# Accelerating Proximal Gradient Descent via Silver Stepsizes

arXiv: https://arxiv.org/abs/2412.05497 · HTML: https://mlsanity.com/paper/2412.05497/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Dec 2024.

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

- [Negative Stepsizes Make Gradient-Descent-Ascent Converge](https://mlsanity.com/paper/2505.01423/) (2025)
- [Projected gradient methods for nonconvex and stochastic smooth optimization: new complexities and auto-conditioned stepsizes](https://mlsanity.com/paper/2412.14291/) (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)

## Abstract (arXiv)

Surprisingly, recent work has shown that gradient descent can be accelerated without using momentum -- just by judiciously choosing stepsizes. An open question raised by several papers is whether this phenomenon of stepsize-based acceleration holds more generally for constrained and/or composite convex optimization via projected and/or proximal versions of gradient descent. We answer this in the affirmative by proving that the silver stepsize schedule yields analogously accelerated rates in these settings. These rates are conjectured to be asymptotically optimal among all stepsize schedules, and match the silver convergence rate of vanilla gradient descent (Altschuler and Parrilo, 2024, 2025), namely O(varepsilon^- logρ 2) for smooth convex optimization and O(kappalogρ 2 log frac1varepsilon) under strong convexity, where varepsilon is the precision, kappa is the condition number, and ρ = 1 + √2 is the silver ratio. The key technical insight is the combination of recursive gluing -- the technique underlying all analyses of gradient descent accelerated with time-varying stepsizes -- with a certain Laplacian-structured sum-of-squares certificate for the analysis of proximal point updates.