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

Hao Yu; Zheng Li; Dayiheng Liu; Jianwei Zhang

cs.CLAug 28, 2026

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

Authors: Hao Yu, Zheng Li, Dayiheng Liu, Jianwei Zhang

Organizations: Qwen Team, Alibaba Inc.

Abstract

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 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.

Figures & tables

Panel of Figure 1 — Figure 1: Gate-projection layer-wise validation on Qwen3-4B-Instruct. Both metrics are normalized by the Min-Max baseline at each layer. Although Weight-Guided Scale gives the lowest unweighted weight MSE, H-Scale closely tracks the Best Scale oracle for layer output error, demonstrating the mismatch between weight reconstruction and output preservation.

Method	Avg.	AIME24	AIME25	MMLU-R	LiveBench	LCBench
BF16 Baseline	81.06	89.52	78.10	90.85	76.23	70.58
RTN	80.03	86.76	77.61	90.54	75.14	70.12
+ H-Scale	80.68	89.33	77.98	90.65	74.72	70.74
4over6	79.99	87.44	77.04	90.23	75.04	70.20
+ H-Scale	80.49	87.82	77.84	90.72	75.53	70.56
ArcQuant	79.73	87.91	74.53	90.42	75.74	70.04

Table 1: Performance evaluation on Qwen3-30A3-Thinking. H-Scale generally improves average performance across PTQ pipelines, though not every individual benchmark improves for every method.

Method	Avg.	C-Eval	LiveBench	MMLU-R	AIME25	LCBench	ARC-C	BBH	GPQA
BF16 Baseline	65.58	63.43	69.27	89.11	59.90	48.42	69.05	70.50	54.99
RTN	64.04	62.51	68.34	89.23	56.91	47.56	67.35	68.33	52.07
+ H-Scale	64.78	62.73	68.23	89.14	57.69	47.61	68.10	67.47	57.24
4over6	64.25	62.44	68.04	88.47	57.88	47.63	68.50	67.96	53.11
+ H-Scale	64.65	62.77	68.49	89.19	57.65	48.19	68.37	68.03	54.54
ArcQuant	63.45	62.59	66.82	89.02	55.03	46.55	68.16	67.45	52.00

Table 2: Evaluation on Qwen3-30A3-Instruct. H-Scale usually improves average accuracy across eight diverse tasks, with mixed effects on individual benchmarks.

Method	Avg.	C-Eval	LiveBench	MMLU-R	AIME25	LCBench	ARC-C	BBH	GPQA
BF16 Baseline	57.76	55.54	63.16	83.98	46.16	36.63	60.72	73.50	42.41
RTN	55.38	52.48	60.66	82.87	40.37	34.32	60.28	71.76	40.29
+ H-Scale	56.26	52.43	61.08	83.13	43.24	35.65	60.69	72.39	41.44
4over6	55.38	52.20	60.43	82.74	43.35	36.16	60.10	70.16	37.91
+ H-Scale	56.12	54.70	59.95	82.73	43.51	35.46	59.30	72.40	40.95
ArcQuant	54.49	52.42	58.87	81.27	40.19	33.80	58.41	70.58	40.42

Table 3: Evaluation on Qwen3-4B-Instruct. H-Scale yields higher observed average scores across the tested NVFP4 pipelines, while individual benchmark gains remain method-dependent.

Method	Avg.	C-Eval	LiveBench	MMLU-R	ARC-C	GSM8K	BBH	GPQA
BF16 Baseline	53.02	51.05	19.24	66.91	51.83	76.89	68.24	36.97
RTN	48.79	45.64	18.17	63.07	49.08	70.92	62.41	32.21
+ H-Scale	49.53	48.15	18.77	64.30	49.70	72.52	61.43	31.84
4over6	48.86	46.70	18.63	61.68	49.73	72.19	62.24	30.87
+ H-Scale	50.05	48.21	19.16	63.16	50.21	74.28	64.02	31.31
ArcQuant	48.40	46.30	17.92	62.74	49.94	72.84	60.32	28.72

Table 4: Cross-family evaluation on LLaMA-3.1-8B-Instruct. Avg. is computed over the seven listed benchmarks. H-Scale yields higher observed average scores for all tested NVFP4 baselines.

