Post-Training Quantization Techniques

• Post-training quantization is a model optimization technique that converts FP32 networks to lower-bit integer representations, enhancing efficiency with minimal accuracy loss.
• It employs various operators, such as symmetric uniform quantization and error correction methods, to tailor quantization parameters for diverse neural architectures.
• Recent approaches integrate joint parameter tuning and loss-aware optimization to address quantization error and hardware constraints in low-bit deployments.

Post-training quantization (PTQ) is a technique in which a neural network, trained in full precision (usually FP32), is converted to a lower bit-width (such as 8, 4, or even 1–2 bits) representation for weights and/or activations after training, without using backpropagation or access to the original training data. The PTQ process aims to minimize the loss in accuracy resulting from quantization while maximizing efficiency in storage, memory, and compute—enabling deployment of large-scale networks on resource-constrained devices and accelerators. Modern PTQ methodologies address not only standard computer vision models but also LLMs, diffusion models, and BCI/classic pipelines. Advancements include sophisticated optimization of quantization parameters, joint parameter tuning, quantization error analysis, hardware-aware constraints, global prediction-based calibration, and provable error bounds.

1. Fundamental Principles and Quantization Operators

PTQ targets efficient model deployment by transforming pre-trained full-precision parameter tensors to lower-precision integer representations. The canonical quantization operator for a scalar $x$ with step size $\Delta$ and bit-width $M$ (using symmetric uniform quantization) is

$$Q_{\Delta,M}(x) = \begin{cases} -2^{M-1}\Delta & x < -2^{M-1}\Delta \ \mathrm{round}(x/\Delta)\cdot\Delta & |x| \leq 2^{M-1}\Delta \ +2^{M-1}\Delta & x > +2^{M-1}\Delta \end{cases}$$

For activations (e.g., in ReLU networks), the quantization range is typically $[0, c]$ with $\Delta = c/2^{M-1}$ (Nahshan et al., 2019).

Beyond basic symmetric quantization, PTQ has evolved toward methods such as:

• Piecewise Linear Quantization, which allocates more levels to dense regions of the value distribution (Fang et al., 2020).
• Cross-quantization, using per-row and per-column scaling to minimize the quantization kernel (set of elements mapped to zero) in LLM activations (Liu et al., 10 Oct 2024).
• Binary quantization with group-wise and Hessian-aware clustering, especially for extreme low-precision regimes (Song et al., 7 Apr 2025).
• Quantization with error correction or compensation, e.g., channel-wise affine post-compensation (Tang et al., 27 May 2025).

The choice of quantization operator and range estimation (e.g., channel-wise, layer-wise, global, percentile-based) critically affects the tradeoff between quantization error and hardware efficiency.

2. Loss Landscape, Parameter Optimization, and Joint Quantization

The structure of the loss landscape after quantization is central to PTQ performance. While high-bit quantization often produces a flat, separable loss surface—allowing independent layer-wise optimization with minimal cross-layer interaction—low-bit quantization (≤4 bits) introduces high curvature and cross-layer coupling: $$\Delta\mathcal{L} \approx (\nabla\mathcal{L})^T\epsilon + \tfrac{1}{2}\epsilon^T H\epsilon$$ Here, $H$ is the Hessian with respect to the quantization steps. The quadratic "Quantization Interaction Term" becomes dominant at low bitwidths, motivating joint optimization of quantization parameters across layers to account for interaction and curvature (Nahshan et al., 2019).

Modern PTQ frameworks—such as Loss-Aware PTQ (LAPQ) (Nahshan et al., 2019), QFT (Finkelstein et al., 2022), sensitivity-aware PTQ (Zheng et al., 6 Sep 2025), and others—solve or approximate the following optimization: $$\min_{\{\Delta_\ell\}} \mathcal{L}_{\text{PTQ}}(W_{\text{quant}}(\{\Delta_\ell\}))$$ where $W_\text{quant}$ are quantized weights as a function of per-layer or per-channel steps $\Delta_\ell$. Algorithmic strategies include:

• Layer-wise L$_p$-norm minimization with candidate selection
• Quadratic interpolation and gradient-free Powell optimization (Nahshan et al., 2019)
• End-to-end gradient-based joint finetuning of all quantization degrees-of-freedom, potentially with simulation graphs that enforce hardware-aware constraints (Finkelstein et al., 2022)
• Greedy path-following with provable error decay (e.g., GPFQ) (Zhang et al., 2022)
• Sensitivity-aware ordering guided by Taylor expansion and Hessian-based error estimation, quantizing high-sensitivity parameters first and updating shared Hessian inverses to accelerate computation (Zheng et al., 6 Sep 2025)

Bias correction, global affine compensation, and blockwise or groupwise parameter tuning further counteract systematic quantization error.

