An Algorithm for Quadratic $\ell_1$-Regularized Optimization with a Flexible Active-Set Strategy

TL;DR

This paper introduces a flexible active-set algorithm for quadratic  -regularized optimization that adaptively chooses between proximal gradient and conjugate gradient steps, with proven convergence and demonstrated effectiveness.

Contribution

It proposes a novel adaptive active-set method for  -regularized problems that combines different step types based on subgradient analysis, with convergence guarantees.

Findings

The method converges globally with established rates.

Work complexity estimates are provided for two variants.

Numerical experiments show the method's effectiveness on test problems.

Abstract

We present an active-set method for minimizing an objective that is the sum of a convex quadratic and $\ell_1$ regularization term. Unlike two-phase methods that combine a first-order active set identification step and a subspace phase consisting of a \emph{cycle} of conjugate gradient (CG) iterations, the method presented here has the flexibility of computing a first-order proximal gradient step or a subspace CG step at each iteration. The decision of which type of step to perform is based on the relative magnitudes of some scaled components of the minimum norm subgradient of the objective function. The paper establishes global rates of convergence, as well as work complexity estimates for two variants of our approach, which we call the iiCG method. Numerical results illustrating the behavior of the method on a variety of test problems are presented.