Revisiting Error Feedback for Model Quantization

Wai-Yiu Keung; Chi Cui; Wing-Kin Ma; Jesús Cid-Sueiro; Hei Victor Cheng

cs.LGOct 13, 2026

Revisiting Error Feedback for Model Quantization

Authors: Wai-Yiu Keung, Chi Cui, Wing-Kin Ma, Jesús Cid-Sueiro, Hei Victor Cheng

Organizations: The Chinese University of Hong Kong, Hong Kong SAR of China · Universidad Carlos III de Madrid, Spain · Aarhus University, Denmark

Abstract

We study causal error-feedback coefficients as design variables for model quantization. Modeling instantaneous quantization errors as independent, zero-mean variables with a common fixed variance, we formulated a metric-dependent trace problem for designing feedback coefficients. The formulation resembles fixed-order GPTQ/LDLQ feedback and supports Jacobian-aware designs that account for downstream error propagation. For a positive-definite metric $\boldsymbol{H}$ , we prove that the joint rotation-feedback trace minimum is $n\det(\boldsymbol{H})^{1/n}$ and present an attaining pair using the geometric mean decomposition. Experiments on synthetic networks, trained MLPs, and language models show benefits from downstream-sensitive feedback, but also reveal that trace optimality need not improve finite-level performance. Small QAT experiments further illustrate direct loss-driven feedback tuning with fixed or trainable network weights.

Figures & tables

	Fixed $\alpha$ : mean test distortion									Per-network $\alpha$ selection
$M$	$0$	$0.01$	$0.03$	$0.1$	$0.2$	$0.4$	$0.6$	$0.8$	$0.99$	Mean distortion	Reduction	Win/Tie/Loss
2	1.985016	1.978903	2.024264	2.032677	2.045072	2.038097	2.036754	2.032965	2.046547	1.691480	14.79%	75/20/5
4	0.136493	0.137417	0.133098	0.133740	0.134332	0.133041	0.131872	0.130650	0.131327	0.112534	17.55%	75/21/4
8	0.022783	0.022621	0.022686	0.022051	0.021853	0.022372	0.022070	0.021908	0.022043	0.018986	16.67%	72/24/4
12	0.008864	0.008892	0.008792	0.008754	0.008883	0.008964	0.009062	0.008925	0.008988	0.007633	13.88%	73/20/7
16	0.004611	0.004536	0.004499	0.004568	0.004534	0.004555	0.004587	0.004654	0.004631	0.003855	16.39%	72/27/1

Table 1: Test-set output distortion over

100

independently generated networks. The fixed-

\alpha

columns report mean distortion when the same

\alpha

is used for every network;

\alpha=0

is the GPTQ-style baseline. In the final three columns,

\alpha

is selected separately for each network using calibration data only. Reduction is relative to

\alpha=0

, and Win/Tie/Loss counts networks on which the selected method has lower, equal, or higher test distortion than that baseline. Boldface in the fixed-

\alpha

columns marks the lowest mean in each row.

Dataset	Method	$M=2$	$M=4$	$M=8$	$M=16$
MNIST	RTN	$8.606688/32.57$	$0.382275/0.92$	$0.056979/0.11$	$0.011988/0.03$
	Local EFQ	$1.695767/5.12$	$0.084901/\mathbf{0.20}$	$0.013702/0.04$	$0.002940/\mathbf{-0.0008}$
	Local EFQ + RHT	$3.921692/15.91$	$0.139418/0.27$	$0.023697/0.04$	$0.004970/0.01$
	Obj. EFQ	$\mathbf{1.637242}/\mathbf{4.83}$	$\mathbf{0.082757}/0.21$	$\mathbf{0.013351}/\mathbf{0.03}$	$\mathbf{0.002856}/-0.0003$
	Obj. EFQ + RHT	$3.839215/15.08$	$0.135396/0.25$	$0.023065/0.04$	$0.004850/0.01$
	Obj. joint ${\bm{R}},{\bm{G}}$	$8.672428/44.34$	$0.140144/0.49$	$0.014719/0.03$	$0.002961/0.00$

Table 2: Post-training quantization of MNIST- and CIFAR-10-trained MLPs. Each entry reports test logit MSE / accuracy drop in percentage points, averaged over 100 trained checkpoints. Mean full-precision accuracy is

97.0539\%

on MNIST and

47.5550\%

on CIFAR-10. Lower is better for both metrics. Boldface marks the lowest unrounded mean within each dataset and

