在线置信区间生成中，如何在保证覆盖率的同时最小化区间长度？

We study the problem of conformal prediction in a novel online framework that directly optimizes efficiency. In our problem, we are given a target miscoverage rate $α> 0$, and a time horizon $T$. On each day $t \le T$ an algorithm must output an interval $I_t \subseteq [0, 1]$, then a point $y_t \in [0, 1]$ is revealed. The goal of the algorithm is to achieve coverage, that is, $y_t \in I_t$ on (close to) a $(1 - α)$-fraction of days, while maintaining efficiency, that is, minimizing the average volume (length) of the intervals played. This problem is an online analogue to the problem of constructing efficient confidence intervals. We study this problem over arbitrary and exchangeable (random order) input sequences. For exchangeable sequences, we show that it is possible to construct intervals that achieve coverage $(1 - α) - o(1)$, while having length upper bounded by the best fixed interval that achieves coverage in hindsight. For arbitrary sequences however, we show that any algorithm that achieves a $μ$-approximation in average length compared to the best fixed interval achieving coverage in hindsight, must make a multiplicative factor more mistakes than $αT$, where the multiplicative factor depends on $μ$ and the aspect ratio of the problem. Our main algorithmic result is a matching algorithm that can recover all Pareto-optimal settings of $μ$ and number of mistakes. Furthermore, our algorithm is deterministic and therefore robust to an adaptive adversary. This gap between the exchangeable and arbitrary settings is in contrast to the classical online learning problem. In fact, we show that no single algorithm can simultaneously be Pareto-optimal for arbitrary sequences and optimal for exchangeable sequences. On the algorithmic side, we give an algorithm that achieves the near-optimal tradeoff between the two cases.