# Exact worst-case convergence rates of the proximal gradient method for composite convex minimization

arXiv: https://arxiv.org/abs/1705.04398 · HTML: https://mlsanity.com/paper/1705.04398/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 May 2017.

## Built on by (most cited first, 5 of 5)

- [PEPit: computer-assisted worst-case analyses of first-order optimization methods in Python](https://mlsanity.com/paper/2201.04040/) (2022)
- [Bias-Optimal Bounds for SGD: A Computer-Aided Lyapunov Analysis](https://mlsanity.com/paper/2505.17965/) (2025)
- [Restarted contractive operators to learn at equilibrium](https://mlsanity.com/paper/2506.13239/) (2025)
- [Online Distributed Learning with Quantized Finite-Time Coordination](https://mlsanity.com/paper/2307.06620/) (2023)
- [Tight Nonasymptotic Local Convergence of Sinkhorn-Knopp](https://mlsanity.com/paper/2608.11760/) (2026)

## Abstract (arXiv)

We study the worst-case convergence rates of the proximal gradient method for minimizing the sum of a smooth strongly convex function and a non-smooth convex function whose proximal operator is available. We establish the exact worst-case convergence rates of the proximal gradient method in this setting for any step size and for different standard performance measures: objective function accuracy, distance to optimality and residual gradient norm. The proof methodology relies on recent developments in performance estimation of first-order methods based on semidefinite programming. In the case of the proximal gradient method, this methodology allows obtaining exact and non-asymptotic worst-case guarantees that are conceptually very simple, although apparently new. On the way, we discuss how strong convexity can be replaced by weaker assumptions, while preserving the corresponding convergence rates. We also establish that the same fixed step size policy is optimal for all three performance measures. Finally, we extend recent results on the worst-case behavior of gradient descent with exact line search to the proximal case.