Dual iterative hard thresholding

Xiao Tong Yuan, Bo Liu, Lezi Wang, Qingshan Liu, Dimitris N. Metaxas

Research output: Contribution to journalArticlepeer-review


Iterative Hard Thresholding (IHT) is a popular class of first-order greedy selection methods for loss minimization under cardinality constraint. The existing IHT-style algorithms, however, are proposed for minimizing the primal formulation. It is still an open issue to explore duality theory and algorithms for such a non-convex and NP-hard combinatorial optimization problem. To address this issue, we develop in this article a novel duality theory for '2-regularized empirical risk minimization under cardinality constraint, along with an IHT-style algorithm for dual optimization. Our sparse duality theory establishes a set of sufficient and/or necessary conditions under which the original non-convex problem can be equivalently or approximately solved in a concave dual formulation. In view of this theory, we propose the Dual IHT (DIHT) algorithm as a super-gradient ascent method to solve the non-smooth dual problem with provable guarantees on primal-dual gap convergence and sparsity recovery. Numerical results confirm our theoretical predictions and demonstrate the superiority of DIHT to the state-of-the-art primal IHT-style algorithms in model estimation accuracy and computational efficiency.

Original languageEnglish (US)
JournalJournal of Machine Learning Research
StatePublished - Jun 2020

All Science Journal Classification (ASJC) codes

  • Software
  • Control and Systems Engineering
  • Statistics and Probability
  • Artificial Intelligence


  • Duality theory
  • Iterative hard thresholding
  • Non-convex optimization
  • Sparsity recovery

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