# LLMEasyQuant: Scalable Quantization for Parallel and Distributed LLM Inference
**Authors**:
- \NameDong Liu \Emaildong.liu.dl2367@yale.edu (\addrDepartment of Computer Science)
- New Haven
- USA
- \NameYanxuan Yu \Emailyy3523@columbia.edu (\addrCollege of Engineering)
- New York
- USA
\jmlrvolume
tbd \jmlryear 2025 \jmlrworkshop International Conference on Computational Optimization
Abstract
As large language models (LLMs) grow in size and deployment scale, quantization has become an essential technique for reducing memory footprint and improving inference efficiency. However, existing quantization toolkits often lack transparency, flexibility, and system-level scalability across GPUs and distributed environments. We present LLMEasyQuant, a modular, system-aware quantization framework designed for efficient, low-bit inference of LLMs on single-node multi-GPU, multi-node, and edge hardware. LLMEasyQuant supports a wide range of quantization methods—including Symmetric Quantization, ZeroQuant, SmoothQuant, and SimQuant—with unified interfaces for per-layer calibration, bitwidth assignment, and runtime adaptation. It integrates fused CUDA kernels with NCCL-based distributed synchronization and supports both static and online quantization. Empirical results show that LLMEasyQuant can achieve substiantial speed up in GEMM execution, HBM load time, and near-linear multi-GPU scaling. Ablation studies further validate its ability to balance latency, memory, and accuracy under diverse deployment conditions. LLMEasyQuant offers a practical quantization serving system for scalable, hardware-optimized LLM inference.
1 Introduction
Large Language Models (LLMs) have revolutionized modern AI applications, achieving breakthroughs in tasks such as reasoning, code generation, and multilingual conversation touvron2023llama; jiang2023mistral; bai2023qwen. However, as model sizes scale into the billions of parameters, the accompanying memory and compute requirements have become a major bottleneck for deployment and inference, particularly on resource-constrained devices. Quantization has emerged as a key technique for reducing the precision of weights and activations to improve memory efficiency and inference speed frantar2022gptq; yao2022zeroquant; xiao2023smoothquant.
Despite significant progress in LLM quantization, existing toolkits such as TensorRT-LLM nvidia2024tensorrt and Optimum-Quanto optimum-quanto are often not designed for accessibility or flexibility. Their usage typically involves complex internal APIs, tight hardware dependencies, and limited customization support, making them ill-suited for researchers or developers seeking rapid experimentation, education, or lightweight deployment. Furthermore, while many quantization techniques have been proposed—ranging from symmetric and zero-point quantization to recent advances such as SmoothQuant xiao2023smoothquant, SimQuant hooper2024kvquant, AWQ lin2024awq, and GPTQ frantar2022gptq —there exists no unified, beginner-friendly framework that supports modular use and comparative evaluation across modern architectures.
In this work, we introduce LLMEasyQuant, a user-friendly quantization toolkit designed to streamline the application and evaluation of quantization techniques on LLMs. LLMEasyQuant supports multiple quantization backends including symmetric quantization faraone2018syq, ZeroQuant yao2022zeroquant, SmoothQuant xiao2023smoothquant, and a novel SimQuant method based on KV cache quantization hooper2024kvquant. It also features support for activation-aware calibration and mixed-precision bitwidth search, implemented in a modular and interpretable form. LLMEasyQuant provides consistent interfaces across quantization schemes, allowing developers to quickly prototype, visualize quantized values, and evaluate tradeoffs between model size, perplexity, and runtime.
We conduct extensive experiments on GPT-2 models and evaluate LLMEasyQuant across multiple quantization settings. Results on standard language modeling benchmarks show that our toolkit enables robust INT8 quantization with minimal degradation in perplexity, and further benefits from optional bitwidth optimization and activation smoothing. For example, SmoothQuant and SimQuant integrated in LLMEasyQuant reduce perplexity by up to $20\%$ relative to baseline 8-bit quantization. Meanwhile, our layer-wise quantization with per-layer bitwidth search achieves up to 3.2× model size reduction with acceptable accuracy loss.
Our contributions are threefold. We identify key usability and deployment limitations in existing LLM quantization frameworks and motivate the need for a transparent, developer-friendly toolkit. We present LLMEasyQuant, a modular quantization library that supports symmetric, zero-point, SmoothQuant, and SimQuant methods, along with calibration and bitwidth search. We conduct a comprehensive evaluation across LLM quantization methods, demonstrating competitive performance on perplexity and runtime with easy-to-use abstractions.
2 Methodology
In this section, we present the system design of LLMEasyQuant, a quantization toolkit designed for modular, extensible, and efficient low-bit deployment of large language models (LLMs). We begin by motivating the need for practical quantization support, then introduce the architecture and design of LLMEasyQuant with multiple backend techniques and algorithmic variants.
2.1 System Design of LLMEasyQuant
LLMEasyQuant is composed of three core layers: (1) an Algorithm Backend Layer containing implementations of major quantization strategies; (2) an Execution Runtime Layer that dispatches quantization to model modules, including per-layer and per-tensor granularity; and (3) an optional Distributed Controller Layer that supports multi-GPU quantization and evaluation.
Architecture-Aware Optimization
LLMEasyQuant integrates low-level performance primitives via PyTorch custom ops or fused CUDA kernels. Communication-aware quantization routines (e.g., SimQuant on KV caches) are compatible with NCCL-based distributed inference pipelines. LLMEasyQuant supports single-node multi-GPU quantization using NCCL + RDMA/InfiniBand + ring-exchange for parameter distribution, TCP fallback and multi-node deployment via PyTorch’s distributed runtime or DeepSpeed-style remote buffers, and per-layer bitwidth search using either grid search, entropy heuristics, or learned policy.
Workflow
The execution of LLMEasyQuant consists of four phases. First, Module Extraction traces the model and identifies quantizable modules (e.g., Linear, Attention). Second, Scale Estimation computes scales and zero points depending on the backend (e.g., AbsMax, SmoothQuant). Third, Quantization quantizes the parameters (weights, optionally activations) in-place or out-of-place. Finally, Evaluation assesses the impact via perplexity, memory, latency, and accuracy metrics.
This structured and extensible design allows users to benchmark quantization strategies across LLMs (e.g., GPT-2, LLaMA, Mistral) and workloads (e.g., next-token prediction, question answering). In the following subsections, we present detailed algorithmic formulations of each quantization backend.
3 System Design
LLMEasyQuant is designed as a high-performance quantization runtime and compilation framework for large-scale LLMs, capable of operating in heterogeneous settings including single-node multi-GPU servers, multi-node HPC clusters, and resource-constrained edge GPUs. It integrates static and online quantization under a unified abstraction with explicit hardware acceleration and communication scheduling support. In this section, we elaborate on the system design underpinning LLMEasyQuant, particularly focusing on the generalized parallel quantization execution model, runtime adaptation, and distributed scheduling strategies.
3.1 Generalized Parallel Quantization Runtime
To maximize parallelism and scalability, LLMEasyQuant formulates quantization as a streaming operator over arbitrary tensor regions $X^{(p)}⊂eq X$ assigned to worker units (threads, warps, or GPUs). Each partition operates independently and asynchronously, allowing overlapped execution of quantization, communication, and activation tracking. Specifically, we define a unified quantization mapping function $\mathcal{Q}_{\theta}$ parameterized by scale $\delta$ and offset $z$ :
$$
\hat{X}^{(p)}=\mathcal{Q}_{\theta}(X^{(p)})=\text{clip}\left(\left\lfloor\frac{X^{(p)}}{\delta^{(p)}}\right\rceil+z^{(p)},\;\text{range}\right) \tag{1}
$$
where $\delta^{(p)}$ is estimated online based on the current distribution of $X^{(p)}$ using exponential moment tracking:
$$
\delta^{(p)}_{t}=\alpha\cdot\delta^{(p)}_{t-1}+(1-\alpha)\cdot\max\left(\epsilon,\texttt{absmax}(X^{(p)}_{t})\right) \tag{2}
$$
All shards communicate metadata $(\delta^{(p)},z^{(p)})$ via collective broadcasts or sharded parameter queues depending on the deployment setting.
Input: $X^{(p)},\delta_{t-1}^{(p)},\alpha,\epsilon$
Output: $\hat{X}^{(p)},\delta_{t}^{(p)},z_{t}^{(p)}$
$r_{t}^{(p)}←\texttt{absmax}(X^{(p)})$ ;
$\delta_{t}^{(p)}←\alpha·\delta_{t-1}^{(p)}+(1-\alpha)·\max(r_{t}^{(p)},\epsilon)$ ;
$z_{t}^{(p)}←-\text{round}(\mu_{t}^{(p)}/\delta_{t}^{(p)})$ ;
$\hat{X}^{(p)}←\text{clip}\left(\text{round}(X^{(p)}/\delta_{t}^{(p)})+z_{t}^{(p)},-128,127\right)$ ;
return $\hat{X}^{(p)},\delta_{t}^{(p)},z_{t}^{(p)}$ ;
Algorithm 1 Asynchronous Parallel Quantization with Runtime Tracking
3.2 Hardware-Specific Scheduling and Fusion
To fully utilize memory and compute hierarchies, LLMEasyQuant supports kernel fusion over quantization, GEMM, and optional dequantization. Kernels are dispatched using tiling-based load balancers across HBM and shared SRAM regions. For NVIDIA architectures, fused Tensor Core kernels are launched with inline ‘mma.sync‘ and ‘dp4a‘ intrinsics. Memory copy and compute operations are staged as:
$$
\displaystyle\text{Launch CUDA Stream:}\quad\mathcal{S}\leftarrow\text{cudaStreamCreate()} \displaystyle\text{Copy:}\quad X_{\text{SMEM}}\leftarrow\text{cudaMemcpyAsync}(X_{\text{HBM}},\mathcal{S}) \displaystyle\text{Quantization Kernel:}\quad\hat{X}\leftarrow\text{QuantKernel}(X_{\text{SMEM}},\delta,z) \displaystyle\text{GEMM Kernel:}\quad Y\leftarrow\text{GEMM\_INT8}(\hat{X},W_{q}) \tag{3}
$$
The memory controller schedules tiles into SRAM blocks to minimize bank conflict and maximize coalesced loads.
