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arXiv · 2203.00564

A unified analysis of convex and non-convex lp-ball projection problems

Joong-Ho Won·Kenneth Lange·Jason Xu
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Abstract

The task of projecting onto $\ell_p$ norm balls is ubiquitous in statistics and machine learning, yet the availability of actionable algorithms for doing so is largely limited to the special cases of $p = \left\{ 0, 1,2, \infty \right\}$. In this paper, we introduce novel, scalable methods for projecting onto the $\ell_p$ ball for general $p>0$. For $p \geq1 $, we solve the univariate Lagrangian dual via a dual Newton method. We then carefully design a bisection approach for $p<1$, presenting theoretical and empirical evidence of zero or a small duality gap in the non-convex case. The success of our contributions is thoroughly assessed empirically, and applied to large-scale regularized multi-task learning and compressed sensing.

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Joong-Ho Won, Kenneth Lange, Jason Xu. 2022-03-01. A unified analysis of convex and non-convex lp-ball projection problems. https://arxiv.org/abs/2203.00564

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