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H-Scale: Hessian-Guided Scale Refinement for NVFP4 Sub-Byte LLM Inference

Hao Yu, Zheng Li, Dayiheng Liu, Jianwei Zhang

arXiv:2608.28113v1cs.CL

TL;DR

NVFP4’s fine-grained scale space makes per-group scale selection an underexplored calibration problem, while existing PTQ methods primarily refine quantized values. H-Scale uses calibration-activation-derived diagonal second-order weighting to select hardware-valid scales, and generally improves diverse NVFP4 baselines while narrowing the gap to BF16 in several settings. Its refinement is lightweight and adds no inference cost.

  • Problem

    NVFP4’s fine-grained per-group scales are highly sensitive, while existing PTQ methods primarily refine weight values rather than this scale-selection step.

    Source: §1 · p. 1

  • Method

    H-Scale selects hardware-valid per-group scales by minimizing a diagonal-Hessian-weighted reconstruction objective computed from calibration activations.

    Source: §1 · p. 2, §3.2 · p. 3

  • Results

    H-Scale generally yields higher average performance across diverse NVFP4 baselines and narrows the gap to BF16 in several complex reasoning settings.

    Source: §1 · p. 2, §4.2 · p. 8, §5 · p. 9

  • Takeaways & Limitations

    H-Scale provides a plug-and-play NVFP4 scale-refinement step with zero additional inference cost.

    Source: §5 · p. 9

    The study is limited to NVFP4 with g = 16 and text-only LLMs; transfer to other formats, group sizes, or multimodal models remains open.

    Source: §5 · p. 9

Abstract

from arXiv · show

The NVIDIA Blackwell architecture, with native support for the ultra-fine-grained NVFP4 format, opens new opportunities for accelerating large language model (LLM) inference. NVFP4's micro-block design, such as a group size of 16, offers strong representational flexibility for capturing local weight distributions and isolating outliers, but it also introduces a large and highly sensitive space of per-group scaling factors. Existing post-training quantization (PTQ) methods primarily focus on refining quantized weight values, leaving this scale-selection step underexplored. To address this gap, we propose \textbf{H-Scale}, a lightweight post-processing method for NVFP4 per-group scale refinement. Instead of minimizing plain weight reconstruction error, H-Scale selects hardware-valid group scales using a diagonal second-order proxy derived from calibration activations, thereby targeting layer output perturbation more directly. It is designed as a drop-in replacement for RTN-style scale selection in diverse NVFP4 pipelines, requires only modest offline calibration, and introduces strictly zero overhead at inference time. Under a fixed evaluation protocol, experiments on mainstream LLMs show that H-Scale generally improves a broad range of NVFP4 baselines and brings several variants closer to the BF16 reference.

1 Introduction

NVFP4’s fine-grained scales improve local representation but make scale calibration a major accuracy factor. H-Scale addresses this with activation-derived Hessian-guided scale selection as a lightweight, zero-inference-cost refinement.

Motivation

  • NVFP4’s 16-value micro-blocks capture local weight distributions and isolate outliers, but per-group scale miscalibration can strongly affect accuracy.Each group couples E2M1 4-bit values with a shared E4M3 scale.Source: §1 · p. 1
  • Existing PTQ methods mainly reduce quantization error by reconstructing or regularizing low-bit weight values, leaving scale selection underexplored.H-Scale treats scaling factors as a separate calibration target.Source: §1 · p. 1

Method

  • H-Scale chooses per-group scales with a diagonal-Hessian-weighted reconstruction objective derived from calibration activations.The objective shifts from plain weight reconstruction toward output-aware weighted reconstruction.Source: §1 · pp. 1–2

Practicality

  • H-Scale is designed as a lightweight post-processing step that can be grafted onto existing NVFP4 baselines, runs in minutes, and adds zero inference cost.It is positioned as a drop-in replacement for baseline scale selection.Source: §1 · pp. 1–2

Method

  • H-Scale enumerates neighboring hardware-valid FP8 local scales around a baseline and retains the candidate minimizing Hessian-weighted reconstruction error.The quantized weights remain in native NVFP4, so inference cost is unchanged.Source: §1 · p. 2

Results

  • Across diverse evaluations, H-Scale generally improves average performance and narrows the NVFP4-to-BF16 gap in several complex reasoning settings.The reported comparison is under a fixed evaluation protocol.Source: §1 · p. 2

2 Related Work

Prior work addresses low-bit quantization through weight reconstruction, activation-aware scaling, transforms, and fine-grained NVFP4-specific designs. H-Scale instead refines final per-group NVFP4 scales using an output-aware second-order objective, complementing those approaches.

