Accelerating Proximal Gradient Descent via Silver Stepsizes

TL;DR

This paper demonstrates that using a specific 'silver stepsize' schedule can accelerate proximal gradient descent methods for constrained and composite convex optimization, matching optimal convergence rates without momentum.

Contribution

The authors prove that the silver stepsize schedule accelerates proximal gradient methods in constrained and composite convex optimization, extending previous results to broader settings.

Findings

Accelerated convergence rates with silver stepsize schedule.

Optimality conjecture for the proposed stepsize schedule.

Extension of acceleration results to proximal and constrained settings.

Abstract

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 $O(\varepsilon^{- \log_{\rho} 2})$ for smooth convex optimization and $O(\kappa^{\log_\rho 2} \log \frac{1}{\varepsilon})$ under strong convexity, where $\varepsilon$ is the precision, $\kappa$ is the condition number, and $\rho = 1 + \sqrt{2}$ 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.