Tequila: Trapping-free Ternary Quantization for Large Language Models
Authors: Hong Huang
Decheng Wu
Rui Cen
Guanghua Yu
Zonghang Li
Kai Liu
Jianchen Zhu
Peng Chen
Xue Liu
Dapeng Wu
Venue: ICLR 2026 Poster (2026)
Keywords: Ternary Quantization
Large Language Models
Edge Computing
Primary area: foundation or frontier models, including LLMs

Abstract: Quantization techniques are essential for the deployment of Large Language Models (LLMs) on edge devices. However, prevailing methods often rely on mixed-precision multiplication that lacks efficient hardware support, making it not feasible. Ternary weight quantization addresses this by constraining weights to {-1, 0, 1}, replacing expensive multiplications with hardware-efficient additions. However, such aggressive compression leads to significant accuracy degradation, even after costly quantization-aware training with massive data.  We identify the core issue as _**deadzone trapping**: a large number of weights are trapped at the deadzone boundary._ This occurs because these weights receive only noisy, less informative gradients, preventing stable escape from the deadzone and severely impeding model capacity and optimization. To address this issue, we propose **Tequila**, a trapping-free quantization optimization method that reactivates deadzone-trapped weights by repurposing them as dynamic biases. This allows the repurposed weights to provide a continuous signal in the forward pass and, critically, receive direct, meaningful gradient signals during backpropagation, thereby enhancing model capacity and optimization with nearly _zero_ inference overhead. Extensive evaluations demonstrate that Tequila outperforms state-of-the-art (SOTA) ternary quantization methods across five benchmarks. Specifically, on the ARC benchmark, it achieves $>4$% accuracy gain over the SOTA baseline, nearly matching full-precision performance (within $<1$% gap) with an $3.0\times$ inference speedup. Consequently, Tequila offers a highly practical and efficient implementation for the deployment of advanced LLMs in resource-constrained environments. The code is available at https://github.com/Tencent/AngelSlim/tree/tequila/TernaryQuant .

Published as a conference paper at ICLR 2026
Y TEQUILA: TRAPPING-FREE TERNARY QUANTIZA- TION FOR LARGE LANGUAGE MODELS
Hong Huang! Decheng Wu Rui Cen
Guanghua Yu² Zonghang Li
Kai Liu? Jianchen Zhu Peng Chen² Xue Liu" Dapeng Wu
City University of Hong Kong TTencent *MBZUAI *McGill University
ABSTRACT
Quantization techniques are essential for the deployment of Large Language Mod- els (LLMs) edge devices. However, prevailing methods often rely mixed- precision multiplication that lacks efficient hardware support, making it not feasi- ble. Ternrry weight quantization addresses this by constraining weights to {-1, O, 1), replacing expensive multiplications with hardware-efficien additions. How- ever, such aggressive compression leads significant accuracy degradation, even after costly quantization- aware training with massive data. We identify the core is- sue deadzone trapping: large number of weights are trapped at the deadzone boundary. This occurs because these weights receive only noisy, less informa- tive gradients, preventing stable escape from the deadzone and severely impeding model capacity and optimization. To address this issue, we propose Tequila, a trapping-free quantization optimization method that reactivates deadzone-trapped weights by repurposing them as dynamic biases. This allows the repurposed weights to provide a continuous signal in the forward pass and, critically, re- ceive direct, meaningful gradient signals during backpropagation, thereby enhanc- ing model capacity and optimization with nearly zero inference overhead. Ex- tensive evaluations demonstrate that Tequila outperforms state-of-the-art (SOTA) ternary quantization methods across five benchmarks. Specifically, on the ARC benchmark. it achieves 4% accuracy gain over the SOTA baseline, nearly matching full-precision performance (within < 1% gap) with a 3.0x inference speedup. Consequently, Tequila offers a practical and efficient implementation for the deployment of advanced LLMs in resource-constrained environments. The code available at /https://qi thub. ctt////eeeeeeeeeeeggeeeeee
INTRODUCTION
Recent advancements in large language models (LLMs) (Wu et al. 2023] Floridi & Chiriatti 2020 Zhang al.) 2022 have demonstrated remarkable success across wide range of applications, from conversational chatbots to creative writing. However, growing concerns over data privacy, the need for offline functionality, and the high cost of large- scale cloud deployment (Yao al.] [024] Liagkou et al. 2024) have necessitated the deployment of these models directly on edge devices, which are usually resource-constrained.
Quantization (Lin et al. 2023 Frantar et al. 2022) has emerged as a promising technique to achieve this goal, reducing model size and computational requirements by representing model weights with lower precision. However, most existing quantization methods (Kwon et al. [2022] Lin et al. 2023 Frantar et al. 2022) are primarily designed for server-grade GPUs that support specialized hard- features, such as mixed- precision multiplication (Lin et al. 2023). These methods are often incompatible with edge and mobile hardware, highlighting a critical need for hardware-friendee quantization approaches that remain effective across diverse and resource-constrained devices.
mixed-precision
Ternary quantization (Li et al. 2016 Liu & Liu 2023
Ma et al. 2025 Wang et al. 2023 2025a
offers promising path for de-iev deployment of LLMs by constraining weights 0.
Work with Tencent

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Published as a conference paper at ICLR 2026
Deadzone
Deadzone
Limited Capacity
Deadzone Trapping
200K
Dead weights are pruned, providing no information
weights are trapped at the boundary, disrupting optimization Large Acc gap 5%
Training with Ternary Quantization
Dead weights obtain noisy,
less informative gradients
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The reactive weights can escape stably, enhancing optimization Slight Acc gap 1%
Training with Our Tequila
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adaptive blases
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Roundary of deadrone
Weights quantized to
Wrights quanthed to
Weights converted to blas
Figure 1: (Top) Deadzone Trapping in Ternary Quantization: Dead weights are trapped around the deadzone boundary due to noisy and less informative gradients, impeding model capacity and optimization, causing a significant accuracy drop (> 5%) versus the full-precision. (Bottom) Reac- tivation Strategy of Tequila: Our Tequila reactivates dead weights as biases, providing meaningful gradients for stable escapes, enhancing model optimization, achieving a minor accuracy gap (< 1%).
This method reduces matrix multiplication to efficient additions, illustrated in Fig. 2 which widely supported by most hardware. However, such aggressive compression introduces significant information loss, often leading to severe accuracy degradation even after costly Quantization-ANa Aware Training (QAT) on massive datasets. For instance. BitNet (Ma et al., 22025 Wang et al. 2025a) consumes 4T tokens during QAT but still fails match full-preciston performance. Thus, the dual problems of performance degradation and prohibitive training overheads persist as the fundamental barriers to the development of effective ternrry LLMs.
In this paper, we identify the key source of these challenges deadzone trapping, where a large number of weights are trapped at the deadzone boundary. Deadzone trapping arises from the aggres- sive nature of ternary quantization, which creates a vast deadzone that quantizes a large proportion of weights to zero. During training. these "dead" weights receive only noisy, less informative gradients from the Straigh-TThrouhh Estimator (STE), preventing effective optimization. Lacking consistent directional signals, these weights are hard to escape the deadzone stably and are accumulated at the deadzone boundary, shown in Fig. | (Top). This results in cycle of ineffective oscillation, rendering these weights forg-term inactive and severely impeding model capacity and optimization.
To address the deadzone trapping issue, we propose Tequila, trapping-free Ternryy quantization method for large language models. Our key idea is to reactivate dead weights by repurposing them biases. This provides continuous signals to the output, enhancing the model capacity, as shown in Fig. 2 (c). More importantly, these weights receive direct and informative gradients via the bias terms, enabling them to escape the deadzone stably, shown Fig. | (Bottom). Crucially, these biases can be computed offline, introducing nearly zero inference overnead.
We evaluate Tequila on five common benchmarks using LLaMA 3.2 (Touvron al.) Qwen3 (Bai et al.) (023) models. Our experiments demonstrate that Tequila outperforms all state- of-the-art (SOTA) ) ternary methods across all benchmarks while requiring only limited training data. For instance, when trained on just 10B tokens, Tequila achieves > 4% accuracy gain over the SOTA baseline on the ARC benchmark, nearly matching full-precision performance. Furthermore, it delivers significant 3x inference speedup a practical and efficient solution for deploying LLMs on resource-constrained devices.
