Dual-Smoothed Fine-Grained Quantization

• Dual-Smoothed Fine-Grained Quantization is a technique that partitions neural network weights into finely grouped blocks and applies local scaling to minimize reconstruction error during quantization.
• It employs mixed-precision strategies, including ternary quantization and block-wise methods, leveraging Fisher information and error compensation to preserve model accuracy.
• The approach integrates hardware-aware optimizations, such as integer scale amplification, to boost inference speed and reduce energy consumption in large-scale neural models.

Dual-Smoothed Fine-Grained Quantization is a set of methodologies for post-training quantization of deep neural networks that combine high-resolution grouping (fine granularity) with local smoothing or dual precision mechanisms—typically for both weights and activations—to minimize accuracy loss while maximizing efficiency. These techniques have evolved to merge mathematical insights about parameter distributions with hardware-aware strategies for block-wise mixed precision, error compensation, and scaling optimizations, making them particularly relevant for low-power, high-throughput deployment of large neural models.

1. Mathematical Formulation and Core Methodology

Dual-Smoothed Fine-Grained Quantization approaches begin by decomposing neural network weight tensors into disjoint groups or blocks, then quantizing each independently with local scaling and thresholding. The fundamental operation is group-wise ternarization or quantization, often formalized as:

• For weights $W$ grouped into $k$ partitions, each sub-vector $W^{(i)}$ is represented as $W^{(i)} \approx \alpha_i \cdot \hat{W}^{(i)}$, where $\hat{W}^{(i)} \in \{-1, 0, +1\}$ for ternary quantization.
• The optimal scaling and threshold per group are determined by minimizing the reconstruction error: $\|W^{(i)} - \alpha_i \cdot \hat{W}^{(i)}\|^2$ These can be solved analytically for common distributions (e.g., exponential, Gaussian), yielding threshold formulas such as $\Delta^* \approx \frac{1}{\lambda}$ where $\lambda$ is estimated from $\sum |W_i| / n$.

For mixed-precision variants, each group or block can be assigned a bit-width, e.g., using optimization over block sensitivities (Fisher information for loss perturbation):

$I_\mathcal{L}(v) = \sum_i g_i^2 (\Delta v_i)^2$

Blocks with higher $I_\mathcal{L}$ are retained at high precision, others reduced to low precision.

Activations are similarly quantized with per-group or per-layer scaling, occasionally employing data-driven or distribution alignment losses as in FDDA (see below).

2. Dual Smoothing: Theory and Algorithmic Design

The "dual-smoothed" concept incorporates two orthogonal smoothing processes:

• Local smoothing within weight/activation groups: Assigning per-group scaling factors optimally fitted to the distribution in that segment, not globally across the full tensor.
• Dual smoothing across quantization modalities: Smoothing the trade-off between accuracy and compression via continuous optimization or Lagrangian relaxation (as in differentiable fine-grained quantization (Cheng et al., 2018)).

In practice, differentiable relaxation is used, such as defining quantized outputs as softmax-weighted sums over candidate bitwidths:

$$q_i = \frac{\sum_j \exp(\alpha_{ij}) \mathcal{B}(q_{ij})}{\sum_j \exp(\alpha_{ij})}$$

where $\mathcal{B}$ may represent batch normalization.

Other techniques leverage stochastic smoothing, as in SDQ (Huang et al., 2022), where per-layer bitwidth assignment is governed by differentiable probability parameters, optimized with Gumbel-softmax reparameterization. This yields smooth gradient flow and interpolates quantization decisions at both the parameter and architectural (bitwidth selection) levels.

3. Block-wise and Outlier-aware Mixed Precision

Recent block-wise fine-grained methods, e.g., FGMP (Hooper et al., 19 Apr 2025), divide weights and activations into small blocks (sub-vectors or clusters), assigning precision levels using sensitivity metrics. Fisher-information-weighted perturbation measures determine which blocks are most vulnerable to quantization-induced loss changes.

For weights $v$ and gradient $g$ per element,

$$I'_\mathcal{L}(v) = \sum_i g_i^2 (\Delta_{p_h \rightarrow p_l} v_i)^2$$

Blocks with higher $I'_\mathcal{L}$ are kept at high precision (e.g., FP8), others in low precision (e.g., NVFP4). Sensitivity-weighted clipping is then applied to optimize the quantization scale within each block, minimizing

$\min_s \sum_i g_i^2 (\Delta^{s} v_i)^2$

Cluster-wise approaches, such as FineQ (Xie et al., 28 Apr 2025), further refine block-level granularity—partitioning each channel into clusters (of 3 weights). Outlier values within clusters are adaptively protected using increased bit-width encoding (3 bits for outliers, 2 bits for regular values), with encoding schemes that maintain aligned memory access in support of efficient hardware decoding.

4. Scaling, Inference Efficiency, and Hardware Integration

A critical bottleneck in block-wise quantization is the compute overhead associated with multiplying each accumulated INT32 result by group-wise floating-point scale factors (C-scales). Integer Scale (Li et al., 23 May 2024) resolves this by amplifying these scales to integer values using a layer-wise amplifier $\alpha$, thereby avoiding expensive INT-to-FP32 type conversions in GEMM kernels. The formula is:

$$O_i = s_{a_i} \times \text{FLOAT}\left\{ \sum_g \left(X_{g_i} W_{g_i}^T \right) \times \text{INT}(s_{g_i} \alpha) \right\} / \alpha$$

where $s_{a_i}$ is the activation scale, $s_{g_i}$ the group scale, and $\alpha$ is chosen so that all $s_{g_i} \times \alpha \geq 1$. This approach offers up to $2.31\times$ end-to-end speed boosts on modern LLMs without calibration or fine-tuning.

