# Primal-Dual Block Frank-Wolfe

## TL;DR

This paper introduces a Primal-Dual Block Frank-Wolfe algorithm that efficiently solves sparse and low-rank optimization problems, including Elastic Net and regularized SVMs, with reduced per-iteration cost and proven linear convergence.

## Contribution

It presents a novel Frank-Wolfe variant that lowers computational cost per iteration while maintaining convergence speed, applicable to various structured optimization problems.

## Key findings

- Outperforms state-of-the-art methods on classification tasks
- Reduces per-iteration computational cost based on solution structure
- Maintains linear convergence rate

## Abstract

We propose a variant of the Frank-Wolfe algorithm for solving a class of sparse/low-rank optimization problems. Our formulation includes Elastic Net, regularized SVMs and phase retrieval as special cases. The proposed Primal-Dual Block Frank-Wolfe algorithm reduces the per-iteration cost while maintaining linear convergence rate. The per iteration cost of our method depends on the structural complexity of the solution (i.e. sparsity/low-rank) instead of the ambient dimension. We empirically show that our algorithm outperforms the state-of-the-art methods on (multi-class) classification tasks.