Optimal K-dependence in training-conditional lower bounds for full conformal prediction

Determine the optimal dependence on the number of intervals K in training-conditional lower bounds for full conformal prediction over the algorithm class P_K, where each prediction set is a union of at most K intervals. Specifically, establish the tight K-scaling (as a function of K and sample size n) for the worst-case training-conditional coverage error in the offline regime, beyond the current bound of order \(\widetilde{\Omega}(\min\{1/\sqrt{K},\,1/\sqrt{n}\})\).

Background

In the offline consequence of the online lower-bound analysis, the authors establish a baseline training-conditional lower bound for full conformal prediction when prediction sets are unions of at most K intervals (the class P_K).

They prove a general result (Proposition 4) that any algorithm in P_K suffers worst-case training-conditional coverage error at least on the order of $$\widetilde{\Omega}(\min\{1/\sqrt{K},\,1/\sqrt{n}\})$$.

They note that this result characterizes a baseline dependence on K but does not settle the optimal K-scaling; identifying the exact optimal K-dependence remains unresolved.