# Inexact Block Coordinate Descent Algorithms for Nonsmooth Nonconvex Optimization

## TL;DR

This paper introduces an inexact block coordinate descent algorithm tailored for large-scale nonsmooth nonconvex optimization, emphasizing flexibility, fast convergence, and low complexity, with proven convergence guarantees.

## Contribution

The paper presents a novel inexact block coordinate descent method that allows flexible approximation functions and guarantees convergence even without Lipschitz continuous gradients.

## Key findings

- Algorithm demonstrates fast convergence in large-scale problems.
- Applicable to signal processing and machine learning tasks.
- Converges to stationary points even with inexact subproblem solutions.

## Abstract

In this paper, we propose an inexact block coordinate descent algorithm for large-scale nonsmooth nonconvex optimization problems. At each iteration, a particular block variable is selected and updated by inexactly solving the original optimization problem with respect to that block variable. More precisely, a local approximation of the original optimization problem is solved. The proposed algorithm has several attractive features, namely, i) high flexibility, as the approximation function only needs to be strictly convex and it does not have to be a global upper bound of the original function; ii) fast convergence, as the approximation function can be designed to exploit the problem structure at hand and the stepsize is calculated by the line search; iii) low complexity, as the approximation subproblems are much easier to solve and the line search scheme is carried out over a properly constructed differentiable function; iv) guaranteed convergence of a subsequence to a stationary point, even when the objective function does not have a Lipschitz continuous gradient. Interestingly, when the approximation subproblem is solved by a descent algorithm, convergence of a subsequence to a stationary point is still guaranteed even if the approximation subproblem is solved inexactly by terminating the descent algorithm after a finite number of iterations. These features make the proposed algorithm suitable for large-scale problems where the dimension exceeds the memory and/or the processing capability of the existing hardware. These features are also illustrated by several applications in signal processing and machine learning, for instance, network anomaly detection and phase retrieval.