# Perturbed Fenchel duality and first-order methods

## TL;DR

This paper introduces a unified framework based on perturbed Fenchel duality that explains the convergence rates of various first-order optimization algorithms, including gradient and proximal methods.

## Contribution

It establishes a canonical perturbed Fenchel duality inequality that unifies the analysis of multiple first-order methods, providing new insights into their convergence behavior.

## Key findings

- Unified derivation of convergence rates for popular first-order algorithms
- Applicable to gradient, proximal, and universal gradient methods
- Simplifies analysis through a canonical duality inequality

## Abstract

We show that the iterates generated by a generic first-order meta-algorithm satisfy a canonical perturbed Fenchel duality inequality. The latter in turn readily yields a unified derivation of the best known convergence rates for various popular first-order algorithms including the conditional gradient method as well as the main kinds of Bregman proximal methods: subgradient, gradient, fast gradient, and universal gradient methods.