# Task-Oriented Quantization for Quadratic Scheduling: Centroid Water-Filling and Power-Diagram Encoders

arXiv: https://arxiv.org/abs/2609.20882 · HTML: https://mlsanity.com/paper/2609.20882/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 9 references that are in the map.
- First posted on arXiv 16 Sep 2026.

## Builds on (most cited first, 4 of 4)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Beyond Transmitting Bits: Context, Semantics, and Task-Oriented Communications](https://mlsanity.com/paper/2207.09353/) (2022)
- [Goal-Oriented Quantization: Analysis, Design, and Application to Resource Allocation](https://mlsanity.com/paper/2209.15347/) (2022)
- [Goal-oriented compression for Lp-norm-type goal functions: Application to power consumption scheduling](https://mlsanity.com/paper/2405.07808/) (2024)

## Abstract (arXiv)

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.