恒定步长是s-可组合的：梯度下降的精确插值证书

中文摘要 AI 辅助

Grimmer、Shu和Wang提出了一个开放问题：在每个时间范围内，平衡的恒定调度是否是$s$-可组合的。更精确地说，对于整数$n\geq 1$，令$\bar{h}=1+r$，其中$r\in(0,1)$是方程$$r^n\bigl(1+n(1+r)\bigr)=1$$的唯一解，并以归一化步长$\bar{h}$运行梯度下降$n$步。已知$n=1,2$的情形，而一般情形$n\geq3$仍未被解决。我们证明了该猜想对所有$n$成立。我们的证明给出了光滑凸插值不等式的显式、无维数的非负线性组合。该证书由支持在连续索引区间上的矩阵组装而成。其非对角元素自动为正，其二次部分是对角的，其余乘子归结为两个标量族。我们推导了这些族的闭式表达式，并利用严格凹性和初等有理不等式证明了其正性。因此，对于每个$L$-光滑凸函数，恒定调度满足原始论文中猜想的尖锐混合终端Lyapunov不等式，以及相关的同步目标间隙和梯度范数界。所有例外的时间范围和边界索引都被显式处理。

英文摘要

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.

发表机构

• University of California, San Diego（加州大学圣迭戈分校）

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