Establish necessity of the structural complexity term

Determine whether the structural search complexity term c_0 is necessary for identifying the best leaf in piecewise tree-Lipschitz multi-fidelity bandit instances.

Background

The paper proves a partial lower bound showing that expensive leaf-level certification is unavoidable, up to logarithmic factors, when a violation cell contains the optimum. However, the lower-bound construction does not establish a corresponding necessity result for c_0, the cost of structural localization through cheap probes over smooth cells. The authors explicitly leave that aspect unresolved.

References

The bound says nothing about the structural component $c_0$, whose necessity remains open.

— Canopy: Exploiting Piecewise Smooth Tree Priors for Multi-Fidelity Bandits  (2609.30017 - Jerge et al., 24 Sep 2026) in Proposition 1 (labelled prop:lower), Section 3, subsection “Guarantees”