Papers › Semismooth Newton Algorithm for Efficient Projections onto ℓ_(1, ∞)-norm Ball

Semismooth Newton Algorithm for Efficient Projections onto ℓ_(1, ∞)-norm Ball

1 Jan 2020ICML 2020 1archive 2025-07-28

Dejun Chu, Chang-Shui Zhang, Shiliang Sun, Qing Tao

Structured sparsity-inducing ℓ_(1, ∞)-norm, as a generalization of the classical ℓ₁-norm, plays an important role in jointly sparse models which select or remove simultaneously all the variables forming a group. However, its resulting problem is more difficult to solve than the conventional ℓ₁-norm constrained problem. In this paper, we propose an efficient algorithm for Euclidean projection onto ℓ_(1, ∞)-norm ball. We tackle the projection problem via semismooth Newton algorithm to solve the system of semismooth equations. Meanwhile, exploiting the structure of Jacobian matrix via LU decomposition yields an equivalent algorithm which is proved to terminate after a finite number of iterations. Empirical studies demonstrate that our proposed algorithm outperforms the existing state-of-the-art solver and is promising for the optimization of learning problems with ℓ_(1, ∞)-norm ball constraint.

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