# Acceleration by Stepsize Hedging II: Silver Stepsize Schedule for Smooth Convex Optimization

arXiv: https://arxiv.org/abs/2309.16530 · HTML: https://mlsanity.com/paper/2309.16530/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Sep 2023.

## Built on by (most cited first, 20 of 20)

- [Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiency](https://mlsanity.com/paper/2402.15926/) (2024)
- [Negative Stepsizes Make Gradient-Descent-Ascent Converge](https://mlsanity.com/paper/2505.01423/) (2025)
- [Anytime Acceleration of Gradient Descent](https://mlsanity.com/paper/2411.17668/) (2024)
- [Minimax Optimal Convergence of Gradient Descent in Logistic Regression via Large and Adaptive Stepsizes](https://mlsanity.com/paper/2504.04105/) (2025)
- [Open Problem: Anytime Convergence Rate of Gradient Descent](https://mlsanity.com/paper/2406.13888/) (2024)
- [CyclicReflex: Improving Reasoning Models via Cyclical Reflection Token Scheduling](https://mlsanity.com/paper/2506.11077/) (2025)
- [Convergence Rates for Gradient Descent on the Edge of Stability in Overparametrised Least Squares](https://mlsanity.com/paper/2510.17506/) (2025)
- [Gradient Descent's Last Iterate is Often (slightly) Suboptimal](https://mlsanity.com/paper/2604.13870/) (2026)
- [A lower bound for stepsize-based acceleration of gradient descent](https://mlsanity.com/paper/2608.10418/) (2026)
- [Gradient Methods with Online Scaling Part I. Theoretical Foundations](https://mlsanity.com/paper/2505.23081/) (2025)
- [On the convergence of adaptive first order methods: proximal gradient and alternating minimization algorithms](https://mlsanity.com/paper/2311.18431/) (2023)
- [Improved Gradient Descent Lower Bounds Beyond Nesterov](https://mlsanity.com/paper/2609.02855/) (2026)
- [Tight Bounds for Logistic Regression with Large Stepsize Gradient Descent in Low Dimension](https://mlsanity.com/paper/2602.12471/) (2026)
- [Dynamics of Gradient Descent with Large Step Size Near a Manifold of Flat Minima](https://mlsanity.com/paper/2607.08380/) (2026)
- [Silver Rate Is (Almost) Optimal for Gradient Descent](https://mlsanity.com/paper/2609.09152/) (2026)
- [Empirical and computer-aided robustness analysis of long-step and accelerated methods in smooth convex optimization](https://mlsanity.com/paper/2506.09730/) (2025)
- [Exponential Convergence of (Stochastic) Gradient Descent for Separable Logistic Regression](https://mlsanity.com/paper/2602.18946/) (2026)
- [A short proof of near-linear convergence of adaptive gradient descent under fourth-order growth and convexity](https://mlsanity.com/paper/2604.13393/) (2026)
- [A Lower Bound for the Heavy-Ball Method on Smooth Convex Functions](https://mlsanity.com/paper/2609.08656/) (2026)
- [The Row Normalization Puzzle in Muon](https://mlsanity.com/paper/2609.39114/) (2026)

## Abstract (arXiv)

We provide a concise, self-contained proof that the Silver Stepsize Schedule proposed in Part I directly applies to smooth (non-strongly) convex optimization. Specifically, we show that with these stepsizes, gradient descent computes an ε-minimizer in O(ε^-logρ 2) = O(ε-0.7864) iterations, where ρ = 1+√2 is the silver ratio. This is intermediate between the textbook unaccelerated rate O(ε-1) and the accelerated rate O(ε-1/2) due to Nesterov in 1983. The Silver Stepsize Schedule is a simple explicit fractal: the i-th stepsize is 1+ρv(i)-1 where v(i) is the 2-adic valuation of i. The design and analysis are conceptually identical to the strongly convex setting in Part I, but simplify remarkably in this specific setting.