Best-first Search Algorithm for Non-convex Sparse Minimization

Shinsaku Sakaue; Naoki Marumo

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arXiv · 1910.01296

Best-first Search Algorithm for Non-convex Sparse Minimization

Shinsaku Sakaue·Naoki Marumo

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Abstract

Non-convex sparse minimization (NSM), or $\ell_0$-constrained minimization of convex loss functions, is an important optimization problem that has many machine learning applications. NSM is generally NP-hard, and so to exactly solve NSM is almost impossible in polynomial time. As regards the case of quadratic objective functions, exact algorithms based on quadratic mixed-integer programming (MIP) have been studied, but no existing exact methods can handle more general objective functions including Huber and logistic losses; this is unfortunate since those functions are prevalent in practice. In this paper, we consider NSM with $\ell_2$-regularized convex objective functions and develop an algorithm by leveraging the efficiency of best-first search (BFS). Our BFS can compute solutions with objective errors at most $\Delta\ge0$, where $\Delta$ is a controllable hyper-parameter that balances the trade-off between the guarantee of objective errors and computation cost. Experiments demonstrate that our BFS is useful for solving moderate-size NSM instances with non-quadratic objectives and that BFS is also faster than the MIP-based method when applied to quadratic objectives.

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Shinsaku Sakaue, Naoki Marumo. 2019-10-03. Best-first Search Algorithm for Non-convex Sparse Minimization. https://arxiv.org/abs/1910.01296

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