Anchor to the Receiver: Path-Dependent Communication of Quantized Updates

Quantized models deployed on many devices are adapted and re-delivered as new data arrives. When each receiver stores a single fixed-grid integer model, every update is a patch against the codes it already holds, and every code change is paid for when it is sent, including changes that a later update undoes. The cost of a stream of updates therefore depends on the path of delivered models, not only on its endpoint. We show that the reference a weight regularizer remembers determines this path. Anchoring to the sender's current quantization bin locks in boundary crossings before they are delivered, whereas anchoring to the installed model pulls delivered changes back. We propose QCTR, which anchors continuous sender weights to the receiver's last-delivered codes, advances the anchor only when a patch has been applied, and keeps the sender's within-cell residual. Changing only the reference lowers traffic at every regularization strength we test, by up to 59%. On a Llama-3.2-1B classification stream, QCTR reaches 87% accuracy with 3.5$\times$ less traffic than the strongest alternative. Over ten deliveries it sends 40% less than either alternative reference at similar accuracy and 10.8$\times$ less than straight-through fine-tuning. The advantage transfers to Llama-3.2-3B without retuning, holds on a generative stream, and composes with replay and low-rank updates.