Consistency bounds and support recovery of d-stationary solutions of sparse sample average approximations

Miju Ahn ()
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Miju Ahn: Southern Methodist University

Journal of Global Optimization, 2020, vol. 78, issue 3, No 1, 397-422

Abstract: Abstract This paper studies properties of the d(irectional)-stationary solutions of sparse sample average approximation problems involving difference-of-convex sparsity functions under a deterministic setting. Such properties are investigated with respect to a vector which satisfies a verifiable assumption to relate the empirical sample average approximation problem to the expectation minimization problem defined by an underlying data distribution. We derive bounds for the distance between the two vectors and the difference of the model outcomes generated by them. Furthermore, the inclusion relationships between their supports, sets of nonzero valued indices, are studied. We provide conditions under which the support of a d-stationary solution is contained within, and contains, the support of the vector of interest; the first kind of inclusion can be shown for any given arbitrary set of indices. Some of the results presented herein are generalization of the existing theory for a specialized problem of $$\ell _1$$ ℓ 1 -norm regularized least squares minimization for linear regression.

Keywords: Non-convex optimization; Sparse learning; Difference-of-convex program; Directional stationary solution (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10898-019-00857-z

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