M

Model	$M$	Local EFQ	Local EFQ+RHT	Local joint ${\bm{R}},{\bm{G}}$	Obj.-EFQ	Obj.-EFQ+RHT	Obj. joint ${\bm{R}},{\bm{G}}$
GPT-2	8	118.54 / 24.21 / 257.89	105.41 / 24.60 / 221.42	112.66 / 24.26 / 186.16	119.03 / 24.45 / 222.64	105.38 / 24.40 / 206.28	106.94 / 24.65 / 182.59
	16	99.65 / 25.50 / 85.48	100.69 / 25.30 / 34.73	98.41 / 26.14 / 31.10	99.60 / 25.40 / 72.64	99.95 / 25.50 / 28.13	98.25 / 26.09 / 57.73
Llama-2-7B	8	11.312 / 50.22 / 3.989	9.120 / 53.15 / 2.077	9.893 / 51.70 / 2.794	11.002 / 50.81 / 3.745	8.704 / 53.73 / 1.739	9.338 / 52.55 / 2.253
	16	7.808 / 55.69 / 0.791	7.450 / 56.34 / 0.347	7.563 / 56.06 / 0.498	7.736 / 55.84 / 0.685	7.431 / 56.44 / 0.309	7.521 / 56.18 / 0.453
Llama-2-13B	8	8.209 / 54.56 / 0.842	7.585 / 55.73 / 0.619	8.238 / 54.52 / 0.806	7.929 / 55.13 / 0.765	7.501 / 55.97 / 0.579	7.989 / 54.86 / 0.733
	16	6.714 / 58.00 / 0.201	6.611 / 58.25 / 0.137	6.671 / 58.03 / 0.182	6.630 / 58.15 / 0.168	6.593 / 58.27 / 0.126	6.675 / 58.04 / 0.168

Table 3: Post-training quantization on five language models. Each entry reports PPL / next-token accuracy (%) / final-logit MSE. Lower PPL and MSE and higher accuracy are better. Boldface marks the best of the six methods for each model and bit width. “Local” and “objective-aware” specify the design metric; EFQ, RHT+EFQ, and joint

{\bm{R}},{\bm{G}}

specify the construction.

Method	$M=2$	$M=4$	$M=8$
Full precision	0.01391	0.01391	0.01391
RTN	0.63739	0.06576	0.02395
Local PTQ	0.48603	0.05473	0.02078
Feedback QAT(G)	0.28594	0.04055	0.01899
Weight-only QAT; ${\bm{G}}={\bm{I}}$ fixed	0.25458	0.04118	0.01914
Joint QAT; ${\bm{G}}_{0}={\bm{I}}$	0.24524	0.04013	0.01784

Table 4: Mean test MSE over

100

paired synthetic regression trials; lower is better. Feedback-only tuning holds the network fixed and starts from covariance-designed feedback. Both displayed QAT arms start from

{\bm{G}}={\bm{I}}

. Boldface marks the lowest mean among the quantized methods shown.

Method	$M=2$	$M=4$	$M=8$	$M=16$
Init. Full precision	12.967	12.967	12.967	12.967
Cont. Full precision	12.374	12.374	12.374	12.374
RTN	18.282	13.046	12.986	12.970
Local PTQ	13.629	12.999	12.969	12.970
Feedback QAT ( ${\bm{G}}$ )	13.074	12.842	12.865	12.910
Weight QAT ( ${\bm{W}}$ )	12.561	12.404	12.390	12.382

Table 5: Held-out next-character perplexity on the Shakespeare character dataset, averaged over ten paired seeds. Perplexity is computed for each seed before averaging; lower is better. Evaluation uses the held-out second half of the validation file. Boldface marks the lowest mean among quantized methods.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Compensation criterion	Relationship to our feedback design
GPTQ / QuIP-LDLQ [ 10 , 6 ]	Layer-output quadratic $\operatorname{tr}({\bm{E}}_{\ell}{\bm{C}}_{\ell}{\bm{E}}_{\ell}^{\top})$ .	The closest linear-feedback precedent. With the input covariance as metric, our fixed-isotropic-error design surrogate is proportional to $\operatorname{tr}({\bm{G}}_{\ell}^{\top}{\bm{C}}_{\ell}{\bm{G}}_{\ell})$ .
GuidedQuant [ 16 ]	End-loss-gradient-weighted row quadratics, $\sum_{r}{\bm{e}}_{\ell,r}^{\top}{\bm{F}}_{\ell,r}{\bm{e}}_{\ell,r}$ .	The criterion can be evaluated on ${\bm{e}}_{\ell,r}({\bm{G}}_{\ell})$ . Its weighting uses end-loss gradients rather than our output-Jacobian metric; this objective-level connection does not identify the two quantization algorithms.
YAQA [ 31 ]	A structured quadratic approximation to full-model output-distribution KL.	Its criterion can be restricted to $\operatorname{vec}({\bm{E}}_{\ell}({\bm{G}}_{\ell}))$ . Its adaptive rounding also uses output-side feedback, beyond a single shared input-coordinate ${\bm{G}}_{\ell}$ .
QEP [ 1 ]	Compensation for quantization errors propagated from preceding layers.	Upstream-quantized states can be included in $\Theta_{\ell}$ , permitting an upstream-error-aware objective for ${\bm{G}}_{\ell}$ . This is a restriction to our quantizer, not an equivalence with QEP’s procedure.
Qronos [ 34 ]	Mismatched-input layer reconstruction, $\\|{\bm{W}}_{\ell}{\bm{X}}_{\ell}-\widehat{{\bm{W}}}_{\ell}\widetilde{{\bm{X}}}_{\ell}\\|_{F}^{2}$ .	Taking $\widehat{{\bm{W}}}_{\ell}=\widehat{{\bm{W}}}_{\ell}({\bm{G}}_{\ell})$ gives $f_{\rm M}$ above. Qronos uses causal diffusion to future coordinates, but its full procedure also corrects for the upstream-input mismatch.

