Competitive On-line Interval Prediction

Interval predictions are widely studied in conformal prediction but have never been studied in competitive on-line prediction. Perhaps the reason is that it is very easy to achieve zero (or very small) loss with the loss function

$$
  \Lambda(\omega,\gamma)
  =
  \begin{cases}
    1 & \text{if $\omega\in\gamma$}\\
    0 & \text{otherwise}
  \end{cases}
$$

(make $\gamma$ as large as possible). It might be interesting to prove competitive on-line results for interval prediction considering the loss function

$$
  \lambda(\omega,\gamma)
  =
  \Lambda(\omega,\gamma)
  +
  L(\gamma),
$$

where $\Lambda$ is as defined above and $L(\gamma)$ is a measure of size of $\gamma$ (0 if $\gamma$ is a one-element set). It is likely that this will require randomized prediction algorithms (to make the loss function convex, as in the transition from the simple prediction game to the absolute loss game).