2023 and
Intel 8263C CPU, verifying that Tequila offers
2
BACKGROUND AND CHALLENGE
2.1 TERNARY QUANTIZATION
Teraryy quantization is an extreme compression technique that constrains model weights to ternary values {-1,0, +1). This representation converts the computationally expensive weight-input matrix
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Weights are dead
Input
Dead weights are reactivated
Input
Online Computing
Ternary Weight
Output
Quatermary Weight
Output
bias like term
X
+ xg
0+
X
X/2
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1
0
-1
X2
X4
0+ -1
0_
-1
Xx
+
(Addition-onlyy)
(assuming x are positive)
(a) Ternary quantized layer (Ace 60.3%)
(b) Activating dead weights as minima (Ace 61.5%
Input
Plug Plug-and-1 and-play
Ternary Weight
X
Reactivated weights
Output
Dynamic bias
0
X
+ sum
Aws
AwA
Aw2 Awa
X3
AwHs
Awg + Awa
Adaptive differentiable reactivation Offline computing
(c) Tequila adaptively reactivate dead weights as dynamic hiases (Ace 64.5% 1T)
Figure 2: (a) Prior Ternary Quantization replaces multiplications with efficient additions but suffers from severe information loss and limited capacity due to deadzone-trappet weights. (b) Minima Reactivation assigns signed minima to dead weights, improving capacity but yielding only marginal accuracy gains. (c) Tequila reactivates dead weights adaptive dynamic biases via differentiable function, achieving significant accuracy improvements with nearly zero inference overhead. For simplicity, omit the scaling operation in the Figure.
multiplication into input-inner addition, shown in Fig. (a). offering significant hardware advan- tages. Given full-precision weight vector W = (W1,- w,), the general form of the ternary quantization function (-)) defined as:
+1, if w 2 A: 0, if [wsl < A;
Q(W) = Wa, w
1, if w. S-A,
scaling factor and A threshold parameter. A significant body of research focuses on determining optimal values for a and A. For instance, the TWN (Li et al.] 2016) assumes the weight distribution follows a standard Gaussian distribution.
where W -i (,) is ternary weights,
It approximates the optimal threshold as A 0.75 Σ== /w:] and derives a closed-form solution
for a by minimizing ||W - 22I12 Subsequent methods (Chen et al. 2024 Liu et al. 2025 Zhu al., (2166) forgo this distributional hypothesis and instead treat or A trainable parameters
learned during optimization. In recent open-source ternary LLMs (Ma et al. 2025 Team et al.)
2025 Kaushal et al. (025), the static absmean quantization method has gained wider adoption due to its training stability, where the a and A are defined by
A= a
a=
w
(2)
Due the aggressive nature of this compression, quantized models often require Quantization- Aware Training (QAT) to recover accuracy. During QAT, full-precision weights are dynamically quantized using quantization function Q(-) in Eq. [or the forward pass, while the backward pass operates on full-precision gradients. This process maintains a full-precision copy of the weights W accumulate gradient updates, as detailed Appendix C Due of Q(-), the gradients for W are approximated using the Straight-Through Estimator (STE) (Zhu (ct al., 2016 Chen et al. 2024), leading to the following forward pass and backpropagation with input vector X [21..... ):
oinddifferentiabee function
Y Q") XWW OL TYJ
if i > A; if w A. I
8L By OL
Dw:
where I. denotes the loss of the model prediction. After training, the full-precision weights W are discarded. The ternary weights W and the scaling factor are packed for inference. During infer- ence, the ternary multiplication of X W is computed first, following the efficient process shown in Fig. [ (a). This eliminates the need for mixed-precision matrix multiplication, replacing it with hardware-efficient addition. The related work and details can be found in Appendix and E.4
3

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Nalve Ternary Quantization
Minima Reactivation
Our Tequila
Active Weights
Average GSNR Value
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Average GSNR Value
Unstable
Average GSNR Value
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Figure 3: The average Gradient Signal-to-Noise Ratio (GSNR) for (left) Naive Ternary Quantization, (middle) Minima Reactivation, and (right) our Tequila method during training.
2.2
DEADZONE TRAPPING
Ternary quantization incurs significant information loss and performance degradation, necessitating extensive retraining to recover model accuracy. This problem exacerbated in LLMs, where the scale of parameters amplifies the difficulty of retraining. For instance, BitNet (Ma et al., 2025) trains from scratch on 4T tokens, cost rivaling standard pre-training. Similarly, BitCPM4 (Tem) et al., p025) uses 100B tokens of training even when starting from pre-trained model. To our knowledge, no existing work has achieved competitive performance with less than 100B tokens for lernary-quantized LLMSs
We identify the core cause of this inefficiency as deadzone trapping, where a large number of weights become trapped at the deadzone boundary. This issue originates from the aggressive nature of ternary quantization, which creates a deadzone within the range (-A, A) where weights are quantized to zero. During Training, these "dead" weights (t;; = 0) and their corresponding inputs T; are continually pruned in the forward pass, contributing nearly no information to the output Y and the loss L. Consequently, the upstream signal BY reflects less sensitivity to wy and æ4, making the overall gradients in Eq. [ noisy and less informative. This problem is exacerbated by the non- differentiable quantization function Q(-), as the required use of the STE injects significant noise into these gradients. As shown in Fig. (left), the Gradient Signal-to-Noise Ratio (GSNR) for naive ternary quantization confirms this analysis: in early training, the GSNR for dead weights is much lower than for active weights (see Appendix E.2 for more details and analysis).
ØL
These noisy gradients prevent dead weights from accumulating consistent update signals, trapping them within the deadzone. When large number of weights remain at deadzone for an extended period, the model easily becomes trapped in a suboptimal state. When some weights occasionally break free, their abrupt changes often trigger opposing gradients that pull them back. This dynamic results in weights accumulating at the deadzone boundary, trapped in a cycle of ineffective oscil- lation, as shown in Fig. (top) and Fig. [0] Ultimately, deadzone trapping renders substantial portion of the model weights long-term inactive, severely impairing both model capacity and train- ing efficiency, evidenced in Fig. (left).
3 TEQUILA: DEADZONE-FREE TERNARY QUANTIZATION
3.1 MITIGATING THE DEADZONE TRAPPING BY MINIMA REACTIVATION
We identify the fundamental limitation of deadzone trapping in ternary quantization as the fact that the dead weights provide no meaningful signal to the model output the forward pass. This creates a vicious cycle where trapped weights cannot contribute to learning and struggle to escape the deadzone effectively, significantly impeding convergence.
Our core motivation is to allow dead weights to contribute to the forward output, even with minimal but informative values, which can break this trapping by receiving a more direct and informative gradient signal. Therefore, we propose to repurpose dead weights to provide informative signals, which enhance model capacity and establish clean gradient pathways.
To implement this, intuitively propose Minima Reactivation, preserving the sign information of dead weights, reactivating them as distinct values 0_ and 0,, representing negative and positive minima, respectively, shown in Fig. (b). This creates quaternary weight representation úi
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Published as a conference paper at ICLR 2026
{-1,0-,04- -1}. Crucially, to isolate the impact of increasing the representation capability from 3 to 4 values while preserving only sign information, we replace multiplication with a constant- magnitude mapping. For any imput z, the operation yields value =
I 0+ sign(r)), x 0- sagn(
Formally, let W (w11,-. ........ ,i, represent the quaternary weight vector, and define the set of indices in the deadzone as D = {il A < w; < A.. The forward pass can then be converted to:
Y = XWa a Σ sign(w:) te
Σ (sigg)))) SDD
(5)
iED
original output
online bius- like term
where the 1 denotes the set of indices not the deadzone. This formulation reveals that previously dead weights now contribute meaningfully to the output through the bias-like term. Consequently, these weights wi receive informative gradients from backpropagation, denoted as
aT.
OT. E D.
= e sign(x;) 0Y'
(6)
Ow;
Compared to previous naive ternary quantization, where dead weights receive essentially random gradients in Eq. this approach enables these dead weights to contribute meaningful, non-zero signal that directly influences the final output. This allows them to receive more informative gradient signals proportional
the downstream loss, thereby providing As shown by the GSNR measurements in Fig. (middle)) Minima Reactivation provides reactivated weights with higher GSNR, thereby mitigating the dead-zone trapping issue and enabling more efficient optimization.
effective optimization direction.
Limitations: While this Minima Reactivation method demonstrates the theoretical viability of deadzone repurposing, we identify two practical limitations: (1) Noisy gradients and unstable training: The gradients for reactivated weights Eq. 3 still rely on the STE for the sign(-) oper- ation, introducing noise and unstable training, shown in Fig. (middle), yielding only marginal accuracy gains. (2) Non-negligible inference overhead: The additional bias-like term, which is input-dependent, introduces non-negligible inference overhead, as it requires computation for every forward pass.
