Constant Steps Are s-Composable: An Exact Interpolation Certificate for Gradient Descent

Abstract

Grimmer, Shu, and Wang asked whether a balanced constant schedule is $s$-composable at every horizon. More precisely, for an integer $n\geq 1$, let $\bar{h}=1+r$, where $r\in(0,1)$ is the unique solution of $$r^n\bigl(1+n(1+r)\bigr)=1,$$ and run gradient descent for $n$ steps with normalized stepsize $\bar{h}$. The cases $n=1,2$ were known, while the general case $n\geq3$ was left open. We prove the conjecture for every $n$. Our proof gives an explicit, dimension-free nonnegative linear combination of the smooth convex interpolation inequalities. The certificate is assembled from matrices supported on contiguous index intervals. Its off-diagonal entries are automatically positive, its quadratic part is diagonal, and its remaining multipliers reduce to two scalar families. We derive closed forms for those families and prove positivity using strict concavity and elementary rational inequalities. Consequently, for every $L$-smooth convex function, the constant schedule satisfies the sharp mixed terminal Lyapunov inequality conjectured in the original paper, together with the associated simultaneous objective-gap and gradient-norm bounds. All exceptional horizons and boundary indices are treated explicitly.