Achieving valid conditional coverage in conformal prediction is challenging due to the theoretical difficulty of satisfying pointwise constraints in finite samples. Building upon the characterization of conditional coverage through marginal moment restrictions, we introduce Minimax Optimization Predictive Inference (MOPI), a framework that generalizes prior work by optimizing over a flexible class of set-valued mappings during the calibration phase, rather than simply calibrating a fixed sublevel set. This minimax formulation effectively circumvents the structural constraints of predefined score functions, achieving superior shape adaptivity while maintaining a principled connection to the minimization of mean squared coverage error. Theoretically, we provide non-asymptotic oracle inequalities and show that the convergence rate of the coverage error attains the optimal order under regular conditions. The MOPI also enables valid inference conditional on sensitive attributes that are available during calibration but unobserved at test time. Empirical results on complex, non-standard conditional distributions demonstrate that MOPI produces more efficient prediction sets than existing baselines.

翻译：在共形预测中实现有效的条件覆盖具有挑战性，原因在于有限样本下满足逐点约束存在理论困难。基于通过边际矩约束刻画条件覆盖的思想，我们提出极小极大优化预测推断（MOPI）框架。该框架通过在校准阶段对灵活类别集值映射进行优化（而非简单校准固定子水平集），推广了先前研究工作。这种极小极大公式有效规避了预定义评分函数的结构限制，在保持与均方覆盖误差最小化原则性关联的同时，实现了更优的形状自适应性。理论上，我们提供了非渐近预言不等式，并证明在常规条件下覆盖误差的收敛速度达到了最优阶次。MOPI还能针对在校准时可用但测试时未观测的敏感属性进行有效条件推断。在复杂非标准条件分布上的实证结果表明，MOPI生成的预测集比现有基准方法更高效。