 
Summary: Fast learning rates in statistical inference through
aggregation
J.Y. Audibert1,2
1Certis  Ecole des Ponts  Paris Est
2Willow  ENS/INRIA
Abstract
We develop minimax optimal risk bounds for the general learning
task consisting in predicting as well as the best function in a reference
set G up to the smallest possible additive term, called the convergence
rate. When the reference set is finite and when n denotes the size of
the training data, we provide minimax convergence rates of the form
C log G
n
v
with tight evaluation of the positive constant C and with
exact 0 < v 1, the latter value depending on the convexity of the
loss function and on the level of noise in the output distribution.
The risk upper bounds are based on a sequential randomized algo
rithm, which at each step concentrates on functions having both low
risk and low variance with respect to the previous step prediction func
