Last-Iterate Performance of Gradient Descent and Relaxed Proximal Point via s-Composability
Abstract
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).
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BibTeX
@misc{chikhi2026lastiterate,
title = {{Last-Iterate Performance of Gradient Descent and Relaxed Proximal Point via $s$-Composability}},
author = {Salah Chikhi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13765},
eprint = {2609.13765},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13765},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Last-Iterate Performance of Gradient Descent and Relaxed Proximal Point via s-Composability
AU - Salah Chikhi
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.13765
UR - https://arxiv.org/abs/2609.13765
AB - 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).
ER -