Source-linked AI summary
Iterative Reweighted Minimization Methods for $l_p$ Regularized Unconstrained Nonlinear Programming
Zhaosong Lu
TL;DR
The paper addresses how to solve general l_p-regularized unconstrained optimization problems while handling nonsmooth, nonconvex regularization and obtaining sparse solutions. It develops stationarity bounds, reweighted l_1/l_2 methods, and a locally Lipschitz ε-approximation; the resulting IRL1 methods achieve first-order stationarity for fixed ε below a computable threshold and are generally more stable computationally.
Problem
General l_p-regularized unconstrained optimization needs methods that handle its nonsmooth nonconvex regularizer while seeking sparse solutions.
Method
The paper derives stationary-point lower bounds, extends IRL1 and IRL2 methods, and constructs a locally Lipschitz ε-approximation used to develop new IRL1 algorithms.
Results
Any accumulation point generated by the new IRL1 methods is first-order stationary when ε is below a computable threshold, and the new IRL1 method is generally more stable in objective value and CPU time.
Takeaways & Limitations
The approximation-based IRL1 approach avoids dynamically sending ε to zero while providing a generally more stable method than existing IRL1 methods.
Takeaways & Limitations
One proposed IRLα variant requires solving reweighted l_α subproblems that may not be cheaply computable, motivating closed-form subproblem variants for some α values.
Abstract
from arXiv · showhide
In this paper we study general $l_p$ regularized unconstrained minimization problems. In particular, we derive lower bounds for nonzero entries of first- and second-order stationary points, and hence also of local minimizers of the $l_p$ minimization problems. We extend some existing iterative reweighted $l_1$ (IRL1) and $l_2$ (IRL2) minimization methods to solve these problems and proposed new variants for them in which each subproblem has a closed form solution. Also, we provide a unified convergence analysis for these methods. In addition, we propose a novel Lipschitz continuous $ε$-approximation to $\|x\|^p_p$. Using this result, we develop new IRL1 methods for the $l_p$ minimization problems and showed that any accumulation point of the sequence generated by these methods is a first-order stationary point, provided that the approximation parameter $ε$ is below a computable threshold value. This is a remarkable result since all existing iterative reweighted minimization methods require that $ε$ be dynamically updated and approach zero. Our computational results demonstrate that the new IRL1 method is generally more stable than the existing IRL1 methods [21,18] in terms of objective function value and CPU time.
1 Introduction
The paper studies general l_p-regularized unconstrained optimization for sparse solutions, positioning it between l_0 sparsity seeking and l_1 convex relaxation. It extends reweighted methods and introduces an approximation-based IRL1 approach with a fixed sufficiently small parameter.
- Motivation: The l_p model is designed to find sparse vectors while minimizing a smooth objective function.As p approaches 0, it approaches l_0 minimization; as p approaches 1, it approaches l_1 minimization.
- Motivation: The l_p model can produce sparser solutions than l_1 while achieving similar objective-function values.This motivates using l_p regularization as an intermediate model between l_0 and l_1 formulations.
- Related work: Existing work includes recovery conditions and iterative reweighted l_1 and l_2 methods for related sparse optimization problems.Prior studies also considered smooth approximations, SQP, trust-region Newton, and interior-point methods for related nonconvex models.
- Contributions: The paper derives lower bounds for nonzero stationary-point entries, extends IRL1 and IRL2 methods, and provides unified convergence analysis.It also proposes new variants whose subproblems can have closed-form solutions.
- Contributions: A locally Lipschitz continuous ε-approximation yields new IRL1 methods whose accumulation points are first-order stationary when ε is below a computable threshold.Unlike existing methods, the approximation parameter need not be dynamically driven toward zero.
2 Technical results
The paper develops stationarity bounds and a locally Lipschitz approximation for l_p regularized optimization. These results connect stationary points of the approximation to stationary points of the original problem when the approximation parameter is sufficiently small.
- Stationarity bounds: Lower bounds are derived for nonzero entries of first- and second-order stationary points, and therefore of local minimizers.The analysis covers general l_p-regularized unconstrained problems rather than only a special quadratic case.
- Stationarity bounds: Every local minimizer is a first-order stationary point and, under twice continuous differentiability, also a second-order stationary point.The second-order statement requires f to be twice continuously differentiable near the local minimizer.
- Lipschitz approximation: The proposed F_ε is a nonsmooth but locally Lipschitz continuous approximation to F, with Clarke subdifferentials defined everywhere.The approximation is constructed for every ε > 0 and supports analysis of the corresponding approximation problem.
- Lipschitz approximation: When ε is below a computable threshold, a suitable first-order stationary point of the approximation problem is also first-order stationary for the original problem.Its nonzero entries also satisfy the original problem’s first-order lower bound.
