A unified analysis of convex and non‑convex ‑ball projection problems

The task of projecting onto $$ norm balls is ubiquitous in statistics and machine learning, yet the availability of actionable algorithms for doing so is largely limited to the special cases of $$. In this paper, we introduce novel, scalable methods for projecting onto the $$-ball for general $$. For $$, we solve the univariate Lagrangian dual via a dual Newton method. We then carefully design a bisection approach For $$, 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.