Baseline	Variant	Avg.	C-Eval	LiveBench	MMLU-R	AIME25	LCBench	ARC-C	BBH	GPQA
4over6	Baseline	64.25	62.44	68.04	88.47	57.88	47.63	68.50	67.96	53.11
	Scale-only (w/o $\mathbf{H}$ )	63.55	64.92	66.65	87.33	56.58	46.72	67.15	66.53	52.52
	+ H-Scale	64.53	62.77	68.49	89.19	57.65	48.19	68.37	68.03	53.54
ArcQuant	Baseline	63.45	62.59	66.82	89.02	55.03	46.55	68.16	67.45	52.00
	Scale-only (w/o $\mathbf{H}$ )	63.10	62.98	66.55	87.54	54.58	47.06	66.81	66.80	52.51
	+ H-Scale	64.24	63.03	67.24	88.70	55.68	47.13	67.58	67.49	57.04

Table 5: Ablation on Qwen3-30A3-Instruct: scale with vs. without Hessian. “Scale-only” uses identity weighting in Eq. ( 6 ).

Method	Runtime	Method	Runtime
4over6	0.2 h	GPTQ	9.6 h
4over6 + H-Scale	0.4 h	GPTQ + H-Scale	9.9 h

Table 6: Runtime comparison between each baseline and its H-Scale variant using the same hardware and calibration data.

Panel of Figure 2 — Figure 2: Scale-shift statistics on the three MLP projections of Qwen3-4B-Instruct ( $1.68\times 10^{8}$ groups). Positive $\delta$ is a larger scale. (a) About half of all groups prefer a scale larger than max-abs. (b) Mean per-group reduction of the Hessian-weighted rounding error $\sum_{j}h_{j}d_{j}^{2}$ for groups that selected each $\delta$ , relative to $\delta=0$ . Nearby shifts give a modest reduction, while rare deep contractions give the largest.

$U$ / $K$	$\delta$ range	Median (%)	P10 (%)
0 0 / 16	$[-15,\phantom{+0}0]$	0 64.63	57.94
0 4 / 16	$[-11,\phantom{0}+4]$	0 90.29	87.46
0 5 / 16	$[-10,\phantom{0}+5]$	0 98.59	97.83
0 6 / 16	$\mathbf{[\phantom{0}-9,\phantom{0}+6]}$	100.00	99.91
0 7 / 16	$[\phantom{0}-8,\phantom{0}+7]$	100.00	99.89
0 8 / 16	$[\phantom{0}-7,\phantom{0}+8]$	100.00	99.84

Table 7: Candidate-window sweep on all

144

linear layers of Qwen3-4B-Instruct. Median and P10 are the layer-wise fraction (%) of the max-abs

\to

full-E4M3

L_{\mathrm{full}}

gain recovered by each

(U,K)

window. Default in bold.

\dagger

marks a wider budget.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Scale choice	Objective	Scale	$\hat{\mathbf{w}}$	Weight / Output error
Min-max	Preserve range	0.813	$[-3.253,-1.627,4.880,4.880]$	$0.573$ / $3.927$
Weight-centric search	$\min\\|\mathbf{w}-\hat{\mathbf{w}}\\|_{2}^{2}$	0.801	$[-3.204,-1.602,4.805,4.805]$	$\mathbf{0.559}$ / $3.448$
Hessian-guided search	$\min\\|\mathbf{X}(\mathbf{w}-\hat{\mathbf{w}})^{\top}\\|_{2}^{2}$	0.895	$[-3.579,-1.789,3.579,5.368]$	$0.820$ / $\mathbf{0.079}$

Appendix

Table 8: A calibration microcase showing why scale search should be output-aware.

Panel of Figure 3 — Figure 3: Additional normalized layer output error on Qwen3-4B-Instruct. All curves are normalized by the Min-Max baseline at each layer. Best Scale is the oracle reference and gives the lowest output error, while H-Scale is the strongest practical scale-search method, reducing average error over Min-Max by 24% for up_proj and 30% for down_proj .

Panel of Figure 4 — Figure 4: Additional normalized weight MSE on Qwen3-4B-Instruct. Weight-Guided Scale generally minimizes unweighted reconstruction error, while H-Scale gives lower output error. For down_proj , H-Scale can exceed the Min-Max weight MSE at early layers and the final layer while still reducing output error, confirming that weight MSE is an unreliable proxy for layer output fidelity.

\mathbf{W}_{:,i+B:n}\leftarrow\mathbf{W}_{:,i+B:n}-\mathbf{E}_{i}\mathbf{H}^{-1/2}_{i:i+B-1,i+B:n}.