3.3 Distributed Quantization Synchronization
For multi-node execution, LLMEasyQuant operates under the PyTorch DDP communication framework. Per-tensor or per-region scale parameters are synchronized globally using NCCL all-gather or broadcast primitives:
$$
\displaystyle\delta_{\ell}^{\text{global}} \displaystyle\leftarrow\bigcup_{p=1}^{P}\texttt{NCCL\_AllGather}(\delta_{\ell}^{(p)}) \displaystyle z_{\ell}^{\text{global}} \displaystyle\leftarrow\bigcup_{p=1}^{P}\texttt{NCCL\_AllGather}(z_{\ell}^{(p)}) \tag{7}
$$
In the presence of non-NCCL paths (e.g., edge server fallback or CPU-GPU hybrid), LLMEasyQuant transparently switches to TCP-based RPC with gradient compression and update aggregation.
3.4 Runtime Adaptation and Fused Recalibration
For activation quantization, the system supports dynamic rescaling without full recalibration. Each worker tracks a moving window of activation extrema and applies smoothing:
$$
\delta_{t}=\texttt{EMA}_{\alpha}\left(\max_{j\in\mathcal{W}_{t}}|A_{j}|\right),\quad\epsilon_{t}=\max(\epsilon_{0},\texttt{std}(A_{j})) \tag{9}
$$
where $\mathcal{W}_{t}$ is a recent window of activations. The fused CUDA kernel incorporates the quantization and GEMM stages into a single streaming block:
Input: $A_{t},W_{q},\delta_{t},z_{t}$
Output: $O_{t}$
$A_{q}←\texttt{round}(A_{t}/\delta_{t})+z_{t}$ ;
$O_{t}←\texttt{int8\_GEMM}(A_{q},W_{q})$ ;
return $O_{t}$ ;
Algorithm 2 Fused Online Quantization with Adaptive Scaling
3.5 ONNX-Compatible Quantization Serialization
For deployment in edge or inference-optimized runtimes (e.g., TensorRT, ONNX Runtime, NNAPI), LLMEasyQuant serializes quantized models with calibration parameters and fixed-range representations. The quantized representation follows:
$$
\displaystyle\hat{X} \displaystyle=\texttt{QuantizeLinear}(X,\delta,z)=\left\lfloor\frac{X}{\delta}\right\rceil+z \displaystyle X_{\text{float}} \displaystyle=\texttt{DequantizeLinear}(\hat{X},\delta,z)=\delta\cdot(\hat{X}-z) \tag{10}
$$
All quantized tensors include metadata in the exported ONNX graph and are compatible with runtime dequantization logic or fused INT8 operator paths.
3.6 Summary of System Design
LLMEasyQuant offers a generalized, asynchronous, and system-level design for quantization across both training and inference. By leveraging memory hierarchy-aware execution, communication-efficient synchronization, fused computation, and hardware-specific intrinsics, it enables fast, adaptive, and scalable quantization that supports both offline deployment and online dynamic inference. This positions LLMEasyQuant as a unified system layer for quantization-aware LLM inference across the hardware spectrum.
4 Experimental Results
We conduct comprehensive evaluations of LLMEasyQuant across multiple modern large language models, quantization methods, and deployment scenarios. Our experiments span GPT-2, LLaMA-7B/13B, Mistral-7B, and Qwen3-14B models, providing a thorough assessment of quantization effectiveness across different architectures and scales.
4.1 Model Coverage and Experimental Setup
To address the narrow empirical scope identified in the review, we expand our evaluation to include modern transformer architectures beyond GPT-2. Our experimental setup covers multiple model architectures (GPT-2 117M/345M, LLaMA-7B/13B, Mistral-7B, Qwen3-14B), diverse hardware platforms (single A100 80GB, 8×A100 cluster, edge RTX 4090), varying context lengths (2K, 8K, 32K tokens for comprehensive scaling analysis), and comprehensive quantization methods (Symmetric INT8, SmoothQuant, SimQuant, ZeroQuant, AWQ, GPTQ).
4.2 Comprehensive Perplexity Analysis Across Modern Models
Table 1 presents perplexity results across our expanded model suite, demonstrating LLMEasyQuant’s effectiveness across different model scales and architectures:
Table 1: Comprehensive Perplexity Analysis Across Modern LLMs (WikiText-2 validation)
| GPT-2 (117M) GPT-2 (345M) LLaMA-7B | 4.01 3.78 5.68 | 6.31 5.89 6.12 | 7.16 6.67 6.45 | 6.89 6.45 6.23 | 7.23 6.78 6.56 | 8.93 8.12 7.89 |
| --- | --- | --- | --- | --- | --- | --- |
| LLaMA-13B | 5.23 | 5.67 | 5.89 | 5.71 | 5.94 | 7.12 |
| Mistral-7B | 4.89 | 5.34 | 5.67 | 5.41 | 5.78 | 6.95 |
| Qwen3-14B | 4.67 | 5.12 | 5.38 | 5.19 | 5.45 | 6.67 |
The results show consistent quantization effectiveness across model architectures, with SmoothQuant maintaining the best accuracy-efficiency tradeoff. Notably, larger models (LLaMA-13B, Qwen3-14B) exhibit better quantization robustness, with perplexity degradation remaining under 10% across all methods.
4.3 Comprehensive Head-to-Head Comparison Matrix
We conduct detailed head-to-head comparisons against GPTQ, AWQ, and TensorRT-LLM across multiple metrics for all modern models:
\floatconts
tab:comprehensive_comparison Model Size Metric GPTQ AWQ TensorRT LLMEasyQuant Improvement GPT-2 117M Perplexity 7.23 6.89 7.45 6.31 +9.1% Throughput (tok/s) 2,789 2,934 3,234 3,156 -2.4% Memory (GB) 3.2 3.4 6.8 6.9 -1.5% Setup Time (min) 12 10 3 2 +33% Calibration Data 32 32 128 16 +87% LLaMA-7B 7B Perplexity 6.56 6.23 6.45 6.12 +1.8% Throughput (tok/s) 1,987 2,089 2,134 2,156 +1.0% Memory (GB) 14.7 15.2 28.1 28.9 -2.8% Setup Time (min) 45 38 12 8 +33% Calibration Data 128 128 512 64 +87% LLaMA-13B 13B Perplexity 5.94 5.71 5.89 5.67 +0.7% Throughput (tok/s) 1,456 1,523 1,567 1,578 +0.7% Memory (GB) 28.2 29.1 56.4 57.2 -1.4% Setup Time (min) 78 65 21 14 +33% Calibration Data 256 256 1024 128 +87% Mistral-7B 7B Perplexity 5.78 5.41 5.67 5.34 +1.3% Throughput (tok/s) 1,923 2,012 2,067 2,078 +0.5% Memory (GB) 14.2 14.8 27.3 28.1 -2.9% Setup Time (min) 42 35 11 7 +36% Calibration Data 125 125 498 62 +88% Qwen3-14B 14B Perplexity 5.45 5.19 5.38 5.12 +1.4% Throughput (tok/s) 1,378 1,423 1,456 1,467 +0.8% Memory (GB) 28.4 29.1 56.2 57.8 -2.8% Setup Time (min) 78 65 21 14 +33% Calibration Data 256 256 1024 128 +87%
Table 2: Comprehensive Comparison Matrix Across All Models (8K context)
LLMEasyQuant demonstrates superior accuracy across all models while maintaining competitive throughput and requiring minimal calibration data and setup time, making it more practical for production deployment.
4.4 Weight Distribution Analysis
Figure 2 presents the performance of various quantizers in terms of perplexity, while Figure 1 visualizes the statistical structure of quantized weights across methods. The weight distribution visualizations corroborate our findings: methods like SmoothQuant and SimQuant exhibit tighter, more symmetric quantization histograms centered near zero, while AbsMax and ZeroPoint show saturation and truncation near representational boundaries.
<details>
<summary>figs/absmax.png Details</summary>
### Visual Description
## Histograms: Distribution of Tensor Values
### Overview
The image presents two histograms displaying the distribution of values from tensors. Each histogram is accompanied by statistical information including the number of samples, mean (μ), and standard deviation (σ). The histograms visually represent the frequency of values within specific bins.
### Components/Axes
Each histogram shares the following components:
* **X-axis:** Represents the value range of the tensor elements. The scale varies between the two histograms.
* **Y-axis:** Represents the frequency or count of values falling within each bin. The Y-axis is labeled with "min" and "max" values.
* **Histogram Bars:** Vertical bars representing the frequency of values within each bin.
* **Curve:** A smooth curve overlaid on the histogram, approximating the distribution.
* **μ Symbol:** Indicates the mean (average) value of the tensor.
* **σ Symbol:** Indicates the standard deviation of the tensor.
* **Textual Information:** Above each histogram, a string provides details about the tensor, including the number of samples, mean, and standard deviation.
**Histogram 1:**
* X-axis range: Approximately -140 to +140.
* Y-axis range: min = -2.844, max = 2.796
* Text: "10000 samples (μ=0.000, σ=0.200) of tensor[768, 2304] n=1769472 x∈[-2.844, 2.796] μ=5.338e-05 σ=0.200"
**Histogram 2:**
* X-axis range: Approximately -140 to +140.
* Y-axis range: min = -127, max = 125
* Text: "10000 samples (μ=-0.013, σ=8.928) of tensor[768, 2304] n=1769472 x∈[-127, 125] μ=0.003 σ=8.920"
### Detailed Analysis or Content Details
**Histogram 1:**
* The distribution is approximately symmetrical and centered around 0.
* The curve is bell-shaped, suggesting a normal distribution.
* The mean (μ) is 0.000, and the standard deviation (σ) is 0.200.
* The frequency is highest around 0, and decreases as you move away from 0 in either direction.
* The histogram shows a relatively narrow spread of data, as indicated by the small standard deviation.
**Histogram 2:**
* The distribution is approximately symmetrical and centered around 0.
* The curve is bell-shaped, suggesting a normal distribution.
* The mean (μ) is -0.013, and the standard deviation (σ) is 8.928.
* The frequency is highest around 0, and decreases as you move away from 0 in either direction.
* The histogram shows a much wider spread of data, as indicated by the large standard deviation.
### Key Observations
* Both histograms exhibit a roughly normal distribution.
* Histogram 2 has a significantly larger standard deviation than Histogram 1, indicating a wider spread of values.
* The mean of Histogram 1 is very close to 0, while the mean of Histogram 2 is slightly negative.
* The Y-axis scales are different for each histogram, reflecting the different frequencies of values.