These insights motivate our final Tequila method, which retains the core concept of deadzone reac- tivation while introducing key optimizations to overcome these limitations, as detailed in the next subsection.
3.2 TEQUILA: REPURPOSING DEAD WEIGHT AS DYNAMIC BIAS
This section introduces Tequila, a trapping-free quantization method that reactivates deadzone- trapped weights enhance model capacity and restore optimization potential without sacrificing hardware efficiency. Tequila's core innovation lies in repurposing the deadzone from a fundamental limitation into source of adaptability through the following three key designs.
Differentiable Reactivation: To address the noisy gradient and unstable training problem in Min- ima Reactivation, we replace the non-differentiable mapping to a constant the with scaling for dead weight w,, allowing the computation of reactivation values as Awa, resulting in a smooth, differentiable reactivation function. That is, in forward pass, the Eq. 5 is converted as:
== XWa a Σ sign(w;) + Σ sign(x.)w.
RED
iED
Crucially, this design bypasses the STE, providing direct and informative gradients that enable fective optimization of previously trapped weights.
To eliminate the non-negligible inference overhead of Minima Reactivation, repurpose dead weights as actual biases, thereby converting online com- putation into an offline one, i.c., setting EieD sign(x;)us > EieD w; in Eq. 7 This simplifi- cation is justified by two reasons. First, it remains faithful to motivation of enabling dead
Repurposing Dead Weights as Biases:
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Published as a conference paper at ICLR 2026
weights contribute to the output; bias term achieves this in activation-fiee manner, perfectly aligning with our hypothesis while minimizing overhead. Second, we find empirical support for this approximation: the cosine similarity between the original bias-like vector and the simplified bias vector high (> 70%), as shown in Fig. validating that our simplification retains the essential functional behavior.
Hybrid Roles of Reactivated Weights: While converting dead weights to pure biases (EeD Awn)) provides clean gradients, it discards valuable input information. Tequila overcomes this limitation by assigning reactivated weights to hybrid roles. In addition to functioning as an dynamic bias, these partici-
1.0
Average Cosine Similarity
0.0
W
0.6
weights simultaneously maintained pants in the ternary matrix multiplication. This dual role creates mixed gradient from both the standard ternary pathway and the direct bias pathway. Con-
0.4
sequently, the optimization process preserves cru- cial input information while benefiting from a direct. informative gradient signal, driving more effective training.
0.0
200
800
1000
Training Steps
Figure 4: The average cosine similarity of bias-like term in Eq. and pure bias term Eq. 8 during training of Tequila.
With these three key designs, the Tequila forward pass combines efficient ternary operations with dy- namic biases:
Y XQ(W) C(W) = XWa+ Σ Xm0
ND
where the bias term C(W) EiED Aw; acts as residual connection for weights within the dead- zone. This formulation directly yields superior gradients for these dead weights:
0L
ØL
ØL aY
dw, = DY
ViED,
thereby preserving input-dependent information and delivering a direct, informative gradient signal to enable effective optimization. As shown by the GSNR measurements in Fig. (right), Tequila provides high and stable GSNR during the training, thereby enabling more efficient optimization.
Ad Advantages: Tequila provides five key advantages over existing ternary quantization methods:
(1) Enhanced Model Capacity: Reactivating dead weights effectively expands the model parameter space without increasing computational complexity during inference.
(2)
Trapping-frt Optimization: By providing direct, informative gradients, Tequila enables stable escape from deadzone, achieving trapping-free free weight optimization.
(3) Training Stability: The differentiable reactivation function ensures stable optimization while maintaining quantization constraints, resulting in more consistent, reliable training convergence.
(4) Plug-and-play Design: Tequila is simple and plug-and-pla. module that be easily inte- grated into most existing ternary quantization methods.
(5) Nearly Zero Inference Overhead: The input-agnostic bias term can be precomputed offline and seamlessly fused into the computation kernel, achieving nearly zero inference overhead. This preserves the hardware efficiency of pure ternary quantization.
4 EVALUATION
To validate the efficacy of Tequila, conduct comprehensive experiments evaluating its perfor- mance against sfate-of-the-an ternary quantization methods. All experimental results are averaged three independent runs with random seeds. In all tables, the best and second-best results highlighted in purple and blue color, respectively, and the result of the full- full-precision method is set gray color for reference.

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Published as a conference paper at ICLR 2026
Size
Method
ARC-e
ARC-c
HeIS
PIQA
WinG
GPQA
Average
BF16
0.654
0.313
0.477
0.742
0.603
0.222
0.502
LSQ
0.376
0.177
0.258
0.574
0.506
0.231
0.354
SEQ
0.421
0.180
0.273
0.604
0.510
0.232
0.370
1B
DLT
0,424
0.174
0.256
0.563
0,513
0.277
0.368
TWN
0.407
0.220
0.284
0.601
0,492
0.212
0.363
AbsMedian
0.567
0.251
0.339
0.674
0,533
0.222
0,431
AbsMean
0.603
0.259
0.360
0.683
0.541
0.227
0.445
0.645
0.308
0.393
0.719
0,558
0.237
0.477
Tequila+AbsMean
0.645
0.305
0.391
0.710
0.542
0.232
0.471
BF16
0.745
0,422
0.552
0.768
0,691
0.303
0.580
LSQ
0.431
0,200
0.294
0.599
0,522
0.239
0.354
SEQ
0.498
0.231
0.303
0.645
0.529
0.258
0.411
3B
DLT
0.361
0.161
0.260
0.572
0.496
0.272
0.354
TWN
0.692
0.351
0.462
0.734
0.586
0.237
0.510
AbsMedian
0.636
0.299
0,406
0.713
0,558
0.181
0,466
AbsMean
0.672
0.329
0.439
0.735
0.582
0.301
0.510
Dequila+AbsMeede
0.701
0.345
0.449
0.733
0.611
0.237
0.513
Tequila+AbsMent
0.702
0.346
0.464
0.739
0.627
0.303
0.530
Table Comparison of Tequila method with different ternary quantization methods
4.1 EXPERIMENTAL SETUP
We provide a comprehensive overview our experimental configuration below, with additional implementation details available in the Appendix
Datasets, Models and Evaluation: We utilize the LLLMA-3.2-1BB LLaMA-3.2-3B Toouroo al., 2023) and Qwen3-4B Bai et al. (2023) models as our base architectures, employing group experiments unless otherwise specified. For quantization-awan train- ing, we use 1OB tokens sampled from the UltraFine Web dataset (Wang et al., 2025c). Following
size of 128 throughout
established practices in ternary quantization research (Liu al. 2025| Chen et al.) 2024 Ma al.,
2025), we evaluate model performance with Im-evaluation--arness (Gao et al. (024) on five zero-
shot benchmarks: PIQA (Bisk et al.] 20200), ARC-Easy/Challeng (ARC-e/ARC-o)
(Clark et al.)
2018), HellaSwag (HelS) (Zellers et al. 2019), GPQA-Diamont (Rein et al.
2023 and Wino-
Grande(WinG) (Sakaguchi et 2021).
Details for benchmarks are in Appendix E.1
Baselines: We compare Tequila against several quantization method baselines, which represent the methods used in existing state-of-the-art (SOTA) ternary LLMs. These include two types of quantization methods: (1) static methods: TWN Li et al. (2016). AbsMedian and AbsMean used in BitNet (Ma et al. 2025 Wang et al. 2023). Spectra (Kaushal et al. 2025). and BitCPM (Team) al. 2025); and (2) learnable methods: DLT in TernaryLLM (Chen et al. 2024). LSQ (Esser al., 2019) and SEQ used in ParetoQ (Liu et al., (2025). In addition to comparing quantization methods, we also directly compare against those well-trained ternary LLMs. Further discussion about the baselines is in Appendix E.4
Implementation Details: All experiments are conducted on 16 GPUs for training, with inference performance evaluated on an Intel 8263C CPU. Following established practices (Liu et al. 2025). we quantize all linear layers within the transformer architecture. The sequence length for input and output 1024. The learning rate set as fixed value of 10-4, Given that Tequila designed as plug-and-play solution, the AbsMean in Eq, 2 was selected for Tequila's 's base quantization method due to its prevalence open-source ternary large language models. We set 10-3 for Tequila by default. The LLM trained by Tequila is called TequilaLLM.
