Rethinking Post-Training Quantization: Introducing a Statistical

Pre-Calibration Approach

Alireza Ghaffari

, Sharareh Younesian

, Boxing Chen

, Vahid Partovi Nia

and Masoud Asgharian

Department of Mathematics and Statistics, McGill University, Montreal, Canada

Huawei Noah’s Ark Lab, Montreal, Canada

Keywords:

Post-Training Quantization (PTQ), Model Compression, Adaptive LASSO.

Abstract:

As Large Language Models (LLMs) become increasingly computationally complex, developing efﬁcient de-

ployment strategies, such as quantization, becomes crucial. State-of-the-art Post-training Quantization (PTQ)

techniques often rely on calibration processes to maintain the accuracy of these models. However, while these

calibration techniques can enhance performance in certain domains, they may not be as effective in others.

This paper aims to draw attention to robust statistical approaches that can mitigate such issues. We propose a

weight-adaptive PTQ method that can be considered a precursor to calibration-based PTQ methods, guiding

the quantization process to preserve the distribution of weights by minimizing the Kullback-Leibler diver-

gence between the quantized weights and the originally trained weights. This minimization ensures that the

quantized model retains the Shannon information content of the original model to a great extent, guaranteeing

robust and efﬁcient deployment across many tasks. As such, our proposed approach can perform on par with

most common calibration-based PTQ methods, establishing a new pre-calibration step for further adjusting

the quantized weights with calibration. We show that our pre-calibration results achieve the same accuracy as

some existing calibration-based PTQ methods on various LLMs.

1 INTRODUCTION

Large Language Models (LLMs) have rapidly

evolved, demonstrating unprecedented capabilities in

natural language processing tasks. However, the im-

mense computational resources required for their de-

ployment pose signiﬁcant challenges, particularly in

resource-constrained environments. As these models

become more complex, the need for efﬁcient deploy-

ment strategies becomes increasingly critical. Quan-

tization, a technique that reduces the precision of

the model, has emerged as a promising solution to

this problem by signiﬁcantly reducing the computa-

tional and memory demands of LLMs while striving

to maintain their performance.

Post-training quantization (PTQ) is a widely

adopted approach for implementing quantization af-

ter a model has been fully trained. Traditionally, PTQ

methods rely heavily on calibration processes to ﬁne-

tune the quantized model, ensuring that it retains a

high degree of accuracy. These calibration techniques

have proven effective in various domains, particularly

when the target deployment environment closely re-

sembles the conditions under which the model was

calibrated. However, their efﬁcacy may diminish

in scenarios where the deployment environment di-

verges from the calibration conditions, leading to sub-

optimal performance.

For instance, Table 1 shows this limitation when

quantizing a Code-Llama model using mainstream

PTQ methods such as SpQR (Dettmers et al., 2024b).

To showcase the robustness issue of calibration-based

PTQ method, we evaluated the coding performance

of quantized Code-Llama model (Roziere et al., 2023)

on HumanEval (Chen et al., 2021) and MBPP (Austin

et al., 2021) datasets. HumanEval includes 164

human handwritten programming problems with a

function signature, docstring, body, and several unit

tests, and MBPP consists of around 1,000 crowd-

sourced Python programming problems. Table 1

shows that a robust pre-calibration method outper-

forms SpQR(Dettmers et al., 2024b), demonstrating

that if calibration data does not have the same nature

as the task, using calibration data decreases the per-

formance.

Given these limitations, there is a growing inter-

est in exploring mathematical approaches to enhance

the robustness of PTQ methods. In particular, sta-

Ghaffari, A., Younesian, S., Chen, B., Nia, V. P. and Asgharian, M.

Rethinking Post-Training Quantization: Introducing a Statistical Pre-Calibration Approach.

DOI: 10.5220/0013348800003905

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 14th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2025), pages 159-169

ISBN: 978-989-758-730-6; ISSN: 2184-4313

159

Table 1: Comparison of weight-adaptive pre-calibration re-

sults for Code-Llama models on HumanEval (Chen et al.,

2021) and MBPP (Austin et al., 2021).

Model Method Avg Bits Human Eval MBPP

pass@1 pass@10 pass@1 pass@10

FP16 16.00 29.63 59.84 25.87 63.52

RTN (g128) 4.25 30.13 57.97 28.26 62.42

Code-Llama-7B SpQR

∗

4.63 29.94 57.40 27.59 61.78

Pre-calibration (g128, α=5%) 4.60 30.34 58.60 28.03 62.55

FP16 16.00 34.79 66.50 30.17 67.51

RTN (g128) 4.25 33.70 65.88 29.63 66.00

Code-Llama-13B SpQR

∗

4.63 34.19 65.69 29.74 66.20

Pre-calibration (g128, α=6%) 4.67 34.79 66.02 31.36 66.82

tistical methods that guide the quantization process

itself —prior to any calibration— offer a promising

avenue for improvement. By focusing on preserving

the underlying distribution of model weights, these

approaches can potentially ensure a more consistent

performance across diverse deployment scenarios.