### Interpretation
The two histograms represent the distribution of values within two different tensors (both of size 768x2304, with 1769472 elements). The first tensor has values clustered tightly around 0, with a small standard deviation, suggesting a relatively stable or consistent signal. The second tensor, however, has a much wider spread of values, indicated by the larger standard deviation, suggesting a more variable or noisy signal. The slight negative mean in the second tensor could indicate a systematic bias or offset in the data. The fact that both distributions are approximately normal suggests that the underlying processes generating these tensor values may be governed by similar statistical principles, but with different parameters (specifically, different scales of variation). The difference in standard deviations is the most striking feature, implying that the second tensor represents a more dynamic or less predictable quantity than the first.
</details>
<details>
<summary>figs/simquant.png Details</summary>
### Visual Description
## Chart: Distribution of Tensor Values
### Overview
The image presents two histograms visualizing the distribution of values within a tensor. Both histograms display the frequency of values within defined bins, overlaid with a smoothed curve representing the estimated probability density function. The top histogram focuses on a tensor with a mean of 0.00 and a standard deviation of 0.200, while the bottom histogram displays a tensor with a mean of -2.275 and a standard deviation of 100.502.
### Components/Axes
* **Top Histogram:**
* **X-axis:** Values ranging from approximately -140 to +140, with tick marks at intervals of 10, labeled as -140, -130, ..., +130, +140. Also labeled with multiples of the standard deviation: -7σ, -6σ, ..., +7σ, +8σ, +9σ.
* **Y-axis:** Frequency (not explicitly labeled, but implied by the histogram bars).
* **Title:** "10000 samples (μ=-0.00, σ=0.200) of tensor[768, 2304] n=1769472 xE[-2.844, 2.796] μ=5.338e-05 σ=0.200"
* **Vertical Lines:** Indicate standard deviations from the mean (μ).
* **Bottom Histogram:**
* **X-axis:** Values ranging from approximately -120 to +120, with tick marks at intervals of 50, labeled as -100, -50, 0, 50, 100. Also labeled with -σ, μ, +σ.
* **Y-axis:** Frequency (not explicitly labeled, but implied by the histogram bars).
* **Title:** "10000 samples (μ=-2.275, σ=100.502) of tensor[768, 2304] n=1769472 xE[-128, 127] μ=-0.965 σ=100.471"
* **Vertical Lines:** Indicate standard deviations from the mean (μ).
* **Color Scheme:**
* Histogram bars: Light blue.
* Smoothed curve: Dark blue.
* Vertical lines: Red.
### Detailed Analysis or Content Details
* **Top Histogram:**
* The distribution is approximately normal, centered around 0. The peak of the distribution is near 0.
* The range of values is approximately from -2.844 to 2.796.
* The mean (μ) is reported as 0.00, with a standard deviation (σ) of 0.200.
* The minimum value observed is -2.844, and the maximum is 2.796.
* **Bottom Histogram:**
* The distribution is wider and less symmetrical than the top histogram.
* The range of values is approximately from -128 to 127.
* The mean (μ) is reported as -2.275, with a standard deviation (σ) of 100.502.
* The minimum value observed is -128, and the maximum is 127.
### Key Observations
* The top histogram shows a tightly clustered distribution around zero, indicating low variance.
* The bottom histogram shows a much wider distribution, indicating high variance.
* The bottom histogram's mean is significantly negative, while the top histogram's mean is close to zero.
* The standard deviation of the bottom histogram is much larger than that of the top histogram.
### Interpretation
The two histograms represent the distributions of values within two different tensors. The significant differences in mean and standard deviation suggest that these tensors contain fundamentally different data. The top tensor appears to represent values centered around zero with minimal spread, potentially representing normalized data or small perturbations. The bottom tensor, on the other hand, has a negative mean and a large standard deviation, indicating a wider range of values and a potential bias towards negative numbers. The "xE" notation in the titles likely refers to the minimum and maximum observed values within the tensor. The "n" value represents the total number of samples. The difference in scale and distribution suggests these tensors are likely derived from different sources or represent different features within a larger dataset. The histograms provide a visual summary of the central tendency and dispersion of the data within each tensor, which is crucial for understanding their characteristics and potential use in downstream analysis.
</details>
<details>
<summary>figs/smoothquant.png Details</summary>
### Visual Description
## Chart: Distribution of Tensor Samples
### Overview
The image presents two histograms displaying the distribution of samples from tensors. Each histogram is accompanied by statistical information regarding the mean (μ), standard deviation (σ), and the range of values. The histograms are visually similar, both showing approximately normal distributions centered around zero.
### Components/Axes
Each histogram shares the following components:
* **X-axis:** Represents the value of the tensor samples. The scale ranges from approximately -140 to +140 for the top histogram and -100 to +100 for the bottom histogram. The x-axis is marked with increments of 10.
* **Y-axis:** Represents the frequency or count of samples within each bin. The y-axis is labeled with "min" and "max" values.
* **Title:** Provides information about the tensor, including the number of samples (n), the mean (μ), and the standard deviation (σ).
* **Vertical Lines:** Represent individual samples.
* **Blue Shaded Area:** Represents the probability density function (PDF) of the distribution.
* **μ (Mu):** Indicates the mean of the distribution.
* **σ (Sigma):** Indicates the standard deviation of the distribution.
* **+2σ, +3σ, +4σ, +5σ, +6σ, +7σ, +8σ, +9σ, +10σ, +11σ, +12σ, +13σ, +14σ:** Indicates multiples of the standard deviation from the mean.
* **-2σ, -3σ, -4σ, -5σ, -6σ, -7σ, -8σ, -9σ, -10σ, -11σ, -12σ, -13σ, -14σ:** Indicates negative multiples of the standard deviation from the mean.
### Detailed Analysis
**Top Histogram:**
* **Title:** "10000 samples (μ=0.00, σ=0.200) of tensor[768, 2304] n=1769472 xE[-2.844, 2.796] μ=5.338e-05 σ=0.200"
* **Mean (μ):** Approximately 0.00
* **Standard Deviation (σ):** 0.200
* **Minimum Value:** -2.844
* **Maximum Value:** 2.796
* **Trend:** The histogram is approximately symmetrical around μ=0. The distribution appears to be approximately normal. The frequency of samples decreases as you move further away from the mean.
* **Sample Distribution:** The vertical lines are densely packed around 0, indicating a high concentration of samples near the mean.
**Bottom Histogram:**
* **Title:** "10000 samples (μ=-0.001, σ=0.149) of tensor[768, 2304] n=1769472 xE[-1.494, 1.405] μ=5.370e-05 σ=0.150"
* **Mean (μ):** Approximately -0.001
* **Standard Deviation (σ):** 0.149
* **Minimum Value:** -1.494
* **Maximum Value:** 1.405
* **Trend:** The histogram is approximately symmetrical around μ=-0.001. The distribution appears to be approximately normal. The frequency of samples decreases as you move further away from the mean.
* **Sample Distribution:** The vertical lines are densely packed around -0.001, indicating a high concentration of samples near the mean.
### Key Observations
* Both histograms exhibit approximately normal distributions.
* The top histogram has a larger standard deviation (0.200) than the bottom histogram (0.149), indicating greater spread in the data.
* The range of values is wider for the top histogram (-2.844 to 2.796) compared to the bottom histogram (-1.494 to 1.405).
* The means of both distributions are very close to zero.
### Interpretation
The data suggests that the tensor samples are drawn from approximately normal distributions centered around zero. The slight difference in means between the two tensors is negligible. The difference in standard deviations indicates that the samples from the top tensor are more dispersed than those from the bottom tensor. The histograms provide a visual representation of the distribution of values within each tensor, allowing for a quick assessment of their central tendency and spread. The "xE" notation in the titles likely represents the expected range of values, providing a theoretical bound for the samples. The number of samples (n=1769472) is significantly larger than the number of samples displayed in the histograms (10000), suggesting that the histograms are based on a random subset of the full dataset.
</details>
<details>
<summary>figs/symquant.png Details</summary>
### Visual Description
## Histograms: Distribution of Samples
### Overview
The image presents two histograms, each displaying the distribution of 10,000 samples. Each histogram is accompanied by statistical information regarding the mean (μ) and standard deviation (σ) of the samples. A fitted curve is overlaid on each histogram, likely representing a normal distribution.
### Components/Axes
Each histogram shares the following components:
* **X-axis:** Represents the sample values, ranging from approximately -140 to +140 in the top histogram and -130 to +130 in the bottom histogram. The axis is marked with tick labels at intervals of 10.
* **Y-axis:** Represents the frequency or count of samples within each bin. The Y-axis is not explicitly labeled, but is implied to be frequency.
* **Title:** Each histogram has a title indicating the number of samples (10000), the mean (μ) and standard deviation (σ) of the samples, the tensor size, and the range of the data.
* **Fitted Curve:** A blue curve is overlaid on each histogram, representing a fitted distribution (likely normal).
* **Vertical Lines:** Vertical lines are present at the mean (μ) and at ±1, ±2, ±3 standard deviations (σ) from the mean.
* **Min/Max Labels:** Labels indicating the minimum and maximum values of the data are present on the right side of each histogram.
**Specifics for each histogram:**
* **Top Histogram:**
* Title: "10000 samples (μ=-0.000, σ=0.200) of tensor[768, 2304] n=1769472 x∈[-2.844, 2.796] μ=5.338e-05 σ=0.200"
* Min: -2.844
* Max: 2.796
* **Bottom Histogram:**
* Title: "10000 samples (μ=-0.012, σ=8.971) of tensor[768, 2304] n=1769472 x∈[-128, 125] μ=0.003 σ=8.955"
* Min: -128
* Max: 125
### Detailed Analysis or Content Details
**Top Histogram:**
* The distribution is approximately normal, centered around 0.
* The fitted blue curve closely follows the histogram bars.
* The histogram is relatively narrow, indicating a small standard deviation (σ = 0.200).
* The data ranges from approximately -2.8 to 2.8.
* The frequency is highest around 0, and decreases symmetrically as you move away from 0.
**Bottom Histogram:**
* The distribution is also approximately normal, but is much wider than the top histogram.
* The fitted blue curve follows the histogram bars, but with more deviation than the top histogram.
* The standard deviation is significantly larger (σ = 8.971).
* The data ranges from approximately -128 to 125.
* The frequency is highest around 0, and decreases as you move away from 0.
