Eliminate the logarithmic overhead for passing through the large-gradient regime

Determine whether the large-gradient regime for convex $(L_0,L_1)$-smooth functions can be traversed using $O(1+L_1R)$ first-order oracle calls without line search.

Background

For the radius-only class of convex (L0,L1)(L_0,L_1)-smooth functions, the paper establishes an optimal deterministic first-order oracle complexity of Θ~(1+L0R2/ε+L1R)\widetilde\Theta(1+\sqrt{L_0R^2/\varepsilon}+L_1R). The upper bound incurs a factor of order log⁡(2+L1R)\log(2+L_1R) in the L1RL_1R term because the SHERPA first stage uses the ICEPICK inexact line search, while the lower bound loses a factor of order log⁡(2+L1R)\sqrt{\log(2+L_1R)} through the construction of its chain.

References

We do not know whether the large-gradient regime can be passed in $O(1+L_1R)$ calls without line search.

— 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 “Logarithmic factors”