Task-Oriented Quantization for Quadratic Scheduling: Centroid Water-Filling and Power-Diagram Encoders
Abstract
We distinguish two regimes in task-oriented quantization with a known deterministic oracle action. For an unconstrained interior oracle and a smooth strongly concave utility, quantizing the oracle action by vector Lloyd-Max minimizes a mean-squared-error surrogate and achieves a β/α approximation to the optimal K-level task quantizer. The reduction is exact for isotropic quadratic loss, and the corresponding task rate--distortion function is bracketed by two ordinary rate--distortion functions. Budget-constrained quadratic scheduling is different: the oracle satisfies a variational inequality, so quantizing water-filled actions is not generally optimal. We derive the exact Lloyd-type conditions for this case. The optimal action for a quantizer cell is water-filling evaluated at the cell's conditional-mean load, and the optimal encoder partitions load space into affine power-diagram cells. Thus the correct prescription is to quantize the load and water-fill its centroid. The distinction is material whenever a cell crosses water-filling active-set boundaries.
Related
- A positive solution to the Lp projection centroid conjecture
- Breaking the Fourth Wall: The Planar Centroid Banach--Mazur Diameter Is Strictly Less than $4$
- Same path, different walk: a mechanistic comparison of looped and stacked transformer encoders on 12-lead ECG
- Evaluating cross-encoders for semantic similarity assessment in psychological questionnaires
- It Takes Two Moments: Rigidity and Limits of Continuous Encoders for Over-the-Air Computation
BibTeX
@misc{armstrong2026taskoriented,
title = {{Task-Oriented Quantization for Quadratic Scheduling: Centroid Water-Filling and Power-Diagram Encoders}},
author = {Joss Armstrong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20882},
eprint = {2609.20882},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20882},
}
RIS
TY - UNPB
TI - Task-Oriented Quantization for Quadratic Scheduling: Centroid Water-Filling and Power-Diagram Encoders
AU - Joss Armstrong
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.20882
UR - https://arxiv.org/abs/2609.20882
AB - We distinguish two regimes in task-oriented quantization with a known deterministic oracle action. For an unconstrained interior oracle and a smooth strongly concave utility, quantizing the oracle action by vector Lloyd-Max minimizes a mean-squared-error surrogate and achieves a β/α approximation to the optimal K-level task quantizer. The reduction is exact for isotropic quadratic loss, and the corresponding task rate--distortion function is bracketed by two ordinary rate--distortion functions. Budget-constrained quadratic scheduling is different: the oracle satisfies a variational inequality, so quantizing water-filled actions is not generally optimal. We derive the exact Lloyd-type conditions for this case. The optimal action for a quantizer cell is water-filling evaluated at the cell's conditional-mean load, and the optimal encoder partitions load space into affine power-diagram cells. Thus the correct prescription is to quantize the load and water-fill its centroid. The distinction is material whenever a cell crosses water-filling active-set boundaries.
ER -