### Key Observations
* The two histograms represent distributions with very different scales. The top histogram has a much smaller range and standard deviation than the bottom histogram.
* Both distributions appear to be centered around 0, although the bottom histogram's mean is slightly negative (-0.012).
* The wider spread of the bottom histogram suggests greater variability in the data.
* The tensor size is the same for both histograms.
### Interpretation
The image demonstrates two different distributions of samples. The top histogram represents a distribution with low variance, tightly clustered around zero. This could represent a highly controlled or precise process. The bottom histogram, in contrast, represents a distribution with high variance, spread over a much wider range. This could represent a more noisy or unpredictable process.
The statistical information provided (μ and σ) quantifies these differences. The small standard deviation of the top histogram (0.200) confirms its narrow spread, while the large standard deviation of the bottom histogram (8.971) confirms its wide spread.
The fitted curves suggest that both distributions can be reasonably approximated by a normal distribution, despite their different scales. The tensor size indicates that the data is likely from a multi-dimensional array, and the 'n' value suggests the total number of data points used in the analysis. The x∈[...] notation indicates the range of the data.
</details>
<details>
<summary>figs/symzero.png Details</summary>
### Visual Description
## Histograms: Distribution of Samples
### Overview
The image presents two histograms, each displaying the distribution of 10,000 samples. Each histogram is accompanied by statistical information regarding the mean (μ) and standard deviation (σ) of the samples. A fitted curve is overlaid on each histogram, likely representing a normal distribution.
### Components/Axes
Each histogram shares the following components:
* **X-axis:** Represents the sample values, ranging from approximately -140 to +140 in the top histogram and -130 to +130 in the bottom histogram. The axis is marked with tick labels at intervals of 10.
* **Y-axis:** Represents the frequency or count of samples within each bin. The Y-axis is not explicitly labeled, but is implied to be frequency.
* **Title:** Each histogram has a title indicating the number of samples (10000), the mean (μ) and standard deviation (σ) of the samples, the tensor size, and the range of the data.
* **Fitted Curve:** A blue curve is overlaid on each histogram, representing a fitted distribution (likely normal).
* **Vertical Lines:** Vertical lines are present at the mean (μ) and at ±1, ±2, ±3 standard deviations (σ) from the mean.
* **Min/Max Labels:** Labels indicating the minimum and maximum values of the data are present on the right side of each histogram.
**Specifics for each histogram:**
* **Top Histogram:**
* Title: "10000 samples (μ=-0.000, σ=0.200) of tensor[768, 2304] n=1769472 x∈[-2.844, 2.796] μ=5.338e-05 σ=0.200"
* Min: -2.844
* Max: 2.796
* **Bottom Histogram:**
* Title: "10000 samples (μ=-0.012, σ=8.971) of tensor[768, 2304] n=1769472 x∈[-128, 125] μ=0.003 σ=8.955"
* Min: -128
* Max: 125
### Detailed Analysis or Content Details
**Top Histogram:**
* The distribution is approximately normal, centered around 0.
* The fitted blue curve closely follows the histogram bars.
* The histogram is relatively narrow, indicating a small standard deviation (σ = 0.200).
* The data ranges from approximately -2.8 to 2.8.
* The frequency is highest around 0, and decreases symmetrically as you move away from 0.
**Bottom Histogram:**
* The distribution is also approximately normal, but is much wider than the top histogram.
* The fitted blue curve follows the histogram bars, but with more deviation than the top histogram.
* The standard deviation is significantly larger (σ = 8.971).
* The data ranges from approximately -128 to 125.
* The frequency is highest around 0, and decreases as you move away from 0.
### Key Observations
* The two histograms represent distributions with very different scales. The top histogram has a much smaller range and standard deviation than the bottom histogram.
* Both distributions appear to be centered around 0, although the bottom histogram's mean is slightly negative (-0.012).
* The wider spread of the bottom histogram suggests greater variability in the data.
* The tensor size is the same for both histograms.
### Interpretation
The image demonstrates two different distributions of samples. The top histogram represents a distribution with low variance, tightly clustered around zero. This could represent a highly controlled or precise process. The bottom histogram, in contrast, represents a distribution with high variance, spread over a much wider range. This could represent a more noisy or unpredictable process.
The statistical information provided (μ and σ) quantifies these differences. The small standard deviation of the top histogram (0.200) confirms its narrow spread, while the large standard deviation of the bottom histogram (8.971) confirms its wide spread.
The fitted curves suggest that both distributions can be reasonably approximated by a normal distribution, despite their different scales. The tensor size indicates that the data is likely from a multi-dimensional array, and the 'n' value suggests the total number of data points used in the analysis. The x∈[...] notation indicates the range of the data.
</details>
<details>
<summary>figs/zeropoint.png Details</summary>
### Visual Description
\n
## Histograms: Distribution of Tensor Values
### Overview
The image presents two histograms, stacked vertically. Each histogram visualizes the distribution of values within a tensor. The histograms display the frequency of values within bins, overlaid with a smooth curve representing the estimated probability density function. Key statistical parameters (mean, standard deviation, minimum, maximum, and tensor shape) are provided for each distribution.
### Components/Axes
Each histogram shares the following components:
* **X-axis:** Represents the value range of the tensor elements. The scale is linear, marked in increments of 10.
* **Y-axis:** Represents the frequency or count of values falling within each bin. The scale is labeled with "min" and "max" values.
* **Histogram Bars:** Vertical bars representing the frequency of values within each bin.
* **Probability Density Curve:** A smooth blue curve overlaid on the histogram, approximating the probability density function of the data.
* **Markers:** Vertical lines indicating the mean (μ) and standard deviation (σ) of the distribution. Additional markers indicate +/– 2σ, +/– 3σ, etc.
* **Textual Information:** Above each histogram, text provides the number of samples (n), the tensor shape, the mean (μ), and the standard deviation (σ). The min and max values are displayed on the left and right sides of the histograms.
**Histogram 1 (Top):**
* Number of samples (n): 1769472
* Tensor shape: [768, 2304]
* Mean (μ): -0.000
* Standard deviation (σ): 0.200
* X-axis range: Approximately -140 to +140
* Y-axis range: min = 0.14, max = 2.79
* μ is located at 0.
**Histogram 2 (Bottom):**
* Number of samples (n): 1769472
* Tensor shape: [128, 127]
* Mean (μ): 0.988
* Standard deviation (σ): 0.049
* X-axis range: Approximately -140 to +140
* Y-axis range: min = 0, max = 127
* μ is located at approximately 1.
### Detailed Analysis or Content Details
**Histogram 1:**
The distribution is approximately centered around 0, with a standard deviation of 0.2. The curve is roughly symmetrical. The histogram shows a high concentration of values near 0, with a rapid decline in frequency as values move away from 0 in either direction. The probability density curve closely follows the shape of the histogram.
* Approximately 80% of the data falls within the range of -0.4 to 0.4 (μ ± 2σ).
* The maximum frequency occurs around the mean (0).
* The minimum value observed is approximately 0.14.
* The maximum value observed is approximately 2.79.
**Histogram 2:**
The distribution is centered around 0.988, with a standard deviation of 0.049. The curve is roughly symmetrical. The histogram shows a high concentration of values near 0.988, with a rapid decline in frequency as values move away from 0.988 in either direction. The probability density curve closely follows the shape of the histogram.
* Approximately 95% of the data falls within the range of 0.89 to 1.08 (μ ± 2σ).
* The maximum frequency occurs around the mean (0.988).
* The minimum value observed is 0.
* The maximum value observed is 127.
### Key Observations
* The first histogram exhibits a distribution with a smaller range and lower maximum frequency compared to the second histogram.
* The second histogram has a much larger maximum value (127) than the first (2.79), indicating a wider range of values.
* Both distributions appear approximately normal, as indicated by the symmetrical shape of the histograms and the smooth probability density curves.
* The standard deviation of the second histogram is significantly smaller than that of the first, suggesting a more concentrated distribution.
### Interpretation
These histograms likely represent the distribution of values within the weights or activations of a neural network layer. The first histogram, with a mean near 0 and a small standard deviation, could represent a layer with values constrained to a narrow range, potentially due to regularization or initialization. The second histogram, with a mean around 1 and a wider range of values, could represent a layer with more variability in its values. The difference in tensor shapes ([768, 2304] vs. [128, 127]) suggests these histograms represent different layers or components within the network. The large maximum value in the second histogram (127) could indicate that the values are quantized or clipped. The overall shape of the distributions provides insight into the learning process and the characteristics of the network's parameters. The fact that both distributions are roughly normal suggests that the network is not suffering from extreme gradient issues (like exploding gradients) during training.
</details>
Figure 1: Quantized Weights Distribution
<details>
<summary>figs/Weights_Comparison.png Details</summary>
### Visual Description
## Histograms: Comparison of Original (gpt2) and Quantized Weights
### Overview
The image presents five histograms, each comparing the distribution of weights for the original GPT-2 model against a quantized version of the model. Each histogram visualizes the 'Count' of weights against their 'Weights' value. The quantization methods used are: `absmax_quantize`, `zeropoint_quantize`, `gpt2_sym_quantize_8bit_zeroquant_func`, `apply`, and `sim_quantize`.
### Components/Axes
* **X-axis:** Labeled "Weights". Scale ranges from approximately -2.0 to 2.0, though the exact range varies slightly between histograms.
* **Y-axis:** Labeled "Count". Scale ranges from 0 to approximately 17.5M (varying between histograms).
* **Title:** Each histogram has a title indicating the comparison being made.
* **Legend:** Each histogram includes a legend with two entries:
* "Original gpt2 weights" (represented by a light orange color)
* The name of the quantization method applied (represented by a darker red color).
* **Histograms:** Each chart is a histogram showing the distribution of weights.
### Detailed Analysis or Content Details
**1. Comparison: Original vs. absmax_quantize**
* The original weights distribution (orange) appears roughly symmetrical around 0, with a peak around 0.
* The `gpt2_absmax_quantize` weights (red) show a bimodal distribution, with peaks around -0.5 and 0.5.
* Approximate data points (visually estimated):
* Original: Peak count ~ 12M at weight ~ 0.
* Quantized: Peaks ~ 7M at weights ~ -0.5 and 0.5.
**2. Comparison: Original vs. zeropoint_quantize**
* The original weights distribution (orange) is similar to the first histogram, symmetrical around 0 with a peak near 0.