Appendix

Algorithm 3 Combining GPTQ with H-Scale

Method	Avg.	AIME24	AIME25	MMLU-R	LiveBench	LCBench
BF16 Baseline	0.10	0.38	0.29	0.43	0.38	0.43
RTN	0.08	0.08	0.36	0.32	0.23	0.40
+ H-Scale	0.13	0.22	0.18	0.19	0.53	0.55
4over6	0.22	0.37	0.65	0.79	0.20	0.42
+ H-Scale	0.14	0.64	0.63	0.25	0.47	0.37
ArcQuant	0.14	0.21	0.45	0.19	0.47	0.11

Appendix

Table 9: Run-to-run sample standard deviation (three runs,

n{=}3

, ddof

{=}1

) for every entry of Table 1 (Qwen3-30A3-Thinking, five-task suite). The Avg. column is the standard deviation of the three per-run averages, not a propagation of the per-task values.

Method	Avg.	C-Eval	LiveBench	MMLU-R	AIME25	LCBench	ARC-C	BBH	GPQA
BF16 Baseline	0.10	0.10	0.47	0.37	0.63	0.54	0.38	0.25	0.49
RTN	0.10	0.39	0.51	0.20	0.41	0.57	0.53	0.35	0.46
+ H-Scale	0.27	0.40	0.51	0.54	0.30	0.28	0.41	1.00	0.57
4over6	0.13	0.34	0.22	0.43	0.24	0.47	0.29	0.31	0.41
+ H-Scale	0.25	0.09	0.48	0.53	0.42	0.57	0.59	0.22	0.19
ArcQuant	0.17	0.35	0.93	0.23	0.24	0.18	0.47	0.29	0.14

Appendix

Table 10: Run-to-run sample standard deviation (three runs,

n{=}3

, ddof

{=}1

) for every entry of Table 2 (Qwen3-30A3-Instruct, eight-task suite). The Avg. column is the standard deviation of the three per-run averages, not a propagation of the per-task values.

Baseline	Variant	Avg.	AIME24	AIME25	MMLU-R	LiveBench	LCBench
4over6	Baseline	80.32	87.82	77.04	90.83	75.34	70.56
	Scale-only (w/o $\mathbf{H}$ )	80.05	87.71	76.67	90.39	76.47	69.03
	+ H-Scale	80.35	87.44	77.84	90.72	75.53	70.20
ArcQuant	Baseline	79.73	87.91	74.53	90.42	75.74	70.04
	Scale-only (w/o $\mathbf{H}$ )	79.68	87.08	75.42	90.78	75.74	69.39
	+ H-Scale	80.46	89.03	77.78	90.89	74.65	69.97

Appendix

Table 11: Ablation on Qwen3-30A3-Thinking: scale with vs. without Hessian. “Scale-only” uses uniform unit weights in Eq. ( 6 ).

Method	Avg.	C-Eval	LiveBench	MMLU-R	AIME25	LCBench	ARC-C	BBH	GPQA
BF16 Baseline	65.58	63.43	69.27	89.11	59.90	48.42	69.05	70.50	54.99
GPTQ	62.70	62.08	66.10	87.87	56.25	47.74	67.32	64.75	49.49
+ H-Scale	63.32	62.97	65.56	87.90	55.42	48.14	67.89	66.15	52.53
GPTAQ	63.44	62.75	66.31	87.76	54.33	48.21	67.41	66.20	54.55
+ H-Scale	64.47	63.84	67.41	87.47	58.17	48.21	67.92	67.72	55.05

Appendix

Table 12: Qwen3-30A3-Instruct with a 128-sample calibration set.

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NVFP4 is a recently introduced hardware-supported FP4 format that improves the fidelity of 4-bit quantization through fine-grained block scales. However, existing NVFP4 scale initialization methods still primarily rely on AbsMax initialization, which leaves a noticeable gap to the optimal solution. To address this, we propose ScaleSweep, a simple and efficient scale optimization method that sweeps over feasible block scale candidates and selects the candidate that minimizes a target objective. We further provide a theoretical analysis of NVFP4 quantization and derive both lower and upper bounds for the required sweep range under mean square error (MSE) and weighted mean square error (WMSE) between the original tensor and the quantized reconstructed tensor. The proposed bounds substantially reduce the sweep space while preserving the optimal candidate, enabling negligible overhead compared with the baseline quantization operators. Experiments on Llama and Qwen models demonstrate that ScaleSweep consistently improves quantization performance over existing initialization methods and further narrows the gap to full precision. In particular, under aggressive end-to-end quantization of weights, activations, KV cache, and query states, ScaleSweep preserves more than 93% of the full-precision performance.

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