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Constant s-composable schedule validity for all n

Determine whether, for every integer n ≥ 1, the constant stepsize sequence h_k = \bar h defined by the equation 1/(1 + \bar h n) = (1 − \bar h)^n is s-composable with rate η = 1/(1 + n \bar h).

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Background

The authors define s-composability, which simultaneously balances worst-case guarantees on function suboptimality and gradient norm, and show that constant schedules satisfying 1/(1 + \bar h n) = (1 − \bar h)n are the only candidates for s-composability.

They verify analytically that this constant schedule is s-composable with the stated rate for n = 1 and n = 2, and conjecture—based on numerical PEP solves—that the property extends to all n, leaving a complete proof as future work.

References

One can directly verify analytically that this construction is $s$-composable with rate $\eta = \frac{1}{1+n \bar h}$ for $n=1,2$. We conjecture based on numerical PEP solves that this holds for all $n$. We leave this for future work.

— Composing Optimized Stepsize Schedules for Gradient Descent (2410.16249 - Grimmer et al., 21 Oct 2024) in Example ex:constant_composable, Section “Simple Examples of Composable Patterns”