In this paper, we introduce a novel weight-

adaptive pre-calibration quantization method that

functions as a precursor to traditional calibration-

based techniques. Our method is grounded in a statis-

tical framework that minimizes the Kullback-Leibler

divergence between the original weights and quan-

tized weights, thereby preserving the Shannon infor-

mation content of the model. This pre-calibration

step ensures that the quantized model remains robust

across various tasks, even before any further calibra-

tion is applied.

Note that our approach not only preserves the

accuracy of the quantized model but also sets a

new initial point for subsequent calibration processes.

Through extensive experiments on various LLMs, we

show that our pre-calibration method achieves per-

formance on par with existing calibration-based PTQ

techniques, offering a more reliable and efﬁcient de-

ployment strategy for LLMs in diverse environments.

To summarize, we make the following contribu-

tions

• We introduce a weight-adaptive pre-calibration

method that as a precursor to traditional

calibration-based methods guides the quantization

process to better preserve model information. To

the best of our knowledge, this is the ﬁrst time

a statistical pre-calibration method has been pro-

posed to improve the quantization process.

• Our proposed pre-calibration method classiﬁes

weights and does not adjust them as opposed to

traditional PTQ methods. The proposed method

then uses pseudo activations (i.e. identity ma-

trix) to identify and isolate important weights sim-

plifying the algorithm to soft-thresholding which

makes the pre-calibration computationally efﬁ-

cient.

• The proposed pre-calibration approach ensures

that the quantized model performs consistently

across a variety of deployment environments, ad-

dressing the limitations of calibration-based meth-

ods in domain-speciﬁc scenarios.

• Our work introduces a new pre-calibration step

that can be integrated with existing PTQ calibra-

tion methods, offering a new initial point for the

calibration optimization procedure, enhancing the

overall effectiveness of the PTQ proces.

• We provide a theoretical foundation for our pro-

posed pre-calibration method using information

theory and techniques from statistical machine

learning.

The rest of the paper is organized as follows.

In Section 2 we provide a detailed problem state-

ment and clarify our proposed weight-adaptive pre-

calibration. Section 3 reviews recent works in the

ﬁeld of PTQ and speciﬁes the differences to our pro-

posed weight-adaptive pre-calibration method. Sec-

tion 4 discusses the proposed pre-calibration algo-

rithm in detail. Section 5 delves deeper into the the-

oretical analysis of the algorithm and shows how pre-

calibration can control information loss in quantiza-

tion. Finally, experimental results supporting our pro-

posed methodology and theoretical ﬁndings are pre-

sented in Section 6.

2 PROBLEM STATEMENT

Recently proposed PTQ methods such as (Fran-

tar et al., 2023; Chee et al., 2023) often use

argmin

∥WX −

WX∥

to adjust the quantized

model weights

W with respect to original weights W

, ensuring that the reduction in precision does not sig-

niﬁcantly degrade performance. In contrast, we pro-

pose a fundamentally different approach to PTQ no-

table as pre-calibration, which re-frames the quan-

tization process as a classiﬁcation problem on the

model’s weights. This approach does not rely on any

calibration data, setting it apart from the conventional

PTQ methods. Instead of using calibration for post-

hoc adjustments, our method classiﬁes the model’s

weights into quantization bins in a manner that in-

herently preserves the underlying distribution of the

weights.

2.1 Weight Adaptive Penalization

Let us consider the following optimization problem

argmin

∥WX −

WX∥

+ λD

( f

∥ f

), (1)

where W denotes original weights with f

distribu-

tion and

W denotes quantized weights with f

distri-

ICPRAM 2025 - 14th International Conference on Pattern Recognition Applications and Methods

160

bution.

Note that problem (1) is only used for classiﬁca-

tion, not shrinkage, of weights and the penalty term,

λD

( f

∥ f

) , is used to guide the classiﬁcation in a

way that the distribution of quantized weights closely

follows that of the original weights.

By viewing pre-calibration as a classiﬁcation

problem, we can ensure that the quantization pro-

cess itself is robust, reducing the need for extensive

calibration afterward. This method fundamentally

shifts the focus from calibration after quantization to

optimizing the quantization process from the outset,

thereby enhancing the robustness and generalizability

of the quantized model across various tasks.

2.2 Weight Classiﬁcation, Penalization,

and Saliency Detection

Unlike traditional calibration-based methods that ad-

just quantized weights by solving arg min

∥WX −

WX∥

, our proposed pre-calibration method does not

modify the quantized weight tensor. Instead, the op-

timization problem (1) and its penalty term are em-

ployed solely to classify weights into two categories:

salient weights and non-salient weights.

It is important to clarify that in this context, salient

weights are not simply large values. Rather, our op-

timization framework deﬁnes salient weights as those

that cause the distribution of quantized weights to de-

viate signiﬁcantly from the original distribution. The

penalty term λD

( f

∥ f

) is used speciﬁcally to en-

sure that the classiﬁcation of weights is conducted in

a way that it preserves the overall weight distribution

after quantization.

