# Linearization Algorithms for Fully Composite Optimization

## TL;DR

This paper introduces new first-order algorithms for fully composite optimization that efficiently handle differentiable and non-differentiable components, extending classical methods with improved convergence and practical implementation.

## Contribution

It generalizes classical Frank-Wolfe and Conditional Gradient methods for non-differentiable problems using a stronger linear minimization oracle and provides convergence analysis and acceleration.

## Key findings

- Global convergence rates for convex and non-convex objectives
- An affine-invariant analysis of the basic method
- An accelerated convex optimization algorithm with improved complexity

## Abstract

This paper studies first-order algorithms for solving fully composite optimization problems over convex and compact sets. We leverage the structure of the objective by handling its differentiable and non-differentiable components separately, linearizing only the smooth parts. This provides us with new generalizations of the classical Frank-Wolfe method and the Conditional Gradient Sliding algorithm, that cater to a subclass of non-differentiable problems. Our algorithms rely on a stronger version of the linear minimization oracle, which can be efficiently implemented in several practical applications. We provide the basic version of our method with an affine-invariant analysis and prove global convergence rates for both convex and non-convex objectives. Furthermore, in the convex case, we propose an accelerated method with correspondingly improved complexity. Finally, we provide illustrative experiments to support our theoretical results.