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Xinyu Song

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Preprint Aug 2026

Pre-Disclosure Experiment Menus: Oracle-Relative Risk and Joint Sample--Menu Asymptotics

We study a resolution problem in local asymptotic decision theory: individual risks may admit Gaussian approximations that do not determine their vanishing difference. A finite menu of experiments is installed before context disclosure, although observations may be routed adaptively afterward. A greatest-element Blackwell order collapses adaptive routing to the best installed experiment and reduces the fixed-menu excess to an inverse-information distortion with frontier $A_k$. We develop differentiated, all-prior posterior transfer along a one-dimensional degradation chain and establish $F_{n,k_n}(H_n)=A_{k_n}\{1+o(1)\}$ for every diverging menu sequence with positive frontier and every admissible localization radius, without an additional direct sample-menu restriction. The transfer is exact under Gaussian degradation. Prior-free likelihood-generator conditions imply it for jump generators and are verified for binary attenuation, Poisson thinning, and negative-binomial thinning. If the distortion is uniformly quadratic on an Ahlfors-regular oracle image of dimension $r$, then $A_k\asymp k^{-2/r}$, and the original-scale excess mean squared error is of order $n^{-1}k^{-2/r}$. Calibrated Poisson sensor and radial-qubit measurement menus illustrate the result. A triangular Gaussian counterexample shows why pointwise Gaussian convergence is insufficient.

Xinyu Song · 0 citations

Preprint Aug 2026

Batched and Complete U-Statistics for Trace-Polynomial Estimation from Classical Shadows

We study estimation of the trace polynomial $\operatorname{tr} p(P\rho P)$ from global classical shadows, where $\rho$ is an unknown quantum state and $P$ is a fixed projector. Disjoint batching and complete U-statistics yield unbiased estimators of the same trace moments, but assign different sample-size factors to the degenerate terms in their Hoeffding decompositions. Under the global Clifford protocol, exact degree-two variance formulas show that, on a null projected block of rank $s$, the quadratic degenerate term has order $s^2/N$ under batching and $s^2/N^2$ under complete symmetrization. For a logarithmic-degree polynomial used in entropy approximation, the quadratic coefficient raises the batched variance to at least order $s^2N\log^2N$ at the classical entropy cutoff. For complete U-statistics, we derive a cross-degree covariance identity and an exact variance decomposition for polynomial estimators. We also bound every Hoeffding order at a fixed degree and obtain a growing-dimensional risk bound for a small-spectrum entropy functional. The higher-order bounds retain a polynomial dependence on the ambient dimension and therefore do not cover logarithmically increasing degrees. Monte Carlo experiments confirm the degree-two formulas, and exact calculations illustrate the entropy risks.

Xinyu Song · 0 citations