PTQ for LLMs

  • GPTQ and related PTQ methods compensate low-bit errors through weight reconstruction or regularization, while other methods protect salient channels or mitigate activation outliers.The related methods include GPTQ, AWQ, GPTAQ, and SpinQuant.Source: §2 · p. 2

Fine-Grained Quantization

  • Blackwell’s native 4-bit floating-point support has motivated NVFP4 algorithms that exploit group size 16 while addressing increased scale sensitivity.MR-GPTQ, 4over6, and ArcQuant represent distinct NVFP4-specific strategies.Source: §2 · p. 2

Scale Refinement

  • Scale-refinement methods range from learned scales during fine-tuning to activation-weight scale migration, but they target different pipeline stages.LSQ incurs training costs, whereas SmoothQuant focuses on scale placement between activations and weights.Source: §2 · p. 2

H-Scale’s Positioning

  • H-Scale differs through a diagonal second-order, output-aware reconstruction loss applied at the final per-group scale-selection step.This placement makes it complementary to prior reconstruction and structural quantization methods.Source: §2 · p. 2

3 Method

H-Scale refines NVFP4 per-group scales by replacing plain weight MSE with a diagonal-Hessian, output-aware objective and searching hardware-valid neighboring scales. The resulting drop-in search preserves the native quantization pipeline while targeting layer-output error more directly.

NVFP4 quantization

  • NVFP4 partitions weights into groups of 16, using E2M1 values with per-group E4M3 scales and a tensor-level FP32 global scale.The effective group scale is the product of the global and local scales.Source: §3.1 · p. 3

Hessian-guided objective

  • H-Scale weights reconstruction errors by a diagonal Hessian derived from calibration activations instead of using plain weight MSE.The diagonal proxy retains per-channel activation energy while dropping cross-channel correlation terms.Source: §3.2 · p. 3

Group-wise scale selection

  • For each group, H-Scale independently selects the hardware-valid effective scale that minimizes the corresponding weighted reconstruction error.The candidate set is induced by a bidirectional walk over neighboring E4M3 local scales.Source: §3.2 · p. 3, §3.3 · p. 4

Efficient scale search

  • The deterministic search keeps baseline grouping and the FP4 value grid unchanged, refining only the per-group scale around the baseline initialization.The default window uses nine smaller and six larger E4M3 steps, with endpoint clipping and exact hardware representability.Source: §3.3 · p. 4

Output-aware refinement

  • H-Scale allocates limited FP4 resolution toward weights that matter most for calibration-layer output, while retaining the same hardware-valid scale space.This replaces identity weighting in scale-only search with the diagonal Hessian proxy.Source: §3.2 · p. 3, §3.4 · p. 5

Validation and integration

  • On Qwen3-4B-Instruct, H-Scale reduces activation loss by 32.5% relative to Min-Max, versus 40.8% for the Best Scale oracle.Weight-Guided Scale achieves the lowest unweighted weight MSE but only a 12.6% activation-loss reduction; H-Scale can replace RTN-style selectors without inference changes.Source: §3.4 · p. 5, §3.5 · p. 5

4 Experiments

Experiments evaluate H-Scale across NVFP4 baselines, model sizes, model families, ablations, scale shifts, and runtime. H-Scale generally improves average benchmark performance while preserving inference cost and using a compact candidate window.

Experimental Setup

  • The evaluation tests average performance, transfer, and offline-cost trade-offs across diverse NVFP4 baselines, models, and benchmark suites.Benchmarks cover reasoning, math, knowledge, and coding, using fixed decoding with three independent runs.Source: §4 · p. 6, §4.1 · pp. 6–7

Main Results

  • H-Scale yields higher average scores across Qwen3-30A3-Thinking, Qwen3-30A3-Instruct, Qwen3-4B-Instruct, and LLaMA-3.1-8B-Instruct evaluations.The reported gains extend across model sizes and families, although individual benchmark improvements remain method-dependent.Source: Table 3 · p. 7, Table 4 · p. 7, §4.2 · p. 8

Ablation Analysis

  • H-Scale gives the highest average score in the Hessian ablation, while identity-weighted scale-only optimization is less stable and can hurt 4over6 and ArcQuant.The comparison distinguishes Hessian-weighted scale refinement from scale adjustment alone.Source: §4.3 · p. 8, Table 5 · p. 8

Efficiency and Hardware Deployment

  • Inference cost remains unchanged, while GPTQ runtime rises from 9.6 h to 9.9 h and representative 4over6 runtime rises from 0.2 h to 0.4 h.The added cost is offline scale search and, for some baselines, a one-time activation pass.Source: §4.3 · p. 8, Table 6 · p. 8

