Price of metric universality in vector quantization is at most 0.11 bit

Fast computation of a matrix product W^top X is a workhorse of modern LLMs. To make their deployment more efficient, a popular approach is that of using a low-precision approximation widehat W in place of true W (``weight-only quantization''). Information theory demonstrates that an optimal algorithm for reducing precision of W depends on the (second order) statistics of X and requires a careful alignment of vector quantization codebook with PCA directions of X (a process known as ``waterfilling allocation''). Dependence of the codebook on statistics of X, however, is highly impractical. This paper proves that there exist a universal codebook that is simultaneously near-optimal for all possible statistics of X, in the sense of being at least as good as an X-adapted waterfilling codebook with rate reduced by 0.11 bit per dimension in the case when W is Gaussian. Such universal codebook would be an ideal candidate for the low-precision storage format, a topic of active modern research, but alas the existence proof is non-constructive. Equivalently, our result shows existence of a net in ℝn that is a nearly-optimal covering of a sphere simultaneously with respect to all Hilbert norms.