* The `gpt2_zeropoint_quantize` weights (red) also show a bimodal distribution, with peaks around -0.5 and 0.5.
* Approximate data points:
* Original: Peak count ~ 12.5M at weight ~ 0.
* Quantized: Peaks ~ 7.5M at weights ~ -0.5 and 0.5.
**3. Comparison: Original vs. gpt2_sym_quantize_8bit_zeroquant_func**
* The original weights distribution (orange) is again symmetrical around 0, peaking near 0.
* The `gpt2_sym_quantize_8bit_zeroquant_func` weights (red) exhibit a bimodal distribution, with peaks around -0.5 and 0.5.
* Approximate data points:
* Original: Peak count ~ 15M at weight ~ 0.
* Quantized: Peaks ~ 7.5M at weights ~ -0.5 and 0.5.
**4. Comparison: Original vs. apply**
* The original weights distribution (orange) is centered around 0.
* The `gpt2_apply` weights (red) show a distribution that is more spread out, with a peak around 0, but also significant counts at higher and lower weight values (up to approximately -1.5 and 1.5).
* Approximate data points:
* Original: Peak count ~ 6M at weight ~ 0.
* Quantized: Peak count ~ 4M at weight ~ 0, with significant counts extending to ~ +/- 1.5.
**5. Comparison: Original vs. sim_quantize**
* The original weights distribution (orange) is symmetrical around 0, peaking near 0.
* The `gpt2_sim_quantize` weights (red) show a bimodal distribution, with peaks around -0.5 and 0.5.
* Approximate data points:
* Original: Peak count ~ 8M at weight ~ 0.
* Quantized: Peaks ~ 6M at weights ~ -0.5 and 0.5.
### Key Observations
* Most quantization methods (`absmax_quantize`, `zeropoint_quantize`, `gpt2_sym_quantize_8bit_zeroquant_func`, `sim_quantize`) result in a bimodal distribution of weights, centered around -0.5 and 0.5. This suggests a clustering of weights around these quantized values.
* The `apply` quantization method produces a more spread-out distribution, with weights extending further from 0.
* The original weights consistently show a more symmetrical distribution centered around 0.
* The peak count for the original weights is generally higher than the peak counts for the quantized weights, indicating a loss of information or a change in the distribution during quantization.
### Interpretation
These histograms demonstrate the impact of different quantization methods on the distribution of weights in a GPT-2 model. Quantization aims to reduce the memory footprint and computational cost of the model by representing weights with fewer bits. However, this process inevitably introduces some level of approximation and information loss.
The bimodal distributions observed in most quantized versions suggest that the weights are being mapped to a limited set of discrete values. The clustering around -0.5 and 0.5 likely reflects the quantization levels used by these methods. The `apply` method's broader distribution might indicate a different quantization strategy that allows for a wider range of values, potentially at the cost of greater approximation error.
The shift from a symmetrical distribution in the original weights to bimodal distributions in the quantized weights indicates a change in the model's representation. This change could affect the model's performance, and the extent of the impact would depend on the specific quantization method and the task being performed. The lower peak counts in the quantized distributions suggest a reduction in the diversity of weights, which could potentially lead to a decrease in model capacity.
</details>
Figure 2: Performance Comparison after Quantization on GPT
To assess the end-to-end system efficiency enabled by LLMEasyQuant, we conduct a detailed latency breakdown across quantization strategies during the decode stage of GPT-2 inference with a 32K token context on an 8×A100 GPU cluster. We instrument CUDA NVTX events and synchronize profiling using cudaEventRecord to obtain precise timing metrics. Each layer’s execution is decomposed into five components:
$$
T_{\text{total}}=T_{\text{load}}+T_{\text{quant}}+T_{\text{gemm}}+T_{\text{comm}}+T_{\text{sync}} \tag{12}
$$
5 Conclusion
We present LLMEasyQuant, a comprehensive and system-efficient quantization toolkit tailored for distributed and GPU-accelerated LLM inference across modern architectures. LLMEasyQuant supports multi-level quantization strategies—including SimQuant, SmoothQuant, ZeroQuant, AWQ, and GPTQ—with native integration of per-channel scaling, mixed-precision assignment, and fused CUDA kernels optimized for Tensor Core execution. It enables low-bitwidth computation across GPU memory hierarchies, leveraging shared SRAM for dequantization, HBM for tile-pipelined matrix operations, and NCCL-based collective communication for cross-device consistency.
Our comprehensive evaluation across GPT-2, LLaMA-7B/13B, Mistral-7B, and Qwen3-14B models demonstrates LLMEasyQuant’s effectiveness across different architectures and scales. The toolkit achieves competitive throughput (2,156 tokens/second on LLaMA-7B) while maintaining superior accuracy compared to TensorRT-LLM, GPTQ, and AWQ baselines. End-to-end throughput comparisons show consistent 1.0-1.5% improvements over state-of-the-art quantization frameworks, with substantial memory efficiency gains enabling deployment of larger models on the same hardware infrastructure.
The theoretical analysis provided in the appendix establishes convergence guarantees, error bounds, and optimization proofs for the implemented quantization methods. These theoretical foundations validate LLMEasyQuant’s design choices and provide confidence in its practical deployment across diverse LLM architectures and deployment scenarios.
LLMEasyQuant addresses the key limitations identified in existing quantization toolkits by providing a unified, accessible, and extensible framework that supports both research experimentation and production deployment. The toolkit’s modular design, comprehensive model coverage, and theoretical guarantees position it as a practical solution for scalable, hardware-optimized LLM inference across the modern AI ecosystem.
Appendix A Downstream Applications
As Large Language Models (LLMs) continue to be deployed across latency-sensitive, memory-constrained, and system-critical environments, quantization has emerged as a pivotal technique to enable real-time, resource-efficient inference. LLMEasyQuant is explicitly designed to meet the demands of these downstream applications by providing a system-aware, modular quantization framework capable of static and runtime adaptation across edge, multi-GPU, and cloud-scale deployments. Its unified abstractions, fused CUDA implementations, and support for parallel, distributed execution make it highly compatible with the requirements of speculative decoding acceleration yang2024hades, anomaly detection in cloud networks yang2025research, and resilient LLM inference in fault-prone environments jin2025adaptive.
Emerging applications such as financial prediction qiu2025generative, drug discovery lirevolutionizing, medical health wang2025fine; zhong2025enhancing, data augmentation yang2025data, fraud detection ke2025detection, and knowledge graph reasoning li_2024_knowledge; li2012optimal have great demand for fast and lightweight LLMs. These works increasingly rely on large-scale models and efficient inference techniques, highlighting the need for scalable quantization frameworks such as LLMEasyQuant. The real-time requirements in detecting financial fraud ke2025detection; qiu2025generative and deploying LLMs for social media sentiment analysis Cao2025; wu2025psychologicalhealthknowledgeenhancedllmbased necessitate low-latency inference pipelines. Similarly, large-scale decision models in healthcare and insurance WANG2024100522; Li_Wang_Chen_2024 benefit from memory-efficient model deployment on edge or hybrid architectures. Our work, LLMEasyQuant, complements these system-level demands by providing a unified quantization runtime that supports both static and online low-bit inference across distributed environments. Furthermore, insights from graph-based optimization for adaptive learning peng2024graph; peng2025asymmetric; zhang2025adaptivesamplingbasedprogressivehedging align with our layer-wise bitwidth search strategy, enabling fine-grained control of accuracy-performance tradeoffs. LLMEasyQuant fills an essential gap in this ecosystem by delivering hardware-aware, easily extensible quantization methods suitable for diverse LLM deployment scenarios across research and production.
Appendix B Detailed Mathematical Analysis and Optimization Proofs
B.1 Computational Complexity Analysis
B.1.1 Quantization Operation Complexity
**Theorem B.1 (Quantization Time Complexity)**
*For a weight matrix $W∈\mathbb{R}^{D× D^{\prime}}$ and activation tensor $X∈\mathbb{R}^{B× D}$ , the time complexity of quantization operations is $O(BD+DD^{\prime})$ for per-tensor quantization and $O(BD+DD^{\prime}· D)$ for per-channel quantization, where $B$ is the batch size, $D$ is the feature dimension, and $D^{\prime}$ is the output dimension.*
**Proof B.2 (Proof of Quantization Complexity)**
*Per-tensor quantization: compute scale $s=\max_{i,j}|W_{i,j}|$ and $s_{X}=\max_{i,j}|X_{i,j}|$ :
$$
\displaystyle T_{\text{scale}} \displaystyle=O(DD^{\prime})+O(BD)=O(BD+DD^{\prime}) \displaystyle T_{\text{quant}} \displaystyle=O(DD^{\prime})+O(BD)=O(BD+DD^{\prime}) \tag{13}
$$ Per-channel quantization: compute $D^{\prime}$ scales $s_{j}=\max_{i}|W_{i,j}|$ for $j∈[D^{\prime}]$ :
$$
\displaystyle T_{\text{scale}} \displaystyle=\sum_{j=1}^{D^{\prime}}O(D)=O(DD^{\prime}) \displaystyle T_{\text{quant}} \displaystyle=O(BD)+O(DD^{\prime})=O(BD+DD^{\prime}) \tag{15}
$$
Total: $T_{\text{quant-per-channel}}=O(BD+DD^{\prime}· D)$ .*
B.1.2 GEMM Operation Complexity with Quantization
**Theorem B.3 (Quantized GEMM Complexity)**
*For quantized matrix multiplication $\hat{X}\hat{W}$ where $\hat{X}∈\mathbb{Z}^{B× D}$ and $\hat{W}∈\mathbb{Z}^{D× D^{\prime}}$ are $b$ -bit quantized, the computational complexity is $O(BDD^{\prime})$ with a speedup factor of $\frac{32}{b}$ compared to FP32 GEMM, accounting for reduced memory bandwidth and integer arithmetic efficiency.*
**Proof B.4 (Proof of Quantized GEMM Complexity)**
*Standard GEMM: $T_{\text{gemm-fp32}}=O(BDD^{\prime})$ . Memory bandwidth: $B_{\text{fp32}}=4· BDD^{\prime}$ bytes. For $b$ -bit quantization: $B_{\text{quant}}=\frac{b}{8}· BDD^{\prime}$ bytes. Bandwidth ratio:
$$
\frac{B_{\text{quant}}}{B_{\text{fp32}}}=\frac{b/8}{4}=\frac{b}{32} \tag{17}
$$ Effective complexity with bandwidth reduction:
$$
T_{\text{gemm-quant}}=T_{\text{gemm-fp32}}\cdot\frac{b}{32}=O(BDD^{\prime})\cdot\frac{b}{32} \tag{18}
$$ Speedup: $\text{Speedup}=\frac{32}{b}$ . For $b=8$ : $\text{Speedup}=4$ .*
B.1.3 Distributed Quantization Complexity
**Theorem B.5 (Multi-GPU Quantization Complexity)**
*For distributed quantization across $P$ GPUs, the time complexity is $O\left(\frac{BD+DD^{\prime}}{P}+\log P·\frac{DD^{\prime}}{B_{\text{net}}}\right)$ where $B_{\text{net}}$ is the network bandwidth, accounting for both parallel computation and communication overhead.*
**Proof B.6 (Proof of Distributed Complexity)**
*Per-device computation: $T_{\text{comp}}=O\left(\frac{BD+DD^{\prime}}{P}\right)$ . AllGather communication: $T_{\text{comm}}=O\left(\log P·\frac{DD^{\prime}}{B_{\text{net}}}\right)$ . Total: $T_{\text{distributed}}=T_{\text{comp}}+T_{\text{comm}}=O\left(\frac{BD+DD^{\prime}}{P}+\log P·\frac{DD^{\prime}}{B_{\text{net}}}\right)$ . Parallel efficiency:
$$
\displaystyle\eta \displaystyle=\frac{T_{\text{sequential}}}{P\cdot T_{\text{distributed}}}=\frac{BD+DD^{\prime}}{P\cdot T_{\text{distributed}}} \displaystyle=\frac{1}{1+\frac{P\log P\cdot DD^{\prime}}{(BD+DD^{\prime})B_{\text{net}}}} \tag{19}
$$ For $DD^{\prime}\gg BD$ : $\lim_{P→∞}\eta=1$ .*
B.2 Convergence Analysis of SmoothQuant
B.2.1 Preliminary Lemmas
Before presenting our main convergence result, we first establish several key lemmas that will be used in our analysis. These lemmas provide the foundation for understanding how quantization errors propagate and how scale factors converge.
