Determine the optimal worst-case objective-gap bound

Determine the optimal worst-case objective-gap bound for exact Euclidean ball-proximal point trajectories, including the asymptotically sharp dependence on \(S=D_0^2/t^2\) and the iteration horizon \(K\), beyond the Jensen estimate and the stricter implicit final-distance refinement.

Background

For a constant radius tt, the paper derives the Jensen objective-gap estimate ΔK/Δ0≤((S−K)/(S+K))K\Delta_K/\Delta_0\le ((S-K)/(S+K))^K, where S=D02/t2S=D_0^2/t^2. This estimate is sharp for the scalar relaxation used in its derivation.

A preceding remark shows that incorporating the final gap-distance relation yields a strictly smaller implicit factor for fixed SS and KK. However, it is not established whether that implicit factor is optimal among actual convex ball-proximal trajectories, nor what the asymptotically sharp dependence on SS and KK should be.

References

The second concerns the optimal worst-case objective-gap bound for . The Jensen estimate eq:jensen is sharp for its scalar relaxation, but \Cref{rem:jensen-final-distance} shows that retaining the gap--distance relation gives a strictly smaller factor for fixed $S$ and $K$. Determining the optimal bound for convex trajectories, and the asymptotic sharpness of its dependence on $S$ and $K$, remains open.

eq:jensen:

ΔK≤Δ0(S−KS+K)K≤Δ0exp⁡ ⁣(−2K2S).\Delta_K\le\Delta_0\left(\frac{S-K}{S+K}\right)^K \le\Delta_0\exp\!\left(-\frac{2K^2}{S}\right).

— A Sharper Theory of Ball-Proximal Optimization: Convergence and Radius Selection  (2609.35147 - Richtárik et al., 28 Sep 2026) in Section 6, Discussion