Universal Refinement without Interaction: Order-Optimal 1-Bit Mean Estimation

What the math gives to ML

The paper develops a decoder-side refinement principle: all one-bit queries are fixed before the decoder learns a coarse estimate, yet the same stored bits can later be reinterpreted relative to that estimate. Its periodic-residue construction and shifted random grids produce bounded, correlated measurements whose bias and second moment remain controlled under only a finite k-th central moment, including heavy-tailed regimes with k≤2. A direct neural-network transfer is communication-efficient distributed or federated optimization: workers transmit a non-adaptive one-bit block for gradients, while the server first localizes the average gradient and then selects the appropriate refinement interpretation. The resulting quantizer is especially relevant when gradient coordinates are heavy-tailed and ordinary sign or fixed-range quantization has unstable variance.