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GLOBAL SOLUTIONS TO FOLDED CONCAVE PENALIZED NONCONVEX LEARNING.

Hongcheng Liu, Tao Yao, Runze Li1

  • 1The Pennsylvania State University.

Annals of Statistics
|May 4, 2016
PubMed
Summary

This study introduces Mixed Integer Programming-based Global Optimization (MIPGO) to solve nonconvex learning problems. MIPGO reformulates these problems as quadratic programs, guaranteeing globally optimal solutions for folded concave penalties like SCAD and MCP.

Keywords:
Folded concave penaltiesMCPSCADglobal optimizationhigh dimensional statistical learningnonconvex quadratic programmingsparse recovery

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Area of Science:

  • Optimization
  • Machine Learning
  • Statistical Learning

Background:

  • Nonconvex learning problems with folded concave penalties offer desirable statistical properties.
  • Existing optimization techniques often fail to guarantee global optimality for these problems.

Purpose of the Study:

  • To develop a novel optimization technique that guarantees global optimality for nonconvex learning problems with folded concave penalties.
  • To establish the first global optimization scheme with theoretical guarantees for SCAD and MCP penalties.

Main Methods:

  • Equivalence of a class of nonconvex learning problems to general quadratic programs.
  • Development of Mixed Integer Linear Programming (MILP) reformulations.
  • Introduction of the Mixed Integer Programming-based Global Optimization (MIPGO) technique.

Main Results:

  • MIPGO provides a finite algorithm to find provably global optimal solutions.
  • Demonstrates significant outperformance of MIPGO over existing methods like local linear approximation.
  • First theoretical guarantee for global optimality in folded concave penalized nonconvex learning.

Conclusions:

  • MIPGO offers a robust and theoretically sound approach for solving complex nonconvex learning problems.
  • The technique significantly advances the field of optimization for penalized nonconvex learning.
  • MIPGO represents a breakthrough in achieving global optimality for SCAD and MCP penalized models.