# Silver Rate Is (Almost) Optimal for Gradient Descent

arXiv: https://arxiv.org/abs/2609.09152 · HTML: https://mlsanity.com/paper/2609.09152/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 31 references that are in the map.
- First posted on arXiv 8 Sep 2026.

## Builds on (most cited first, 18 of 18)

- [Smooth Strongly Convex Interpolation and Exact Worst-case Performance of First-order Methods](https://mlsanity.com/paper/1502.05666/) (2015)
- [Provably Faster Gradient Descent via Long Steps](https://mlsanity.com/paper/2307.06324/) (2023)
- [Acceleration by Stepsize Hedging II: Silver Stepsize Schedule for Smooth Convex Optimization](https://mlsanity.com/paper/2309.16530/) (2023)
- [Acceleration by Stepsize Hedging I: Multi-Step Descent and the Silver Stepsize Schedule](https://mlsanity.com/paper/2309.07879/) (2023)
- [Branch-and-Bound Performance Estimation Programming: A Unified Methodology for Constructing Optimal Optimization Methods](https://mlsanity.com/paper/2203.07305/) (2022)
- [Accelerated Objective Gap and Gradient Norm Convergence for Gradient Descent via Long Steps](https://mlsanity.com/paper/2403.14045/) (2024)
- [Anytime Acceleration of Gradient Descent](https://mlsanity.com/paper/2411.17668/) (2024)
- [Acceleration by Random Stepsizes: Hedging, Equalization, and the Arcsine Stepsize Schedule](https://mlsanity.com/paper/2412.05790/) (2024)
- [Open Problem: Anytime Convergence Rate of Gradient Descent](https://mlsanity.com/paper/2406.13888/) (2024)
- [Accelerated Gradient Descent by Concatenation of Stepsize Schedules](https://mlsanity.com/paper/2410.12395/) (2024)
- [Composing Optimized Stepsize Schedules for Gradient Descent](https://mlsanity.com/paper/2410.16249/) (2024)
- [A lower bound for stepsize-based acceleration of gradient descent](https://mlsanity.com/paper/2608.10418/) (2026)
- [Lower Bounds for Anytime Acceleration of Gradient Descent](https://mlsanity.com/paper/2607.02053/) (2026)
- [Improved Gradient Descent Lower Bounds Beyond Nesterov](https://mlsanity.com/paper/2609.02855/) (2026)
- [Stronger Lower Bounds for (Non-)Anytime Acceleration of Gradient Descent](https://mlsanity.com/paper/2609.04032/) (2026)
- [Relaxed Proximal Point Algorithm: Tight Complexity Bounds and Acceleration without Momentum](https://mlsanity.com/paper/2410.08890/) (2024)
- [Optimized methods for composite optimization: a reduction perspective](https://mlsanity.com/paper/2506.23756/) (2025)
- [Stepsize Hedging: an Alternative Mechanism for Accelerating Gradient Descent](https://mlsanity.com/paper/2605.31386/) (2026)

## Built on by (most cited first, 1 of 1)

- [The Row Normalization Puzzle in Muon](https://mlsanity.com/paper/2609.39114/) (2026)

## Abstract (arXiv)

We study how far gradient descent (GD) can be accelerated by predetermined stepsizes in smooth convex optimization. Writing psil=log2(1+√2), we prove an Omegaleft(n^-psil-O(√loglog n/log n)right) non-anytime lower bound. In the anytime setting, every infinite schedule has infinitely many horizons with error Omegaleft(n^-frac2psil1+psil-O(√loglog n/log n)right). Together with the silver-schedule upper bound [Altschuler and Parrilo, 2025] and the anytime upper bound [Zhang et al., 2025], our results determine the optimal polynomial convergence exponents in both settings.