arXiv subjects

Earl Killian

Publications and source records attributed to Earl Killian.

3 recordsLinked to original sources

arXiv2026

SHq: Learnable Grids for LLM Weight Quantization

Block-scaled lookup-table (LUT) formats, which represent weights as indices into a small value set (the grid), have become standard for low-bit LLM quantization, ranging from NF4 and BOF4 to the 3-bit LUT instruction in NVIDIA Rubin. While these formats leave the grid definition flexible, existing implementations rely on fixed grids derived either from numerical data types (INT, FP) or assumed weight distributions (NF, BOF). We demonstrate that no single fixed grid is universally optimal; the best fixed grid varies with bit-width: BOFS at 2 bits, NF at 4 and 5 bits, and INT at 6 bits. We introduce SHq, a closed-form family of grids parameterized by two critical degrees of freedom: a companding strength α that interpolates level spacing from uniform to floating-point-like, and an offset c that introduces asymmetry. This offset allows the sign bit of the FP8 block scale, otherwise unused, to select between the grid and its mirror image, up to doubling the number of distinct values available per block. SHq modifies only the lookup table contents and serves as a drop-in replacement for the grid in any such format. Across five models ranging from 3B to 35B parameters, including a mixture-of-experts architecture, evaluated at 2-6 bits using RTN and GPTQ, SHq achieves the lowest KL divergence in every setting at 3 and 4 bits and in most settings at 2, 5 and 6 bits. It outperforms the best fixed grid by up to 34% and remains superior under mixed-precision configurations. At 4.5 bits per weight, the SHq format reduces the KL divergence of NVFP4 by 41-67% while improving downstream accuracy. Grid families incorporating both spacing and offset parameters perform comparably, whereas those lacking an offset exhibit clearly higher error, confirming that these two degrees of freedom drive the observed gains. We therefore propose SHq as the standard grid for block-scaled lookup-table formats.

cs.LG↗

arXiv2026

A Hardware-Aware, Per-Layer Methodology for Post-Training Quantization of Large Language Models

Scaled Outer Product (SOP) is a post-training quantization methodology for large language model weights, designed to deliver near-lossless fidelity at 4.5--6 bits per weight on hardware with per-layer LUT decode. The methodology combines per-layer search of fixed and dynamic codebook pairs selected by a per-block selection bit, signed per-block scales, activation-weighted cosine selection, and multiple-choice knapsack promotion of sensitive layers with outlier and sparse-residual correction. Fixed codebooks include NF4, BOF4, Split87, and SH4; per-layer optimized codebooks (DD4) are hosted in LUT SRAM. A new hardware-efficient LUT output format (HIF) is proposed to improve performance, energy, and cost. Across six open model families, the recommended FP6 operating point (E2M3sUE4M4, 6.5 bpw) achieves lower weight reconstruction error than the conventional per-layer-POT FP8 baseline (E4M3, 8.0 bpw) at 1.5 bpw lower storage cost, demonstrating that block-scaled small atoms with carefully chosen scale precision can replace conventionally-deployed FP8. Full evaluation across the 4.5--6 bpw range, including layer promotion and sparse residual correction, is reported in a companion paper.

cs.LG↗

arXiv2026

Grid Games: The Power of Multiple Grids for Quantizing Large Language Models

A major recent advance in quantization is given by microscaled 4-bit formats such as NVFP4 and MXFP4, quantizing values into small groups sharing a scale, assuming a fixed floating-point grid. In this paper, we study the following natural extension: assume that, for each group of values, we are free to select the "better" among two or more 4-bit grids marked by one or more bits in the scale value. We formalize the power-of-two-grids (PO2) problem, and provide theoretical results showing that practical small-group formats such as MXFP or NVFP can benefit significantly from PO2 grids, while the advantage vanishes for very large groups. On the practical side, we instantiate several grid families, including 1) PO2(NF4), which pairs the standard NF4 normal grid with a learned grid, 2) MPO2, a grid pair that is fully learned over real weights and activations, 3) PO2(Split87), an explicit-zero asymmetric grid and 4) SFP4, a TensorCore-implementable triple which pairs NVFP4 with two shifted variants. Results for post-training quantization of standard open models and pre-training of Llama-like models show that adaptive grids consistently improve accuracy vs single-grid FP4 under both weight-only and weight+activation. Source code is available at https://github.com/IST-DASLab/GridGames.

cs.LG↗