Summary
Working with computers, especially in AI, often means simplifying numbers to save space and speed up calculations, but this can add noise and reduce accuracy. The authors studied how this noise behaves when multiplying matrices with different number formats, including a type called FP4. They developed a new theory that predicts this noise more precisely and found a mathematical limit for how well transformations can reduce it. Using this understanding, they created a method called KBBQ that gets closer to this limit and improves accuracy without extra work when running AI models. Their tests showed better results across multiple models and formats compared to previous methods.
quantizationmatrix multiplicationfloating-point formatFP4noise variancesignal-to-noise ratiolinear transformparticipation factorAI model accuracynumerical precision
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.