Iteratively reweighted $\ell_1$ algorithms with extrapolation

TL;DR

This paper explores how extrapolation techniques can accelerate iteratively reweighted $ ext{l}_1$ algorithms for nonconvex sparse optimization, providing convergence guarantees and demonstrating improved performance over existing methods.

Contribution

It introduces three extrapolated versions of the iteratively reweighted $ ext{l}_1$ algorithm with convergence conditions and shows their effectiveness through numerical experiments.

Findings

Algorithms outperform existing methods in CPU time and solution quality.

Explicit conditions ensure convergence to stationary points.

Numerical results validate acceleration benefits.

Abstract

Iteratively reweighted $\ell_1$ algorithm is a popular algorithm for solving a large class of optimization problems whose objective is the sum of a Lipschitz differentiable loss function and a possibly nonconvex sparsity inducing regularizer. In this paper, motivated by the success of extrapolation techniques in accelerating first-order methods, we study how widely used extrapolation techniques such as those in [4,5,22,28] can be incorporated to possibly accelerate the iteratively reweighted $\ell_1$ algorithm. We consider three versions of such algorithms. For each version, we exhibit an explicitly checkable condition on the extrapolation parameters so that the sequence generated provably clusters at a stationary point of the optimization problem. We also investigate global convergence under additional Kurdyka-$\L$ojasiewicz assumptions on certain potential functions. Our numerical experiments show that our algorithms usually outperform the general iterative shrinkage and thresholding algorithm in [21] and an adaptation of the iteratively reweighted $\ell_1$ algorithm in [23, Algorithm 7] with nonmonotone line-search for solving random instances of log penalty regularized least squares problems in terms of both CPU time and solution quality.