矩模糊性与鲁棒随机优化的极限

中文摘要 AI 辅助

我们研究了当分布仅通过其精确矩序列已知时，鲁棒随机优化的基本信息论极限。我们开发了一个统一框架，该框架生成具有相同所有矩但导致截然不同最优决策的分布族，从而为矩模糊性下的一系列决策问题建立了强不可能性结果。我们的方法给出了两个显式构造：一个二元构造和一个$N$元构造，表明具有相同矩序列的分布可以表现出任意不同的分位数、顺序统计量和阈值区域，迫使产生不兼容的最优行动。这些分布族实际上在多个随机优化问题中产生了强不可能性结果。首先，对于报童问题，矩等价导致分位数模糊性，使得任何固定或随机订购量都会任意糟糕地失败。其次，对于收益最大化问题，没有任何确定性或随机定价方案能够相对于完全信息基准保证非平凡的近似比。第三，对于具有基数观测的秘书问题，在所有精确矩披露下的最坏情况鲁棒值恰好是经典的$1/e$有限时域值，而不是Gilbert和Mosteller（J. Am. Stat. Assoc., 1966）所提出的具有完全信息的$0.58$成功概率的著名结果。我们还恢复并扩展了Correa等人（STOC, 2026）关于具有矩知识的先知不等式的近期不可能性结果。事实上，精确矩知识最多只能产生$\Theta(1/\log n)$的竞争比，即使与基于期望$r$阶统计量的宽松基准竞争，或者允许算法选择$r$个物品时也是如此。

英文摘要

We study fundamental information-theoretic limits of robust stochastic optimization when the distribution is known only through its exact moment sequence. We develop a unified framework that produces families of distinct distributions sharing all moments yet inducing radically different optimal decisions, thereby establishing strong impossibility results for a range of decision problems under moment ambiguity. Our approach gives two explicit constructions: a binary and an $N$-way construction showing that distributions with identical moment sequences can nevertheless exhibit arbitrarily different quantiles, order-statistics and threshold regions, forcing incompatible optimal actions. These families of distributions yield, in fact, strong impossibility results across several stochastic optimization problems. First, for the newsvendor problem, moment equivalence causes quantile ambiguity, inducing any fixed or randomized order quantity to fail arbitrarily badly. Second, for revenue maximization, no deterministic or randomized posted pricing scheme can secure a nontrivial approximation relative to the full-information benchmark. Third, for the secretary with cardinal observations setting, the worst-case robust value over all exact moment disclosures is exactly the classical $1/e$ finite-horizon value as opposed to the celebrated result of $0.58$ success probability with full-information by Gilbert and Mosteller (J. Am. Stat. Assoc., 1966). We also recover and expand upon the recent impossibility result of Correa et al. (STOC, 2026) for prophet inequalities with moment knowledge. Indeed, exact moment knowledge can yield at best a $Θ(1/\log n)$ competitive ratio, even when competing against relaxed benchmarks based on expected $r$-th order statistics or when the algorithm is allowed to select $r$ items.

发表机构

• EPFL（洛桑联邦理工学院）

机构由 AI 辅助整理，请以论文原文为准。