Optimal high-probability lower bound and sample complexity

Determine the matching lower bound for the optimal $(\varepsilon,\delta)$ sample complexity under monotone insertions, complementing the high-probability upper-bound transfer obtained from the clean setting.

Background

The paper observes that its almost-sure insertion-stability reduction transfers clean high-probability error guarantees to the monotone-insertion model without additional loss. This addresses the upper-bound side of the optimal $(\varepsilon,\delta)$ sample-complexity question posed in prior work.

The corresponding lower bound matching the optimal high-probability sample complexity has not been established. Thus, although high-probability guarantees are available whenever clean guarantees are known, the exact sample complexity under monotone insertions remains unresolved.