4.2 PERFORAMCNE EVALUATION
Comparison of Different Ternary Quantization Methods: To evaluate the effectiveness of Tequila, we conduct QAT with different ternary quantization methods with 10B tokens and eval- uate their performance. Our Tequila is plugged into the AbsMed and AbsMean quantization

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Published as a conference paper at ICLR 2026
Model
Size
#lokens
ARC-e
ARC-c
HeIS
PIQA
WinG
Average
LLaMA3.2
IB
0.654
0.313
0.477
0.742
0.603
0.558
TernaryL.M L
IB
10B
0.424
0.174
0.256
0.563
0.513
0.386
ParetoQ
IB
10B
0.421
0.180
0.273
0.604
0.510
0.398
LLM-QAT
IB
100B
0.360
0.262
0.313
0.551
0.496
0.397
BitNet
1.3B
100B
0.549
0.242
0.377
0.688
0.558
0.483
Spectra
1.IB
100B
0.563
0.246
0.388
0.693
0.555
0,489
TequilaLLLM
1B
10B
0.645
0.305
0.391
0.710
0.542
0.519
LLaMA3.2
3B
0.745
0.422
0.552
0.768
0.691
0.636
TernaryLLM'
3B
10B
0.361
0.161
0.260
0.572
0.496
0.370
ParetoQ
3B
10B
0.498
0.231
0.303
0.645
0.529
0.441
LLM-QAT
3B
100B
0.445
0.307
0.434
0.627
0.506
0.464
BitNet
3B
100B
0.614
0.283
0.429
0.715
0.593
0.527
Spectra
3.9B
100B
0.660
0.319
0.483
0.744
0.631
0.567
TequilaLLM
3B
10B
0.702
0.346
0.464
0.739
0.627
0.576
Table 2: Comparison of TequilalLLM with other ternary LLMs across different model sizes and training token counts (#Tokens); indicates LLMs obtained from our reproduction.
4.0
LSQ
Inference Speed (tokens/sec)
Zero
LLaMA
SEQ
60
Overleea
BitNet
Training Loss
3.0
Absmean
DLT
50
TequilaLLM
Tequila
40
30
Zero
2.0
Overhead
10
3.0x
0
2000
4000
6000
8000
10000
0.78
3B
Training steps
Model size
Figure 5: Evaluation of Tequila on convergence speed compared to SOTA terarry quantization.
Figure 6: Inference speed of TequilaLLM ver- sus BF16 LLaMA and ternary BitNet.
method. The experimental results in Table Table that Tequila outperforms all baselines on both IB and 3B models, achieving an average accuracy gain of 2.6% over SOTA methods. Specifically, on both ARC-Easy Easy and ARC- CRC-Challenge benchmarks, Tequila achieves significant > 4% accuracy gains over SOTA methods, while matching the BF16 performance with only minimal gap (< 1%).
An important observation is that learnable ternary quantization methods generally underperform static ones. We attribute this to the fact that increasing learnable parameters slows convergence and makes optimization more prone getting stuck in local optima. This aligns with the broader trend of using static AbsMean quantization method in open-source ternary LLMs
(Liu et al. 2025 Chen
al. 2024
Ma al. 2025). confirming our design decision.
To further evaluate the effectiveness of Tequila, name the resulting model trained by Tequila as Tequila and compare against existing ternary LLaMA-based LLMs. We reproduce methods Liu et al. ); Chen et al. (2024) with available training code and train them on 1OB tokens from the Uhtaainn Web dataset using identical hyper- parameters for a fair comparison. For models without available implementations, we report results from their original papers or published weights. The GPQA benchmark is excluded from compar- ison as it is not consistently reported across baselines. As shown in Table 2 TequilaLLM achieves the best average accuracy in the benchmarks. Remarkably, Tequila achieves superior performance using significantly fewer training tokens than other well-trained ternary LLMs, demonstrating both faster convergence and higher final accuracy. Specifically, our TequilaLLM-33 model outperforms the SOTA ternary LLM, Spectra-3.9BB 3.99B, by 0.9% in average accuracy while using only 10% of the training tokens. These results robustly validate the effectiveness of Tequila's 's adaptive reactivation strategy in resolving the deadzone trapping problem.
Comparison with Different Ternary LLMs:
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Absmean
LLaMA-3B
Mixed
0.56
Mimima Reactivation
LLaMA-1B
Tequila w/o Mixed Gradient
Gradients 1.4%
Average Accuracy
0.54
Tequila Differentiable
Accuracy
64
Reactivation
Weights
1.1%
0.50
Reactivation
62
1.2%
0.48
0.46
60
0.42
10-1
10-1
10-2
10-2
10-1
Varlous variants Tequila
Reactivation parameters (A)
Figure 7: Ablation study comparing Tequila Figure 8: Sensitivity evaluation for the reactiva- against its variants.
tion parameter
Convergence Analysis: To demonstrate that Tequila's more direct and informative gradients en- able faster information recovery, we compare the training loss convergence Tequila against ternary baselines over 10,000 steps. As shown in Fig. [S] Tequila achieves faster convergence rate than other baselines. This result validates the effectiveness of our core innovation: reactivating dead weights adaptive biases by differentiable reactivation function obtain perior gradient signals, thereby enhancing optimization.
1B model using
Theoretically, Tequila introduces nearly zero inference overhead, as the reactivation bias terms input-independent and can be precomputed offline. The additional com- putational cost for bias addition is negligible, measured at less than 0. %. To empirically validate this claim, evaluate the token generation speed of Tequila against BitNet and a BF16 LLaMA baseline on an Intel 8263C CPU. Due to limitations the compiled BitNet model, we conduct experiments using 0.7B and 3B model sizes. We accelerate both Tequila and BitNet using the effi- cient lookup table paradigm (Wang et al. 20255) to eliminate multiplication operations, as shown in The results in Fig. show that TequilaLLM maintains 3.0x speedup over the LLaMA-3.2, matching the practical inference speed of BitNet, demonstrating that Tequila introduces nearly zero overhead compared to pure ternary methods.
Inference Efficiency:
Fig. 9
Ablation Study: To analyze the individual contributions of Tequila's components in addressing deadzone trapping, we conduct an ablation study on a IB model using the ARC-Easy benchmark. We attribute Tequila's benefits three key aspects: the reactivated forward signal, differentiable reactivation, and the hybrid roles of reactivated weights. To evaluate these aspects, we compare the following variants of Tequila: (1) AbsMean: This baseline disables all reactivation aspects to evaluate Tequila's 's overall effectiveness. (2) Minima Reactivation: This variant reactivates dead weights in-place as signed minima (Sec.[3.1) enabling the reactivated forward signal but still relying on the STE for gradients.
(3) Tequila w/o Mixed Gradients: This variant treats dead weights biases
only by differentiable reactivation. replaces the gradients Eq. with Dur while keeping the forward pass unchanged.
ØL. X ØL for Vi e D, BY
The results in Fig. demonstrate the incremental effectiveness of each aspect. First, Minima Re- activation outperforms the AbsMean, confirming that reactivating dead weights enhances model capacity. Second, Tequila w/o Mixed Gradients surpasses Minima Reactivation, demonstrating that differentiable reactivation is more effective than the STE, as it provides direct backpropagation to mitigate deadzone trapping. Finally, the Tequila achieves superior performance to Tequila w/o Mixed Gradients, validating that mixed gradients from the residual pathway are more effective than the single gradient from input-agnostic bias. This shows that assigning dead weights hybrid role (functioning as both weights and biases) is more suitable than a pure bias assignment. This ablation study conclusively demonstrates the individual and combined importance of the reactivated forward signal, differentiable reactivation, and hybrid roles of reactivated weights.
Impact of the Reactivation Parameter A: The choice of the reactivation parameter is critical for Tequila, An excessively high value of À may cause the bias dominate the output, while value

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Model BF16
ARC-e
ARC-c
HeiS
PIQA
WinG
Average
0.8186
0.5213
0.5423
0.7769
0.6772
0.6673
AWQ 0.2702
0.2244
0.2587
0.5305
0.5122
0.3592
AbsMean
0.6915
0.3507
0.4616
0.7339
0.5856
0.5647
Tequila 0.7538
0.4189
0.4681
0.7383
0.6133
0.5985
Table 4: Performance comparison (accuracy) for Qwen3-4B.
that is too low renders the reactivation ineffective; if 0, Tequila degenerates standard ternary quantization. To analyze the sensitivity of X, we evaluate average accuracy on five benchmarks across range of values: À E {0, 10-5,1000000 10-1, 10-3 10 10-10. The results in Figure 8 indicate that even a small À provides a noticeable gain, and performance is robust across a wide range of effectively adapt the dead weights into useful biases during
values. This suggests the model training, converging to a near-optimal configuration regardless of the value of A. And the optimal correlated with model size, with larger models preferring smaller X.
