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

TL;DR

DiscQuant introduces a data-dependent rounding method based on discrepancy theory that significantly improves neural network quantization accuracy over existing methods, especially for low-bit representations.

Contribution

The paper presents a novel discrepancy theory-based rounding algorithm, DiscQuant, that enhances neural network quantization by minimizing approximation error with theoretical guarantees.

Findings

DiscQuant outperforms GPTQ and RTN in accuracy on benchmark datasets.

Theoretical analysis shows effective rounding with limited data samples and low-rank gradient spaces.

Empirical results demonstrate substantial accuracy improvements in quantized models.

Abstract

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 \emph{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 $m=\mathrm{poly}(1/\epsilon)$ 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 $\le \epsilon$ 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 \emph{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.