Conformal prediction is a framework for providing prediction intervals with distribution-free validity, guaranteeing predictive coverage for data drawn from any distribution. Its two main variants are full conformal prediction and split conformal prediction (also called transductive and inductive). Full conformal prediction is widely considered to be statistically more efficient (since split conformal prediction requires data splitting, and therefore can lead to wider prediction intervals due to the resulting loss in sample size), but its implementation is computationally prohibitive, as it requires the underlying model to be refit for every candidate value in the response space. Existing computational shortcuts, such as using a discrete grid of values to approximate the full conformal prediction construction, frequently lack theoretical guarantees on marginal coverage and can fail in practice. To address this limitation, we introduce a novel class of approximations to the full conformal prediction method, based on the idea of \emph{tournaments}, which enables the construction of prediction sets with a rigorous marginal coverage guarantee of $1-2α$. Under stability conditions, the theoretical coverage guarantee tightens to approximately $1-α$. This new framework generalizes the existing method of leave-one-out cross-conformal prediction, while allowing for flexible use of various existing approximation strategies.

翻译：共形预测是一种提供预测区间的框架，具有无分布有效性，可保证对任意分布数据的预测覆盖性。其两种主要变体为全共形预测和分割共形预测（也称转换型和归纳型）。全共形预测被广泛认为在统计上更高效（因为分割共形预测需要数据分割，导致样本量损失从而产生更宽的预测区间），但其实现计算成本高昂，需针对响应空间中的每个候选值重新拟合基础模型。现有计算捷径（如使用离散值网格近似全共形预测构造）常缺乏边际覆盖率的理论保证，且在实际应用中可能失效。为克服这一局限，我们提出一类基于"竞赛"思想的全共形预测近似新方法，可构建具有严格边际覆盖率保证（$1-2α$）的预测集。在稳定性条件下，理论覆盖率保证可收紧至约$1-α$。该新框架泛化了现有的留一法交叉共形预测方法，同时允许灵活采用多种近似策略。