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  \textbf{\Large{\textsc{STANFORD UNIVERSITY}}}\\[5pt]
  \textbf{\Large{\textsc{DEPARTMENT OF STATISTICS}}}\\[5pt]
  \Large{\textsc{DEPARTMENTAL SEMINAR}}
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  Time of talk, Weekday, Date of talk\\
  4:15 p.m., Tuesday, March 11, 2008\\
  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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  \textsl{Yoram Singer} \\
  Google
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  \subsection*{Efficient Projections Algorithms onto the $\ell_1$ Ball for
 Learning Sparse Representations from High Dimension Data}
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\noindent

We describe efficient algorithms for projecting a vector onto the
$\ell_1$-ball. We present two projection methods. The first method performs
exact projection in $O(n)$ time, where $n$ is the dimension of the space. The
second method works on vectors $k$ of whose elements are perturbed outside the
$\ell_1$-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 statistical learning tasks. We show
that variants of gradient projection methods augmented with our efficient
projection procedures outperform interior point methods, which are considered
state-of-the-art optimization techniques. For least squares problems we show
that the algorithm is competitive with a coordinate descent algorithm that was
tailored to the problem. We also show that in online settings gradient updates
with $\ell_1$ projections outperform the entropic descent and exponentiated
gradient algorithms while obtaining models with high degrees of sparsity. To
conclude, we present general analysis for online convex programming with
strongly convex functions and show that all algorithms we discuss attain
logarithmic regret.
  
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