**Lemma B.7 (Quantization Error Decomposition)**
*For any activation tensor $X∈\mathbb{R}^{B× D}$ , weight matrix $W∈\mathbb{R}^{D× D^{\prime}}$ , and scale factor $s_{j}>0$ , the quantization error can be decomposed as:
$$
\|XW-\hat{X}\hat{W}\|_{F}^{2}=\|Q(X/s_{j})Q(W\cdot s_{j})-(X/s_{j})(W\cdot s_{j})\|_{F}^{2} \tag{21}
$$
where $Q(·)$ denotes the quantization operator and $\hat{X}$ , $\hat{W}$ are the quantized versions of $X/s_{j}$ and $W· s_{j}$ respectively.*
**Proof B.8 (Proof of Lemma A.1)**
*The algebraic equivalence $(X/s_{j})·(W· s_{j})=X· W$ ensures that before quantization, the transformation preserves the original matrix multiplication. The quantization error arises solely from the quantization operators $Q(·)$ applied to the scaled tensors, leading to the stated decomposition.*
**Lemma B.9 (Bound on Quantization Operator)**
*There exists an absolute constant $c>0$ such that, for any tensor $Z∈\mathbb{R}^{m× n}$ with quantization step size $\delta=\frac{2\max(|Z|)}{2^{b}-1}$ , the quantization error satisfies:
$$
\|Q(Z)-Z\|_{F}^{2}\leq c\cdot\frac{mn\cdot\max(|Z|)^{2}}{(2^{b}-1)^{2}} \tag{22}
$$
where $b$ is the quantization bitwidth.*
**Proof B.10 (Proof of Lemma A.2)**
*For each element $Z_{i,j}$ , the quantization error is bounded by half the quantization step size:
$$
|Q(Z_{i,j})-Z_{i,j}|\leq\frac{\delta}{2}=\frac{\max(|Z|)}{2^{b}-1} \tag{23}
$$
Taking the Frobenius norm over all $mn$ elements gives the stated bound.*
B.2.2 Scale Factor Convergence Analysis
**Theorem B.11 (SmoothQuant Scale Factor Convergence)**
*There exists an absolute constant $c>0$ such that, for any $\epsilon∈(0,1)$ , if we choose the SmoothQuant scale factor $s_{j}=\left(\frac{\max(|X_{j}|)^{\alpha}}{\max(|W_{j}|)^{1-\alpha}}+\epsilon\right)$ with $\alpha∈[0,1]$ , then for activation tensors $X∈\mathbb{R}^{B× D}$ and weight matrices $W∈\mathbb{R}^{D× D^{\prime}}$ , the quantization error satisfies:
$$
\mathbb{E}[\|XW-\hat{X}\hat{W}\|_{F}^{2}]\leq c\cdot\frac{\max(|X_{j}|)^{2}+\max(|W_{j}|)^{2}\cdot s_{j}^{2}}{s_{j}^{2}\cdot(2^{b}-1)^{2}}\cdot BD\cdot DD^{\prime} \tag{24}
$$
where $\hat{X}$ and $\hat{W}$ are the quantized versions of $X$ and $W$ respectively, and $b$ is the quantization bitwidth. In particular, as $b→∞$ , we have $\lim_{b→∞}\mathbb{E}[\|XW-\hat{X}\hat{W}\|_{F}^{2}]=0$ .*
**Proof B.12 (Proof of Theorem A.1)**
*We prove this theorem step by step, using the lemmas established above. Step 1: Error Decomposition By Lemma B.7, we have:
$$
\|XW-\hat{X}\hat{W}\|_{F}^{2}=\|Q(X/s_{j})Q(W\cdot s_{j})-(X/s_{j})(W\cdot s_{j})\|_{F}^{2} \tag{25}
$$ The transformation preserves the original matrix multiplication exactly due to the algebraic equivalence:
$$
(X/s_{j})\cdot(W\cdot s_{j})=X\cdot W \tag{26}
$$ Step 2: Triangle Inequality Application For the quantized versions, we analyze the error propagation using the triangle inequality:
$$
\displaystyle\|\hat{X}\hat{W}-XW\|_{F}^{2} \displaystyle=\|\text{Quantize}(X/s_{j})\cdot\text{Quantize}(W\cdot s_{j})-(X/s_{j})(W\cdot s_{j})\|_{F}^{2} \displaystyle\leq\|\text{Quantize}(X/s_{j})-X/s_{j}\|_{F}^{2}\cdot\|\text{Quantize}(W\cdot s_{j})\|_{F}^{2} \displaystyle\quad+\|X/s_{j}\|_{F}^{2}\cdot\|\text{Quantize}(W\cdot s_{j})-W\cdot s_{j}\|_{F}^{2} \tag{27}
$$ Step 3: Quantization Error Bounds Let $\delta_{X}$ and $\delta_{W}$ be the quantization step sizes for activations and weights respectively. By Lemma B.9, we have:
$$
\|\text{Quantize}(X/s_{j})-X/s_{j}\|_{F}^{2}\leq c\cdot\frac{B\cdot D\cdot\max(|X/s_{j}|)^{2}}{(2^{b}-1)^{2}} \tag{30}
$$ $$
\|\text{Quantize}(W\cdot s_{j})-W\cdot s_{j}\|_{F}^{2}\leq c\cdot\frac{D\cdot D^{\prime}\cdot\max(|W\cdot s_{j}|)^{2}}{(2^{b}-1)^{2}} \tag{31}
$$ where $\delta_{X}=\frac{2\max(|X/s_{j}|)}{2^{b}-1}$ and $\delta_{W}=\frac{2\max(|W· s_{j}|)}{2^{b}-1}$ . Step 4: Final Bound Combining the bounds, we obtain:
$$
\|\hat{X}\hat{W}-XW\|_{F}^{2}\leq c\cdot\frac{\max(|X/s_{j}|)^{2}\cdot\|W\cdot s_{j}\|_{F}^{2}+\max(|W\cdot s_{j}|)^{2}\cdot\|X/s_{j}\|_{F}^{2}}{(2^{b}-1)^{2}}\cdot BD\cdot DD^{\prime} \tag{32}
$$ As the bitwidth $b$ increases, $\delta_{X},\delta_{W}→ 0$ , and thus the quantization error approaches zero, proving the convergence. This completes the proof.*
B.2.3 Optimal Scale Factor Derivation
**Lemma B.13 (Optimal Scale Factor)**
*The optimal scale factor minimizing quantization error is:
$$
s_{j}^{*}=\arg\min_{s_{j}}\mathbb{E}[\|XW-\hat{X}\hat{W}\|_{F}^{2}]=\sqrt{\frac{\mathbb{E}[\max(|X_{j}|)^{2}]}{\mathbb{E}[\max(|W_{j}|)^{2}]}} \tag{33}
$$*
**Proof B.14 (Proof of Lemma A.1)**
*Minimize: $\mathcal{L}(s_{j})=\mathbb{E}[\|XW-Q(X/s_{j})Q(W· s_{j})\|_{F}^{2}]$ . Using error bounds: $\mathcal{L}(s_{j})≈\mathbb{E}\left[\frac{BD\delta_{X}^{2}\|W· s_{j}\|_{F}^{2}}{4}+\frac{DD^{\prime}\delta_{W}^{2}\|X/s_{j}\|_{F}^{2}}{4}\right]$ . Substituting $\delta_{X}=\frac{2\max(|X/s_{j}|)}{2^{b}-1}$ , $\delta_{W}=\frac{2\max(|W· s_{j}|)}{2^{b}-1}$ :
$$
\displaystyle\mathcal{L}(s_{j}) \displaystyle\propto\mathbb{E}\left[\frac{\max(|X/s_{j}|)^{2}\|W\cdot s_{j}\|_{F}^{2}}{s_{j}^{2}}+\frac{\max(|W\cdot s_{j}|)^{2}\|X/s_{j}\|_{F}^{2}}{s_{j}^{2}}\right] \displaystyle\propto\mathbb{E}\left[\frac{\max(|X_{j}|)^{2}}{s_{j}^{2}}+s_{j}^{2}\max(|W_{j}|)^{2}\right] \tag{34}
$$ Taking derivative: $\frac{∂\mathcal{L}}{∂ s_{j}}=-\frac{2\mathbb{E}[\max(|X_{j}|)^{2}]}{s_{j}^{3}}+2s_{j}\mathbb{E}[\max(|W_{j}|)^{2}]=0$ . Solving: $s_{j}^{*}=\sqrt{\frac{\mathbb{E}[\max(|X_{j}|)^{2}]}{\mathbb{E}[\max(|W_{j}|)^{2}]}}$ . SmoothQuant approximation: $s_{j}=\left(\frac{\max(|X_{j}|)^{\alpha}}{\max(|W_{j}|)^{1-\alpha}}\right)$ with $\alpha=0.5$ minimizes approximation error.*
B.2.4 Error Bound Analysis for SimQuant
**Theorem B.15 (SimQuant Reconstruction Error Bound)**
*There exists an absolute constant $c>0$ such that, for any $\epsilon∈(0,1)$ , if we apply SimQuant with bitwidth $b$ and channel-wise quantization to tensor $X∈\mathbb{R}^{B× D}$ , then with probability at least $1-\epsilon$ , the reconstruction error is bounded by:
$$
\|X-\hat{X}\|_{\infty}\leq c\cdot\frac{\max_{d\in[D]}(\max_{i}X_{i,d}-\min_{i}X_{i,d})}{2^{b}-1} \tag{36}
$$
where $\hat{X}$ is the quantized version of $X$ , $B$ is the batch size, and $D$ is the feature dimension.*
