Fundamental dependence on the candidate-set size

Determine whether the linear dependence on the candidate-set size in the horizon-dependent regret bound of HE-CV-UCB is fundamental, or whether it can be improved to a dependence comparable to the square-root candidate-set dependence achieved by single-criterion elimination.

Background

The cumulative regret bound proved for HE-CV-UCB contains a factor proportional to the number of candidate surrogates, |Theta|, in its horizon-dependent terms. This is worse than the square-root dependence obtained for single-criterion elimination methods.

The additional factor arises from the two-sided elimination mechanism: optimistic and pessimistic candidates are tested separately, and an elimination on one side can create gaps in the residual sequence used by the other side. The paper does not establish whether this proof-dependent loss is unavoidable or whether a sharper analysis or alternative elimination procedure can reduce it.

References

Our regret bound also incurs a factor $|\Theta|$ due to the two-sided elimination procedure, compared to the $\sqrt{|\Theta|}$ dependence obtained for single-criterion elimination; whether this gap is fundamental remains open.

— Bayesian Optimization of CVaR with Unknown Hyperparameters  (2610.18306 - Margraf et al., 13 Oct 2026) in Section 6, paragraph “Limitations and future work”