The Impact of Quantization Granularity: Quantization granularity presents a fundamental trade-off between a model's efficiency and its performance. Per-token quantization enables high acceleration but introduces significant quantization error. Conversely, per-group quantization mitigates this error the cost of reduced efficiency, due to the overhead of storing and applying scaling matrices. We evaluate Tequila across various granularities: per-token, per-channel, and per-group with group size 128. The results in Table 3 indicate that Tequila exhibits minimal performance loss across different granularities. This robustness stems from its ability to use reactivated biases to compensate for quantization errors.
Tequila per-tensor
Average Acc
0,463
per-channel per-group
0.471
0.471
Table 3:
Average accuracy of
Tequila across quantization gran- ularities on a IB Model
Experiments Qwen3-4B: To validate the scalability and generalization of Tequila, we extend our evaluation to the Qwen3-4B model Bai et al. (2023). We compare Tequila against AbsMean and standard Post-Training Quantization (PTQ) approach using AWQ Lin et al. (2023). The results shown Table 4 The AWQ suffers a catastrophic performance drop, which underscores the necessity of QAT for aggressive ternary quantization in LLMs. Tequila consistently outperforms other baselines across all tasks, confirming that our proposed reactivation mechanism delivers a consistent advantage by providing more informative gradients and mitigating the dead zone problem, even at the 4B scale.
5 CONCLUSION
In this paper, we first identified deadzone trapping as a fundamental obstacle to efficient and accu- rate ternary quantization of large language models for on-device deployment. Deadzone trapping. where weights become trapped in ineffective oscillation around the quantization boundary due to less informative gradients, severely diminishes model capacity and impedes optimization.
To overcome this challenge, we introduced Tequila, a novel trapping- frren ternary quantization method. Tequila repurposes trapped weights as adaptive dynamic biases, successfully reactivating them to enhance model expressiveness with nearly zero inference overhead. Crucially, this approach provides direct gradient signals, enabling efficient escape from the deadzone and substantially ac- celerating quantization-awan aware training. Extensive evaluations on five benchmarks demonstrate that Tequila outperforms state-of-the-art ternary methods, closing the gap to cell-precision performance while using limited training data, and preserving the computational benefits of ternary quantization, delivering up to 3x inference speedup.
Looking forward, Tequila establishes of dy dynamically repurposing dead weights opens promising avenues for future research into ex- treme quantization. We believe work contributes practical and scalable path toward bringing advanced LLM capabilities to resource-constraine devices.
direction for efficient model compression. The concept
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6 ACKNOWLEDGEMENTS
The paper is partially supported by Hong Kong Research Grants Council grants #11205424, #1 #11203523, and C1042-23GF.
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Appendix
A THE USAGE OF LLMS
We state that the LLM is used exclusively for polishing the English text and grammar in this manuscript, prompted with: "Please polish and rephrase the following sentences: #input". All tech- nical content, ideas, methodologies, experimental results, analyses, and conclusions are the original work of the authors. The LLM acted solely as writing assistant and did not contribute to the intellectual substance of the research.
B RELATED WORK
B.1 GENERAL QUANTIZATION
Quantization DDttmers et al. 2021 2022 Lin et al. 2023 Frantar et al. 2022) is well-established technique for improving the effiiency of Large Language Models (LLMs) by reducing the precision of weights and activations. However, low-precision quantization (Lin et al. 2023 Frantar et al.) 2022] often leads to mixed-precision matrix multiplication, where weights and activations have different data types. This requires specific hardware support for efficient computation, which is significant limitation for edge and mobile deployment, given the extreme diversity of devices in these environments.
While activation-weigee quantization methods (Dettmers et al. 2022 Xiao et al. 2023 Huang & Wu 2025) attempt mitigate this by using unified low-precision format for both weights and activations, they still face challenges. These methods often require specialized hardware adapta- tion and suffer from high quantization error in activations due outlier issues (Xiao et al. (023), preventing activations from reaching the same effective precision as weights. Consequently, exist- ing general quantization methods struggle enable efficient LLM deployment on diverse edge and mobile platforms.
B.2 TERNARY QUANTIZATION
Ternary quantization Li et al. (2016) Zhu et al. (2016), 1.58bit quantization. presents a com- pelling alternative by constraining weights to temary values, typically {-1, 0,+,0,+13 +1]. Beyond the substantial memory savings, this approach transforms the core matrix multiplication operation into efficient addition operations by replacing multiplications with conditional sign flips. This intrin- sic hardware-friendliinee makes particularly suitable for resource-constraines edge and mobile hardware.
Early research termary quantization Li al. 20166;; Zhu et al. (2016); Leng al. (2018) primar- ily focused on refining the quantization function, particularly the selection of threshold parameters and scaling factors. The foundational Ternary Weight Networks (TWN) (Li et al. (016) assumed Gaussian weight distribution to determine optimal thresholds that minimize the distortion between full-precision and quantized weights. Trained Ternary Quantification (TTQ) (Zhu et al., (016) ad- vanced this paradigm by introducing trainable scaling factors, enabling models to learn optimal ternary representations directly during training. Further extending these ideas, [eng al. (2018) formulated the problem using the Alternating Direction Method of Multipliers (ADMM) to itera- tively optimize both scaling factors and thresholds.
With the emergence of Large Language Models (LLMs) Wu et al. (202)) Floridi & Chiriatti (2020);
Zhang et al. (2022), the limitations of early ternary quantization methods have become apparent. The generative capabilities of LLMS are highly sensitive to precision loss, which often leads to un- acceptable performance degradation under existing ternary techniques. This challenge has spurred two primary research directions. One line of work employs Post-Training Quantization (PTQ) tech- niques Lin et al. (2023); Frantar et al. (2022) to mitigate performance loss efficiently Xiao et al. 2025). However, concerns over PTQ's potential performance drop have led the most of work to explore Quantization- Aware Training (QAT) Chen et al.) (2025) for more robust recovery Within OAT-based ternary methods, a divergence in strategy exists: some approaches, such as those in the
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1. Online preprocess
Inder
X
X
X2
X-3
X5
-x
Input
0001
X X1
2. Lookup
-Xo X1 X3 X5 -x- X4
1100
-2
1101
Unsigned result
Lookupi Table
3. Sign
0. Offline transform
10001:0 101
byte
1010 0000
-Xp
+Xg + X4 X5
-1
Index Weight
2Wa
Awa
Signed result
4. Accumulate
Ternayy Weight
& Scale
Sign Weight
& Add bias
Awy+ Aws
2wg + 2wng
Output
Bias
factor a
Figure 9: The Lookup Table- Based Inference designed for Tequila. Weights are packed into indices, signs, and precomputed bias offline. At runtime, inputs processed to construct lookup table containing various unsigned intermediate values. The system then uses the weight indices to retrieve these values from the table. After that, the 1-bit sign values are applied generate signed results and are accumulated, and the per-channel bias is added to produce the final output.
2025b), utilize the straightforward AbsMean BitNet family Ma al. (025 Wang al. (2023
quantization, while others, like Chen et al. (202)) and Liu et al. (2025), have adapted more sophis- ticated techniques such Learned Step Size Quantization (LSQ) Esser al. (2019) the ternary
setting.
Despite these advancements, these existing ternary LLMs still suffer from the deadzone trapping issue, where a large number of weights become trapped in a cycle of ineffective oscillation around the deadzone boundary, severely impeding model capacity and convergence. Therefore, we propose Tequila to achieve trapping-fres quantization.
C QUANTIZAATION-AWAE TRAINING
Quantization-Awan Training (QAT) Chen et al. 2025 ; Liu et al. (025) is employed recover model performance by incorporating quantization into the training loop. The core idea of QAT to model the effects of quantization during the forward pass, allowing the model to adapt its parameters the subsequent precision loss. This is achieved through a fake quantization process, where full-precision weights and activations are passed through a quantization function Q(-) that mimics the behavior of integer arithmetic. Critically, the backward pass leverages the Straight- Through Estimator (STE) Yin et al. (2019) to approximate gradients through this otherwise non- differentiable operation.
The standard QAT procedure can be summarized as follows:
(1) Maintain a full-precision weight copy W as the master weights for accumulation.
During the forward pass, generate quantized weights Vi- (W) and scaling factor a for computation.
(3) Compute the output Y using the quantized weights W. = QWW,, scaling factor a and activations X.
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(4) Compute the Loss L using output Y and begin backward pass. aL (5) During the backward pass, full-precision gradients aw are computed with the STE, the gradients are passed directly to the full-precision weights: aW aw
aL at.
