An $\alpha$-Law for Scale Multipliers in Low-Bit Post-Training Quantization

Preserving language-model accuracy below four bits per weight motivates understanding how quantizers allocate their limited representable values. We ask whether post-training quantizers with different optimization procedures choose scales that follow a shared, predictive relationship. We compare their choices through a per-group multiplier $\alpha$ that measures scale contraction relative to the group's largest weight magnitude. For AutoRound, GPTQ, and AWQ, its median follows ${\alpha }^{\ast }=q_{\max }/(q_{\max }+c)$, where $q_{\max }$ is the largest positive integer quantization code and $c$ is fitted per method and configuration. A symmetric clipped mean-squared-error approximation interprets this contraction through the trade-off between clipping and rounding error. Across seven models from six families at two, three, and four bits per weight, the law achieves in-sample $R^2>0.97$. Constants fitted on models with at most nine billion parameters predict the sampled median $\alpha$ of Llama-3.1-70B with absolute errors below 0.016 for AutoRound and 0.008 for GPTQ at two and three bits per weight. Among the four tested methods, larger fitted $c$ corresponds to higher two-bit accuracy averaged across the seven models. For Llama-3.1-70B quantized to two bits per weight, AutoRound and its ablation without rounding iterations have similar sampled median scale multipliers yet retain 91% and 45% of the half-precision reference model's ten-task mean accuracy. The $\alpha$-law reveals shared structure in method-specific scale choices and supports prediction across bit widths and, for AutoRound and GPTQ, to a held-out 70B model.