Understanding Scale and Zero Points in Quantization – Part 2

In the first article, we introduced the basics of quantization in deep learning. We discussed its importance and various methods. This article will delve deeper into two fundamental aspects Scale and Zero Points in Quantization.

These elements play a crucial role in the quantization process, affecting both the efficiency and accuracy of the models.

Understanding scale and zero points is essential for implementing effective quantization. They determine how floating-point numbers are mapped to integers and back.

This mapping is key to achieving the balance between model size and performance.

Get the Scale and the Zero Point

Definitions and Roles in Quantization

The quantization operation adjusts the values based on the calculated scale and zero point. Proper quantization parameters are essential for keeping the model accurate

Scale and zero points are fundamental in the quantization process. The scale determines how much the original floating-point numbers are compressed. The zero point adjusts the range of the quantized values.

In quantization, the scale converts floating-point numbers to a smaller range. The zero point shifts this range to ensure it covers the necessary values. Together, they ensure that the quantized model maintains its accuracy and efficiency.

Proper calculation of scale and zero points is crucial. Incorrect values can lead to significant errors in the quantized model, affecting its performance. Therefore, understanding these elements and their roles is essential for any developer working with quantized models.

Mathematical Explanation

The scale is calculated by finding the range of the original tensor values and the range of the quantized values. Here’s the formula:

The zero point adjusts the minimum value of the quantized range to match the original data. Here’s the formula:

These calculations help map the floating-point numbers to the quantized range effectively.

Practical Applications of Scale and Zero Points

Understanding scale and zero points is not just theoretical; it has practical applications in various fields. For instance, in edge computing, where AI models run on devices with limited resources, efficient quantization is crucial.

By properly calculating scale and zero points, developers can deploy robust models on smartphones, IoT devices, and embedded systems.

In addition, scale and zero points play a vital role in cloud-based AI services. Optimizing model size and performance can lead to significant cost savings regarding storage and computational power.

This is especially important for large-scale applications, such as image and speech recognition systems, where models need to process vast amounts of data quickly and efficiently.

Understanding these concepts also helps in enhancing the performance of real-time applications.

For example, autonomous vehicles rely on fast and accurate AI models to make split-second decisions. Properly quantized models can process sensory data more efficiently, improving the vehicle’s response time and safety.

Code Implementation: Calculating Scale and Zero Point

Now, let’s implement the calculation of scale and zero points in PyTorch. This example will guide you through the process.

First, import the necessary libraries and define a dummy tensor:

import torch
# Dummy tensor for testing
test_tensor = torch.tensor([
    [191.6, -13.5, 728.6],
    [92.14, 295.5, -184],
    [0, 684.6, 245.5]
])

Next, find the scale and zero point:

# Define quantization range for int8
q_min = torch.iinfo(torch.int8).min
q_max = torch.iinfo(torch.int8).max
# Find the range of the tensor
r_min = test_tensor.min().item()
r_max = test_tensor.max().item()
# Calculate scale
scale = (r_max - r_min) / (q_max - q_min)
print(f'Scale: {scale}')

# Calculate zero point
zero_point = q_min - (r_min / scale)
zero_point = int(round(zero_point))
print(f'Zero Point: {zero_point}')

Let’s put all of this into a function for reusability:

def get_q_scale_and_zero_point(tensor, dtype=torch.int8):
    q_min, q_max = torch.iinfo(dtype).min, torch.iinfo(dtype).max
    r_min, r_max = tensor.min().item(), tensor.max().item()

    # Calculate scale
    scale = (r_max - r_min) / (q_max - q_min)

    # Calculate zero point
    zero_point = q_min - (r_min / scale)

    # Clip zero point to be within the quantized range
    zero_point = max(min(int(round(zero_point)), q_max), q_min)

    return scale, zero_point

# Test the function
new_scale, new_zero_point = get_q_scale_and_zero_point(test_tensor)
print(f'New Scale: {new_scale}, New Zero Point: {new_zero_point}')

Common Pitfalls and How to Avoid Them

When calculating scale and zero points, several common pitfalls can occur. One mistake is not properly handling the range of the tensor values. If the range is not accurately calculated, the resulting scale and zero point can lead to significant quantization errors.

Finally, rounding errors can occur when converting floating-point values to integers. To minimize these errors, always use the ‘round’ function and ensure that the resulting zero point is cast to an integer.

By being aware of these pitfalls and following best practices, you can ensure accurate and efficient quantization of your models.