(6) Update the full-precision weights W using the approximated gradients via standard opti- mizer (c.g., SGD, Adam).
By maintaining and updating full-preee cull-precision master weights, QAT provides stable optimization process while training a model that is robust to the quantization emros will encounter during inference. This makes it the preferred method for achieving high accuracy with ultra-low precision, such as ternary or binary quantization.
D INFERENCE DESIGN
Tequila scamlessly integrates with the lookup table paradigm to enable efficient, multiplication-fiet inference. As illustrated in Figure our system operates in two phases: an offline packing stage and efficient online inference stage.
During the offline phase, the ternary weights and reactivation biases are packed into compact data structures, including index weights, sign weights, and channel-wise biases. To maximize efficiency, every three weights are packed into a 4-bit index and a 1-bit sign value.
At inference time, the input values preprocessed into lookup table within segments. For each segment, the corresponding weight index is used to retrieve the results from the lookup table. This process entirely replaces multiplication operations with efficient table lookups. The retrieved result is then combined with the sign weight to determine the final polarity, and subsequent accumulation across segments is followed by the addition of the channel-wise bias term. This results in highly optimized inference path that maintains the theoretical hardware efficiency of ternary quantization while requiring only minimal modifications existing inference frameworks.
E
MORE EXPERIMENTAL DETAILS
E.
EVALUTATON BENCHMARKS
The evaluation of language models has evolved beyond simple word prediction to assessing their ability to understand and apply knowledge in human-like like manner. This requires benchmarks that probe deeper cognitive capabilities, such as commonsense reasoning, logical deduction, and special- ized knowledge. This section introduces suite of prominent benchmarks designed for this purpose: PIQA, ARC-Easy, ARC-Challenge, HellaSwag, GPQA, and WinoGrande.
PIQA (Physical Interaction Question Answering) Bisk al. (2020) focuses on physical common- sense reasoning, testing a model's understanding of how the everyday physical world works. The benchmark presents questions about the mechanics of physical actions (e.g.. "How do you stabilize a wobbly table?") and requires choosing the correct solution from two options. Success on PIQA indicates that a model possesses foundational knowledge of physical laws and object interactions.
The ARC (AI2 Reasoning Challenge) dataset Clark et al. 2018) is divided into two tiers to assess scientific knowledge and reasoning. The ARC-Easy set contains grade-school-leve science ques- tions that are often answerable through simple fact retrieval. In contrast, the ARC-Challenge set specifically curated to be difficult, consisting questions that require complex reasoning and deeper understanding of scientific concepts, posing significant challenge for even advanced mod- els.
HellaSwag Zellers et al. (2019) is benchmark for evaluating contextual commonsense reasoning. presents beginning of a situation (e.g.. "A person is folding a paper towel") and challenges the model to select the most plausible continuation from four options. The distractors are generated by adversarial models, making them deceptively plausible and ensuring the task cannot be solved by simple word association, but rather requires nuanced understanding of event dynamics.
GPQA Rein et al. (2023) represents a significant leap in difficulty, designed as a "graduate-Ieve" benchmark for highly specialized knowledge. The questions, written by domain experts in biology,
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physics, and chemistry, are exceptionally challenging and "Google-pin "roogle-proof,"" meaning they difficult to answer by simply searching the web. On GPQA-Diamonde the set of GPQA's 198 most difficult questions, PhD experts achieve 65% accuracy, while skilled non-experts with web access only reach 34%.
WinoGrande Sakaguchi al. 2021 is large-scale dataset for assessing commonsense reasoning through pronoun resolution. Inspired by the Winograd Schema Challenge, it presents sentences with ambiguous pronouns (e.g., "The trophy didn't fit in the suitcase because it was too big.") and requires the model to determine the referent of "it." WinoGrande is designed to be adversarial, with focus on reducing spurious statistical biases present in earlier datasets, forcing models to rely on genuine commonsense understanding.
Together, these benchmarks provide multifaceted evaluation framework, testing language models on everything from everyday physical intuition (PIQA, HellaS and general scientific knowledge (ARC) to expert-level understanding (GPQA) and nuanced linguistic reasoning (WinoGrande).
E.2 GRADIENT SIGNAL-TO-NOISS RATIO (GSNR) MEASUREMENT
The Gradient Signal-to-Nont Noise Ratio (GSNR) Liu et al. 2020) is a crucial metric for quantifying the reliability of gradient updates for model weights. is defined as the ratio of the squared expectation to the variance of a weight's gradients over the data distribution. Formally, for a weight wy, its GSNR is calculated as:
GSNR(w;) (ExxDD Veo L))
(10)
Varx V
where Vw; L. is the gradient of the loss with respect to wy for data sample x. high GSNR indicates stable and consistent gradient signal across the dataset, whereas low GSNR suggests noisy, uninformative updates that hinder effective optimization.
To evaluate the impact of our method on we measure the GSNR specifically for dead weights, active weight and reactive weights that are trained by AbsMean, Minima Reactivation and Tequila. We estimate the expectation and variance usSiing a 1024 training data samples and compute the average GSNR. This comparative analysis, shown in Figure 3 directly demonstrates that our reactivation mechanism provides clearer optimization signal by significantly elevating the GSNR of previously dead weights.
E.3 GENERALIZABILITY TO OTHER QUANTIZATION SCHEMES
This paper primarily addresses the deadzone trapping problem, a fundamental challenge in ternary quantization that arises from uniquely fixing the zero point to eliminate multiplications. In higher- -bi regimes (e.g., 2-b b-t),, which are multiplication-baat based, this specific issue is effectively mitigated through techniques like zero-point shifting Liu et al. (2025). Consequently, the primary contribution of our work the identification and resolution a problem inherent to the ternary paradigm.
However, the core innovation of Tequila that repurposes a subset of weights to form dynamic bias term is a general mechanism to enhance model capacity and gradient quality. To validate Tequila's generalizability beyond the ternary setting, we adapt Tequila to the following higher bit schemes.
Low-bit Quantization Scheme Without Deadzone. To show the benefit of the weight repurpos- ing method in a non-deadzone quantization scheme, we adapt Tequila into a 2-bit scheme using the levels {-1, -055, +0.5, +1} (SEQ) by introducing repurposed-bias term, modifying the forward pass to Y = XW + A A W, Results in Table 5) confirm the broader applicability of our Tequila. term consistently improves performance across most tasks,
The integration of the repurposed- yielding a noticeable 1.04% gain in the average accuracy. This demonstrates that the benefits of Tequila can be effectively translated to other low-precision formats.
High-bit Quantization Scheme With Deadzone. We further evaluate Tequila's generalization on bigh-bit quantization scheme by applying it to INT4 LSQ quantization. As shown in Figure [] (left), the deadzone trapping effect INT4 LSQ is less severe than in ternary quantization (Fig- ure T.. Despite this, Tequila effectively mitigates the remaining trapping, leading to more stable
17

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BF16
Ternary absmean
Tequila
layers.0.mpp mlp. down_proj
layers.0.att.
q_proj
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layers.9.mlp.
down_proj
630
EC0
041
3.44
KON
0.6
layers.9.att.
q_proj
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-8.0
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layers.15.mlp
down_proj
-054
-002
6.
6.5
066
layers.15.att.
q_proj
-0.0 -504 -000
052
C.34
606
Figure 10: The distribution of weights from randomly selected linear layers in transformers with full-precision (BF16), ternary AbsMean, and our Tequila. The weights from the ternary AbsMean method are trapped at the deadzone boundary, suffering from deadzone trapping. Our Tequila can significantly address this issue. This observation aligns with our findings and demonstrates the effectiveness of Tequila.
weight distribution Figure II (right). The resulting performance gains, reported in Table are sistent but modest. This is because the INT4 LSQ baseline is already highly competitive with the BF16 baseline, leaving little for significant improvement. These results confirm that Tequila generalizes to higher-bit quantization, even when absolute gains are limited by a strong baseline.
E.4 TERNARY QUANTIZATION BASELINES
Recall the general form of the ternary quantization: Given a full- full-precision weight vector (wi,...; Wn), the general form of the ternary quantization function Q(-) is defined as:
+1, un > A
Q(W) = Wa. a. w
0. if [ A
(11)
if ut
where W (d.......u w.,) ) is termary weights, is scaling factor and A is threshold parameter. Due to the non-differentiable function of Q(.), the gradients for W are approximated using the Strought-Through Estimator (STE) (Zhu al. 2016 Chen et al. 2024). leading the following forward pass and backpropagation with input vector ( F.):
al Dw;
8L 0;0,
if [wy > A
Y == x=X((W)) XTWa.