Appendix

Table 6: Related compensation criteria viewed through

f({\bm{G}}_{\ell};\Theta_{\ell})

$M$	Method	Trace ratio	Realized metric error	Test logit MSE
2	Covariance feedback	1.98079	98.60880	1.69577
	Joint rotation-feedback	1.00000	347.72978	9.21850
4	Covariance feedback	1.98079	9.73280	0.08490
	Joint rotation-feedback	1.00000	22.36973	0.15830
8	Covariance feedback	1.98079	1.76087	0.01370
	Joint rotation-feedback	1.00000	1.79542	0.01529

Appendix

Table 7: Trace optimality versus realized finite-level error on MNIST at

\alpha=0

, averaged over 100 paired seeds. The trace ratio is computed for each seed from row-weighted block traces and determinant bounds, then averaged over seeds. Realized metric error is the aggregate blockwise quadratic error; test logit MSE is measured for the fully quantized network. Lower is better.

$M$	Method	Trace ratio	Realized metric error	Test logit MSE
2	Covariance feedback	1.13685	107.21706	1.83842
	Joint rotation-feedback	1.00000	191.77801	3.52017
4	Covariance feedback	1.13685	10.39471	0.10596
	Joint rotation-feedback	1.00000	12.26181	0.12758
8	Covariance feedback	1.13685	1.89431	0.01863
	Joint rotation-feedback	1.00000	2.20850	0.02126

Appendix

Table 8: Trace optimality versus realized finite-level error on CIFAR-10 at

\alpha=0

, averaged over 100 paired seeds. Quantities and aggregation follow Table 7 . Lower is better.

Method	$M=2$	$M=4$	$M=8$
Full precision	0.01391	0.01391	0.01391
RTN	0.63739	0.06576	0.02395
Covariance-feedback PTQ	0.48603	0.05473	0.02078
Feedback-only tuning; network fixed	0.28594	0.04055	0.01899
Weight-only QAT; ${\bm{G}}={\bm{I}}$ fixed	0.25458	0.04118	0.01914
Joint QAT; ${\bm{G}}_{0}={\bm{I}}$	0.24524	0.04013	0.01784

Appendix

Table 9: Mean test MSE over 100 paired synthetic regression trials without a feedback-training regularizer. Lower is better. Boldface indicates the lowest mean among quantized methods in each column. Feedback-only tuning starts from covariance feedback; the identity-initialized QAT arms start from

{\bm{G}}={\bm{I}}

Method	Bits	PPL $\downarrow$	Acc. (%) $\uparrow$	Logit MSE $\downarrow$	PPL $\downarrow$	Acc. (%) $\uparrow$	Logit MSE $\downarrow$
		WikiText-2			C4
Local EFQ	2	$14874.2\pm 1736.0$	$0.426\pm 0.050$	$8.387\pm 0.323$	12077.99	0.420	8.550
Local EFQ+RHT	2	$2975.59\pm 601.50$	$4.833\pm 1.291$	$5.926\pm 0.141$	3212.88	4.467	6.938
Local joint $R,G$	2	$30881.4\pm 14267.5$	$0.090\pm 0.043$	$10.495\pm 1.498$	18447.59	0.442	10.308
Obj.-EFQ	2	$13371.2\pm 728.1$	$0.408\pm 0.028$	$8.083\pm 0.356$	7538.99	1.294	7.388
Obj.-EFQ+RHT	2	$\mathbf{343.58\pm 32.22}$	$\mathbf{15.709\pm 0.554}$	$\mathbf{4.478\pm 0.035}$	486.20	12.647	4.854

Appendix

Table 16: Llama-2-13B quantization results on WikiText-2 and C4. WikiText-2 values are mean

\pm

sample standard deviation over calibration seeds

\{7,8,42\}

; C4 values are from the complete seed-7 cohort. Accuracy is reported in percent. Lower PPL and logit MSE are better; higher accuracy is better. Boldface marks the best quantized result at each bit width within each dataset. The BF16 references are

6.3937

PPL /

58.819\%

accuracy on WikiText-2 and

7.2441

PPL /

56.444\%

accuracy on C4.

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