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  \textbf{\textsc{STANFORD UNIVERSITY}}\\[5pt]
  \textbf{\textsc{DEPARTMENT OF STATISTICS}}\\[5pt]
  \Large{\textbf\textsc{{DEPARTMENTAL SEMINAR}}}
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  4:15 p.m., Tuesday, April 4, 2006\\
  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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  \textsl{  Alexandre Tsybakov  }\\
University of Paris 6 and Miller Institute, UC Berkeley
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  \textbf{Aggregation by mirror averaging    } 
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\noindent
Given a collection of $M$ different estimators or classifiers, we
study the problem of convex or model selection type aggregation, i.e., we
construct a new estimator or classifier, called aggregate, which is
nearly as good as the best among them (or nearly as good as their best
convex combination), with respect to a given risk criterion. We define our
aggregate by mixing of the initial estimators with data-dependent weights
which are computed by a simple recursive procedure called the mirror
averaging algorithm. We show that the mirror averaging aggregate satisfies
sharp oracle inequalities under some general assumptions. They improve
in several cases inequalities known for batch methods and apply
under weaker conditions, because they are not based on the techniques
of empirical processes.
The results allow one to construct in an easy on-line way sharp adaptive
nonparametric estimators for several problems including regression,
classification and density estimation.

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