# Block-coordinate and incremental aggregated proximal gradient methods for nonsmooth nonconvex problems

## TL;DR

This paper develops convergence analysis for block-coordinate and incremental proximal gradient methods applied to nonconvex, nonsmooth optimization problems, introducing the forward-backward envelope as a Lyapunov function and extending results to popular algorithms like Finito/MISO.

## Contribution

It provides the first convergence analysis of block-coordinate and incremental proximal gradient methods for nonconvex, nonsmooth problems using the forward-backward envelope.

## Key findings

- Established global convergence under Kurdyka-ojasiewicz property.
- Proved linear convergence rates for specific problem classes.
- Extended analysis to Finito/MISO algorithm with new convergence results.

## Abstract

This paper analyzes block-coordinate proximal gradient methods for minimizing the sum of a separable smooth function and a (nonseparable) nonsmooth function, both of which are allowed to be nonconvex. The main tool in our analysis is the forward-backward envelope (FBE), which serves as a particularly suitable continuous and real-valued Lyapunov function. Global and linear convergence results are established when the cost function satisfies the Kurdyka-\L ojasiewicz property without imposing convexity requirements on the smooth function. Two prominent special cases of the investigated setting are regularized finite sum minimization and the sharing problem; in particular, an immediate byproduct of our analysis leads to novel convergence results and rates for the popular Finito/MISO algorithm in the nonsmooth and nonconvex setting with very general sampling strategies. This paper analyzes block-coordinate proximal gradient methods for minimizing the sum of a separable smooth function and a (nonseparable) nonsmooth function, both of which are allowed to be nonconvex. The main tool in our analysis is the forward-backward envelope (FBE), which serves as a particularly suitable continuous and real-valued Lyapunov function. Global and linear convergence results are established when the cost function satisfies the Kurdyka-\L ojasiewicz property without imposing convexity requirements on the smooth function. Two prominent special cases of the investigated setting are regularized finite sum minimization and the sharing problem; in particular, an immediate byproduct of our analysis leads to novel convergence results and rates for the popular Finito/MISO algorithm in the nonsmooth and nonconvex setting with very general sampling strategies.