Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Minimization Over the Nonconvex Sparsity Constraint Using A Hybrid First-order method

Xiangyu Yang, Hao Wang, Yichen Zhu, Xiao Wang

Abstract

We investigate a class of nonconvex optimization problems characterized by a feasible set consisting of level-bounded nonconvex regularizers, with a continuously differentiable objective. We propose a novel hybrid approach to tackle such structured problems within a first-order algorithmic framework by combining the Frank-Wolfe method and the gradient projection method. The Frank-Wolfe step is amenable to a closed-form solution, while the gradient projection step can be efficiently performed in a reduced subspace. A notable characteristic of our approach lies in its independence from introducing smoothing parameters, enabling efficient solutions to the original nonsmooth problems. We establish the global convergence of the proposed algorithm and show the $O(1/\sqrt{k})$ convergence rate in terms of the optimality error for nonconvex objectives under reasonable assumptions. Numerical experiments underscore the practicality and efficiency of our proposed algorithm compared to existing cutting-edge methods. Furthermore, we highlight how the proposed algorithm contributes to the advancement of nonconvex regularizer-constrained optimization.

Minimization Over the Nonconvex Sparsity Constraint Using A Hybrid First-order method

Abstract

We investigate a class of nonconvex optimization problems characterized by a feasible set consisting of level-bounded nonconvex regularizers, with a continuously differentiable objective. We propose a novel hybrid approach to tackle such structured problems within a first-order algorithmic framework by combining the Frank-Wolfe method and the gradient projection method. The Frank-Wolfe step is amenable to a closed-form solution, while the gradient projection step can be efficiently performed in a reduced subspace. A notable characteristic of our approach lies in its independence from introducing smoothing parameters, enabling efficient solutions to the original nonsmooth problems. We establish the global convergence of the proposed algorithm and show the convergence rate in terms of the optimality error for nonconvex objectives under reasonable assumptions. Numerical experiments underscore the practicality and efficiency of our proposed algorithm compared to existing cutting-edge methods. Furthermore, we highlight how the proposed algorithm contributes to the advancement of nonconvex regularizer-constrained optimization.

Paper Structure

This paper contains 22 sections, 15 theorems, 74 equations, 4 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Literature and Our Contributions
  3. Organization
  4. Notation and Preliminaries
  5. A Hybrid First-order Algorithm for Solving \ref{['eq: main_Opt']}
  6. Main Algorithm Framework
  7. Frank-Wolfe Block
  8. Gradient Projection Block
  9. Computing the FW Subproblem Stepsize
  10. Convergence Analysis
  11. Optimality Conditions of Subproblems
  12. Global Convergence
  13. Convergence Rate Analysis
  14. Numerical Experiments
  15. Euclidean Projection onto the $\ell_p$ Ball
  16. ...and 7 more sections

Key Result

Proposition 1.6

Consider eq: main_Opt_General. It holds for any $\bar{\bm{x}}$ that Consequently, the subdifferential regularity of $\Phi$ holds at every $\bm{x} \in \mathbb{R}^{n}$.

Figures (4)

  • Figure 1: Performance comparison of various considered algorithms. Left: The empirical probability of success versus $m$. Right: The elapsed wall-clock time versus $m$.
  • Figure 2: Performance for the Cauchy loss. Left: The empirical probability of success versus $m$. Right: The elapsed wall-clock time versus $m$.
  • Figure 3: The first three columns are the reconstructed image with $p \in \{0.1,0.5,0.9\}$ and the last column corresponds to the original images.
  • Figure :

Theorems & Definitions (39)

  • Example 1.1: Projection and compressive sensing
  • Example 1.2: Supervised sparse learning
  • Example 1.3: Adversarial examples generation
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Proposition 1.6
  • proof
  • Proposition 1.7
  • ...and 29 more