Improving floating point quantization noise prediction in matrix multiplication
KBBQ: A Predictive Noise Law and the Limits of Spectrum Flattening in FP4 Quantization
Machine LearningArtificial Intelligence
Summary
When computers do many calculations with limited precision numbers, they introduce tiny errors called quantization noise. The authors created a new mathematical way to understand how this noise behaves in matrix multiplication when using different types of number formats. They derived a formula that predicts the noise's effect and found the best possible limit on how much noise could be reduced through certain transformations. Using this insight, they developed a new technique called KBBQ that better controls this noise in floating-point 4-bit formats, leading to more accurate results without slowing down the computation.
What this means in practice
- •For machine learning engineers: Optimize 4-bit floating point computations in neural network inference to improve accuracy without added runtime cost.
- •For hardware accelerator designers: Design quantization modules that approach theoretical noise reduction limits to enhance precision in matrix multiplication hardware.
Authors
Lexington Whalen, Yuki Ito, Ryo Sakamoto
Abstract
We develop a second-order theory of quantization noise in matrix multiplication in which the quantization format is characterized by the variance it assigns to each element. The constant variance profile of integer quantization recovers existing integer-noise theory, while the multiplicative profile of floating-point rounding reduces the data dependence to a scalar, the participation factor $κ$, yielding a closed-form signal-to-noise-ratio law. The resulting functional also admits a closed-form upper bound $κ^{*}$ that no function-preserving linear transform can exceed and that is attained by a recent state-of-the-art method. Building on this analysis, we introduce KBBQ (\textbf{K}appa-\textbf{B}raked \textbf{B}lockwise \textbf{Q}uantization), which parameterizes the extent to which a transform approaches this ceiling. At W4A4, across four base models and two FP4 formats, KBBQ outperforms the prior state of the art without additional deployment-time computation.