精确极小极大一位无偏压缩：重尾必要性与有限随机性逼近
Exact Minimax One-Bit Unbiased Compression: Heavy-Tail Necessity and Finite-Randomness Approximation
- Key Laboratory of System Software (Chinese Academy of Sciences)（系统软件重点实验室（中国科学院））
- State Key Laboratory of Computer Science（计算机科学国家重点实验室）
- Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所）
- School of Computer Science and Technology, University of Chinese Academy of Sciences（中国科学院大学计算机科学与技术学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究证明一位无偏压缩的极小极大二阶矩下确界，揭示高斯极小极大最优性需重尾分布，并提出有限随机性逼近方案，在优化中匹配理论下界。
AI中文摘要：
逐点无偏的一位压缩器在期望意义上重建每个实数输入的同时仅传输一位。对于具有累积分布函数 $F$、均值 $m$ 和 $\mathcal J(P)=\int_{\mathbb R}\sqrt{F(r)(1-F(r))}\\,dr$ 的标量源 $P$，我们证明在所有在 $\mathbb R$ 上无偏的公共随机一位码上，源平均重建二阶矩的下确界为 $m^2+\mathcal J(P)^2$。对于正则全支撑源，一种以分布为中心的随机阈值码达到该值；对任意随机二元编码器的逆命题以及等式分析刻画了所有达到该值的码，直至零集、位重标记和公共种子细化。对于具有 $|\mu|\le c\sigma$ 的高斯位置族 $\mathcal N(\mu,\sigma^2)$，端点均值上的等先验是最不利的，极小极大值为 $\sigma^2\Lambda_c^2$。精确的高斯极小极大最优性强制要求临界重尾：在端点均值处，绝对矩仅在 $p<3$ 时有限，且 $\Pr(W>t)=\Theta(t^{-3}/\sqrt{\log t})$。柯西混合鲁棒化将二阶矩最多膨胀 $1/(1-\eta)$ 倍，同时使所有正阶绝对矩有限。具有有限解码器均值的有限支撑公共随机性无法在 $\mathbb R$ 上实现精确无偏性，但使用恰好 $R$ 个共享随机位的有界输出逼近具有显式偏差和二阶矩界，收敛到极小极大常数。最后，坐标分配在每次高斯梯度查询时精确通信 $B$ 位。在 Kim 的连续二次困难族上，期望优化保证在 $(\sigma,d,B,\varepsilon)$ 的依赖关系上与下界匹配，且有限方差高概率界仅产生对数置信因子。
英文摘要：
A pointwise-unbiased one-bit compressor reconstructs every real input in expectation while transmitting one bit. For a scalar source $P$ with CDF $F$, mean $m$, and $\mathcal J(P)=\int_{\mathbb R}\sqrt{F(r)(1-F(r))}\,dr$, we prove that the infimum of the source-averaged reconstruction second moment over all public-coin one-bit codes unbiased on $\mathbb R$ is $m^2+\mathcal J(P)^2$. For regular full-support sources, a distribution-centered random-threshold code attains this value; a converse over arbitrary randomized binary encoders and an equality analysis characterize every attaining code up to null sets, bit relabeling, and public-seed refinement. For the Gaussian location family $\mathcal N(μ,σ^2)$ with $|μ|\le cσ$, the equal prior on the endpoint means is least favorable and the minimax value is $σ^2Λ_c^2$. Exact Gaussian minimax optimality forces a critical heavy tail: at the endpoint means, absolute moments are finite exactly for $p<3$, and $\Pr(W>t)=Θ(t^{-3}/\sqrt{\log t})$. A Cauchy-mixture robustification inflates the second moment by at most $1/(1-η)$ while making every positive-order absolute moment finite. Finite-support public randomness with finite decoder means cannot achieve exact unbiasedness on $\mathbb R$, but a bounded-output approximation using exactly $R$ shared random bits has explicit bias and second-moment bounds converging to the minimax constant. Finally, coordinate allocation communicates exactly $B$ bits per Gaussian-gradient query. On Kim's continuous quadratic hard family, the expected optimization guarantee matches the lower bound in its dependence on $(σ,d,B,\varepsilon)$, and a finite-variance high-probability bound incurs only a logarithmic confidence factor.