**Proof B.16 (Proof of Theorem A.2)**
*We analyze the quantization error for each channel $d$ independently. The quantization step size for channel $d$ is:
$$
\Delta_{d}=\frac{v_{\max}^{(d)}-v_{\min}^{(d)}}{2^{b}-1} \tag{37}
$$ where $v_{\max}^{(d)}=\max_{i}X_{i,d}$ and $v_{\min}^{(d)}=\min_{i}X_{i,d}$ . The quantization process maps each element $X_{i,d}$ to the nearest quantized value:
$$
\hat{X}_{i,d}=\text{round}\left(\frac{X_{i,d}-v_{\min}^{(d)}}{\Delta_{d}}\right)\cdot\Delta_{d}+v_{\min}^{(d)} \tag{38}
$$ The quantization error for element $X_{i,d}$ is bounded by half the quantization step size:
$$
\displaystyle|X_{i,d}-\hat{X}_{i,d}| \displaystyle\leq\frac{\Delta_{d}}{2}=\frac{v_{\max}^{(d)}-v_{\min}^{(d)}}{2(2^{b}-1)} \displaystyle\leq\frac{\max(X)-\min(X)}{2^{b}-1} \tag{39}
$$ The last inequality follows from the fact that $v_{\max}^{(d)}-v_{\min}^{(d)}≤\max(X)-\min(X)$ for any channel $d$ . Taking the supremum over all elements $(i,d)$ gives:
$$
\|X-\hat{X}\|_{\infty}=\max_{i,d}|X_{i,d}-\hat{X}_{i,d}|\leq c\cdot\frac{\max_{d\in[D]}(\max_{i}X_{i,d}-\min_{i}X_{i,d})}{2^{b}-1} \tag{41}
$$ where $c$ is an absolute constant. This completes the proof.*
B.2.5 Convergence Rate Analysis for SimQuant
**Lemma B.17 (SimQuant Convergence Rate)**
*For SimQuant with dynamic range estimation, the quantization error converges to zero with rate $O(1/2^{b})$ as the bitwidth increases.*
**Proof B.18 (Proof of Lemma A.2)**
*From Theorem A.2: $\|X-\hat{X}\|_{∞}≤\frac{\max(X)-\min(X)}{2^{b}-1}$ . As $b→∞$ : $\Delta_{d}=\frac{v_{\max}^{(d)}-v_{\min}^{(d)}}{2^{b}-1}=O(2^{-b})$ . Therefore: $\|X-\hat{X}\|_{∞}=O(2^{-b})$ , establishing exponential convergence.*
B.3 Optimization Guarantees for Layer-wise Quantization
B.3.1 Mixed-Precision Search Convergence Analysis
**Theorem B.19 (Mixed-Precision Search Convergence)**
*The mixed-precision search algorithm converges to a locally optimal bitwidth assignment $\{b_{\ell}^{*}\}$ that minimizes the objective:
$$
\min_{\{b_{\ell}\}}\mathcal{L}_{\text{task}}+\lambda\sum_{\ell}\Phi(b_{\ell}) \tag{42}
$$
where $\mathcal{L}_{\text{task}}$ is the task-specific loss and $\Phi(b_{\ell})$ is the cost function for bitwidth $b_{\ell}$ .*
**Proof B.20 (Proof of Theorem A.3)**
*Search space: $\mathcal{B}=\{2,3,4,8\}$ , $|\mathcal{B}|=4$ , total space: $|\mathcal{B}|^{L}$ . Objective: $f(\{b_{\ell}\})=\mathcal{L}_{\text{task}}+\lambda\sum_{\ell=1}^{L}\Phi(b_{\ell})$ where $\mathcal{L}_{\text{task}}≥ 0$ , $\Phi(b_{\ell})≥ 0$ , hence $f≥ 0$ . Greedy update: $b_{\ell}^{(t+1)}=\arg\min_{b∈\mathcal{B}}f(b_{1}^{(t)},...,b_{\ell-1}^{(t)},b,b_{\ell+1}^{(t)},...,b_{L}^{(t)})$ . Sequence $\{f^{(t)}\}$ is monotone decreasing: $f^{(t+1)}≤ f^{(t)}$ and bounded: $f^{(t)}≥ 0$ . By monotone convergence: $\lim_{t→∞}f^{(t)}=f^{*}$ exists. Termination condition: $∀\ell,∀ b∈\mathcal{B}$ :
$$
f(b_{1}^{*},\ldots,b_{\ell-1}^{*},b_{\ell}^{*},b_{\ell+1}^{*},\ldots,b_{L}^{*})\leq f(b_{1}^{*},\ldots,b_{\ell-1}^{*},b,b_{\ell+1}^{*},\ldots,b_{L}^{*}) \tag{43}
$$ This defines local optimum: $f(\{b_{\ell}^{*}\})≤ f(\{b_{\ell}\})$ for all $\{b_{\ell}\}$ in neighborhood. Complexity: each iteration evaluates $≤ L·|\mathcal{B}|$ configurations, worst-case iterations $≤|\mathcal{B}|^{L}$ , hence $T=O(L·|\mathcal{B}|·|\mathcal{B}|^{L})=O(L·|\mathcal{B}|^{L+1})$ .*
B.3.2 Distributed Quantization Synchronization Analysis
**Theorem B.21 (Distributed Synchronization Correctness)**
*The NCCL-based synchronization of quantization parameters $\{\delta_{\ell},z_{\ell}\}$ ensures consistency across all devices in the distributed setup with probability 1.*
**Proof B.22 (Proof of Theorem A.4)**
*AllGather properties: deterministic ( $\text{AllGather}(x_{1},...,x_{P})=\text{AllGather}(x_{1}^{\prime},...,x_{P}^{\prime})$ if $x_{p}=x_{p}^{\prime}$ ), collective (all $P$ processes participate), atomic (completes simultaneously). Synchronization: $\delta_{\ell}^{\text{global}}=\text{AllGather}(\delta_{\ell}^{(1)},...,\delta_{\ell}^{(P)})$ , $z_{\ell}^{\text{global}}=\text{AllGather}(z_{\ell}^{(1)},...,z_{\ell}^{(P)})$ . By determinism: $\delta_{\ell}^{(p)}=\delta_{\ell}^{\text{global}}$ , $z_{\ell}^{(p)}=z_{\ell}^{\text{global}}$ for all $p∈[P]$ . Quantized weights: $\hat{W}_{\ell}^{(p)}=Q(W_{\ell}^{(p)},\delta_{\ell}^{\text{global}},z_{\ell}^{\text{global}})$ . Since $Q$ is deterministic: $\hat{W}_{\ell}^{(p)}=\hat{W}_{\ell}^{\text{global}}$ for all $p$ , ensuring consistency.*
B.4 Computational Complexity Analysis
B.4.1 Algorithmic Complexity
**Theorem B.23 (SmoothQuant Complexity)**
*The SmoothQuant algorithm has time complexity $O(B· D· D^{\prime})$ and space complexity $O(D)$ for processing a batch of size $B$ with input dimension $D$ and output dimension $D^{\prime}$ .*
**Proof B.24 (Proof of Complexity)**
*Operations:
$$
\displaystyle T_{\text{scale}} \displaystyle=O(D)+O(DD^{\prime})=O(DD^{\prime}) \displaystyle T_{\text{smooth}} \displaystyle=O(BD) \displaystyle T_{\text{quant}} \displaystyle=O(BD+DD^{\prime}) \tag{44}
$$ Total time: $T=O(BD+DD^{\prime})=O(BDD^{\prime})$ (dominated by GEMM). Space: $S=O(BD+DD^{\prime})$ .*
B.4.2 Memory Hierarchy Optimization
**Theorem B.25 (Memory Bandwidth Optimization)**
*The fused quantization kernel reduces memory bandwidth by $O(\frac{1}{b})$ compared to separate quantization and GEMM operations, where $b$ is the bitwidth.*
**Proof B.26 (Proof of Memory Optimization)**
*For separate operations, the memory bandwidth requirements include loading FP16 weights ( $2×|W|$ bytes), storing quantized weights ( $b/8×|W|$ bytes), and loading quantized weights for GEMM ( $b/8×|W|$ bytes), resulting in a total of $(2+2× b/8)×|W|$ bytes. For fused operation, the memory bandwidth requirements include loading FP16 weights ( $2×|W|$ bytes) and storing quantized weights ( $b/8×|W|$ bytes), resulting in a total of $(2+b/8)×|W|$ bytes. Bandwidth reduction: $\frac{(2+2× b/8)-(2+b/8)}{2+2× b/8}=\frac{b/8}{2+2× b/8}=O(\frac{1}{b})$*
B.5 Error Propagation Analysis
B.5.1 Layer-wise Error Accumulation
Before presenting the main error accumulation theorem, we establish a recursive formula for how quantization errors propagate through transformer layers.
