Determine the optimal predetermined Heavy-Ball rate

Determine the optimal worst-case last-iterate convergence rate of the classical Heavy-Ball method with arbitrary predetermined, horizon-dependent nonnegative step sizes and momenta in [0,1) on the class of smooth convex objectives; in particular, establish whether predetermined momentum can improve on the Silver polynomial exponent or whether the lower bound with exponent (1+√5)/2 can be strengthened toward that exponent.

Background

The paper proves that every predetermined, horizon-dependent Heavy-Ball schedule has a smooth convex objective on which the last-iterate error is at least on the order of T{-α}/log T, where α=(1+√5)/2≈1.618. The paper also notes that the Heavy-Ball method contains predetermined gradient descent as the special case β_t=0, for which the Silver stepsize schedule achieves an O(T{-s}) rate with s=log₂(1+√2)≈1.272.

Consequently, the paper leaves a gap between the established Heavy-Ball lower bound and the best known predetermined gradient-descent upper bound. The unresolved issue is whether momentum improves upon the Silver exponent or whether the Heavy-Ball lower bound can be strengthened toward it.

References

The optimal rate of the Heavy-Ball method with predetermined schedules therefore remains unresolved. In light of the GD lower bound reported by \citet{yeliu2026silver}, the central question is whether predetermined momentum can improve on the Silver polynomial exponent, or whether the Heavy-Ball lower bound can be strengthened toward it.

— A Lower Bound for the Heavy-Ball Method on Smooth Convex Functions  (2609.08656 - Ma et al., 8 Sep 2026) in Section Conclusion

Compared with the lower bound $\Omega(1/K{(1+\sqrt{5})/2}\log K)$ in , we conjecture that this lower bound is not tight and that the convergence rate $\mathcal O(1/K{3/2})$ is already optimal for the heavy-ball method.

— Heavy-Ball Method under Randomized Schedules  (2609.09743 - He et al., 9 Sep 2026) in Section 5, Concluding Remarks