Acceleration for LMO-based methods on sets with constant smoothness

Ascertain whether any first-order methods that use a linear minimization oracle can achieve accelerated convergence rates when optimizing over β-smooth constraint sets S with β being a constant (independent of the target accuracy), beyond the non-accelerated guarantees established for the modestly smooth regime β = Ω(1/√ε).

Background

The authors extend span-based lower bounds to β-smooth sets via Minkowski sums and show that for modest smoothness (β = Ω(1/√ε)), no acceleration is possible relative to known optimal complexities for convex or strongly convex sets.

They explicitly leave unresolved whether stronger smoothness (constant β) permits acceleration for LMO-based methods, noting that preliminary numerical evidence does not clearly identify acceleration in small-iteration regimes.

References

We leave open whether acceleration is possible when sets possess constant levels of smoothness.

— Lower Bounds for Linear Minimization Oracle Methods Optimizing over Strongly Convex Sets  (2602.22608 - Grimmer et al., 26 Feb 2026) in Section “An Extension of Lower Bounds to Modestly Smooth Sets”

However, to the best of the author's knowledge, the gap between the existing achievable $(L/T2)$ rate and the performance limit $(L/T3)$ has not yet been closed in the literature. Closing this gap for general convex smooth optimization has been explicitly proposed as an open problem in .

— An $O(1/T^3)$ algorithm for minimizing convex quadratic functions over the $L_1$ ball  (2609.10314 - Ouyang, 9 Sep 2026) in Section 1, Introduction