NF4 Isn't Information Theoretically Optimal (and that's Good)

This note shares some simple calculations and experiments related to absmax-based blockwise quantization, as used in Dettmers et al., 2023. Their proposed NF4 data type is said to be information theoretically optimal for representing normally distributed weights. I show that this can’t quite be the case, as the distribution of the values to be quantized depends on the block-size. I attempt to apply these insights to derive an improved code based on minimizing the expected L1 reconstruction error, rather than the quantile based method. This leads to improved performance for larger quantization block sizes, while both codes perform similarly at smaller block sizes.