# Forward-backward splitting under the light of generalized convexity

arXiv: https://arxiv.org/abs/2503.18098 · HTML: https://mlsanity.com/paper/2503.18098/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Mar 2025.

## Built on by (most cited first, 2 of 2)

- [Rethinking PCA Through Duality](https://mlsanity.com/paper/2510.18130/) (2025)
- [Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization](https://mlsanity.com/paper/2606.27767/) (2026)

## Abstract (arXiv)

In this paper we present a unifying framework for continuous optimization methods grounded in the concept of generalized convexity. Utilizing the powerful theory of Phi-convexity, we propose a conceptual algorithm that extends the classical difference-of-convex method, encompassing a broad spectrum of optimization algorithms. Relying exclusively on the tools of generalized convexity we develop a gap function analysis that strictly characterizes the decrease of the function values, leading to simplified and unified convergence results. As an outcome of this analysis, we naturally obtain a generalized PL inequality which ensures q-linear convergence rates of the proposed method, incorporating various well-established conditions from the existing literature. Moreover we propose a Phi-Bregman proximal point interpretation of the scheme that allows us to capture conditions that lead to sublinear rates under convexity.