# DiscQuant: A Quantization Method for Neural Networks Inspired by Discrepancy Theory

arXiv: https://arxiv.org/abs/2501.06417 · HTML: https://mlsanity.com/paper/2501.06417/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 38 references that are in the map.
- First posted on arXiv 11 Jan 2025.

## Builds on (most cited first, 24 of 24)

- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Phi-3 Technical Report: A Highly Capable Language Model Locally on Your Phone](https://mlsanity.com/paper/2404.14219/) (2024)
- [SmoothQuant: Accurate and Efficient Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2211.10438/) (2022)
- [AWQ: Activation-aware Weight Quantization for LLM Compression and Acceleration](https://mlsanity.com/paper/2306.00978/) (2023)
- [SparseGPT: Massive Language Models Can Be Accurately Pruned in One-Shot](https://mlsanity.com/paper/2301.00774/) (2023)
- [A Simple and Effective Pruning Approach for Large Language Models](https://mlsanity.com/paper/2306.11695/) (2023)
- [KIVI: A Tuning-Free Asymmetric 2bit Quantization for KV Cache](https://mlsanity.com/paper/2402.02750/) (2024)
- [QuaRot: Outlier-Free 4-Bit Inference in Rotated LLMs](https://mlsanity.com/paper/2404.00456/) (2024)
- [Up or Down? Adaptive Rounding for Post-Training Quantization](https://mlsanity.com/paper/2004.10568/) (2020)
- [BRECQ: Pushing the Limit of Post-Training Quantization by Block Reconstruction](https://mlsanity.com/paper/2102.05426/) (2021)
- [SpinQuant: LLM quantization with learned rotations](https://mlsanity.com/paper/2405.16406/) (2024)
- [OmniQuant: Omnidirectionally Calibrated Quantization for Large Language Models](https://mlsanity.com/paper/2308.13137/) (2023)
- [QuIP: 2-Bit Quantization of Large Language Models With Guarantees](https://mlsanity.com/paper/2307.13304/) (2023)
- [QuIP#: Even Better LLM Quantization with Hadamard Incoherence and Lattice Codebooks](https://mlsanity.com/paper/2402.04396/) (2024)
- [SpQR: A Sparse-Quantized Representation for Near-Lossless LLM Weight Compression](https://mlsanity.com/paper/2306.03078/) (2023)
- [Optimal Brain Compression: A Framework for Accurate Post-Training Quantization and Pruning](https://mlsanity.com/paper/2208.11580/) (2022)
- [SqueezeLLM: Dense-and-Sparse Quantization](https://mlsanity.com/paper/2306.07629/) (2023)
- [Extreme Compression of Large Language Models via Additive Quantization](https://mlsanity.com/paper/2401.06118/) (2024)
- [QTIP: Quantization with Trellises and Incoherence Processing](https://mlsanity.com/paper/2406.11235/) (2024)
- [GPTVQ: The Blessing of Dimensionality for LLM Quantization](https://mlsanity.com/paper/2402.15319/) (2024)
- [MARLIN: Mixed-Precision Auto-Regressive Parallel Inference on Large Language Models](https://mlsanity.com/paper/2408.11743/) (2024)
- [QuantEase: Optimization-based Quantization for Language Models](https://mlsanity.com/paper/2309.01885/) (2023)
- [A Greedy Algorithm for Quantizing Neural Networks](https://mlsanity.com/paper/2010.15979/) (2020)
- [Sparse Fine-tuning for Inference Acceleration of Large Language Models](https://mlsanity.com/paper/2310.06927/) (2023)

## Built on by (most cited first, 3 of 3)

- [Model-Preserving Adaptive Rounding](https://mlsanity.com/paper/2505.22988/) (2025)
- [G2PTQ: Improving LLM Post-Training Quantization with Generalized Gradient Compensation](https://mlsanity.com/paper/2609.31009/) (2026)
- [Product-Aware Deterministic Rounding for Quantized Matrix Multiplication](https://mlsanity.com/paper/2609.31641/) (2026)

## Abstract (arXiv)

Quantizing the weights of a neural network has two steps: (1) Finding a good low bit-complexity representation for weights (which we call the quantization grid) and (2) Rounding the original weights to values in the quantization grid. In this paper, we study the problem of rounding optimally given any quantization grid. The simplest and most commonly used way to round is Round-to-Nearest (RTN). By rounding in a data-dependent way instead, one can improve the quality of the quantized model significantly. We study the rounding problem from the lens of discrepancy theory, which studies how well we can round a continuous solution to a discrete solution without affecting solution quality too much. We prove that given m=poly(1/ε) samples from the data distribution, we can round all but O(m) model weights such that the expected approximation error of the quantized model on the true data distribution is ≤ ε as long as the space of gradients of the original model is approximately low rank (which we empirically validate). Our proof, which is algorithmic, inspired a simple and practical rounding algorithm called DiscQuant. In our experiments, we demonstrate that DiscQuant significantly improves over the prior state-of-the-art rounding method called GPTQ and the baseline RTN over a range of benchmarks on Phi3mini-3.8B and Llama3.1-8B. For example, rounding Phi3mini-3.8B to a fixed quantization grid with 3.25 bits per parameter using DiscQuant gets 64% accuracy on the GSM8k dataset, whereas GPTQ achieves 54% and RTN achieves 31% (the original model achieves 84%). We make our code available at https://github.com/jerry-chee/DiscQuant.