Simulating Quantization Effects During Training

Post-Training Quantization (PTQ) can lead to accuracy degradation, particularly at lower bit depths. To mitigate this issue, Quantization-Aware Training (QAT) offers an alternative approach. Instead of quantizing a fully trained model, QAT introduces the effects of quantization during the training or fine-tuning process itself. The core idea is not to perform the entire training using low-precision arithmetic directly, which poses challenges for gradient calculation. Instead, QAT simulates the impact of quantization within the standard floating-point training framework.

The Role of Fake Quantization

The mechanism enabling this simulation is often called "fake quantization" or "simulated quantization". It involves inserting special nodes or operations into the model's computation graph during training. These fake quantization operations perform the following steps in the forward pass:

Quantize: They take the incoming floating-point tensor (representing either weights or activations) and apply the chosen quantization scheme (e.g., mapping the float values to an 8-bit integer range using calculated scale factors and zero-points).
Dequantize: Immediately after quantization, they convert the low-precision integer representation back to floating-point values, using the same scale factor and zero-point.

The output of a fake quantization node is therefore still a floating-point tensor. However, its values are now constrained; they represent only those specific floating-point numbers that can be exactly represented by the target low-precision data type (e.g., INT8).

Mathematically, if $x$ is the input floating-point tensor, the fake quantization process $x_{fq}$ can be represented as:

x_{fq} = \text{dequantize}(\text{quantize}(x, \text{scale}, \text{zero\_point}), \text{scale}, \text{zero\_point})

Here, $\text{quantize}$ maps $x$ to the low-precision domain (like INT8), and $\text{dequantize}$ maps it back to the floating-point domain. The point is that $x_{fq}$ contains the error or information loss inherent in the quantization process.

Forward Pass with Fake Quantization

During the forward pass of training, these fake quantization operations are placed strategically within the model architecture. Common locations include:

After weight tensors: Before they are used in operations like matrix multiplications (e.g., in Linear layers).
After activation functions: To simulate the quantization of intermediate results passed between layers.

The diagram below illustrates how a fake quant node alters the forward pass for an activation tensor flowing between two layers.

Flow comparing standard training and QAT forward pass. In QAT, the fake quantization node simulates the effect of quantization on the activations before they are passed to the next layer.

By performing calculations with these slightly altered, "quantization-error-aware" tensors ( $x_{fq}$ ), the model learns to adjust its weights during training. The optimization process (like Stochastic Gradient Descent) implicitly minimizes the loss function considering the precision limitations that will be present after true quantization. The model adapts its parameters to become more resilient to the noise and information loss introduced by the quantization process.

The Backward Pass Challenge

While the forward pass simulation is relatively straightforward, the backward pass presents a challenge. The quantization function itself (the mapping from float to integer) involves rounding or flooring operations, which have zero gradients almost everywhere. This non-differentiability means standard backpropagation cannot compute gradients through the quantize step within the fake quantization node.

To overcome this, QAT relies on an approximation technique for calculating gradients during the backward pass. This technique, known as the Straight-Through Estimator (STE), effectively allows gradients to bypass the non-differentiable quantization step, enabling the model's weights to be updated. We will examine the STE in detail in the next section.

In summary, simulating quantization effects during training via fake quantization nodes is the core mechanism of QAT. It allows the model to proactively adapt to the constraints of low-precision arithmetic, often leading to significantly better accuracy in the final quantized model compared to PTQ, especially for aggressive quantization (e.g., 4-bit).

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References

Quantization and Training of Neural Networks for Efficient Integer-Arithmetic-Only Inference, Benoit Jacob, Skirmantas Kligys, Bo Chen, Menglong Zhu, Matthew Tang, Andrew Howard, Hartwig Adam, Dmitry Kalenichenko, 2018 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE) DOI: 10.1109/CVPR.2018.00096 - A foundational paper introducing Quantization-Aware Training (QAT) and the concept of fake quantization for integer-only inference.
Learning binary weights for efficient implementation of deep neural networks, Matthieu Courbariaux, Yoshua Bengio, and Jean-Pierre David, 2015 Advances in Neural Information Processing Systems (NeurIPS) (Neural Information Processing Systems Foundation) DOI: 10.5591/978-1-55860-845-8.3123 - This paper popularizes the Straight-Through Estimator (STE) for training neural networks with low-precision weights, directly relevant to the backward pass challenge in QAT.
Quantization Recipes, PyTorch Authors, 2024 (PyTorch Foundation) - Official documentation providing practical implementation details for Quantization-Aware Training (QAT) in PyTorch, including the use of fake quantization modules.