# Convex-Concave Backtracking for Inertial Bregman Proximal Gradient Algorithms in Non-Convex Optimization

## TL;DR

This paper introduces a novel double convex-concave backtracking method for inertial Bregman proximal gradient algorithms, enabling adaptive step size and extrapolation control, with proven convergence and improved performance in non-convex problems.

## Contribution

It proposes a new double backtracking procedure controlling both step size and extrapolation in inertial Bregman proximal methods, with theoretical convergence guarantees.

## Key findings

- Convergence to critical points is proven for the proposed algorithms.
- Numerical experiments show improved performance in image processing and machine learning tasks.
- The method effectively balances inertial step and backtracking for non-convex optimization.

## Abstract

Backtracking line-search is an old yet powerful strategy for finding a better step sizes to be used in proximal gradient algorithms. The main principle is to locally find a simple convex upper bound of the objective function, which in turn controls the step size that is used. In case of inertial proximal gradient algorithms, the situation becomes much more difficult and usually leads to very restrictive rules on the extrapolation parameter. In this paper, we show that the extrapolation parameter can be controlled by locally finding also a simple concave lower bound of the objective function. This gives rise to a double convex-concave backtracking procedure which allows for an adaptive choice of both the step size and extrapolation parameters. We apply this procedure to the class of inertial Bregman proximal gradient methods, and prove that any sequence generated by these algorithms converges globally to a critical point of the function at hand. Numerical experiments on a number of challenging non-convex problems in image processing and machine learning were conducted and show the power of combining inertial step and double backtracking strategy in achieving improved performances.