2.3 Pre-Calibration and Pseudo

Activations

An inherent challenge that emerges from the opti-

mization problem (1) is that activations X are inher-

ently tied to the input of a layer, implying a need

for calibration. To address this issue, we remove the

necessity for calibration by utilizing pseudo activa-

tions. For example, when XX

⊤

= bI, we can leverage

speciﬁc mathematical properties to simplify the KL-

divergence using a straightforward soft-thresholding

approach.

3 RELATED WORKS

In the ﬁeld of low-precision deep learning, three ex-

isting notable categories are (i) low-precision or quan-

tized training, (ii) quantization-aware training (QAT),

and (iii) post-training quantization (PTQ). While our

proposed method can be applied to both low-precision

training (e.g. (Banner et al., 2018; Zhang et al., 2020;

Zhu et al., 2020; Zhao et al., 2021; Ghaffari et al.,

2022)) and QAT (e.g. (Zhu et al., 2023; Dettmers

et al., 2024a; Liu et al., 2023) ), our primary focus

is PTQ of LLMs which is found to be more challeng-

ing in the literature. As such, we conﬁne our attention

to PTQ of LLMs in this section.

Historically, PTQ methods were common for

computer vision models with small number of param-

eters, some notable methods are AdaRound (Nagel

et al., 2020), OBQ (Frantar and Alistarh, 2022),

AdaQuant (Hubara et al., 2021), and BRECQ (Li

et al., 2021). However, these methods were found

to be either compute-intensive or inaccurate for large

language models.

LLM.int8() (Dettmers et al., 2022) and ZeroQuant

(Yao et al., 2022) are among the ﬁrst PTQ techniques

that were designed for LLMs. LLM.int8() separates

the outlier activations and keeps them in ﬂoating-

point number format while quantizing weights and

non-outlier activations to 8-bit integers. LLM.int8()

separates the outlier activations based on their mag-

nitude. On the other hand, ZeroQuant uses a

ﬁne-grained hardware-friendly quantization scheme

as well as layer-by-layer knowledge distillation for

quantizing both weight and activations. However,

both LLM.int8() and ZeroQuant are not efﬁcient for

quantizing LLMs to extreme low-precision number

formats such as 3-bit integers.

OPTQ (Frantar et al., 2023) is a PTQ algorithm

for LLMs that can quantize weights to 3- or 4-bit inte-

gers. OPTQ adapted a calibration algorithm inspired

by (Hassibi and Stork, 1992) that minimizes the ℓ

loss of the quantized layer output with the original

output. SpQR (Dettmers et al., 2024b) uses OPTQ

algorithm while separating the salient weights and

keeping them in FP16 format and further uses double

quantization to reduce the memory. Both SpQR and

OPTQ algorithms require calibration data for quanti-

zation.

SmoothQuant (Xiao et al., 2023) performs 8-bit

integer quantization of weights and activation by of-

ﬂine migration of the quantization difﬁculty from ac-

tivations to weights. Likewise, AWQ (Lin et al.,

2023), quantized weights by applying per-channel

scales that protect the salient weights by observing the

activation. SmoothQuant and AWQ algorithms also

require calibration data to perform quantization.

QuarRot (Ashkboos et al., 2024), AQLM

(Egiazarian et al., 2024) and QServe (Lin et al., 2024)

are among the most recent PTQ approaches. Quarot

Rethinking Post-Training Quantization: Introducing a Statistical Pre-Calibration Approach

161

tackles the outlier problem by rotating LLMs in a way

that removes outliers from the hidden state without

changing the output. AQLM generalizes the clas-

sic Additive Quantization (AQ) approach for LLMs.

QServe introduces a quantization algorithm with 4-bit

weight, 8-bit activation, and 4-bit KV cache. More-

over, QServe introduces SmoothAttention to effec-

tively mitigate the accuracy degradation incurred by

4-bit KV quantization.

The key feature of our proposed pre-calibration

algorithm lies in its ability to classify weights to

improve quantization accuracy without performing

any calibration. Furthermore, our proposed method

uniquely classiﬁes and isolates outlier weights solely

through analysis of the model weight tensors ensuring

more robust quantization that does not depend on the

calibration dataset. While our proposed methodology

is a precursor to PTQ algorithms, it can also be com-

bined with calibration based method to improve the

accuracy.

4 METHODOLOGY

The core intuition behind our approach is rooted in the

goal of matching the distribution of quantized weights

to that of the original weights as explained by op-

timization problem (1). A straightforward way to

achieve this is by ensuring that each quantized weight

is as close as possible to its corresponding original

weight i.e. ˆw

= w

ˆw

− 1 = 0 s.t. w

̸= 0. This

proximity naturally preserves the overall distribution,

minimizing the divergence between the original and

quantized weight distributions.

