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Quantization

Aims to reduce the precision of the model weights and activations. Usually by changing the data type from floating point to integer.

Benefits:

• Reduced memory usage
• Faster inference
• Reduced model size

Types:

• Symmetric: same quantization for positive and negative values
• Asymmetric: different quantization for positive and negative values

Symmetric Quantization

Asymmetric Quantization

GPU acceleration

When computing $X*W+B$ each row of $X$ is multiplied by each column of $W$ and then added to $B$. The GPU can perform this computation in parallel for each row of $X$ and each column of $W$ by using Multiply-Accumulate (MAC) operations.

How to choose the range of the quantization?

• Use the minimum and maximum value of the weights and activations
• Use the mean-square error (MSE) of the weights and activations
• Cross-entropy

Code

Create a simple tensor with random items

import numpy as np

# Suppress scientific notation
np.set_printoptions(suppress=True)

# Generate randomly distributed parameters
params = np.random.uniform(low=-50, high=150, size=20)

# Make sure important values are at the beginning for better debugging
params[0] = params.max() + 1
params[1] = params.min() - 1
params[2] = 0

# Round each number to the second decimal place
params = np.round(params, 2)

# Print the parameters
print(params)

[135.27 -40.19   0.    71.58   0.15  89.11 -39.19 -12.81 -14.72 128.6
 -28.3  113.96 134.27 -25.94  36.49  53.55  24.1  133.    32.29  97.02]

Define the quantization methods and quantize

def clamp(params_q: np.array, lower_bound: int, upper_bound: int) -> np.array:
    params_q[params_q < lower_bound] = lower_bound
    params_q[params_q > upper_bound] = upper_bound
    return params_q

def asymmetric_quantization(params: np.array, bits: int) -> tuple[np.array, float, int]:
    # Calculate the scale and zero point
    alpha = np.max(params)
    beta = np.min(params)
    scale = (alpha - beta) / (2**bits-1)
    zero = -1*np.round(beta / scale)
    lower_bound, upper_bound = 0, 2**bits-1
    # Quantize the parameters
    quantized = clamp(np.round(params / scale + zero), lower_bound, upper_bound).astype(np.int32)
    return quantized, scale, zero

def asymmetric_dequantize(params_q: np.array, scale: float, zero: int) -> np.array:
    return (params_q - zero) * scale

def symmetric_dequantize(params_q: np.array, scale: float) -> np.array:
    return params_q * scale

def symmetric_quantization(params: np.array, bits: int) -> tuple[np.array, float]:
    # Calculate the scale
    alpha = np.max(np.abs(params))
    scale = alpha / (2**(bits-1)-1)
    lower_bound = -2**(bits-1)
    upper_bound = 2**(bits-1)-1
    # Quantize the parameters
    quantized = clamp(np.round(params / scale), lower_bound, upper_bound).astype(np.int32)
    return quantized, scale

def quantization_error(params: np.array, params_q: np.array):
    # calculate the MSE
    return np.mean((params - params_q)**2)

(asymmetric_q, asymmetric_scale, asymmetric_zero) = asymmetric_quantization(params, 8)
(symmetric_q, symmetric_scale) = symmetric_quantization(params, 8)

print(f'Original:')
print(np.round(params, 2))
print('')
print(f'Asymmetric scale: {asymmetric_scale}, zero: {asymmetric_zero}')
print(asymmetric_q)
print('')
print(f'Symmetric scale: {symmetric_scale}')
print(symmetric_q)

Original:
[135.27 -40.19   0.    71.58   0.15  89.11 -39.19 -12.81 -14.72 128.6
 -28.3  113.96 134.27 -25.94  36.49  53.55  24.1  133.    32.29  97.02]

Asymmetric scale: 0.6880784313725491, zero: 58.0
[255   0  58 162  58 188   1  39  37 245  17 224 253  20 111 136  93 251
 105 199]

Symmetric scale: 1.0651181102362206
[127 -38   0  67   0  84 -37 -12 -14 121 -27 107 126 -24  34  50  23 125
  30  91]

Dequantize the parameters back to 32 bits

params_deq_asymmetric = asymmetric_dequantize(asymmetric_q, asymmetric_scale, asymmetric_zero)
params_deq_symmetric = symmetric_dequantize(symmetric_q, symmetric_scale)

print(f'Original:')
print(np.round(params, 2))
print('')
print(f'Dequantize Asymmetric:')
print(np.round(params_deq_asymmetric,2))
print('')
print(f'Dequantize Symmetric:')
print(np.round(params_deq_symmetric, 2))

Original:
[135.27 -40.19   0.    71.58   0.15  89.11 -39.19 -12.81 -14.72 128.6
 -28.3  113.96 134.27 -25.94  36.49  53.55  24.1  133.    32.29  97.02]

Dequantize Asymmetric:
[135.55 -39.91   0.    71.56   0.    89.45 -39.22 -13.07 -14.45 128.67
 -28.21 114.22 134.18 -26.15  36.47  53.67  24.08 132.8   32.34  97.02]

Dequantize Symmetric:
[135.27 -40.47   0.    71.36   0.    89.47 -39.41 -12.78 -14.91 128.88
 -28.76 113.97 134.2  -25.56  36.21  53.26  24.5  133.14  31.95  96.93]

Calculate the quantization error

print(f'{"Asymmetric error: ":>20}{np.round(quantization_error(params, params_deq_asymmetric), 2)}')
print(f'{"Symmetric error: ":>20}{np.round(quantization_error(params, params_deq_symmetric), 2)}')

  Asymmetric error: 0.03
   Symmetric error: 0.06