For every integer n≥ 1, let r_n∈(0,1) be the unique solution of r_n^n bigl(1+n(1+r_n)bigr)=1, and set bar h_n=1+r_n. For every L-smooth convex function f and gradient-descent iterates x_k+1=x_k-(bar h_n/L)nabla f(x_k), determine whether the constant schedule (bar h_n,…,bar h_n) is s-composable at horizon n, equivalently whether it satisfies the sharp mixed terminal Lyapunov inequality conjectured by Grimmer, Shu, and Wang. The supplied source does not state the Lyapunov inequality explicitly.

Constant-schedule s-composability conjecture

For every integer $n\geq 1$ , let $r_n\in(0,1)$ be the unique solution of $r_n^n\bigl(1+n(1+r_n)\bigr)=1$ , and set $\bar h_n=1+r_n$ . For every $L$ -smooth convex function $f$ and gradient-descent iterates $x_{k+1}=x_k-(\bar h_n/L)\nabla f(x_k)$ , determine whether the constant schedule $(\bar h_n,\ldots,\bar h_n)$ is $s$ -composable at horizon $n$ , equivalently whether it satisfies the sharp mixed terminal Lyapunov inequality conjectured by Grimmer, Shu, and Wang. The supplied source does not state the Lyapunov inequality explicitly.

References

Primary source

arXiv

Additional references

Constant Steps Are s-Composable: An Exact Interpolation Certificate for Gradient Descent — arXiv — Jinze Zhao

convex-analysis optimization

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 preprint claims to settle the conjecture for every horizon, but the result has not been refereed.

The conjecture asks whether the balanced constant schedule satisfies the sharp mixed terminal Lyapunov inequality at every horizon. Before this announcement, it was known only for the first two horizons.

Known results

The first two horizons were previously established; the available source gives no mathematician or year.

September 2026 claimed proof

Jinze Zhao's preprint Constant Steps Are s-Composable: An Exact Interpolation Certificate for Gradient Descent claims the balanced constant schedule is $s$ -composable for all horizons, thereby settling the conjecture. The preprint is unrefereed, so the claimed resolution remains unverified.

Current status (as of September 2026): The conjecture is claimed solved for all horizons by Jinze Zhao's unrefereed preprint, but the proof has not been independently verified.

Sources

No solutions have been posted yet.