🤖 AI Summary
研究了任务导向量化中，通过向量Lloyd-Max和质心注水法解决确定性oracle动作下的二次调度问题，提出了一种针对预算约束情况的有效编码方法。
📝 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.