However, directly matching each quantized

weight to its original counterpart may not always be

possible, especially when the quantization process in-

troduces signiﬁcant changes in the weight values. To

achieve parsimony, we may decide, according to the

”importance” of each weight, how close a quantized

weight should be matched with its original counter-

part. Given this basic intuition, we are led to the clas-

siﬁcation of weights. The question then arises as to

how such classiﬁcation should be done. To this end,

we consider penalization methods for classiﬁcations

where the penalty on each quantized weight is gauged

and guided by its original weight, the available gold

standard. One penalty that serves such purpose well

is called Adaptive LASSO (Zou, 2006) in statistical

machine learning literature. Adpative Lasso is the

penalty of choice when a gold standard exists.

argmin

∥WX −

WX∥

+ λ

∑



ˆw



, (2)

The mathematical proof of how problem (2) is a

proxy solution to problem (1) is presented in Sec-

tion 5. We emphasize that the penalization method is

only used for classiﬁcation of weights into salient and

non-salient, in statistics language active and inactive,

weights, not for shrinkage.

4.1 Pseudo Activations

A key challenge arising from the optimization prob-

lem (2) is that activations X are intrinsically linked

to the input of a layer, suggesting a requirement for

calibration. To overcome this, we eliminate the need

for calibration by employing pseudo activations. Let

us assume X is an orthogonal matrix i.e. XX

⊤

= bI,

where b is a constant and I is the identity matrix. By

expanding (2) we have

L =



WX −



WX −



⊤

+ λ

∑



ˆw



(3)

= b∥W∥

− 2b

⊤

+ b∥

W∥

+ λ

∑



ˆw



and since W, weights of the original model are con-

stant, the Adaptive LASSO loss becomes

L = −2b

⊤

+ b∥

W∥

+ λ

∑



ˆw



(4)

∑



− 2bw

ˆw

+ b ˆw

+ λ



ˆw





By taking the derivative with respect to ˆw

, and setting

it equal to zero, it is easy to see

ˆw

= sign(w

)ReLU(|w

| −

′

), (5)

where λ

′

= λ/2b and ReLU() denotes the positive

part i.e. ReLU(x) = max(x, 0). Equation (5) shows

that Adaptive LASSO is a simple soft-thresholding

method that is very efﬁcient to be implemented in the

currently available commodity hardware.

4.2 Proposed Pre-Calibration

Algorithm

To match the distributions of original and quantized

weights using KL divergence, we employ adaptive

lasso penalty term, and the combination of Adaptive

LASSO with pseudo activations naturally leads to a

soft-thresholding approach as shown in (5). Through

using soft-thresholding for classiﬁcations of weights,

we achieve a quantized model that not only retains

the key characteristics of the original but also ensures

robust performance across diverse tasks. Algorithm

ICPRAM 2025 - 14th International Conference on Pattern Recognition Applications and Methods

162

1 shows this classiﬁcation procedure. Note that in

Algorithm 1, none of the weights are shrunk to zero

and equation (5) is only used to classify weights to

salient and non-salient weights. Here, salient weights

are deﬁned as those weights that cause the distribution

of quantized weights to deviate signiﬁcantly from the

original distribution.

Algorithm 1: Pre-Calibration algorithm.

Input: Layer weight tensor W, Outlier percentage α

Step 1. Start from a large λ

′

>> 0 in (5) then reduce

it until α percent of weights are selected as outlier.

(Class 1)

Step 2. Classify all other weights as common

weights. (Class 2)

Step 3. Quantize Class 1 weights and Class 2 weights

using minmax quantization

5 THEORETICAL

CONSIDERATIONS

In this section, we establish the theoretical foundation

that underpins our approach, speciﬁcally demonstrat-

ing that the Adaptive LASSO serves as a proxy so-

lution to minimizing the KL divergence between the

original and quantized weight distributions. By rig-

orously analyzing the relationship between adaptive

lasso regularization and KL divergence, we show that

the adaptive lasso effectively guides the quantization

process toward preserving the original model’s weight

distribution.

Suppose f

is twice continuously differentiable.

Let f

′

and f

′′

denote the ﬁrst and second derivatives

of f

. Suppose the mean µ

and variance σ

of the

quantization error δ are small. Then

Claim 1:

( f

∥ f

) ≈ µ

∑

′

) + µ

∑

′′

)( ˆw

− w

)

Claim 2:



∑

′′

)( ˆw

− w

)



≤ C



∑



ˆw



+ 1



where C is

a constant.

Proof: Assuming a quantization error δ

, each

original weight relates to the quantized weight such

that ˆw

= w

+ δ

. Let us also assume errors δ are

independent of the weights values. Therefore, quan-

tized weight distribution is a convolution of original

weights distribution and quantization error distribu-

tion such that

( ˆw) = ( f

∗ f

)( ˆw) =

∞

−∞

( ˆw −x) f

(x)dx (6)

= f

( ˆw) +

∞

−∞

( f

( ˆw −x) − f

( ˆw)) f

(x)dx.