3. Calibration, Global Metrics, and Robustness to Data Shifts

Optimal calibration of quantization parameters is nontrivial, especially with limited data. While early PTQ methods focused on local metrics (MSE or cosine similarity between quantized and full-precision layer outputs), recent work has shifted toward global, task-driven metrics, such as:

• Minimizing prediction difference (PD) between quantized and full-precision model outputs (using metrics like KL divergence or task loss) (Liu et al., 2022).
• Employing global affine calibration to counter cumulative distributional shifts introduced by quantization and folding (especially after BatchNorm) (Zhu et al., 12 Jun 2025).
• Using sparsity-aware quantization to adaptively allocate bit budgets dynamically at the activation or group level (Shomron et al., 2021).

Robustness under distribution shift, data noise, or calibration set imbalance is insufficiently addressed by classic PTQ methods; average accuracy may remain stable while worst-case subgroup/class performance significantly degrades. Systematic evaluation frameworks have been proposed to benchmark and stimulate progress in PTQ reliability for real-world scenarios (Yuan et al., 2023).

4. Hardware-Aware Design and Deployment Considerations

PTQ methods increasingly target integer-only arithmetic, power-of-two thresholds, and symmetric quantization grids to maximize compatibility and efficiency on edge accelerators and specialized AI hardware (Habi et al., 2021). Typical requirements for hardware-friendly PTQ:

• Use of uniform, symmetric quantization (zero-point $z=0$), i.e., $Q(x) = s \cdot (x_\text{int})$
• Power-of-two thresholds ($t=2^M$), enabling scale factors to be implemented as shifts
• Per-channel quantization for weights, per-tensor or per-channel for activations
• Incorporation of compensation terms, e.g., channelwise affine scaling/offset modules absorptive in hardware quantization multipliers (Tang et al., 27 May 2025)

Hardware-friendly PTQ ensures that compensation, bias correction, and all arithmetic are applied in quantized or integer domain, avoiding floating-point or high-precision bottlenecks during inference.

Specialized frameworks for diffusion models (Ding et al., 10 Mar 2025), LLMs (focusing on activation quantization kernel size) (Liu et al., 10 Oct 2024), and video matting (Zhu et al., 12 Jun 2025) incorporate blockwise calibration, grouping, or custom residual suppression kernels and re-parameterization strategies, often with hardware-accelerated primitives such as leading zero suppression in activation quantization (Kim et al., 30 Sep 2025).

5. Error Analysis, Provable Guarantees, and Theoretical Underpinnings

Several PTQ algorithms have received rigorous theoretical analysis. For example, for greedy path-following and iterative rounding algorithms:

• The error in the quantized output $Xw - Xq$ can be tightly bounded in both $\ell_2$ and $\ell_\infty$ norm, in terms of calibration data, quantization step $\delta$, and, if applicable, the conditioning and ordering of features (Zhang et al., 2022, Zhang et al., 6 Aug 2025).
• For deterministic algorithms like OPTQ/GPTQ, the reconstruction error is controlled by cumulative rounding errors and the geometry of the projected feature space. Ordering features by descending norm can be formally justified (Zhang et al., 6 Aug 2025).
• Stochastic quantization variants offer stronger $\ell_\infty$ error control, preventing catastrophic outliers from propagating to downstream nonlinearities or layers.
• Theoretical support for model expansion techniques, in which post-training Hadamard transformations increase model capacity and the nullspace available for quantization error, has emerged as a novel co-design axis for balancing bit-width and model size (Franco et al., 21 Mar 2025).

Explicit penalization or error allocation to outlier and least significant bits preserves information-rich activations (e.g., via residual truncation and zero suppression in QuaRTZ (Kim et al., 30 Sep 2025)).

6. Empirical Performance, Limitations, and Future Directions

Comprehensive empirical results establish that state-of-the-art PTQ methods can achieve:

• $<2\%$ accuracy drop at 4-bit quantization on deep image classification backbones using loss-aware joint optimization (Nahshan et al., 2019)
• Robust quantization for LLMs, closing the performance gap with full precision when the quantization kernel for activations is kept below critical thresholds (Liu et al., 10 Oct 2024)
• FID scores on diffusion models with 4-bit PTQ matched or improved compared to methods requiring mixed-precision branches (Kim et al., 30 Sep 2025)
• Quantization time speedups up to $200\times$ with sensitivity-aware row-parallel methods, with negligible accuracy loss (Zheng et al., 6 Sep 2025)
• Significant storage and energy savings for portable BCI and edge vision applications with minimal accuracy degradation (Latotzke et al., 2022, Cecotti et al., 10 Oct 2024)

Major limitations include remaining performance drops in ultra-low bit regimes (≤2 bits), sensitivity to calibration set shifts, and increased error in the presence of nonlinear inter-layer dependencies. Open research challenges include:

• Efficient hybrid-precision allocation under fixed compute/memory budgets
• Universal PTQ pipelines for non-vision domains (e.g. video, BCI, LLMs, diffusion models)
• Further reduction of quantization kernel without retraining
• Designs for data- and task-agnostic robustness under adversarial and real-world deployment conditions.

This synthesis captures the principal developments, technical formulations, algorithmic strategies, and empirical evidence underlying post-training quantization in contemporary neural network deployment.