用于改进先知不等式的核方法

中文摘要 AI 辅助

单选择先知不等式是一种典型的贝叶斯在线选择问题，其中独立非负数值依次到达，决策者必须不可撤销地选择至多一个。经典的单阈值保证在最坏情况下是紧的，但证明其紧性的困难实例高度不规则：先知的优势由最大值的罕见、极大实现值驱动。我们通过对先知价值的相对方差施加约束来改进这种最坏情况图景，即Var(max_{i∈[n]}X_i)/E[max_{i∈[n]}X_i]^2。这产生了一种非参数复杂度度量，可在可恢复全部先知价值的确定性实例与无约束最坏情况机制之间进行插值。我们的主要技术贡献是一种用于单阈值先知不等式的通用核方法。该方法通过最大值的分位数函数表示实例，并将阈值的收益重写为该分位数的线性核泛函。这将最坏情况分析转化为无限维凸规划，恢复了分位数空间中的强极小极大对偶性，并将有界方差对手的问题简化为单参数变分族。应用该框架，我们获得了独立同分布（IID）有界方差曲线的精确刻画、渐近最优有限时间阈值、固定阶非同分布模型的闭式表达式，以及先知-秘书下界规划，且在每个正有限方差约束下与独立同分布基准存在严格分离。作为同一核视角的进一步应用，我们在时间跨度概率生成函数（pgf）的凸性条件下推导了独立同分布随机时间跨度的精确公式，其中包括单调风险率时间跨度，凸显了该单阈值设置新技术的广泛适用性。

英文摘要

The single-selection prophet inequality is a canonical Bayesian online selection problem in which independent nonnegative values arrive sequentially and the decision-maker must irrevocably select at most one. Classical single-threshold guarantees are tight in the worst case, but the hard instances that prove tightness are highly irregular: the prophet's advantage is driven by rare, very large realizations of the maximum. We refine this worst-case picture by imposing a bound on the relative variance of the prophet's value, $\mathrm{Var}(\max_{i\in[n]}X_i)/\mathbb E[\max_{i\in[n]}X_i]^2$. This yields a nonparametric complexity measure that interpolates between deterministic instances, where the full prophet value can be recovered, and the unrestricted worst-case regime. Our main technical contribution is a general kernel method for single-threshold prophet inequalities. The method represents an instance by the quantile function of the maximum and rewrites the payoff of a threshold as a linear kernel functional of this quantile. This turns the worst-case analysis into an infinite-dimensional convex program, restores strong minimax duality in quantile space, and reduces the bounded-variance adversary's problem to a one-parameter variational family. Applying this framework, we obtain an exact characterization of the IID bounded-variance curve and asymptotically optimal finite-horizon thresholds, a closed-form expression for the fixed-order non-identical model, and a prophet-secretary lower-bound program together with a strict separation from the IID benchmark at every positive finite variance constraint. As a further application of the same kernel viewpoint, we derive an exact formula for IID random horizons under a convexity condition on the horizon pgf, which includes monotone-hazard-rate horizons, highlighting the broad applicability of this new technique for single threshold settings.

发表机构

• Inria（法国国家信息与自动化研究所）
• Tel-Aviv University（特拉维夫大学）
• CREST（经济与统计研究中心）
• ENSAE（巴黎统计与经济管理学院）
• IP Paris（巴黎理工学院）
• CRITEO AI Team（Criteo人工智能团队）
• Rice University（莱斯大学）

机构由 AI 辅助整理，请以论文原文为准。