Using the mean value theorem for f

( ˆw − x) −

( ˆw), we have

( ˆw) = f

( ˆw) +

∞

−∞

(−x) f

′

(ξ

ˆw

(x)) f

(x)dx

is small

≈ f

( ˆw) −

∞

−∞

x f

′

( ˆw) f

(x)dx, (7)

and thus,

( ˆw)

≈ 1 −

∞

−∞

x f

′

( ˆw)

(x)dx (8)

= 1 −

′

( ˆw)

∞

−∞

x f

(x)dx = 1 − µ

′

( ˆw)

where µ

is the mean of the quantization error δ. Then



( ˆw)



≈ ln



1 − µ

′

( ˆw)



(9)

|µ

| is small

≈ −µ

′

( ˆw)

By plugging the equation (10) in KL divergence, we

have

( f

∥ f

) = −

∑

( ˆw

)ln



( ˆw

)

( ˆw

)



(10)

≈ µ

∑

′

( ˆw

Then, it follows from Taylor’s expansion around

the original weight w

, i.e. f

′

( ˆw

) ≈ f

′

) +

′′

)( ˆw

− w

) that

( f

∥ f

) ≈ µ

∑

′

) +µ

∑

′′

)( ˆw

− w

(11)

which proves Claim 1.

To prove Claim 2, since w

and f

′′

) are

bounded, i.e. |w

| ≤ A and | f

′′

)| ≤ B in which A

and B are constants, using triangular inequality

|µ

∑

′′

)( ˆw

− w

)| =

|µ

∑

|| f

′′





ˆw

− 1





≤ C

∑



ˆw



+ 1

(12)

in which C = |µ

|AB. This completes the proof.

Since in PTQ, original weights, and their distribu-

tion are known, the ﬁrst term in equation (11) is con-

stant. Therefore, minimizing D

( f

∥ f

) is almost

like minimizing µ

∑

′′

)( ˆw

− w

). Thus, follow-

ing inequality (12), we may replace D

( f

∥ f

)

with

∑

ˆw

| in minimization problem (1) which shows

Adaptive LASSO is a proxy solution to minimization

problem (1).

Rethinking Post-Training Quantization: Introducing a Statistical Pre-Calibration Approach

163

6 EXPERIMENTAL RESULTS

This section provides experimental results supporting

our proposed methodology for pre-calibration quan-

tized LLMs. Note that in our results, we use a

row-wise group quantization technique in conjunction

with the soft-thresholding method as explained in Al-

gorithm 1.

Average Bits: The average bit presented in the

results of our proposed pre-calibration is calculated

based on three factors, (i) the number of non-salaient

weights and their bit-width, (ii) the number of salient

weights and their bitwidth and (iii) location index

of the salient weights. Since pre-calibration classi-

ﬁes the salient weights in an unstructured manner,

tracking the location index is essential to deal with

quantized salient and non-salient weights separately.

While maintaining a mask is straightforward, it would

add an extra bit per weight, which is inefﬁcient in

terms of memory consumption. To tackle this is-

sue, we chose to retain the location index of salient

weights within each group when using group quanti-

zation. Retaining the index of salient weights leads

to a lower average bit since in our method the salient

weight ratio α, is at most 10%. This approach results

in fewer bits compared to using a mask, i.e. it requires

log

g bits only for each salient weight where g is the

group size. Moreover, we store scales and zero-points

in 16-bit ﬂoating-point format. In summary, the aver-

age number of bits per weight is computed as

avg

= (13)



2 ×16



× (1 − α) +



+ log

g +

2 ×16



× α

where g is the group size, α is the percentage of out-

lier weights, and b

and b

are the bit-widths of out-

lier and non-outlier weights respectively.

Clipping Non-Outlier Weights: We also used

clipping to further reduce the b

avg

while maintain-

ing the accuracy in our 3-bit results. The clipping is

done because in 3-bit quantization, maintaining quan-

tization accuracy requires a higher ratio of salient

weights. On the other hand, increasing the ratio would

increase b

avg

due to index tracking of outliers. We

observed that applying a clipping range of 90-95%

to non-salient weights yields similar accuracy com-

pared to increasing the salient ratio. This conﬁrms

that our proposed pre-calibration method can also be

combined with other known quantization techniques

to achieve better results.

Perplexity: We evaluated perplexity of quantized

LLaMA models on WikiText2 (Merity et al., 2016)

and C4 (Raffel et al., 2020) datasets when sequence

length is 2048. Table 3 shows the results compar-

ing perplexity scores for FP16, Round to Nearest

(RTN), GPTQ (Frantar et al., 2023), AWQ(Lin et al.,

2023), SpQR,(Dettmers et al., 2024b) and our pro-

posed pre-calibration method. Pre-calibration outper-

forms AWQ and RTN consistently in terms of per-

plexity scores. Furthermore, pre-calibration exhibits

perplexity scores that closely follow those of SpQR

and OmniQuant, particularly for larger models. These

results highlight the pre-calibration ability to achieve

competitive accuracy while offering a signiﬁcant ad-

vantage in terms of quantization time efﬁciency and

robustness. These results can be used as an initial

point to further optimize the model using calibration.

Refer to Appendix 8 for more perplexity results on

Falcon (Almazrouei et al., 2023) and OPT (Zhang

et al., 2022) models.

Quantization Time: Beneﬁting from our sim-

ple soft-thresholding technique, our proposed pre-

calibration method signiﬁcantly reduces the quanti-

zation time compared to existing methods. The pro-

posed method achieves at least 10× faster quantiza-

tion speed than AWQ (Lin et al., 2023) and surpasses

SpQR (Dettmers et al., 2024b) quantization time by a

factor of 100× as shown in Table 2.

