Efficient Projections onto the ℓ1-Ball for Learning in High Dimensions

Efficient Projections onto the ℓ1-Ball for Learning in High Dimensions

2008 | John Duchi, Shai Shalev-Shwartz, Yoram Singer, Tushar Chandra
The paper presents efficient algorithms for projecting a vector onto the $\ell_1$-ball, which is crucial for learning in high-dimensional spaces. Two methods are introduced: an exact projection in $O(n)$ expected time and a more efficient method for vectors with only $k$ non-zero elements, achieving $O(k \log(n))$ time complexity. These algorithms are particularly useful for online learning in sparse feature spaces, such as text categorization. The authors demonstrate the effectiveness of these algorithms through experiments on synthetic and real datasets, showing that they outperform state-of-the-art optimization techniques like interior point methods and the exponentiated gradient algorithm in terms of sparsity, convergence speed, and lower regret. The paper also discusses the theoretical foundations and provides detailed algorithms for both methods, including a reduction to Euclidean projection onto the simplex and an efficient search for the pivot value in sparse feature spaces.The paper presents efficient algorithms for projecting a vector onto the $\ell_1$-ball, which is crucial for learning in high-dimensional spaces. Two methods are introduced: an exact projection in $O(n)$ expected time and a more efficient method for vectors with only $k$ non-zero elements, achieving $O(k \log(n))$ time complexity. These algorithms are particularly useful for online learning in sparse feature spaces, such as text categorization. The authors demonstrate the effectiveness of these algorithms through experiments on synthetic and real datasets, showing that they outperform state-of-the-art optimization techniques like interior point methods and the exponentiated gradient algorithm in terms of sparsity, convergence speed, and lower regret. The paper also discusses the theoretical foundations and provides detailed algorithms for both methods, including a reduction to Euclidean projection onto the simplex and an efficient search for the pivot value in sparse feature spaces.
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[slides and audio] Efficient projections onto the l1-ball for learning in high dimensions