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Area of Science:

  • Statistics
  • Machine Learning
  • Optimization

Background:

  • Folded concave penalized sparse linear regression (FCPSLR) is a popular sparse recovery method.
  • Global optimization for FCPSLR is NP-complete, necessitating analysis of local solutions.
  • Existing research on local minimizers has not fully clarified their statistical sufficiency or independence from specific algorithms.

Purpose of the Study:

  • To determine if local solutions of FCPSLR are sufficient for statistical performance.
  • To investigate if statistical performance is independent of specific computational procedures.
  • To analyze the conditions under which local solutions guarantee desirable statistical properties.

Main Methods:

  • Analysis of local solutions (stationary points) of FCPSLR under folded concave penalties.
  • Introduction and application of the significant subspace second-order necessary condition (S³ONC).
  • Application of S³ONC to FCPSLR with minimax concave penalty (MCP) under restricted eigenvalue conditions.

Main Results:

  • Local solutions are shown to be sparse estimators under specific penalty parameter conditions.
  • S³ONC, a weaker condition than second-order KKT, ensures bounded approximation error and high-probability oracle recovery.
  • For MCP, S³ONC solutions offer strong oracle properties and model error comparable to optimal estimators when sample size is sufficient.

Conclusions:

  • Local solutions of FCPSLR, particularly those satisfying S³ONC, offer significant statistical guarantees.
  • The study demonstrates consistency between improving statistical performance and optimizing the non-convex formulation of FCPSLR.
  • S³ONC provides a viable criterion for achieving high-performance sparse recovery with computational tractability (FPTAS).