Zero-Shot Task Evaluation: We also evaluated

the accuracy of LLaMA 1 (Touvron et al., 2023a)

and LLaMA 2 (Touvron et al., 2023b) models on

5 zero-shot common-sense reasoning tasks includ-

ing ARC(easy and challenge) (Clark et al., 2018),

HellaSwag (Zellers et al., 2019), WinoGrande (Sak-

aguchi et al., 2021) and PIQA (Bisk et al., 2020) using

LM Evaluation Harness (Gao et al., 2021). As shown

in Table 4, our proposed pre-calibration outperforms

SpQR (Dettmers et al., 2024b) in both 4-bit and 3-

bit quantization, showing correctly classifying salient

weight in pre-calibration step can improve the quality

of PTQ to a great extent.

Table 2: Quantization time comparison.

Model Method Avg Bits Quantization Time (s) ↓

AWQ (g128) 4.25 838

LLaMA-7B SpQR 4.63 10901

Pre-calibration (g128, α=8%) 4.81 57

AWQ (g128) 4.25 1608

LLaMA-13B SpQR 4.63 20502

Pre-calibration (g128, α=6%) 4.67 116

AWQ (g128) 4.25 3740

LLaMA-30B SpQR 4.63 24069

Pre-calibration (g128, α=5%) 4.60 470

ICPRAM 2025 - 14th International Conference on Pattern Recognition Applications and Methods

164

Table 3: Comparison of the pre-calibration perplexity results of 4-bit & 3-bit on WikiText2.

4-bit 3-bit

Model Method Quantization setting Avg Bits Wiki2 ↓ Quantization setting Avg Bits Wiki2 ↓

LLaMA-7B

FP16 - 16.00 5.67 - 16.00 5.67

RTN 4bit-g128 4.25 5.96 3bit-g128 3.25 7.01

OPTQ 4bit-g128 4.25 5.83 3bit-g128 3.25 6.58

AWQ 4bit-g128 4.25 5.78 3bit-g128 3.25 6.35

OmniQuant 4bit-g128 4.25 5.77 3bit-g128 3.25 6.15

SpQR Refer to Appendix 8 4.63 5.73 Refer to Appendix 8 3.98 5.87

Pre-calibration (4bit-g128, α = 8%) 4.81 5.78 (3bit-g128, α = 9%, b

=4) 3.97 6.07

LLaMA-13B

FP16 - 16.00 5.09 - 16.00 5.09

RTN 4bit-g128 4.25 5.25 3bit-g128 3.25 5.88

OPTQ 4bit-g128 4.25 5.20 3bit-g128 3.25 5.70

AWQ 4bit-g128 4.25 5.18 3bit-g128 3.25 5.52

OmniQuant 4bit-g128 4.25 5.17 3bit-g128 3.25 5.44

SpQR Refer to Appendix 8 4.63 5.13 Refer to Appendix 8 3.97 5.22

Pre-calibration (4bit-g128, α = 6%) 4.67 5.15 (3bit-g128, α = 9%, b

=4) 3.97 5.32

LLaMA-30B

FP16 - 16.00 4.10 - 16.00 4.10

RTN 4bit-g128 4.25 4.23 3bit-g128 3.25 4.88

OPTQ 4bit-g128 4.25 4.22 3bit-g128 3.25 4.74

AWQ 4bit-g128 4.25 4.21 3bit-g128 3.25 4.61

OmniQuant 4bit-g128 4.25 4.19 3bit-g128 3.25 4.56

SpQR Refer to Appendix 8 4.63 4.14 Refer to Appendix 8 3.90 4.25

Pre-calibration (4bit-g128, α = 5%) 4.60 4.16 (3bit-g128, α = 8%, b

=4) 3.89 4.31

LLaMA2-7B

FP16 - 16.00 5.47 - 16.00 5.47

RTN 4bit-g128 4.25 5.72 3bit-g128 3.25 6.66

OPTQ 4bit-g128 4.25 5.61 3bit-g128 3.25 6.38

AWQ 4bit-g128 4.25 5.60 3bit-g128 3.25 6.24

OmniQuant 4bit-g128 4.25 5.58 3bit-g128 3.25 6.03

SpQR Refer to Appendix 8 4.63 5.52 Refer to Appendix 8 3.98 5.66

Pre-calibration (4bit-g128, α = 8%) 4.81 5.60 (3bit-g128, α = 9%, b

=4) 3.97 5.83

LLaMA2-13B

FP16 - 16.00 4.88 - 16.00 4.88

RTN 4bit-g128 4.25 4.98 3bit-g128 3.25 5.52

OPTQ 4bit-g128 4.25 4.99 3bit-g128 3.25 5.42

AWQ 4bit-g128 4.25 4.97 3bit-g128 3.25 5.32

OmniQuant 4bit-g128 4.25 4.95 3bit-g128 3.25 5.28

SpQR Refer to Appendix 8 4.63 4.92 Refer to Appendix 8 3.96 5.01

Pre-calibration (4bit-g128, α = 6%) 4.67 4.93 (3bit-g128, α = 9%, b

=4) 3.97 5.05

Table 4: Comparison of the pre-calibration results on zero-shot tasks using LM Evaluation Harness (Gao et al., 2021).

