Characterize the intermediate-dimensional complexity regime

Characterize the oracle complexity of convex $(L_0,L_1)$-smooth optimization in the intermediate regime where the dimension is neither fixed nor proportional to the number of oracle calls.

Background

The arbitrary-method lower bounds in the paper use dimensions proportional to the number of oracle calls, whereas the fixed-dimension appendix shows that, for fixed L0L_0, L1L_1, and normalized accuracy, the complexity grows only logarithmically with RR. Thus, the established results leave unresolved how oracle complexity interpolates between fixed dimension and dimension growing with the lower-bound chain length.

References

The intermediate regime has not been studied.

— Descending Everest with First-Order Oracles: Oracle Complexity of Convex Optimization under $(L_0,L_1)$-Smoothness  (2610.21461 - Rudomanov et al., 15 Oct 2026) in Section 5, Conclusion and open questions, bullet “Dimension”