com/paper/2412.15204/) (2024)
- [QuIP#: Even Better LLM Quantization with Hadamard Incoherence and Lattice Codebooks](https://mlsanity.com/paper/2402.04396/) (2024)
- [LLM-QAT: Data-Free Quantization Aware Training for Large Language Models](https://mlsanity.com/paper/2305.17888/) (2023)
- [Optimal Brain Compression: A Framework for Accurate Post-Training Quantization and Pruning](https://mlsanity.com/paper/2208.11580/) (2022)

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

- [Why Gated DeltaNet Survives 4-Bit Quantization: NVFP4 W4A4 for the Recurrent Half of a Hybrid 27B LLM](https://mlsanity.com/paper/2609.04098/) (2026)

## Abstract (arXiv)

As large language model inference shifts toward lower precision, post-training quantization (PTQ) becomes increasingly brittle, making quantization-aware training (QAT) essential for preserving model quality. However, QAT has a structural mismatch: gradient updates are applied to latent full-precision weights, while the loss and gradients are computed on lossy reconstructions of those weights. This mismatch can lead to suboptimal training trajectories and a higher loss floor. Second-order PTQ methods address a similar problem by minimizing loss-aware reconstruction error, but applying such expensive reconstruction repeatedly during QAT as the weights evolve is impractical. We introduce QUASAR, a QAT method that brings lightweight, loss-aware reconstruction into the training loop. At each training step, QUASAR reconstructs the latent weights by searching over a small set of clipping ranges and fitting dequantization parameters through saliency-weighted least squares. We use an exponential moving average of squared gradients as the per-parameter saliency signal. Our theoretical analysis shows that optimizing QUASAR's reconstruction objective tightens both the convergence and final-loss bounds of QAT. We evaluate QUASAR across four model families and across INT4, INT3, INT2, and NVFP4 quantization formats. QUASAR consistently achieves lower training and evaluation loss than competitive QAT methods and outperforms QAT and PTQ baselines on downstream benchmarks. At INT2, QUASAR improves average accuracy over the best QAT baseline by 13.3 points with quantization-aware distillation and by 10.9 points with QAT on mathematical reasoning data. Notably, after distillation on only about 600M tokens, QUASAR's INT4 Gemma-4 E4B checkpoint outperforms the corresponding QAT checkpoint released by Google, with 66% lower KL divergence and 1.8 points higher average accuracy.