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Price of universality in vector quantization is at most 0.11 bit

Alina Harbuzova, Or Ordentlich, Yury Polyanskiy

TL;DR

This work addresses universal vector quantization for weight matrices in inner-product computations by proving the existence of a universal codebook that nearly matches the waterfilling rate-distortion benchmark for all input statistics $\Sigma_X$. Using a random-coding construction with an isotropic Gaussian codebook and encoder-selected scale $\tau$, the authors show a per-coordinate distortion within $\mathbf{D}_{\mathrm{rc}}(\mathrm{spec}(\Sigma_X),R^{\star})+\text{small}$ uniformly over all $\Sigma_X$ with $\mathrm{tr}(\Sigma_X)=n$, with rate $R$ at most $R^{\star}+\varepsilon$. They further prove a worst-case overhead to the oracle waterfilling of at most $0.11$ bits per coordinate for distortions away from extreme values, highlighting that universality incurs only a small, universal rate penalty. The results imply the existence of a universal, near-optimal low-precision storage format for $W$ applicable across diverse input statistics, though the construction is non-constructive and explicit codebooks remain an open challenge. Overall, the paper connects universal quantization to Gaussian rate-distortion theory and dense nets over covariance families, offering a foundational understanding of universality in high-dimensional quantization under 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.

Price of universality in vector quantization is at most 0.11 bit

TL;DR

This work addresses universal vector quantization for weight matrices in inner-product computations by proving the existence of a universal codebook that nearly matches the waterfilling rate-distortion benchmark for all input statistics . Using a random-coding construction with an isotropic Gaussian codebook and encoder-selected scale , the authors show a per-coordinate distortion within uniformly over all with , with rate at most . They further prove a worst-case overhead to the oracle waterfilling of at most bits per coordinate for distortions away from extreme values, highlighting that universality incurs only a small, universal rate penalty. The results imply the existence of a universal, near-optimal low-precision storage format for applicable across diverse input statistics, though the construction is non-constructive and explicit codebooks remain an open challenge. Overall, the paper connects universal quantization to Gaussian rate-distortion theory and dense nets over covariance families, offering a foundational understanding of universality in high-dimensional quantization under Hilbert norms.

Abstract

Fast computation of a matrix product is a workhorse of modern LLMs. To make their deployment more efficient, a popular approach is that of using a low-precision approximation in place of true ("weight-only quantization''). Information theory demonstrates that an optimal algorithm for reducing precision of depends on the (second order) statistics of and requires a careful alignment of vector quantization codebook with PCA directions of (a process known as "waterfilling allocation''). Dependence of the codebook on statistics of , however, is highly impractical. This paper proves that there exist a universal codebook that is simultaneously near-optimal for all possible statistics of , in the sense of being at least as good as an -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 that is a nearly-optimal covering of a sphere simultaneously with respect to all Hilbert norms.
Paper Structure (40 sections, 12 theorems, 213 equations, 1 figure)

This paper contains 40 sections, 12 theorems, 213 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Paper organization.
  3. Main results and discussion
  4. Discussion and Open Questions.
  5. Related literature.
  6. Technical Overview
  7. Codebook Distribution
  8. Geometric Intuition
  9. Sketch of Proof
  10. Preliminaries and Notation
  11. Notation.
  12. Random Coding: Worst-Case $W$
  13. General Rate-Distortion Function.
  14. Proof of Theorem \ref{['thm:fixed_sigma']}
  15. Quantization Scheme.
  16. ...and 25 more sections

Key Result

Theorem 1

Let us assume that $W\sim \mathcal{N}(0,I_n)$ and let $\mathbf{R}_{\mathrm{wf}}(\Sigma_X, D)$ denote the information-theoretic (waterfilling) lower bound on rate needed to achieve distortion at most $D$ in the oracle setting, where codebook is optimized for a fixed $\Sigma_X$. There exists a univers

Figures (1)

  • Figure 1: Maximum rate gap found numerically at each $R = \mathbf{R}_{\mathrm{rc}}(\lambda, D^{\star})$ in Claim \ref{['claim:numerical']}.

Theorems & Definitions (29)

  • Theorem 1: Informal: Universality costs $\le 0.11$ Bits
  • Proposition 2.1: Waterfilling
  • proof
  • Theorem 2: Main Result I: Universal Quantization Scheme for Gaussian Input
  • Corollary 2.1
  • Theorem 3: Main Result II: Worst-Case Rate Gap to Oracle Setting
  • Proposition A.1: Hanson-Wright Concentration Inequality
  • Theorem 4: Achievability of Random-Coding Rate-Distortion: Nonasymptotic Guarantee
  • Remark B.1
  • Claim B.1: Bound on $\tau$.
  • ...and 19 more