# Convergence of the Forward-Backward Algorithm: Beyond the Worst Case with the Help of Geometry

arXiv: https://arxiv.org/abs/1703.09477 · HTML: https://mlsanity.com/paper/1703.09477/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Mar 2017.

## Built on by (most cited first, 7 of 7)

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- [Polyak's Heavy Ball Method Achieves Accelerated Local Rate of Convergence under Polyak-Lojasiewicz Inequality](https://mlsanity.com/paper/2410.16849/) (2024)
- [Controlling the Flow: Stability and Convergence for Stochastic Gradient Descent with Decaying Regularization](https://mlsanity.com/paper/2505.11434/) (2025)
- [Bias-Optimal Bounds for SGD: A Computer-Aided Lyapunov Analysis](https://mlsanity.com/paper/2505.17965/) (2025)
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## Abstract (arXiv)

We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or Lojasiewicz properties. These geometrical notions are usually local by nature, and may fail to describe the fine geometry of objective functions relevant in inverse problems and signal processing, that have a nice behaviour on manifolds, or sets open with respect to a weak topology. Motivated by this observation, we revisit those geometric notions over arbitrary sets. In turn, this allows us to present several new results as well as collect in a unified view a variety of results scattered in the literature. Our contributions include the analysis of infinite dimensional convex minimization problems, showing the first Lojasiewicz inequality for a quadratic function associated to a compact operator, and the derivation of new linear rates for problems arising from inverse problems with low-complexity priors. Our approach allows to establish unexpected connections between geometry and a priori conditions in inverse problems, such as source conditions, or restricted isometry properties.