(12)
aL by ai
if [w <
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Published as a conference paper at ICLR 2026
INT4 LSQ
Tequila
200%
200k
-8.100
-1100
6.100
Figure 11: The weight distribution of INT4 (left) LSQ and (right) Tequila
Method
Bits
ARC-e
ARC-c
HeIS
PIQA
WinG
Average
LLaMA-IB
16
0.654
0.313
0.477
0.742
0.603
0.557
SEQ
0.591
0.245
0.365
0.671
0.531
0.481
2
Tequila SEQ
0.604
0.258
0.363
0.703
0.528
0.491
LSQ
0.670
0.310
0.458
0.741
0.587
0.553
4
Tequila LSQ
0.672
0.313
0.459
0.747
0.593
0.557
Table Performance comparison of the higher bit scheme with and without the Tequila.
where L denotes the loss of the model prediction.
Previous methods for optimizing ternary quantization can be broadly categorized into two ap- proaches: (1) reducing quantization error, and (2) enhancing the model's expressive capacity.
E.4.1 REDUCING QUANTIZATION ERROR
A primary line of work forcus on optimizing the threshold A and scaling factor a to reduce the quantization error. It is exemplified by estimation-baser methods like Ternary Weight Networks (TWN) (Li et al. 2016), which aim to minimize the reconstruction error between full-precision and quantized weights. This is formalized by the objective:
min W - aaii,
(13)
A.
which has a closed-form solution for given a fixed threshold A:
a²
Σ |ush.
(14)
IED
where δ is the set of weights whose absolute value exceeds A. However, finding the optimal threshold A that minimizes the overall objective is challenging. To circumvent this, TWN hy- pothesizes that the weights follow a standard Gaussian distribution, leading to the approximation A 0.7 B[]]]
This Gaussian assumption often does not hold in modern deep learning models, particularly in LLM, where weight distributions can be highly non-Gaussian. Consequently, the estimated a becomes biased, degrading performance. To address this, subsequent methods like Learned Step Size Quan- tization (LSQ) (Esser et al., (019) and DLT (Chen et al.) (024) propose making the scaling factor trainable parameter, while typically retaining the heuristic estimation for A.
However, optimizing a and alone cannot resolve the fundamental issue of deadzone trapping- Weights within the deadzone A- A) remain long-term pruned during the forward pass and con- tinue to receive only noisy, less informative gradients via the Straight- which prevents effective recovery. Moreover, in the context of LLMs, simply introducing additional trainable parameters (e.g., for a) increases optimization complexity and can make the model more susceptible to converging poor local optima. As results in Fig. 5 demonstrate, these methods exhibit significantly slower convergence and higher final loss compared to our approach, underscor- ing their inherent limitations.
Estimator (STE),
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Published as a conference paper at ICLR 2026
E.5 ENHANCING THE MODEL'S EXPRESSIVE CAPACITY
Previous work has attempted to enhance model capacity by incorporating bias terms, For instance, DLT Li et al. 2016) introduces learnable bias during dequantization:
Q(W) Wa + b, == X + b) XWa Xb,
(15)
where b learnalee bias. However, this approach breaks the computational efficiency of ternary quantization by introducing the dense full-precision scaling X Similarly, SEQ Liu et al. (2025)
introduces bias by reassigning the zero point to a non-zero value ab:
+1,
if Wi 2
ab,
if Wi < A if W/ S -
(16)
This also destroys efficiency, the resulting operations are longer multiplication-free.
While these methods may potentially mitigate the deadzone trapping issue by reactivating the dead- zone, they fundamentally break the hardware efficiency of ternary quantization. In contrast, method, Tequila, introduces an input-agnostic bias that is precomputed offline. This design directly addresses the deadzone trapping issue while perfectly preserving the computational efficiency of terary quantization.
20


Reviews:
[{"review_id": "LUETHabqVP", "reviewer": "Reviewer_GbUj", "summary": "This paper presents Tequila, a ternary quantization method for large language models (LLMs) targeting edge deployment. It tackles the “deadzone trapping” problem, where many weights become inactive due to noisy gradients. Tequila reuses these inactive weights as adaptive biases, restoring useful forward and backward signals. Experiments on five benchmarks and two LLaMA sizes show that Tequila achieves higher accuracy and efficiency than existing ternary methods, approaching full-precision performance with much lower hardware and data costs.", "questions": "Q1: In Section 3.2, the paper states, “Given the approximately symmetric distribution of inputs (activations) in transformer architectures.” Is there any prior literature or empirical evidence supporting this assumption?\n\nQ2: Also in Section 3.2, the paper mentions, “In addition to functioning as an adaptive bias.” However, shouldn’t $\\sum_{i\\in D}\\lambda w_i$ be input-independent? Could the authors clarify this reasoning?\n\nQ3: The paper claims “a learnable reactivation parameter λᵢ for each deadzone weight wᵢ,” but in experiments a fixed λ value is used. Could the authors elaborate on this inconsistency?\n\nQ4: It would strengthen the paper if the authors could include additional comparisons between Tequila and PTQ-based ternary quantization algorithms in terms of quantization efficiency and downstream task performance?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Clear identification of the core problem: The paper accurately pinpoints “deadzone trapping” as a key obstacle to effective ternary quantization, and Figure 8 clearly demonstrates the effectiveness of the proposed Tequila method in addressing this issue.\nPracticality and efficiency: Tequila achieves near–full-precision performance while remaining hardware-friendly, improving accuracy by over 4% compared to SOTA methods and maintaining comparable inference speed to classic ternary quantization approaches such as BitNet.\nHigh-quality and well-presented figures: For example, Figure 1 (“Deadzone Trapping in Ternary Quantization”) and Figure 2 (“Tequila Reactivation Strategy”) intuitively illustrate the problem and solution, while Figure 3 clearly shows faster convergence.\nGood reproducibility and transparency: Experimental details and code implementations are publicly available, enabling community verification and reproduction.", "weaknesses": "Limited discussion of related work: As a QAT approach, the paper overlooks several concurrent efficient ternary quantization methods such as OneBit[1], as well as recent PTQ ternary approaches like PTQTP[2].\nLimited model scale and architectural generalization: Experiments focus mainly on smaller LLaMA models (1B and 3B). Validation on larger models (e.g., 7B, 13B) or different architectures is missing, leaving generalization untested.\nInsufficient analysis of λ sensitivity: Although Figure 6 is provided, the paper does not explore how λ interacts with other hyperparameters or training configurations. Like, whether model size affects the optimal λ during training.\n[1]Xu Y, Han X, Yang Z, et al. Onebit: Towards extremely low-bit large language models[J]. Advances in Neural Information Processing Systems, 2024, 37: 66357-66382.\n[2]Xiao H, Yang R, Yang Q, et al. PTQTP: Post-Training Quantization to Trit-Planes for Large Language Models[J]. arXiv preprint arXiv:2509.16989, 2025.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "SR3yFq8dQH", "reviewer": "Reviewer_ckpi", "summary": "This paper introduces Tequila, a precision enhancement strategy specifically designed for ternary Quantization-Aware Training (QAT) of Large Language Models (LLMs). The paper argues that the primary cause of accuracy degradation in ternary quantization is the phenomenon where many weights are quantized to zero, leading to a \"deadzone trapping\" issue during gradient updates. This trapping prevents these weights from being effectively updated. To address this, the authors propose reutilizing these weights trapped in the deadzone by treating them as dynamic biases, thereby making their gradient updates more effective and meaningful. Through this approach, the authors claim their method can surpass current SOTA techniques for ternary quantization. Furthermore, the authors provide ablation studies in a progressive manner to illustrate the effectiveness of their proposed techniques.", "questions": "The main questions focus on the paper's theoretical analysis, as detailed in weakness 1. I'll briefly summarize here to facilitate the author's rebuttal:\n\n1.  Can the authors provide a more reasonable and complete analysis explaining *why* gradient updates within the deadzone are noisy, uninformative, and potentially lead to \"making model's parameters **permanently inactive**\" (line 161)?\n2.  Can the authors provide a more logical reasoning process for Section 3.1, primarily addressing the origin and justification of the bias-like design term?\n3.  Can the authors add more rigorous, step-by-step deductions in Section 3.2 when deriving the final Tequila formula to ensure soundness?\n\nAdditionally, there are a few other minor questions:\n\n4.  In Table 1, were the experiments conducted using exactly the same datasets across all compared methods?\n5.  The results in Table 2 are particularly impressive. It's striking that LLM-QAT, BitNet, and Spectra, trained on 100B tokens, perform worse on most tasks than TequilaLLM trained on only 10B tokens. Have the authors considered and can they discuss the fundamental reasons behind this impressive phenomenon?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 1, "presentation": 2, "contribution": 3, "strengths": "1.  **Valuable Research Problem:** Improving the accuracy of ternary quantization is highly beneficial for model acceleration during inference, as it allows replacing complex multiplications with additions. This aligns well with current trends in model quantization research. Furthermore, the paper's experimental results demonstrate a significant accuracy improvement compared to other existing ternary quantization works of similar scale.