Revisiting Adaptive Rounding with Vectorized Reparameterization for LLM Quantization

TL;DR

This work tackles post-training quantization of large language models by rethinking adaptive rounding. It introduces VQRound, a parameter-efficient framework that reparameterizes the rounding matrix through a compact vector-quantized codebook, directly targeting the element-wise $L_\infty$ error to better handle heavy-tailed weight distributions. A Hessian-aware residual initialization and an end-to-end finetuning pipeline enable cross-layer error compensation with only 128 calibration samples, achieving faster convergence and competitive or superior perplexities while using around $0.2\%$ of the trainable parameters. Extensive experiments across OPT, LLaMA, LLaMA2, and Qwen3 demonstrate scalability, compatibility with existing PTQ tools, and improved stability over prior adaptive rounding methods. The approach offers a practical path to scalable, accurate low-bit quantization for modern LLMs, with code available for integration into existing pipelines.

Abstract

Adaptive Rounding has emerged as an alternative to round-to-nearest (RTN) for post-training quantization by enabling cross-element error cancellation. Yet, dense and element-wise rounding matrices are prohibitively expensive for billion-parameter large language models (LLMs). We revisit adaptive rounding from an efficiency perspective and propose VQRound, a parameter-efficient optimization framework that reparameterizes the rounding matrix into a compact codebook. Unlike low-rank alternatives, VQRound minimizes the element-wise worst-case error under $L_\infty$ norm, which is critical for handling heavy-tailed weight distributions in LLMs. Beyond reparameterization, we identify rounding initialization as a decisive factor and develop a lightweight end-to-end finetuning pipeline that optimizes codebooks across all layers using only 128 samples. Extensive experiments on OPT, LLaMA, LLaMA2, and Qwen3 models demonstrate that VQRound achieves better convergence than traditional adaptive rounding at the same number of steps while using as little as 0.2% of the trainable parameters. Our results show that adaptive rounding can be made both scalable and fast-fitting. The code is available at https://github.com/zhoustan/VQRound.

Figures (6)

• Figure 1: WikiText-2 perplexity of LLaMA-7B and LLaMA2-7B under 3-bit quantization. VQRound delivers competitive performance and can be further combined with existing methods to reduce quantization error.
• Figure 2: Convergence comparison between AdaRound and VQRound on OPT-350M under the same end-to-end finetuning steps. The marker size indicates the number of trainable parameters. VQRound converges faster and achieves lower perplexity with significantly fewer trainable parameters.
• Figure 3: Comparison of different rounding strategies including AdaRound adaround, LoRA Round ding2025cbqcrossblockquantizationlarge and VQRound. Rounding matrix is initialized by $W/s$ and its floored $floor(W/s)$, and $h^{-1}(\cdot)$ is the inverse rectified sigmoid transform. In VQRound, only the codebook $\mathcal{C}\in \mathbb{R}^{k\times d}$ needs to be updated. This is more parameter-efficient than AdaRound and LoRA Round.
• Figure 4: Initialization of rounding matrix based on residual before inverse rectified sigmoid.
• Figure 5: Comparison of error density distributions between VQRound, LoRA, and Kronecker on parameter matrix $A$ and rounding matrix $H$ after initialization. VQRound (blue) demonstrates a significantly higher density near zero, indicating superior element-wise error control.

Theorems & Definitions (10)

• Theorem 1: Lipschitz contraction of $h$
• Corollary 1: Tail transfer under Lipschitz contraction
• Remark 1
• Theorem 2: Clipping threshold via margin-to-boundary
• Corollary 2: Probability of clipping-induced saturation
• Remark 2
• proof
• proof
• proof
• proof