A Quantization Problem Posed by Adaptive Streaming

What the math gives to ML

The paper recasts a one-sided, threshold-based representation ladder as a nonstandard scalar quantizer: each cell reconstructs at its left edge, so the local error is first-order rather than squared-error. Its asymptotically optimal allocation places representation thresholds with density proportional to \(\sqrt{p(x)Q'(x)}\), producing an \(O(1/n)\) quality gap and an explicit constant. This transfers naturally to activation or weight quantization when downstream utility is monotone but reconstruction must be conservative, such as low-bit inference or thresholded routing. A practical implementation is a calibration-time quantizer whose codebook is obtained from the empirical activation density and a task-derived utility slope, rather than from standard MSE or uniform spacing.