# Composing Optimized Stepsize Schedules for Gradient Descent

arXiv: https://arxiv.org/abs/2410.16249 · HTML: https://mlsanity.com/paper/2410.16249/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Oct 2024.

## Built on by (most cited first, 4 of 4)

- [Anytime Acceleration of Gradient Descent](https://mlsanity.com/paper/2411.17668/) (2024)
- [A lower bound for stepsize-based acceleration of gradient descent](https://mlsanity.com/paper/2608.10418/) (2026)
- [Improved Gradient Descent Lower Bounds Beyond Nesterov](https://mlsanity.com/paper/2609.02855/) (2026)
- [Silver Rate Is (Almost) Optimal for Gradient Descent](https://mlsanity.com/paper/2609.09152/) (2026)

## Abstract (arXiv)

Recent works by Altschuler and Parrilo and the authors have shown that it is possible to accelerate the convergence of gradient descent on smooth convex functions, even without momentum, just by picking special stepsizes. In this paper, we provide a general theory for composing stepsize schedules capturing all recent advances in this area and more. We propose three notions of ``composable'' stepsize schedules with elementary associated composition operations for combining them. From these operations, in addition to recovering recent works, we construct three highly optimized sequences of stepsize schedules. We first construct optimized stepsize schedules of every length generalizing the exponentially spaced silver stepsizes. We then construct highly optimized stepsizes schedules for minimizing final objective gap or gradient norm, improving on prior rates by constants and, more importantly, matching or beating the numerically computed minimax optimal schedules. We conjecture these schedules are in fact minimax (information theoretic) optimal. Several novel tertiary results follow from our theory including recovery of the recent dynamic gradient norm minimizing short stepsizes and extending them to objective gap minimization.