- Lipschitz approximation: Under the same threshold condition, local minimizers of the approximation inherit first- and second-order stationarity and the corresponding lower bounds.The second-order conclusion additionally assumes twice continuous differentiability near the point.
3 A unified analysis for some existing iterative reweighted minimization methods
The paper extends existing IRL1 and IRL2 methods to general l_p problems, introduces closed-form subproblem variants, and gives unified convergence results. Under diminishing smoothing parameters, accumulation points are stationary for the target problem.
- Existing IRL1 and IRL2 methods are extended from specialized models to the general problem (1).
- Any accumulation point of the first IRLα method or its variant is a first-order stationary point of the approximated problem (27).
- The paper presents IRLα methods and variants whose subproblems can have closed-form solutions for α = 1 or 2.The original reweighted lα subproblems may not be cheaply computable, motivating the simpler variants.
- Each outer iteration has finitely many inner iterations under the stated termination criterion.
- If δ_k → 0 and ε_k → 0, any accumulation point generated by the methods is a first-order stationary point of problem (1).
- The second IRLα method likewise requires non-increasing positive ε_k tending to zero for accumulation-point stationarity of problem (1).
4 New iterative reweighted l1 minimization for (1)
The paper develops new IRL1 methods for general l_p regularized unconstrained optimization using a fixed ε approximation. Under a computable threshold condition, accumulation points are first-order stationary points of the original problem.
- Fixed-ε approximation: Unlike existing iterative reweighted methods, the proposed methods do not require ε to be dynamically decreased toward zero.The paper presents this fixed-ε convergence result as the central novelty of the methods.
- Fixed-ε approximation: The methods apply IRL1 directly to a locally Lipschitz continuous ε-approximation of the original problem, with ε chosen once and fixed.The approximation is designed so that suitable stationary points also solve the original problem.
- Convergence analysis: Any accumulation point generated by the new IRL1 method is a first-order stationary point of problem (1) and satisfies the first-order bound on nonzero entries.The result assumes ε satisfies the stated computable threshold condition.
- Closed-form variant: The same accumulation-point guarantee holds for the closed-form variant, whose nonzero entries also satisfy the first-order bound.The variant is motivated by the potentially costly solution of the original weighted l1 subproblems.
- Algorithm: The new IRL1 method repeatedly solves weighted l1 minimization subproblems and updates the iterate until the termination criterion is satisfied.The algorithm starts from an arbitrary x0 and a threshold-compatible ε, then increments the outer iteration counter.
- Convergence analysis: Each outer iteration has finitely many inner iterations before its termination criterion is met.Theorem 4.2 establishes finite inner termination for every outer iteration.
5 Computational results
The experiments compare three IRL1 variants on randomly generated instances of problem (5) for p = 0.1 and p = 0.5. IRL1-3 is reported as generally more stable in objective value and CPU time.
- Experimental setup: The experiments compare IRL1-1, IRL1-2, and IRL1-3 on randomly generated instances using MATLAB implementations.The reported CPU times exclude the time required to obtain the initial point x0.
- First experiment: For p = 0.1 in the first experiment, all three methods produce similar objective function values.Table 1 reports the objective values for the three variants.
- First experiment: For p = 0.1 in the first experiment, IRL1-1 and IRL1-3 require nearly the same CPU time, substantially less than IRL1-2.The comparison is based on the CPU-time columns accompanying Table 1.
- First experiment: For p = 0.5 in the first experiment, IRL1-1 and IRL1-3 achieve better objective function values than IRL1-2 while requiring less CPU time.This comparison is reported from Table 2.
- Second experiment: For p = 0.5 in the second experiment, all three methods achieve similar objective function values, while IRL1-2 and IRL1-3 have similar overall CPU times below IRL1-1.The second experiment uses randomly generated A and b with λ = 3e-3.
- Overall comparison: Across the two experiments, IRL1-3 is generally more stable than IRL1-1 and IRL1-2 in objective function value and CPU time.This is the paper’s overall computational comparison of the three variants.
6 Concluding remarks
The paper develops and analyzes iterative reweighted methods for l_p-regularized unconstrained minimization, including a new Lipschitz continuous ε-approximation. The new IRL1 methods achieve first-order stationarity under a fixed, computable ε threshold and are generally more stable computationally than existing IRL1 methods.
- The paper derives lower bounds for nonzero entries of first- and second-order stationary points, including local minimizers.
- It extends existing IRL1 and IRL2 methods, proposes new variants with closed-form subproblem solutions, and provides unified convergence analysis.
- The paper introduces a novel Lipschitz continuous ε-approximation to the l_p regularizer.
- Any accumulation point generated by the new IRL1 methods is first-order stationary when ε is below a computable threshold, rather than dynamically approaching zero.
- Computational results indicate that the new IRL1 method is generally more stable than existing IRL1 methods in objective function value and CPU time.