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Iterative Hard Thresholding (IHT) is a class of projected gradient descentmethods for optimizing sparsity-constrained minimization models, with the bestknown efficiency and scalability in practice. As far as we know, the existingIHT-style methods are designed for sparse minimization in primal form. Itremains open to explore duality theory and algorithms in such a non-convex andNP-hard problem setting. In this paper, we bridge this gap by establishing aduality theory for sparsity-constrained minimization with $\ell_2$-regularizedloss function and proposing an IHT-style algorithm for dual maximization. Oursparse duality theory provides a set of sufficient and necessary conditionsunder which the original NP-hard/non-convex problem can be equivalently solvedin a dual formulation. The proposed dual IHT algorithm is a super-gradientmethod for maximizing the non-smooth dual objective. An interesting finding isthat the sparse recovery performance of dual IHT is invariant to the RestrictedIsometry Property (RIP), which is required by virtually all the existing primalIHT algorithms without sparsity relaxation. Moreover, a stochastic variant ofdual IHT is proposed for large-scale stochastic optimization. Numerical resultsdemonstrate the superiority of dual IHT algorithms to the state-of-the-artprimal IHT-style algorithms in model estimation accuracy and computationalefficiency.

Dual Iterative Hard Thresholding: From Non-convex Sparse Minimization to Non-smooth Concave Maximization