Price of universality in vector quantization is at most 0.11 bit

TL;DR

This paper proves the existence of a universal vector quantization codebook that is nearly optimal for all data distributions, reducing the rate by at most 0.11 bits per dimension, which has implications for efficient low-precision model deployment.

Contribution

It establishes the theoretical existence of a universal codebook for vector quantization that performs near-optimally across all data statistics, independent of specific data distribution.

Findings

Universal codebook exists with at most 0.11 bit per dimension loss.

Universal codebook is simultaneously near-optimal for all data statistics.

Existence of a net in bR^n that nearly covers all Hilbert norms.

Abstract

Fast computation of a matrix product $W^\top X$ is a workhorse of modern LLMs. To make their deployment more efficient, a popular approach is that of using a low-precision approximation $\widehat W$ in place of true $W$ ("weight-only quantization''). Information theory demonstrates that an optimal algorithm for reducing precision of $W$ depends on the (second order) statistics of $X$ and requires a careful alignment of vector quantization codebook with PCA directions of $X$ (a process known as "waterfilling allocation''). Dependence of the codebook on statistics of $X$, however, is highly impractical. This paper proves that there exist a universal codebook that is simultaneously near-optimal for all possible statistics of $X$, in the sense of being at least as good as an $X$-adapted waterfilling codebook with rate reduced by 0.11 bit per dimension. Such universal codebook would be an ideal candidate for the low-precision storage format, a topic of active modern research, but alas the existence proof is non-constructive. Equivalently, our result shows existence of a net in $\mathbb{R}^n$ that is a nearly-optimal covering of a sphere simultaneously with respect to all Hilbert norms.