\n2.  **Sufficient Experiments and Impressive Results:** Although the authors conducted experiments on relatively small model scales (1B/3B) and a token scale of 10B, they compared their method extensively against numerous other ternary quantization works of the same scale and achieved impressive accuracy gains. Additionally, they performed comprehensive ablation studies concerning method design, parameter selection, and quantization granularity, yielding complete results. Their experimental outcomes are exceptionally good; other ternary-quantized models of similar scale, even after training on 100B tokens, often fail to match the performance on most tasks achieved by TequilaLLM trained on only 10B tokens.", "weaknesses": "1.  **Theoretical Analysis is Unconvincing and Contains Flaws:** \n    1) In Section 2, the paper primarily states the drawbacks of traditional gradient computation for ternary quantization problems. However, the gradient formula (3) presented in line 132 is difficult to further derive into the authors' statement about the gradient computation flaw (line 152: *\"Because they contribute no information to the prediction loss L, they receive uninformative gradients during backpropagation\"*), and the authors do not provide further theoretical analysis supporting this statement. Formula 3 for gradient calculation seems to originate from the reference (Chen et al., 2024). However, if using a traditional STE estimation scheme, the calculation of the original weight gradient ∂L/∂wᵢ should be identical to the quantized weight gradient ∂L/∂Ŵ, which is (∂L/∂Y) * xᵢ * α. In this case, when the original weight lies within the range (–δ, δ) (the \"deadzone\" scenario claimed in the paper), the weight gradient would be the same as outside the deadzone. Therefore, the paper's derivation of the \"DEADZONE TRAPPING\" scenario based solely on this formula in Section 2.2 is unconvincing. Furthermore, line 161 uses exaggerated terms like \"permanently inactive\" and \"drastically\" to describe the harm of the deadzone, which is unpersuasive. \n    2) Section 3's introduction of the bias-like correction term is even more confusing. Firstly, regarding the first variant, Minima Reactivation, the so-called \"informative\" value mentioned in line 192 is merely sign(x)*ε from a formulaic perspective, and the authors do not provide complete analysis proving this design is effective. In my view, this approach seems more like splitting zero into a small positive and a small negative value to expand capacity, effectively simulating 4-bit quantization, which naturally performs better than 3-bit.\n    3) Section 3.2's proposal of the Tequila method is essentially a relaxation of STE approximation, a concept already present in quantization literature. The step-by-step simplification process from STE -> differentiable reactivation -> activation-free bias (using dead weights as bias) -> letting these weights act as both bias and participate in multiplication (dual role) seems reasonable at each step, but lacks persuasive power and involves strong logical leaps. For example, in line 246, the authors claim \"bias-like term can be precomputed offline, reducing inference overhead to nearly zero,\" but do not explain: Why can this term be completely decoupled from the input? Under different input distributions, is it truly \"approximately constant\"? This is an empirical assumption without mathematical or experimental support. Ultimately, in the derived gradient correction formula, the authors claim in line 165 that this gradient for dead weights is \"superior,\" but provide no quantitative analysis proving that adding a constant bias term xᵢ in formula (8) results in a better gradient estimation than normal STE.\n2.  **Exaggeration is Prevalent in the Text:** For example, line 152 asserts that weights within (–∆, ∆) under STE have gradients that \"contribute **no information** to the precision loss L\" and line 153 uses **'uninformative'**, line 220 directly claims \"**fundamentally** enable the efficient recovery of previously trapped weights, **dramatically** accelerating convergence\", and the advantages given in line 270 seem to be summarizing for the sake of summarizing, with little relation to the actual logical flow of the text. The paper uses a large number of highly positive adjectives to describe their method (fundamentally, informative, superior), but lacks specific data or theoretical support, giving the impression of \"more promotion than argumentation.\"\n3.  **Insufficient Analysis of Deadzone Generality:** According to the paper's logic, the deadzone phenomenon should exist for quantization at any bit-width, as some weights will be quantized to zero. If the authors could provide a more detailed analysis of this deadzone problem across different bit-width quantization scenarios, it would be better.\n4.  **Lack of Specific Details on the Mixed-Precision Framework Design:** It appears that during the quantization training process, apart from the online quantization of weights to 3-bit, other components such as main weights, activations, gradients, and optimizer states should remain at higher precision. The authors should briefly state this in the main text or at least in the appendix, explaining that maintaining these parts at high precision is also necessary to ensure the final ternary-quantized model achieves good results during inference, to avoid misleading readers unfamiliar with this field.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "xRRkbk5AUq", "reviewer": "Reviewer_7ZxG", "summary": "The paper builds upon the observation that in extreme quantization settings (i.e., weights taking only values -1,0,1) there is a certain area of values that are mapped to zero. The paper then proposes to re-purpose those weights as biases, through which they receive a more stable gradient signal.", "questions": "Are the experimental results only based on a single run or have they been repeated with different random seeds?", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "Experimental results are very promising. The proposed approach excels in all eval benchmarks for the analyzed 3B model, and in all but one case for the 1B model.\n\nThe paper further provides analysis of loss trajectory (constant improvement) resource requirements (no extra resources needed).\n\nThe visuals of the paper are excellent; very clear and accessible.\n\nA sensitivity analysis of the reactivation threshold is included.\n\nAblation study considering all components of the approach shows consistent gains.", "weaknesses": "Results limited to 1B and 3B models. Other research in quantization-aware training (\"The era of 1-bit LLMs\") suggests that – at ascale – performance decrease is marginal anyways.\n\nThere are other approaches that tackle the same (or highly related) issue of deadzones, e.g., AbsMedian (arXiv:2407.09527) – which do not harm the soundness of the paper but could be considered.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "C4F1Qg6X05", "reviewer": "Reviewer_N2uB", "summary": "This paper proposes Tequila, a training-time method to mitigate the “deadzone trapping” problem inherent in ternary quantization of large language models (LLMs). The authors argue that a large number of weights are sparsified to 0 (the “deadzone”) and receive only noisy straight-through estimator (STE) gradients, which leads to ineffective updates and loss of model capacity. Tequila addresses this by reinterpreting these dead weights as dynamic biases, enabling stronger gradients and providing forward signal restoration. The method is nearly overhead-free during inference and is evaluated across multiple benchmarks (ARC, PIQA, HellaSwag, etc.) using LLaMA-based models, outperforming previous state-of-the-art ternary quantization approaches with significantly fewer training tokens and maintaining ternary inference efficiency.", "questions": "1. Can Tequila be extended to 2-bit or 4-bit quantization scenarios where the weight values are {−1, +1} or {−2, −1, +1, +2} and there is no explicit “deadzone”? If not, can you provide a justification or limitations on generalization?\n\n2. Have the authors considered adding similar learnable correction biases globally or per-group even to non-zero weights in ternary quantization?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper presents a novel and intuitive perspective into the deadzone problem in ternary quantization and proposes a clean and effective approach (Tequila) to solve this challenge.\n\n2. The proposed method is practical and hardware-friendly, as it maintains the computational advantages of ternary quantization and requires negligible inference-time complexity.\n\n3. Convincing empirical results across multiple benchmarks and model sizes (1B and 3B LLaMA) show significant and consistent improvements over both static and learnable ternary quantization baselines, including BitNet and Spectra.\n\n4. The paper is well-written, with clear figures and mathematical exposition. The insight into trapped weights and how reinterpreting them as biases yields optimization benefits is well motivated.", "weaknesses": "1. Tequila risks being a niche solution as this method is tightly coupled with ternary quantization, where the 0 weight level — the \"deadzone\" — is prominent. It is unclear how or whether Tequila can be extended to other quantization schemes, especially 2-bit symmetric quantization (4-value) or binary quantization (2-value), where such a deadzone may not exist.\n\n2. Since the current mechanism focuses only on reactivating dead weights (quantized to 0), it is not evident whether a similar strategy can be applied to weights that are quantized to ±1 in ternary quantization, or other levels in higher-bit setups.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)