Quantization and Dequantization with Calculated Scale and Zero Point

In linear quantization, the scale and zero point are crucial to mapping the original values. The factor and zero point help in transforming floating point numbers into an integer range. Once we have the scale and zero point, we can quantize and dequantize the tensor. Here’s how to do it using the calculated values:

First, define the functions for quantization and dequantization:

def linear_q_with_scale_and_zero_point(tensor, scale, zero_point, dtype=torch.int8):
    # Scale and shift the tensor
    scaled_tensor = tensor / scale + zero_point
   
    # Round and clamp the values to stay within the range of the dtype
    q_min, q_max = torch.iinfo(dtype).min, torch.iinfo(dtype).max
    quantized_tensor = torch.round(scaled_tensor).clamp(q_min, q_max).to(dtype)
   
    return quantized_tensor

def linear_dequantization(quantized_tensor, scale, zero_point):
    # De-quantize the tensor
    dequantized_tensor = scale * (quantized_tensor.float() - zero_point)
   
    return dequantized_tensor

Next, apply these functions to the test tensor:

# Quantize the tensor
quantized_tensor = linear_q_with_scale_and_zero_point(test_tensor, new_scale, new_zero_point)
print(f'Quantized Tensor:\n{quantized_tensor}')
# De-quantize the tensor
dequantized_tensor = linear_dequantization(quantized_tensor, new_scale, new_zero_point)
print(f'De-quantized Tensor:\n{dequantized_tensor}')

Finally, let’s plot the quantization errors to visualize the accuracy:

import matplotlib.pyplot as plt
def plot_quantization_errors(original_tensor, quantized_tensor, dequantized_tensor):
    # Calculate the quantization error
    quantization_error = (dequantized_tensor - original_tensor).abs()
    # Plot the errors
    fig, ax = plt.subplots()
    cax = ax.matshow(quantization_error, cmap='viridis')
    fig.colorbar(cax)
    plt.title('Quantization Error')
    plt.show()
# Plot the quantization errors
plot_quantization_errors(test_tensor, quantized_tensor, dequantized_tensor)

# Calculate mean squared error
mse = (dequantized_tensor - test_tensor).square().mean().item()
print(f'Mean Squared Error: {mse}')

Quantization converts data types from a floating point range to a signed int8 range, which allows models to use less memory while maintaining performance.

This implementation shows how to quantize and dequantize a tensor using a calculated scale and zero points. It also helps visualize the quantization error, providing insights into the precision of the quantized model. 

Comparing Quantization Methods

Linear quantization is just one method among many. Other methods, like logarithmic and dynamic quantization, offer different trade-offs and use cases.

• Logarithmic Quantization: Using a logarithmic scale, this method maps floating-point numbers to integers. It’s useful for data that spans several orders of magnitude. However, implementing it can be more complex and may introduce non-linear errors.
• Dynamic Quantization: This method dynamically adjusts the scale and zero points during model inference. It’s particularly useful for models with variable input data. While it can offer higher accuracy, it requires more computational resources during inference.

Each method has its advantages and disadvantages. The choice depends on the specific requirements of your application, such as the need for speed, accuracy, or resource efficiency.

Put Everything Together: Your Own Linear Quantizer

Now, let’s combine everything into a single linear quantizer function. This function will calculate the scale and zero point, quantize the tensor, and then dequantize it. Linear quantization is the process of mapping real numbers, often represented as floating point numbers, into discrete integer values to optimize model performance.

First, define the linear quantization function:

def linear_quantization(tensor, dtype=torch.int8):
    # Get scale and zero point
    scale, zero_point = get_q_scale_and_zero_point(tensor, dtype=dtype)
   
    # Quantize the tensor
    quantized_tensor = linear_q_with_scale_and_zero_point(tensor, scale, zero_point, dtype=dtype)
   
    return quantized_tensor, scale, zero_point

Choosing the right quantization scheme depends on the type of quantizer you’re using and the specific needs of your machine learning model.

Next, test this function on a random matrix:

# Generate a random tensor
random_tensor = torch.randn((4, 4))
print(f'Original Random Tensor:\n{random_tensor}')

# Quantize the random tensor
quantized_tensor, scale, zero_point = linear_quantization(random_tensor)
print(f'Quantized Random Tensor:\n{quantized_tensor}')
print(f'Scale: {scale}, Zero Point: {zero_point}')

# De-quantize the random tensor
dequantized_tensor = linear_dequantization(quantized_tensor, scale, zero_point)
print(f'De-quantized Random Tensor:\n{dequantized_tensor}')

# Plot the quantization errors
plot_quantization_errors(random_tensor, quantized_tensor, dequantized_tensor)

# Calculate mean squared error
mse = (dequantized_tensor - random_tensor).square().mean().item()
print(f'Mean Squared Error: {mse}')

This function integrates the calculation of scale and zero point, quantization, and dequantization. Testing it on a random tensor demonstrates its effectiveness. Once the scale and zero point are applied, you can now test the quantized model to evaluate how well the model quantization has reduced memory and maintained performance.

Final thoughts

In this article, we explored the importance of scale and zero points in quantization. We discussed their definitions, roles, and mathematical foundations. Understanding these elements is crucial for effective quantization, as they impact both model efficiency and accuracy.

We also implemented functions to calculate scale and zero points. Using these values, we demonstrated the processes of quantization and dequantization in PyTorch. We provided a practical example of their application by integrating these concepts into a single linear quantizer function.

Grasping the concepts of scale and zero points is vital for optimizing machine learning models. These techniques help balance performance and resource efficiency.

In the next article, we will delve into symmetric and asymmetric quantization modes, exploring their definitions, advantages, and practical use cases.

Stay tuned as we continue our deep dive into the world of quantization in deep learning.