Resolve the fixed-confidence versus full-tail minimax question

Determine whether an algorithm can be minimax optimal for fixed confidence and in expectation while failing to be minimax optimal along the entire confidence tail for finite hypothesis spaces.

Background

The paper distinguishes three notions of minimax optimality for finite dictionaries: optimality in expectation, optimality for a fixed input confidence level, and optimality along the whole tail without taking confidence as an input. It states that tail optimality implies the other notions, while the converse implications generally fail, but leaves one intermediate possibility unresolved.

References

Minimax optimality in expectation also does not imply minimax optimality for fixed confidence . Whether there are algorithms that are minimax optimal for fixed confidence and in expectation, but not along the tail, remains open.

— Reconciling Universal and Uniform Learning with $Q$-Aggregation  (2609.05041 - Høgsgaard et al., 4 Sep 2026) in Section 2, paragraph following Definition of minimax optimality for finite hypothesis spaces