Compress Now, Contaminate Later: One Quantizer for Every Huber Level

In binary hypothesis testing with compressed observations, the encoder may have to be fixed before anyone knows a bound on the contamination. We study this setting where the decoder learns a common Huber contamination bound only after encoding, and can then choose its test but not undo the encoder's merges. For known finite laws, $O(1+\log (1/J))$ output symbols suffice to keep a constant fraction of robust Hellinger information at every feasible level, where $J$ is comparable to clean squared Hellinger distance. We also characterize the cost of using fewer symbols, which can be steep. On a geometric family with $L$ evidence scales, the best uniform sample overhead ${\mathcal{R}}_K$ satisfies $\log (1+{\mathcal{R}}_K)\asymp 1+L/K$, even though a bit selected separately for each level has constant overhead. An adjustable-growth version of our construction attains this order. The exponential penalty applies to one identical memoryless channel where a known schedule of different one-bit encoders has $O(L)$ overhead on the same family. Learning the encoder from clean observations is harder still. No finite training budget depending only on clean difficulty preserves every raw-feasible level with a fixed output budget. A fitted test does succeed above an estimation-error threshold, and independent calibration certifies tests through simultaneous moment bounds. Experiments measure the alphabet–information trade-off, compare designs that target clean information, and quantify what certification costs.