Necessity of the Hölder-curvature term below the half-Hölder threshold

Establish whether the additional term \(n^{1+\nu}T^{-2-2\nu}\) in the constant-stepsize, last-iterate convergence bound for random reshuffling is necessary when the average objective has a \(\nu\)-Hölder-continuous Hessian with \(0<\nu<\tfrac12\).

Background

The paper proves an upper bound for random reshuffling on smooth finite sums whose strongly convex average objective has a $\nu$-Hölder-continuous Hessian. Besides the terms $T^{-2}$ and $n^2T^{-3}$, the bound contains the curvature contribution $n^{1+\nu}T^{-2-2\nu}$. When $\nu\ge\tfrac12$, this term is dominated by the quadratic-instance term, and existing quadratic lower bounds establish sharpness in the $(n,K)$-dependence for constant stepsizes and the raw last epoch iterate.

For $\nu<\tfrac12$, the curvature contribution is not dominated by $n^2T^{-3}$. The paper explicitly leaves unresolved whether this term reflects an actual worst-case obstruction or is merely an artifact of the analysis; a matching lower bound or an improved upper bound would resolve the issue.