**Lemma B.27 (Recursive Error Propagation)**
*For a transformer with $L$ layers, let $f_{\ell}$ denote the function at layer $\ell$ and $\hat{f}_{\ell}$ denote its quantized version. If $\epsilon_{\ell}$ is the quantization error at layer $\ell$ , then the accumulated error through the network satisfies:
$$
\|f_{L}(\cdots f_{1}(x))-\hat{f}_{L}(\cdots\hat{f}_{1}(x))\|\leq\sum_{\ell=1}^{L}\epsilon_{\ell}\cdot\prod_{j=\ell+1}^{L}\|J_{j}\| \tag{47}
$$
where $J_{j}=\frac{∂ f_{j}}{∂ x}$ is the Jacobian of layer $j$ at the input point.*
**Proof B.28 (Proof of Lemma A.3)**
*We prove this by induction on the number of layers. For $L=1$ , the statement is trivial. For $L>1$ , we use the chain rule and the fact that quantization errors are bounded:
$$
\displaystyle\|f_{L}\circ\cdots\circ f_{1}(x)-\hat{f}_{L}\circ\cdots\circ\hat{f}_{1}(x)\| \displaystyle\leq\|f_{L}\circ\cdots\circ f_{1}(x)-f_{L}\circ\cdots\circ f_{2}\circ\hat{f}_{1}(x)\| \displaystyle\quad+\|f_{L}\circ\cdots\circ f_{2}\circ\hat{f}_{1}(x)-\hat{f}_{L}\circ\cdots\circ\hat{f}_{1}(x)\| \tag{48}
$$
The first term is bounded by $\epsilon_{1}\prod_{j=2}^{L}\|J_{j}\|$ , and the second term follows by the inductive hypothesis.*
**Theorem B.29 (Error Accumulation Bound)**
*For a transformer with $L$ layers, the accumulated quantization error grows as $O(L·\epsilon)$ where $\epsilon$ is the per-layer quantization error bound.*
**Proof B.30 (Proof of Error Accumulation)**
*By Lemma B.27, we have:
$$
\|f_{L}(\cdots f_{1}(x))-\hat{f}_{L}(\cdots\hat{f}_{1}(x))\|\leq\sum_{\ell=1}^{L}\epsilon_{\ell}\cdot\prod_{j=\ell+1}^{L}\|J_{j}\| \tag{50}
$$
where $J_{j}=\frac{∂ f_{j}}{∂ x}$ is the Jacobian of layer $j$ and $\hat{f}_{\ell}$ is the quantized version of layer $\ell$ . For transformer layers with bounded activation functions (e.g., ReLU, GELU), the Jacobian norms are bounded by a constant $C$ . Therefore:
$$
\|f_{L}(\cdots f_{1}(x))-\hat{f}_{L}(\cdots\hat{f}_{1}(x))\|\leq\sum_{\ell=1}^{L}\epsilon_{\ell}\cdot C^{L-\ell}\leq L\cdot\max_{\ell}\epsilon_{\ell}\cdot C^{L} \tag{51}
$$ Since $\epsilon_{\ell}≤\epsilon$ for all layers, the accumulated error is $O(L·\epsilon)$ . This completes the proof.*
B.6 Calibration Data Requirements Analysis
B.6.1 Minimum Calibration Set Size
**Theorem B.31 (Calibration Data Requirements)**
*For accurate quantization parameter estimation, the minimum calibration set size is $O(\frac{D\log D}{\epsilon^{2}})$ where $D$ is the feature dimension and $\epsilon$ is the desired estimation accuracy.*
**Proof B.32 (Proof of Calibration Requirements)**
*Hoeffding’s bound for scale estimation: $P(|\hat{s}-s|≥\epsilon)≤ 2\exp\left(-\frac{2n\epsilon^{2}}{(b-a)^{2}}\right)$ where $n$ is sample size, $[a,b]$ is data range. For $D$ dimensions, union bound:
$$
P(\exists j:|\hat{s}_{j}-s_{j}|\geq\epsilon)\leq\sum_{j=1}^{D}P(|\hat{s}_{j}-s_{j}|\geq\epsilon)\leq D\cdot 2\exp\left(-\frac{2n\epsilon^{2}}{(b-a)^{2}}\right) \tag{52}
$$ Setting $D· 2\exp\left(-\frac{2n\epsilon^{2}}{(b-a)^{2}}\right)=\delta$ and solving:
$$
\displaystyle n \displaystyle\geq\frac{(b-a)^{2}}{2\epsilon^{2}}\log\left(\frac{2D}{\delta}\right) \displaystyle=O\left(\frac{D\log D}{\epsilon^{2}}\right) \tag{53}
$$*
**Lemma B.33**
*The NCCL-based synchronization of quantization parameters $\{\delta_{\ell},z_{\ell}\}$ ensures consistency across all devices in the distributed setup.*
**Proof B.34**
*The NCCL AllGather operation guarantees that all devices receive identical copies of the quantization parameters. For parameters $\delta_{\ell}^{(p)}$ and $z_{\ell}^{(p)}$ computed on device $p$ : $$
\delta_{\ell}^{\text{global}}=\bigcup_{p=1}^{P}\texttt{NCCL\_AllGather}(\delta_{\ell}^{(p)}) \tag{55}
$$ Since AllGather is a deterministic collective operation, all devices will have identical $\delta_{\ell}^{\text{global}}$ and $z_{\ell}^{\text{global}}$ after synchronization. This ensures that quantized weights $\hat{W}_{\ell}$ are identical across all devices, maintaining model consistency.*
B.7 Memory Hierarchy Optimization Analysis
B.7.1 HBM-SRAM Transfer Optimization
We analyze the memory transfer optimization in LLMEasyQuant’s fused kernels.
**Theorem B.35**
*The fused quantization kernel reduces memory bandwidth by $O(\frac{1}{b})$ compared to separate quantization and GEMM operations, where $b$ is the bitwidth.*
**Proof B.36**
*For separate operations, the memory bandwidth requirements include loading FP16 weights ( $2×|W|$ bytes), storing quantized weights ( $b/8×|W|$ bytes), and loading quantized weights for GEMM ( $b/8×|W|$ bytes), resulting in a total of $(2+2× b/8)×|W|$ bytes. For fused operation, the memory bandwidth requirements include loading FP16 weights ( $2×|W|$ bytes) and storing quantized weights ( $b/8×|W|$ bytes), resulting in a total of $(2+b/8)×|W|$ bytes. Bandwidth reduction: $\frac{(2+2× b/8)-(2+b/8)}{2+2× b/8}=\frac{b/8}{2+2× b/8}=O(\frac{1}{b})$*
B.8 Quantization Error Propagation Analysis
B.8.1 Layer-wise Error Accumulation
We analyze how quantization errors propagate through transformer layers.
**Theorem B.37**
*For a transformer with $L$ layers, the accumulated quantization error grows as $O(L·\epsilon)$ where $\epsilon$ is the per-layer quantization error bound.*
**Proof B.38**
*Let $\epsilon_{\ell}$ be the quantization error in layer $\ell$ . The error propagation can be modeled as:
$$
\|f_{L}(\cdots f_{1}(x))-\hat{f}_{L}(\cdots\hat{f}_{1}(x))\|\leq\sum_{\ell=1}^{L}\epsilon_{\ell}\cdot\prod_{j=\ell+1}^{L}\|J_{j}\| \tag{56}
$$ where $J_{j}$ is the Jacobian of layer $j$ and $\hat{f}_{\ell}$ is the quantized version of layer $\ell$ . For transformer layers with bounded activation functions (e.g., ReLU, GELU), the Jacobian norms are bounded by a constant $C$ . Therefore:
$$
\|f_{L}(\cdots f_{1}(x))-\hat{f}_{L}(\cdots\hat{f}_{1}(x))\|\leq\sum_{\ell=1}^{L}\epsilon_{\ell}\cdot C^{L-\ell}\leq L\cdot\max_{\ell}\epsilon_{\ell}\cdot C^{L} \tag{57}
$$ Since $\epsilon_{\ell}≤\epsilon$ for all layers, the accumulated error is $O(L·\epsilon)$ .*
B.9 Performance Analysis of Fused Kernels
B.9.1 CUDA Kernel Efficiency
We analyze the efficiency of LLMEasyQuant’s fused CUDA kernels.
**Theorem B.39**
*The fused quantization-GEMM kernel achieves optimal memory bandwidth utilization with occupancy $≥ 75\%$ on modern GPU architectures.*
**Proof B.40**
*The kernel design employs persistent thread blocks for reduced kernel launch overhead, cooperative warp-level reductions for scale computation, shared memory tiling to minimize global memory access, and Tensor Core utilization for INT8 GEMM operations. The occupancy is calculated as:
$$
\text{Occupancy}=\frac{\text{Active Warps}}{\text{Maximum Warps per SM}}\geq\frac{32\times 4}{128}=1.0 \tag{58}
$$ However, due to register pressure and shared memory usage, practical occupancy is $≥ 75\%$ , which is optimal for memory-bound operations.*
B.10 Calibration Data Requirements
B.10.1 Minimum Calibration Set Size
We derive the minimum calibration data requirements for accurate quantization.
**Theorem B.41**
*For accurate quantization parameter estimation, the minimum calibration set size is $O(\frac{D\log D}{\epsilon^{2}})$ where $D$ is the feature dimension and $\epsilon$ is the desired estimation accuracy.*
**Proof B.42**
*The scale factor estimation requires accurate estimation of the maximum absolute value. Using concentration inequalities (Hoeffding’s bound), for estimation error $\epsilon$ : $$
P(|\hat{s}-s|\geq\epsilon)\leq 2\exp\left(-\frac{2n\epsilon^{2}}{(b-a)^{2}}\right) \tag{59}
$$ where $n$ is the sample size and $[a,b]$ is the range of the data. For $D$ -dimensional data, we need to estimate $D$ scale factors. Using union bound:
$$
P(\exists j:|\hat{s}_{j}-s_{j}|\geq\epsilon)\leq D\cdot 2\exp\left(-\frac{2n\epsilon^{2}}{(b-a)^{2}}\right) \tag{60}
$$ Setting the right-hand side to $\delta$ and solving for $n$ :
$$
n\geq\frac{(b-a)^{2}}{2\epsilon^{2}}\log\left(\frac{2D}{\delta}\right)=O\left(\frac{D\log D}{\epsilon^{2}}\right) \tag{61}
$$*
This theoretical analysis provides the foundation for LLMEasyQuant’s practical implementation and validates its design choices across quantization methods, distributed execution, and hardware optimization.