梯度下降与松弛近端点的末次迭代性能：基于$s$-可组合性

中文摘要 AI 辅助

$s$-可组合性的概念被引入，用于在保持尖锐保证的同时组合优化的步长调度。我们表明，同一联合势能具有第二个用途：它可以直接作为尖锐末次迭代性能的证书。我们针对常数调度和银调度发展了这一观点。对于常数调度，我们证明了Grimmer等人（2025）提出的作为开放问题的$s$-可组合性陈述，在每个有限时域上，通过一个显式的非负光滑凸插值证书。该结果确定了在初始距离约束下，光滑凸梯度下降中最小化最坏情况最终梯度范数的唯一常数步长，从而证明了Taylor等人（2017）猜想3所预测的最优步长、值和唯一性，而无需解析其完整的最坏情况曲线。通过Moreau包络，同一常数也是最小化松弛近端点最坏情况最终残差的唯一常数松弛参数。更一般地，对于每个正的$s$-可组合调度，我们确定了精确的最坏情况最终近端目标间隙常数，并给出了一个匹配的一维示例。将此结果应用于已经$s$-可组合的原始银调度，得到其精确的末次迭代目标间隙常数，从而解决了Wang等人（2025）留下的一个开放问题。

英文摘要

The notion of s-composability was introduced to compose optimized stepsize schedules while preserving sharp guarantees. We show that the same joint potential has a second use: it can serve directly as a certificate of sharp last-iterate performance. We develop this viewpoint for constant and silver schedules. For the constant schedule, we prove the s-composability statement posed as an open question by Grimmer et al. (2025), at every finite horizon, through an explicit nonnegative smooth-convex interpolation certificate. The result identifies the unique constant stepsize minimizing the worst-case final gradient norm of smooth convex gradient descent under an initial-distance bound, thereby proving the optimal stepsize, value, and uniqueness predicted by Conjecture 3 of Taylor et al. (2017) without resolving its full worst-case curve. Through the Moreau envelope, the same constant is also the unique constant relaxation minimizing the worst-case final residual of relaxed proximal point. More generally, for every positive s-composable schedule, we determine the exact worst-case final proximal objective-gap constant and give a matching one-dimensional example. Applying this result to the already s-composable original silver schedule yields its exact last-iterate objective-gap constant, closing a question left open by Wang et al. (2025).

发表机构

• École Polytechnique（巴黎综合理工学院）

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