Model Method Avg Bit ARC-c ARC-e HellaSwag Winogrande PIQA Avg

LLaMA-7B

FP16 16 41.89 75.25 56.95 69.93 78.67 64.54

RTN (g128) 4.25 42.92 74.54 56.29 70.01 78.18 64.39

OPTQ (g128) 4.25 40.78 74.62 56.59 69.22 78.51 63.94

AWQ (g128) 4.25 41.13 75.00 56.44 69.14 77.86 63.91

SpQR

∗

4.63 41.72 75.21 56.65 69.61 79.05 64.45

Pre-calibration (g128, α = 8%) 4.81 42.15 75.34 56.72 70.17 78.56 64.59

LLaMA-13B

FP16 16 46.42 77.36 59.88 72.69 79.16 67.19

RTN (g128) 4.25 45.82 76.77 59.37 72.45 79.71 66.82

OPTQ (g128) 4.25 45.99 77.06 59.22 73.32 78.94 66.91

AWQ (g128) 4.25 45.99 76.89 59.42 72.53 78.78 66.72

SpQR

∗

4.63 45.73 76.85 59.70 73.09 79.22 66.92

Pre-calibration (g128, α = 6%) 4.67 45.99 76.85 59.41 73.01 78.94 66.84

LLaMA-30B

FP16 16 52.90 80.43 63.37 75.85 81.12 70.73

RTN (g128) 4.25 52.05 80.77 62.89 74.19 80.58 70.10

OPTQ (g128) 4.25 51.37 80.47 63.12 75.30 80.79 70.21

AWQ (g128) 4.25 53.41 80.72 63.16 75.45 80.69 70.69

SpQR

∗

4.63 51.45 80.47 63.08 74.74 80.74 70.10

Pre-calibration (g128, α = 5%) 4.60 51.88 80.77 63.07 74.19 80.74 70.13

LLaMA-2-7B

FP16 16 43.43 76.35 57.16 69.14 78.07 64.83

RTN (g128) 4.25 43.09 76.18 56.90 68.67 77.48 64.46

OPTQ (g128) 4.25 41.89 74.96 56.33 69.30 77.97 64.09

AWQ (g128) 4.25 42.58 75.67 56.39 68.35 77.53 64.10

SpQR

∗

4.63 44.28 76.14 56.95 68.51 77.42 64.66

Pre-calibration (g128, α = 8%) 4.81 43.17 76.39 57.12 69.77 77.97 64.88

LLaMA-2-13B

FP16 16 48.46 79.42 60.05 72.38 79.11 67.88

RTN (g128) 4.25 48.12 78.83 59.74 72.69 78.67 67.61

OPTQ (g128) 4.25 47.95 78.79 59.81 72.85 78.56 67.60

AWQ (g128) 4.25 46.59 79.46 59.85 73.32 79.05 67.65

SpQR

∗

4.63 48.46 79.76 59.97 71.90 78.84 67.79

Pre-calibration (g128, α = 6%) 4.67 48.38 79.63 59.89 72.53 78.94 67.87

∗

Refer to Appendix 8 for quantization settings.

Rethinking Post-Training Quantization: Introducing a Statistical Pre-Calibration Approach

165

7 DIVERGING VIEW ON

IMPROVING POST-TRAINING

QUANTIZATION

Traditional post-training quantization (PTQ) meth-

ods typically follow a two-step process: quantiza-

tion followed by calibration. While this approach has

been effective, the calibration step often poses signiﬁ-

cant challenges, as it involves adjusting the quantized

weights to recover as much of the original model’s

accuracy as possible. The difﬁculty of this task is

compounded by the fact that calibration is inherently

a complex optimization problem, and based on op-

timization theory, having a better initial point can

greatly inﬂuence the outcome.

In this paper, we propose a shift in perspective by

introducing a pre-calibration step prior to the tradi-

tional calibration process. We have demonstrated that

pre-calibration can provide a signiﬁcantly improved

starting point for calibration, enhancing the overall

effectiveness of the PTQ process. In fact, in some

cases, this pre-calibration starting point outperforms

even the ﬁnal results of previously introduced calibra-

tion methods.

Another critical aspect of our approach is the clas-

siﬁcation of weights during quantization. While we

utilized KL divergence in conjunction with Adap-

tive LASSO, where the Adaptive LASSO serves as

a proxy solution for minimizing KL divergence, this

framework is ﬂexible. The choice of divergence mea-

sure or regularization penalty is not ﬁxed; one could

employ other f -divergence measures or adapt differ-

ent penalties based on the speciﬁc needs of the PTQ

method. Our proposal is not prescriptive in this regard

but rather encourages the exploration of the best tools

for achieving optimal weight classiﬁcation before the

PTQ procedure.