Ablation Analysis

  • 47.9% of groups select a larger scale, 31.2% retain max-abs initialization, and 20.9% select a smaller scale.These statistics pool 1.68 × 10^8 groups from the three MLP projections of Qwen3-4B-Instruct.Source: §4.3 · p. 8
  • Deep contractions δ ≤−4 reduce Hessian-weighted rounding error by 58.7% on average, while 0.11% of groups account for 8.3% of total error.Nearby shifts reduce error more modestly, including 32.8% at δ = −1.Source: §4.3 · pp. 8–9
  • A candidate window with U ≥6 reaches 100.00% median recovery and keeps P10 above 99.6%, while the full window removes 50.6% of initialization error.The selected default is U = 6 and K = 16; wider windows add little recovery.Source: §4.3 · p. 9, Table 7 · p. 9

5 Conclusion, Limitations, and Future Work

The paper identifies per-group scale selection as an important NVFP4 fidelity bottleneck and presents H-Scale as a plug-and-play diagonal-Hessian refinement. Its scope is limited to NVFP4 with group size 16 and text-only LLMs.

Conclusion

  • H-Scale generally improves average performance across diverse NVFP4 baselines, model sizes, and Qwen and LLaMA model families.The method is presented as a plug-and-play post-processing step.Source: §5 · p. 9

Limitations and Future Work

  • The study is limited to NVFP4 with g = 16 and text-only LLMs, leaving transfer to other formats, group sizes, and multimodal models open.The authors identify multimodal scale tuning as future work.Source: §5 · p. 9

A.1 Calibration Microcase

The microcase shows why output-aware scale selection can favor a different FP4 pattern than weight-centric search. Layer-level curves extend this behavior beyond the microcase, while comparisons distinguish output error from weight MSE.

A.1 Calibration Microcase

  • Output-aware scoring selects FP4 code pattern [−4, −2, 4, 6] despite larger unweighted error, reducing output error to 0.079.The corresponding output errors are 3.927 for Min-Max and 3.448 for weight-centric search.Source: §A.1 · p. 12
  • The gate-projection layer-wise trend also holds for the other MLP projections, motivating additional up-projection and down-projection curves.Figures 3 and 4 report these additional curves.Source: §A.2 · p. 12
  • H-Scale can have higher weight MSE than Min-Max while achieving lower layer-output error, showing that weight MSE is an unreliable output-fidelity proxy.This behavior is reported for down projections at early and final layers.Source: Figure 4 · p. 13

B Implementation Pseudocode

The implementation inserts a parallel, hardware-valid scale search into NVFP4 quantization pipelines. It uses calibration-derived diagonal Hessian weights, searches neighboring FP8 scales, and leaves GPTQ’s blockwise propagation unchanged.

Implementation

  • Calibration activations provide diagonal weights h, computed as diag(X⊤X) or reused from Hessian statistics already available in GPTQ-like baselines.The weights align with each NVFP4 group.Source: B Implementation Pseudocode · p. 13
  • The search is fully parallelizable across row-group pairs and candidates, enabling batched GPU scoring of FP4 projections and element-wise weighted errors.This separability keeps the additional offline computation small.Source: B Implementation Pseudocode · p. 13

GPTQ Integration

  • In GPTQ integration, H-Scale replaces only the local quantization call; blockwise GPTQ error computation, propagation, and final weight assembly remain unchanged.The GPTQ pseudocode retains inverse-Cholesky processing and standard error propagation around the refined local step.Source: B Implementation Pseudocode · pp. 13–14

HScaleGroupQuantize

  • The selected scale is the candidate minimizing the weighted objective, after which the group is quantized once with that hardware-valid scale.The quantized group is returned in native NVFP4 representation.Source: B Implementation Pseudocode · pp. 13–14

C Evaluation Protocol Details

The evaluation uses model-specific benchmark suites and a fixed decoding protocol, with repeated runs to reduce the impact of stochastic variation.

Benchmark suites

Decoding protocol

  • Three independent runs use fixed sampling settings, with mean scores reported and no significance assigned to small deltas or slight BF16 exceedances.Source: C Evaluation Protocol Details · p. 14

D Run-to-Run Variability of the Main Results

Main-text results average three end-to-end evaluations, and suite-level averaging substantially reduces run-to-run fluctuation relative to individual task variability.

E Hessian Ablation on Qwen3-30A3-Thinking

The ablation indicates that Hessian-weighted scale optimization outperforms scale-only optimization across two NVFP4 baselines, while the benefit persists with a smaller calibration set.

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