Semismooth Newton Algorithm for Efficient Projections onto ℓ 1 , ∞ -norm Ball

Jul 12, 2020

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Structured sparsity-inducing ℓ_1, ∞-norm, as a generalization of the classical ℓ_1-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 ℓ_1-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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The International Conference on Machine Learning (ICML) is the premier gathering of professionals dedicated to the advancement of the branch of artificial intelligence known as machine learning. ICML is globally renowned for presenting and publishing cutting-edge research on all aspects of machine learning used in closely related areas like artificial intelligence, statistics and data science, as well as important application areas such as machine vision, computational biology, speech recognition, and robotics. ICML is one of the fastest growing artificial intelligence conferences in the world. Participants at ICML span a wide range of backgrounds, from academic and industrial researchers, to entrepreneurs and engineers, to graduate students and postdocs.

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