Group-wise post-training quantizers for large language models round weights onto a grid that is not refit to the resulting integer codes.
We show that this leaves accuracy on the table: the best grid depends on the codes, input correlations couple the errors of different groups, and useful code changes often involve many codes at once.
We propose JARQ, a plug-in refinement that starts from any group-wise quantizer and alternates a joint least-squares fit of all group scales with bounded Babai proposals that move many codes of a group together on the current grid.
The problem is a bilinear box-constrained mixed-integer least-squares problem; the solver is backpropagation-free, does not increase the layer-wise objective under exact scale solves, and keeps the host's bit width, groups, zero points, and inference cost.
Across Llama-2, Llama-3, and Qwen models with RTN, GPTQ, OmniQuant, and AWQ hosts, JARQ lowers perplexity in 90 of 96 comparisons, cuts three-bit RTN perplexity by up to 36%, raises mean multiple-choice accuracy in 23 of 24 configurations, and improves QEP, QuaRot, and OJBKQ outputs, at under a minute per 7B block.