Prediction sets provide a means of quantifying the uncertainty in predictive tasks. Using held out calibration data, conformal prediction and risk control can produce prediction sets that exhibit statistically valid error control in a computationally efficient manner. However, in the standard formulations, the error is only controlled on average over many possible calibration datasets of fixed size. In this paper, we extend the control to remain valid with high probability over a cumulatively growing calibration dataset at any time point. We derive such guarantees using quantile-based arguments and illustrate the applicability of the proposed framework to settings involving distribution shift. We further establish a matching lower bound and show that our guarantees are asymptotically tight. Finally, we demonstrate the practical performance of our methods through both simulations and real-world numerical examples.

翻译：预测集合为量化预测任务中的不确定性提供了一种方法。利用预留的校准数据，共形预测与风险控制能够以计算高效的方式生成具有统计有效性误差控制的预测集合。然而，在标准框架中，误差控制仅在固定大小的多个可能校准数据集上平均成立。本文将该控制扩展至在任意时间点，对累积增长的数据集，以高概率保持有效性。我们使用基于分位数的论证推导此类保证，并说明所提框架在涉及分布偏移场景下的适用性。我们进一步建立了一个匹配的下界，并证明我们的保证是渐近紧的。最后，我们通过仿真和真实世界数值示例展示了所提方法的实际性能。