TY - GEN
T1 - Efficient projections onto the ℓ1-ball for learning in high dimensions
AU - Duchi, John
AU - Shalev-Shwartz, Shai
AU - Singer, Yoram
AU - Chandra, Tushar
PY - 2008
Y1 - 2008
N2 - We describe efficient algorithms for projecting a vector onto the ℓ1-ball. We present two methods for projection. The first performs exact projection in O(n) expected time, where n is the dimension of the space. The second works on vectors k of whose elements are perturbed outside the ℓ1-ball, projecting in O(k log(n)) time. This setting is especially useful for online learning in sparse feature spaces such as text categorization applications. We demonstrate the merits and effectiveness of our algorithms in numerous batch and online learning tasks. We show that variants of stochastic gradient projection methods augmented with our efficient projection procedures outperform interior point methods, which are considered state-of-the-art optimization techniques. We also show that in online settings gradient updates with ℓ1 projections outperform the exponentiated gradient algorithm while obtaining models with high degrees of sparsity.
AB - We describe efficient algorithms for projecting a vector onto the ℓ1-ball. We present two methods for projection. The first performs exact projection in O(n) expected time, where n is the dimension of the space. The second works on vectors k of whose elements are perturbed outside the ℓ1-ball, projecting in O(k log(n)) time. This setting is especially useful for online learning in sparse feature spaces such as text categorization applications. We demonstrate the merits and effectiveness of our algorithms in numerous batch and online learning tasks. We show that variants of stochastic gradient projection methods augmented with our efficient projection procedures outperform interior point methods, which are considered state-of-the-art optimization techniques. We also show that in online settings gradient updates with ℓ1 projections outperform the exponentiated gradient algorithm while obtaining models with high degrees of sparsity.
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M3 - Conference contribution
AN - SCOPUS:56449092085
SN - 9781605582054
T3 - Proceedings of the 25th International Conference on Machine Learning
SP - 272
EP - 279
BT - Proceedings of the 25th International Conference on Machine Learning
T2 - 25th International Conference on Machine Learning
Y2 - 5 July 2008 through 9 July 2008
ER -