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.