Hardware support is pivotal for the deployment of dual-smoothed block-wise mixed precision. FGMP (Hooper et al., 19 Apr 2025) and FineQ (Xie et al., 28 Apr 2025) showcase the integration of block-level metadata with specialized accelerators (e.g., temporal coding systolic arrays), supporting both mixed precision (FP4, FP8) and dynamic activation quantization units for minimal runtime and energy overhead.

5. Activation Quantization and Data Distribution Alignment

Activation quantization is addressed with dynamic schemes such as logarithmic equalization (FPTQ (Li et al., 2023)), which computes channelwise scales:

$s_i = \max(|x_i|)/\log_2(2+\max(|x_i|))$

Channels are normalized accordingly, and weights adjusted such that

$$W' = \text{diag}(s) \cdot W, \quad X' = X \cdot \text{diag}(s)^{-1}$$

This suppresses outliers and equalizes inter-channel dynamic range, yielding more stable quantization error.

Post-training methods (FDDA (Zhong et al., 2021)) leverage batch normalization statistics (mean, variance) per class and apply dual loss functions (centralization and distortion) using synthetic calibration data:

• Centered loss: $L_\text{centered} = \sum_k || \mu_k - \mu_c ||^2$
• Distorted loss: $$L_\text{distorted} = \sum_{k, l; k\neq l} \max(0, \Delta - ||\mu_k-\mu_l|| )$$ These maintain inter-class separation and intra-class incohesion after quantization.

6. Performance, Trade-offs, and Applications

Empirical evaluations across FGQ (Mellempudi et al., 2017), SDQ (Huang et al., 2022), DGQ (Zhang et al., 2023), FPTQ (Li et al., 2023), FGMP (Hooper et al., 19 Apr 2025), and FineQ (Xie et al., 28 Apr 2025) show clear trends:

• Fine-grained (block/group-wise) quantization minimizes quantization loss and is robust against outliers, yielding accuracy losses often under 1%.
• Post-training quantization with dual smoothing achieves state-of-the-art results without requiring re-training; e.g., Top-1 accuracy within $3.7\%$ of baseline with $N=4$ grouping (Mellempudi et al., 2017), $<1\%$ perplexity degradation for Llama-2-7B (Hooper et al., 19 Apr 2025).
• Integer Scale and DPQ (Gafni et al., 20 May 2025) enable hardware-friendly computations (FP8/INT4) with substantial throughput improvements ($1.85\times$–$3\times$ on large LLMs) and energy reduction (up to $1.79\times$ (Xie et al., 28 Apr 2025)) compared to conventional INT8/FP16 pipelines.

The practical significance ranges from deployment on edge devices (where memory and bandwidth are limited) to high-throughput server-side inference of LLMs, vision-LLMs, and real-time autoregressive generation.

7. Comparisons, Limitations, and Outlook

Dual-smoothed fine-grained quantization strategies have distinguished themselves from coarse-grained and naive mixed-precision approaches via:

• Sensitivity-aware block selection (Fisher information, activation statistics)
• Outlier-aware encoding and protection
• Plug-and-play compatibility with existing quantization toolkits (GPTQ, AWQ, etc.)
• Tight hardware-software co-design for minimal memory and compute overhead

Limitations involve increased complexity in calibration (when synthetic data and dual loss functions are used), as well as constraints imposed by hardware architectures. The need to maintain efficient, aligned accesses in memory and support metadata-driven computation (as in FineQ) persists as a challenge when integrating new smoothing algorithms.

A plausible future direction is the convergence of block-wise mixed precision with automated, differentiable smoothing assignment both for weights and activations, leveraging metrics derived from data statistics and model gradients. Continued integration with integer scale amplification and temporal coding is likely to further improve energy efficiency and scalability, especially as LLMs proliferate in size and deployment scenarios.

Summary Table: Representative Methods

Paper (arXiv id)	Key Technique	Accuracy Loss	Hardware/Energy Benefit
FGQ (Mellempudi et al., 2017)	Group-wise ternary quant.	<4% (Top-1, N=4)	Up to $15\times$ perf.
SDQ (Huang et al., 2022)	Stochastic grad. bitwidth	None/superior	Latency/energy improved
FGMP (Hooper et al., 19 Apr 2025)	Fisher-info block mixed prec.	<1% (Llama-2-7B)	$14\%$ energy, $30\%$ mem
FineQ (Xie et al., 28 Apr 2025)	Intra-cluster outlier protect	Minimal	$1.79\times$ energy, $61\%$ area
DPQ (Gafni et al., 20 May 2025)	W4A8 via Hessian-compensated	Minor	$3\times$ throughput

These references collectively define the landscape of dual-smoothed fine-grained quantization, establishing it as a foundational class of approaches for efficient neural inference in resource-constrained settings and large-scale neural deployment.