Quantization basics

• https://newsletter.maartengrootendorst.com/p/a-visual-guide-to-quantization

Symmetric quantization

• Quantized value for zero = full precision zero
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Absolute maximum (absmax) quantization

• Given a list of values, we take the highest absolute value, $\alpha$, as the range to perform the linear mapping
  • e.g. FP32 → INT8
    • 
• $\nu$ = max representable value of your format
• scale factor $s=\frac{\nu }{\alpha }$
• $x_{\text{quantized}}=round(s\cdot x)$
• $x_{\text{dequantized}}=x_{\text{quantized}}/s$
• quantization error = $x-x_{\text{dequantized}}$ (in original precision)

Asymmetric quantization

• It maps the minimum ($\beta$) and maximum ($\alpha$) values from the float range to the minimum and maximum values of the quantized range.
• Example with zeropoint quantization
  • $\nu$ = max representable value of your format
  • $\eta$ = max representable value of your format
  • scale factor $s=\frac{\nu -\eta }{\alpha -\beta }$
  • zeropoint $z=round(-s\cdot \beta )+\eta$
  • $x_{\text{quantized}}=round(s\cdot x+z)$
    • e.g. $\alpha$ gets mapped to $\nu$
      • ${\alpha }_{\text{quantized}}=\alpha \cdot \frac{\nu -\eta }{\alpha -\beta }+round(-\frac{\nu -\eta }{\alpha -\beta }\cdot \beta )+\eta \approx (\alpha -\beta )\cdot \frac{\nu -\eta }{\alpha -\beta }+\eta =\nu$

Outliers

Clipping

• If your vector has an outlier, then using “naive” quantization can lead to most values being mapped to the same spot in the band

  • 

• We need to clip the outlier e.g. $[-5,5]$ in FP32

• How do you choose the clipping range?

  • For weights and biases

    • Manually choosing a percentile of the input
    • Optimize the mean squared error (MSE) between the original and quantized weights.
    • Minimizing entropy (KL-divergence) between the original and quantized values

  • For activations

    • Unlike weights, activations vary with each input data fed into the model during inference, making it challenging to quantize them accurately.