8 CONCLUSION

We presented a weight-adaptive pre-calibration ap-

proach for PTQ methods. Traditional PTQ techniques

typically rely on a two-step process of quantization

followed by calibration. However, the calibration step

often proves challenging, as it requires careful ad-

justments to quantized weights to regain the model’s

original accuracy. We have demonstrated that pre-

calibration can provide a signiﬁcantly improved start-

ing point for calibration, enhancing the overall ef-

fectiveness of the PTQ process and, in some cases,

even surpassing the ﬁnal performance of previously

introduced calibration methods. This highlights the

importance of starting with a well-prepared initial

point, which can signiﬁcantly impact the success of

the quantization process. Our work rethinks the tradi-

tional PTQ pipeline, advocating for the integration of

a pre-calibration step that enhances the starting condi-

tions for calibration. This shift not only improves the

robustness and effectiveness of the quantization pro-

cess but also opens the door to further innovations in

model efﬁciency and deployment strategies. As the

demand for efﬁcient deployment of Large Language

Models (LLMs) continues to grow, our approach pro-

vides a new perspective on optimizing PTQ for di-

verse applications.

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APPENDIX

Experimental Settings

Seed Sensitivity

Since our proposed method, only uses deterministic

pre-trained weights of the model and performs a soft-

thresholding to identify α percent of outlier weights,

it does not exhibit any stochastic behavior during the

quantization. Furthermore, we do not use any data for

calibration and thus, our algorithm is robust toward

randomness in data selection. We believe this is the

main advantage of our proposed algorithm.

Calibration Datasets and Parameters

We follow the pipelines used in SpQR

and AWQ

of-

ﬁcial implementation to generate calibration datasets.

Random selection of 128 samples of length 2048

form RedPajama (Computer, 2023), C4 and Reﬁned-

Web (Penedo et al., 2023) is used for quantization of

LLaMA 1, LLaMa 2, OPT (Zhang et al., 2022) and

Falcon (Almazrouei et al., 2023) using SpQR. For

AWQ experiments 128 samples from a small subset

of Pile (Gao et al., 2020) dataset is used following the

AWQ’s implementation.

Hyper-Parameters and Conﬁgs

RTN: We implemented RTN quantization method

based on the implementation of AWQ which supports

weight reshaping for group quantization.

AWQ: We used AWQ’s ofﬁcial implementation for

quantizing LLaMA and OPT models.

SpQR: We use SpQR’s ofﬁcial implementation for

quantizing LLaMA, Code-Llama and OPT models.

Table 5 shows the hyper-parameters used for SpQR

quantization.

Table 5: Quantization conﬁguration of SpQR.

Model Calibration Group Weight Scales & Zeros Outlier

Set Size Bits Bits Threshold

LLaMA RedPajama 16 4 3 0.2

RedPajama 16 3 3 0.25-0.28

Code-Llama RedPajama 16 4 3 0.2

OPT C4 16 4 3 0.2

Falcon ReﬁnedWeb 16 4 3 0.2

Hardware Settings

We perform quantization on single NVIDIA V100-

32G GPU. For evaluation using LM Evaluation Har-

ness we use 8×V100-32G GPUs for 30B models.

Extra Experimental Results

Table 6 shows the perplexity results on OPT (Zhang

et al., 2022) and Falcon(Almazrouei et al., 2023)

models.

See https://github.com/Vahe1994/SpQR

See https://github.com/mit-han-lab/llm-awq

ICPRAM 2025 - 14th International Conference on Pattern Recognition Applications and Methods

168

Table 6: Perplexity of 4-bit OPT and Falcon models on

WikiText2 and C4.

Model Method Quantization setting Avg Bits Wiki2 ↓ C4↓

OPT-6.7B

FP16 - 16.00 10.86 11.74

RTN 4bit-g128 4.25 11.15 12.31

AWQ 4bit-g128 4.25 10.95 11.86

SpQR Refer to Appendix 8 4.63 10.91 11.78

Pre-calibration (4bit-g128, α = 6%) 4.67 10.86 11.99

OPT-13B

FP16 - 16.00 10.13 11.20

RTN 4bit-g128 4.25 10.30 11.51

AWQ 4bit-g128 4.25 10.29 11.28

SpQR Refer to Appendix 8 4.27 10.22 11.27

Pre-calibration (4bit-g128, α = 6%) 4.67 10.20 11.31

OPT-30B

FP16 - 16.00 9.55 10.69

RTN 4bit-g128 4.25 9.94 10.94

AWQ 4bit-g128 4.25 9.61 10.74

SpQR Refer to Appendix 8 4.63 9.55 10.71

Pre-calibration (4bit-g128, α = 5%) 4.60 9.64 10.79

Falcon-7B

FP16 - 16.00 6.59 9.50

RTN 4bit-g128 4.25 6.79 9.79

SpQR Refer to Appendix 8 4.44 6.64 9.58

Pre-calibration (4bit-g128, α = 4%) 4.53 6.69 9.63

Falcon-40B

FP16 - 16.00 5.23 7.76

RTN 4bit-g128 4.25 5.31 7.88

SpQR Refer to Appendix 8 4.46 5.26 7.79

Pre-calibration (4bit-g128, α = 5%) 4.60 5.27 7.81

Rethinking Post-Training Quantization: Introducing a Statistical Pre-Calibration Approach

169