# Linear convergence of accelerated conditional gradient algorithms in spaces of measures

## TL;DR

This paper introduces a class of generalized conditional gradient algorithms for optimization in spaces of Radon measures, demonstrating sub-linear convergence generally and linear convergence under certain structural assumptions.

## Contribution

The paper develops a new class of algorithms for measure space optimization and proves their convergence rates, including local linear convergence under specific conditions.

## Key findings

- Achieves a sub-linear $ ext{O}(1/k)$ convergence rate in general cases.
- Under structural assumptions, attains local linear convergence with rate $ ext{O}(
ho^k)$.
- Provides analysis for finite-dimensional subproblem resolution within the algorithm.

## Abstract

A class of generalized conditional gradient algorithms for the solution of optimization problem in spaces of Radon measures is presented. The method iteratively inserts additional Dirac-delta functions and optimizes the corresponding coefficients. Under general assumptions, a sub-linear $\mathcal{O}(1/k)$ rate in the objective functional is obtained, which is sharp in most cases. To improve efficiency, one can fully resolve the finite-dimensional subproblems occurring in each iteration of the method. We provide an analysis for the resulting procedure: under a structural assumption on the optimal solution, a linear $\mathcal{O}(\zeta^k)$ convergence rate is obtained locally.