A unified analysis of convex and non-convex ℓ _p -ball projection problems

The task of projecting onto ℓ _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 ∈{ 0, 1,2, ∞} . In this paper, we introduce novel, scalable methods for projecting onto the ℓ _p -ball for general p>0 . For p ≥ 1 , 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. The code implementing our methods is publicly available on Github.