A Note on TurboQuant and the Earlier DRIVE/EDEN Line of Work

TL;DR

This note clarifies the relationship between TurboQuant and earlier EDEN/DRIVE schemes, highlighting differences in optimality, bias, and experimental performance, and discusses connections in their analytical approaches.

Contribution

It provides a detailed comparison of TurboQuant with EDEN, showing that EDEN generally outperforms TurboQuant in accuracy and explaining the theoretical and experimental distinctions.

Findings

Biased EDEN with optimized S is more accurate than TurboQuant_mse.

Unbiased EDEN significantly outperforms TurboQuant_prod.

EDEN consistently outperforms TurboQuant in various experimental setups.

Abstract

This note clarifies the relationship between the recent TurboQuant work and the earlier DRIVE (NeurIPS 2021) and EDEN (ICML 2022) schemes. DRIVE is a 1-bit quantizer that EDEN extended to any $b>0$ bits per coordinate; we refer to them collectively as EDEN. First, TurboQuant$_{\text{mse}}$ is a special case of EDEN obtained by fixing EDEN's scalar scale parameter to $S=1$. EDEN supports both biased and unbiased quantization, each optimized by a different $S$ (chosen via methods described in the EDEN works). The fixed choice $S=1$ used by TurboQuant is generally suboptimal, although the optimal $S$ for biased EDEN converges to $1$ as the dimension grows; accordingly TurboQuant$_{\text{mse}}$ approaches EDEN's behavior for large $d$. Second, TurboQuant$_{\text{prod}}$ combines a biased $(b-1)$-bit EDEN step with an unbiased 1-bit QJL quantization of the residual. It is suboptimal in three ways: (1) its $(b-1)$-bit step uses the suboptimal $S=1$; (2) its 1-bit unbiased residual quantization has worse MSE than (unbiased) 1-bit EDEN; (3) chaining a biased $(b-1)$-bit step with a 1-bit unbiased residual step is inferior to unbiasedly quantizing the input directly with $b$-bit EDEN. Third, some of the analysis in the TurboQuant work mirrors that of the EDEN works: both exploit the connection between random rotations and the shifted Beta distribution, use the Lloyd-Max algorithm, and note that Randomized Hadamard Transforms can replace uniform random rotations. Experiments support these claims: biased EDEN (with optimized $S$) is more accurate than TurboQuant$_{\text{mse}}$, and unbiased EDEN is markedly more accurate than TurboQuant$_{\text{prod}}$, often by more than a bit (e.g., 2-bit EDEN beats 3-bit TurboQuant$_{\text{prod}}$). We also repeat all accuracy experiments from the TurboQuant paper, showing that EDEN outperforms it in every setup we have tried.